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  1. Write down the number of faces of a cube.
  2. Write down the number of vertices of a cube.
","answerType":"multi","answer":"a) 6 b) 8","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6"},{"label":"b","answer":"8"}],"solution":["A face is a flat surface of the cube. An edge is where two faces meet, and a vertex is a corner where edges meet. Count the hidden faces and corners too.","A cube has 6 square faces.","A cube has 8 vertices, one at each corner."],"diagram":null,"draw":false,"revision":"bac62908c966"},{"id":"g1-3d-shapes-faces-edges-and-vertices-q02","topicId":"g1-3d-shapes-faces-edges-and-vertices","topic":"3D Shapes: Faces, Edges and Vertices","grade":"1","topicGrade":"1","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

A 3D solid has exactly one square face and four triangular faces.

  1. Write down the mathematical name of this solid.
  2. Write down the number of edges of this solid.
","answerType":"multi","answer":"a) square-based pyramid b) 8","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"square-based pyramid"},{"label":"b","answer":"8"}],"solution":["One square face and four triangular faces describes a square-based pyramid.","It has 4 edges around the square base and 4 sloping edges up to the apex, so 4 + 4 = 8 edges."],"diagram":null,"draw":false,"revision":"b9d159d6ec41"},{"id":"g1-3d-shapes-faces-edges-and-vertices-q03","topicId":"g1-3d-shapes-faces-edges-and-vertices","topic":"3D Shapes: Faces, Edges and Vertices","grade":"1","topicGrade":"1","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Here is a triangular prism.

  1. Write down the number of faces of a triangular prism.
  2. Write down the number of edges of a triangular prism.
","answerType":"multi","answer":"a) 5 b) 9","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5"},{"label":"b","answer":"9"}],"solution":["A triangular prism has 2 triangular faces and 3 rectangular faces, so 5 faces in total.","It has 3 edges on each triangular end and 3 edges joining the two ends, so 3 + 3 + 3 = 9 edges."],"diagram":{"type":"solid","kind":"prism","dims":{"b":3,"h":2.6,"len":5}},"draw":false,"revision":"06408d199064"},{"id":"g1-3d-shapes-faces-edges-and-vertices-q04","topicId":"g1-3d-shapes-faces-edges-and-vertices","topic":"3D Shapes: Faces, Edges and Vertices","grade":"1","topicGrade":"1","strand":"G","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"
  1. A 3D solid has one curved surface, no edges and no vertices. Write down the mathematical name of this solid.
  2. A different 3D solid has two flat circular faces and one curved surface. Write down the mathematical name of this solid.
","answerType":"multi","answer":"a) sphere b) cylinder","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"sphere"},{"label":"b","answer":"cylinder"}],"solution":["A single curved surface with no edges or vertices is a sphere.","Two flat circular faces joined by a curved surface is a cylinder."],"diagram":null,"draw":false,"revision":"4da10103d093"},{"id":"g1-3d-shapes-faces-edges-and-vertices-q05","topicId":"g1-3d-shapes-faces-edges-and-vertices","topic":"3D Shapes: Faces, Edges and Vertices","grade":"1","topicGrade":"1","strand":"G","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"
  1. A 3D solid has exactly one circular face and one vertex. Write down the mathematical name of this solid.
  2. In the space below, draw a sketch of a cuboid.
","answerType":"multi","answer":"a) cone b) a correctly sketched cuboid","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"cone"},{"label":"b","answer":"any recognisable 3D sketch of a cuboid"}],"solution":["One circular face and one vertex (the apex) describes a cone.","A cuboid sketch should show a box shape with 6 rectangular faces, usually drawn with hidden edges dashed."],"diagram":{"type":"shape","grid":false,"width":320,"bounds":[[-0.8,-0.8],[6.3999999999999995,4.5]],"polygons":[{"pts":[[0,0],[4,0],[4,2.5],[0,2.5]],"fill":false,"answer":true}],"segments":[{"from":[4,0],"to":[5.6,1.2],"answer":true},{"from":[5.6,1.2],"to":[5.6,3.7],"answer":true},{"from":[5.6,3.7],"to":[4,2.5],"answer":true},{"from":[0,2.5],"to":[1.6,3.7],"answer":true},{"from":[1.6,3.7],"to":[5.6,3.7],"answer":true},{"from":[0,0],"to":[1.6,1.2],"dashed":true,"answer":true},{"from":[1.6,1.2],"to":[5.6,1.2],"dashed":true,"answer":true},{"from":[1.6,1.2],"to":[1.6,3.7],"dashed":true,"answer":true}]},"draw":true,"revision":"8464f6a6d919"},{"id":"g1-3d-shapes-faces-edges-and-vertices-q06","topicId":"g1-3d-shapes-faces-edges-and-vertices","topic":"3D Shapes: Faces, Edges and Vertices","grade":"1","topicGrade":"1","strand":"G","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

A prism has a cross-section that is a pentagon.

  1. Write down the number of faces of the prism.
  2. Write down the number of edges of the prism.
  3. Write down the number of vertices of the prism.
","answerType":"multi","answer":"a) 7 b) 15 c) 10","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"7"},{"label":"b","answer":"15"},{"label":"c","answer":"10"}],"solution":["A pentagonal prism has 2 pentagonal faces and 5 rectangular faces, so 7 faces.","Each pentagonal end has 5 edges and there are 5 edges joining the ends, so 5 + 5 + 5 = 15 edges.","Each pentagonal end has 5 vertices, so 5 + 5 = 10 vertices."],"diagram":null,"draw":false,"revision":"2666b640148f"},{"id":"g1-3d-shapes-faces-edges-and-vertices-q07","topicId":"g1-3d-shapes-faces-edges-and-vertices","topic":"3D Shapes: Faces, Edges and Vertices","grade":"1","topicGrade":"1","strand":"G","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

A prism has a cross-section that is an equilateral triangle with sides of length 4 cm.

The length of the prism is 9 cm.

Work out the total length of all the edges of the prism.

","answerType":"exact","answer":"51 cm","answerDisplay":null,"accept":["51","51cm"],"parts":[],"solution":["The prism has 2 triangular ends, each with 3 edges of 4 cm: 6 × 4 = 24 cm.","It has 3 edges of length 9 cm joining the ends: 3 × 9 = 27 cm.","Total edge length = 24 + 27 = 51 cm."],"diagram":{"type":"solid","kind":"prism","dims":{"b":4,"h":3.464,"len":9},"labels":{"b":"4 cm","slant":"4 cm","len":"9 cm"},"caption":"Cross-section is an equilateral triangle"},"draw":false,"revision":"92209263467d"},{"id":"g1-3d-shapes-faces-edges-and-vertices-q08","topicId":"g1-3d-shapes-faces-edges-and-vertices","topic":"3D Shapes: Faces, Edges and Vertices","grade":"1","topicGrade":"1","strand":"G","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

A prism has a cross-section that is a regular octagon with sides of length 3 cm.

The length of the prism is 10 cm.

  1. Write down the number of edges of the prism.
  2. Work out the total length of all the edges of the prism.
","answerType":"multi","answer":"a) 24 b) 128 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"24"},{"label":"b","answer":"128 cm"}],"solution":["An octagonal prism has 8 edges on each end and 8 edges joining the ends: 8 + 8 + 8 = 24 edges.","The two octagonal ends have 16 edges of 3 cm: 16 × 3 = 48 cm.","The 8 joining edges have length 10 cm: 8 × 10 = 80 cm.","Total edge length = 48 + 80 = 128 cm."],"diagram":{"type":"shape","notAccurate":true,"width":380,"polygons":[{"pts":[[3.621,1.5],[1.5,3.621],[-1.5,3.621],[-3.621,1.5],[-3.621,-1.5],[-1.5,-3.621],[1.5,-3.621],[3.621,-1.5]],"fill":"#ffffff"}],"segments":[{"from":[8.121,4.5],"to":[6,6.621],"dashed":false},{"from":[3.621,1.5],"to":[8.121,4.5],"dashed":false},{"from":[6,6.621],"to":[3,6.621],"dashed":false},{"from":[1.5,3.621],"to":[6,6.621],"dashed":false},{"from":[3,6.621],"to":[0.879,4.5],"dashed":true},{"from":[-1.5,3.621],"to":[3,6.621],"dashed":false},{"from":[0.879,4.5],"to":[0.879,1.5],"dashed":true},{"from":[-3.621,1.5],"to":[0.879,4.5],"dashed":true},{"from":[0.879,1.5],"to":[3,-0.621],"dashed":true},{"from":[-3.621,-1.5],"to":[0.879,1.5],"dashed":true},{"from":[3,-0.621],"to":[6,-0.621],"dashed":true},{"from":[-1.5,-3.621],"to":[3,-0.621],"dashed":true},{"from":[6,-0.621],"to":[8.121,1.5],"dashed":false},{"from":[1.5,-3.621],"to":[6,-0.621],"dashed":false},{"from":[8.121,1.5],"to":[8.121,4.5],"dashed":false},{"from":[3.621,-1.5],"to":[8.121,1.5],"dashed":false}],"texts":[{"at":[0,-4.35],"text":"3 cm"},{"at":[6.5,2.2],"text":"10 cm"}]},"draw":false,"revision":"1a49f89fa1f5"},{"id":"g1-addition-and-subtraction-q01","topicId":"g1-addition-and-subtraction","topic":"Addition and Subtraction","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Work out 38 + 46

","answerType":"exact","answer":"84","answerDisplay":null,"accept":[],"parts":[],"solution":["Write units below units and tens below tens. When the units total reaches 10, exchange ten units for one ten in the next column.","38 + 46: add the units, 8 + 6 = 14, write 4 carry 1","Add the tens, 3 + 4 + 1 = 8","Answer 84"],"diagram":null,"draw":false,"revision":"e7f3d73a787d"},{"id":"g1-addition-and-subtraction-q02","topicId":"g1-addition-and-subtraction","topic":"Addition and Subtraction","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Work out 72 − 35

","answerType":"exact","answer":"37","answerDisplay":null,"accept":[],"parts":[],"solution":["72 - 35: subtract 30 from 72 to get 42","Subtract the remaining 5: 42 - 5 = 37"],"diagram":null,"draw":false,"revision":"5d8e756989ed"},{"id":"g1-addition-and-subtraction-q03","topicId":"g1-addition-and-subtraction","topic":"Addition and Subtraction","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":false,"text":"

Work out 236 + 487

","answerType":"exact","answer":"723","answerDisplay":null,"accept":[],"parts":[],"solution":["Units: 6 + 7 = 13, write 3 carry 1","Tens: 3 + 8 + 1 = 12, write 2 carry 1","Hundreds: 2 + 4 + 1 = 7","Answer 723"],"diagram":null,"draw":false,"revision":"4539588785d9"},{"id":"g1-addition-and-subtraction-q04","topicId":"g1-addition-and-subtraction","topic":"Addition and Subtraction","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":false,"text":"

Work out 503 − 168

","answerType":"exact","answer":"335","answerDisplay":null,"accept":[],"parts":[],"solution":["503 - 100 = 403","403 - 60 = 343","343 - 8 = 335"],"diagram":null,"draw":false,"revision":"717fde142a00"},{"id":"g1-addition-and-subtraction-q05","topicId":"g1-addition-and-subtraction","topic":"Addition and Subtraction","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":2,"tier":"F","calc":false,"text":"

Complete these calculations.

  1. 64 + □ = 100
  2. 0.3 + □ = 1
","answerType":"multi","answer":"a) 36, b) 0.7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"36"},{"label":"b","answer":"0.7"}],"solution":["a) 100 - 64 = 36, since 64 + 36 = 100","b) 1 - 0.3 = 0.7, since 0.3 + 0.7 = 1"],"diagram":null,"draw":false,"revision":"95e6d9f38bc0"},{"id":"g1-addition-and-subtraction-q06","topicId":"g1-addition-and-subtraction","topic":"Addition and Subtraction","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":1,"tier":"F","calc":false,"text":"

Find the number that is exactly halfway between 26 and 58

","answerType":"exact","answer":"42","answerDisplay":null,"accept":[],"parts":[],"solution":["Add the two numbers: 26 + 58 = 84","Halve the total: 84 ÷ 2 = 42","Check: 42 is 16 above 26 and 16 below 58"],"diagram":null,"draw":false,"revision":"386b689b2e9e"},{"id":"g1-addition-and-subtraction-q07","topicId":"g1-addition-and-subtraction","topic":"Addition and Subtraction","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":1,"tier":"F","calc":false,"text":"

Find the number that is exactly halfway between 3.2 and 4.8

","answerType":"exact","answer":"4","answerDisplay":null,"accept":["4.0"],"parts":[],"solution":["Add the two numbers: 3.2 + 4.8 = 8","Halve the total: 8 ÷ 2 = 4","Check: 4 is 0.8 above 3.2 and 0.8 below 4.8"],"diagram":null,"draw":false,"revision":"2912fafcfee9"},{"id":"g1-addition-and-subtraction-q08","topicId":"g1-addition-and-subtraction","topic":"Addition and Subtraction","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":2,"tier":"F","calc":true,"text":"

Amir has £20

He buys a book costing £7.65 and a magazine costing £4.80

Work out how much money Amir has left.

","answerType":"exact","answer":"£7.55","answerDisplay":null,"accept":["7.55","£7.55","7.55 pounds"],"parts":[],"solution":["Total spent: £7.65 + £4.80 = £12.45","Money left: £20 - £12.45 = £7.55"],"diagram":null,"draw":false,"revision":"e3f88fbbd34e"},{"id":"g1-bidmas-order-of-operations-q01","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Work out 3 + 4 × 5

","answerType":"exact","answer":"23","answerDisplay":null,"accept":[],"parts":[],"solution":["The order of operations is a shared convention so everyone reads the same expression the same way. Do brackets, then powers, then multiplication/division from left to right, then addition/subtraction from left to right.","Multiplication comes before addition","4 × 5 = 20","3 + 20 = 23"],"diagram":null,"draw":false,"revision":"b09e2bcbfc4d"},{"id":"g1-bidmas-order-of-operations-q02","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Work out 20 − 12 ÷ 4

","answerType":"exact","answer":"17","answerDisplay":null,"accept":[],"parts":[],"solution":["Division comes before subtraction","12 ÷ 4 = 3","20 - 3 = 17"],"diagram":null,"draw":false,"revision":"6b8131c1f8cd"},{"id":"g1-bidmas-order-of-operations-q03","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":false,"text":"

Work out (9 − 4) × 6

","answerType":"exact","answer":"30","answerDisplay":null,"accept":[],"parts":[],"solution":["Work out the brackets first: 9 - 4 = 5","5 × 6 = 30"],"diagram":null,"draw":false,"revision":"45b6b0cf372d"},{"id":"g1-bidmas-order-of-operations-q04","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

Work out 18 ÷ (2 + 4)

","answerType":"exact","answer":"3","answerDisplay":null,"accept":[],"parts":[],"solution":["Work out the brackets first: 2 + 4 = 6","18 ÷ 6 = 3"],"diagram":null,"draw":false,"revision":"74f67e9d12f6"},{"id":"g1-bidmas-order-of-operations-q05","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":1,"tier":"F","calc":false,"text":"

Work out 5 × 22

","answerType":"exact","answer":"20","answerDisplay":null,"accept":[],"parts":[],"solution":["A power tells you to multiply a number by itself: 2² means 2 × 2. Work out powers before multiplication.","2² = 4","5 × 4 = 20"],"diagram":null,"draw":false,"revision":"ed7434542e6a"},{"id":"g1-bidmas-order-of-operations-q06","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":1,"tier":"F","calc":true,"text":"

Work out 16 − 42 × 3

","answerType":"exact","answer":"2","answerDisplay":null,"accept":[],"parts":[],"solution":["Work out the top of the fraction: 16 - 4 = 12","Work out the bottom of the fraction: 2 × 3 = 6","12 ÷ 6 = 2"],"diagram":null,"draw":false,"revision":"d3de1ef503b4"},{"id":"g1-bidmas-order-of-operations-q07","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":1,"tier":"F","calc":false,"text":"

Tom works out 2 + 3 × 4

Tom's answer is 20

Tom's answer is wrong.

Explain what Tom has done wrong.

","answerType":"open","answer":"Tom added first. Multiplication must be done before addition, so 2 + 3 × 4 = 2 + 12 = 14","answerDisplay":null,"accept":[],"parts":[],"solution":["Tom worked out 2 + 3 = 5 first, then 5 × 4 = 20","Multiplication must be done before addition","The correct answer is 2 + 12 = 14"],"diagram":null,"draw":false,"revision":"802c00107cc8"},{"id":"g1-bidmas-order-of-operations-q08","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":1,"tier":"F","calc":true,"text":"

Sam works out 24 ÷ 4 × 2

Sam's answer is 3

Sam's answer is wrong.

Explain what Sam has done wrong.

","answerType":"open","answer":"Sam worked out 4 × 2 first. Division and multiplication have equal priority and are done left to right, so 24 ÷ 4 × 2 = 6 × 2 = 12","answerDisplay":null,"accept":[],"parts":[],"solution":["Sam worked out 4 × 2 = 8 first, then 24 ÷ 8 = 3","Division and multiplication have equal priority, so work from left to right","24 ÷ 4 = 6, then 6 × 2 = 12"],"diagram":null,"draw":false,"revision":"21673d38e9a0"},{"id":"g1-bidmas-order-of-operations-q09","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":9,"marks":2,"tier":"F","calc":false,"text":"

Work out 32 + 4 × (7 − 2)

","answerType":"exact","answer":"29","answerDisplay":null,"accept":[],"parts":[],"solution":["Brackets first: 7 - 2 = 5","Indices next: 3² = 9","Multiplication: 4 × 5 = 20","Addition: 9 + 20 = 29"],"diagram":null,"draw":false,"revision":"aa80bed5d99c"},{"id":"g1-bidmas-order-of-operations-q10","topicId":"g1-bidmas-order-of-operations","topic":"BIDMAS (Order of Operations)","grade":"1","topicGrade":"1","strand":"N","difficulty":10,"marks":3,"tier":"F","calc":false,"text":"
  1. Work out 8 + 12 ÷ 4
  2. Put a pair of brackets in this statement to make it true.
    8 + 12 ÷ 4 = 5
  3. Put a pair of brackets in this statement to make it true.
    2 + 3 × 4 − 1 = 19
","answerType":"multi","answer":"a) 11, b) (8 + 12) ÷ 4 = 5, c) (2 + 3) × 4 - 1 = 19","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"11"},{"label":"b","answer":"(8 + 12) ÷ 4 = 5"},{"label":"c","answer":"(2 + 3) × 4 - 1 = 19"}],"solution":["a) Division first: 12 ÷ 4 = 3, then 8 + 3 = 11","b) (8 + 12) ÷ 4 = 20 ÷ 4 = 5","c) (2 + 3) × 4 - 1 = 5 × 4 - 1 = 20 - 1 = 19"],"diagram":null,"draw":false,"revision":"00ede9a92da4"},{"id":"g1-coordinates-q01","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

O is the origin of a coordinate grid.

The point A is 4 units to the right of O and 2 units up from O.

The point B is 3 units to the left of O and 5 units up from O.

  1. Write down the coordinates of A.
  2. Write down the coordinates of B.
","answerType":"multi","answer":"a) (4, 2) b) (-3, 5)","answerDisplay":null,"accept":["(4,2) and (-3,5)"],"parts":[{"label":"a","answer":"(4, 2)"},{"label":"b","answer":"(-3, 5)"}],"solution":["Right is the positive x direction and up is the positive y direction.","A is 4 right and 2 up, so A is (4, 2).","B is 3 left and 5 up, so B is (-3, 5)."],"diagram":null,"draw":false,"revision":"ef5fc04835ba"},{"id":"g1-coordinates-q02","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

A is the point with coordinates (2, 4).

B is the point with coordinates (8, 10).

Work out the coordinates of the midpoint of AB.

","answerType":"exact","answer":"(5, 7)","answerDisplay":null,"accept":["(5,7)","5, 7","x = 5, y = 7"],"parts":[],"solution":["Halfway between two values is their average: start at one value and add half the difference. Do this separately for the horizontal and vertical positions.","The midpoint has the mean of the x coordinates and the mean of the y coordinates.","x coordinate: (2 + 8) ÷ 2 = 5.","y coordinate: (4 + 10) ÷ 2 = 7.","The midpoint is (5, 7)."],"diagram":null,"draw":false,"revision":"f4fda3e182ab"},{"id":"g1-coordinates-q03","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

The point A is shown on the grid.

  1. Write down the coordinates of A.
  2. On the grid, plot the point B with coordinates (−2, 3).
  3. On the grid, plot the point C with coordinates (4, 0).
","answerType":"multi","answer":"a) (1, -3) b) point plotted at (-2, 3) c) point plotted at (4, 0)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(1, -3)"},{"label":"b","answer":"point plotted at (-2, 3)"},{"label":"c","answer":"point plotted at (4, 0)"}],"solution":["A is 1 unit right of the origin and 3 units down, so A is (1, -3).","For B, go 2 left and 3 up from the origin, then mark the point.","For C, go 4 right along the x-axis; the y coordinate is 0, so C sits on the x-axis."],"diagram":{"type":"axes","square":true,"x":{"min":-5,"max":5},"y":{"min":-5,"max":5},"points":[{"at":[1,-3],"label":"A"},{"at":[-2,3],"label":"B","answer":true},{"at":[4,0],"label":"C","answer":true}]},"draw":true,"revision":"51c0f5f72c66"},{"id":"g1-coordinates-q04","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"
  1. The line x = 5 and the line y = −1 cross at the point P.
    Write down the coordinates of P.
  2. The line y = x and the line x = 3 cross at the point Q.
    Write down the coordinates of Q.
  3. Write down the coordinates of the point where the line y = 2 crosses the y-axis.
","answerType":"multi","answer":"a) (5, -1) b) (3, 3) c) (0, 2)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(5, -1)"},{"label":"b","answer":"(3, 3)"},{"label":"c","answer":"(0, 2)"}],"solution":["Every point on x = 5 has x coordinate 5 and every point on y = -1 has y coordinate -1, so P is (5, -1).","On y = x the two coordinates are equal, so when x = 3 the point is (3, 3).","On the y-axis the x coordinate is 0, so the line y = 2 crosses it at (0, 2)."],"diagram":null,"draw":false,"revision":"2aebe033a5a0"},{"id":"g1-coordinates-q05","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

C is the point with coordinates (−6, 2).

D is the point with coordinates (4, −8).

Work out the coordinates of the midpoint of CD.

","answerType":"exact","answer":"(-1, -3)","answerDisplay":null,"accept":["(-1,-3)","-1, -3"],"parts":[],"solution":["x coordinate of the midpoint: (-6 + 4) ÷ 2 = -2 ÷ 2 = -1.","y coordinate of the midpoint: (2 + (-8)) ÷ 2 = -6 ÷ 2 = -3.","The midpoint is (-1, -3)."],"diagram":null,"draw":false,"revision":"9d31be3843ca"},{"id":"g1-coordinates-q06","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

A is the point with coordinates (1, 3).

M is the point with coordinates (4, 5).

M is the midpoint of the line segment AB.

Find the coordinates of the point B.

","answerType":"exact","answer":"(7, 7)","answerDisplay":null,"accept":["(7,7)","7, 7"],"parts":[],"solution":["From A to M you go 3 right (1 to 4) and 2 up (3 to 5).","M is the midpoint, so go the same again from M to reach B.","B is (4 + 3, 5 + 2) = (7, 7)."],"diagram":null,"draw":false,"revision":"5e8d114ba4ee"},{"id":"g1-coordinates-q07","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":false,"text":"

Three vertices of a rectangle ABCD are A (1, 2), B (7, 2) and C (7, 6).

  1. Work out the coordinates of the vertex D.
  2. Work out the coordinates of the midpoint of the diagonal AC.
","answerType":"multi","answer":"a) (1, 6) b) (4, 4)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(1, 6)"},{"label":"b","answer":"(4, 4)"}],"solution":["D is directly above A and level with C, so D has the x coordinate of A and the y coordinate of C.","D is (1, 6).","Midpoint of AC: x coordinate (1 + 7) ÷ 2 = 4, y coordinate (2 + 6) ÷ 2 = 4.","The midpoint is (4, 4)."],"diagram":{"type":"axes","square":true,"x":{"min":0,"max":8},"y":{"min":0,"max":7},"points":[{"at":[1,2],"label":"A","labelPos":"below-left"},{"at":[7,2],"label":"B","labelPos":"below-right"},{"at":[7,6],"label":"C"},{"at":[1,6],"label":"D","labelPos":"above-left","answer":true},{"at":[4,4],"label":"(4, 4)","answer":true}],"polygons":[{"pts":[[1,2],[7,2],[7,6],[1,6]],"fill":false,"dashed":true,"answer":true}],"lines":[{"through":[[1,2],[7,6]],"segment":true,"dashed":true,"answer":true}]},"draw":false,"revision":"9b764c07a320"},{"id":"g1-coordinates-q08","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

The points P and Q are shown on the grid.

  1. Write down the coordinates of P.
  2. Write down the coordinates of Q.
  3. Work out the coordinates of the midpoint of PQ.
  4. On the grid, draw the line y = −2.
","answerType":"multi","answer":"a) (-4, 1) b) (2, 5) c) (-1, 3) d) horizontal line through (0, -2)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(-4, 1)"},{"label":"b","answer":"(2, 5)"},{"label":"c","answer":"(-1, 3)"},{"label":"d","answer":"horizontal line drawn through y = -2"}],"solution":["Read the marked points from the grid: P is (-4, 1) and Q is (2, 5).","Midpoint of PQ: x coordinate (-4 + 2) ÷ 2 = -1, y coordinate (1 + 5) ÷ 2 = 3, so (-1, 3).","The line y = -2 is the horizontal line through every point with y coordinate -2."],"diagram":{"type":"axes","square":true,"x":{"min":-6,"max":6},"y":{"min":-6,"max":6},"points":[{"at":[-4,1],"label":"P"},{"at":[2,5],"label":"Q"},{"at":[-1,3],"label":"(−1, 3)","answer":true}],"lines":[{"y":-2,"answer":true,"label":"y = −2","labelAt":[3,-2]}]},"draw":true,"revision":"7d332ae2f19d"},{"id":"g1-coordinates-q09","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":9,"marks":4,"tier":"F","calc":false,"text":"

The diagram shows three identical squares drawn in a row on a coordinate grid.

The bottom left corner of the first square is at the origin O.

The bottom right corner of the third square is at the point (12, 0).

  1. Work out the length of a side of one square.
  2. Write down the coordinates of the top right corner of the first square.
  3. Work out the coordinates of the point where the tops of the second and third squares meet.
","answerType":"multi","answer":"a) 4 b) (4, 4) c) (8, 4)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"4"},{"label":"b","answer":"(4, 4)"},{"label":"c","answer":"(8, 4)"}],"solution":["The three squares together stretch from x = 0 to x = 12, so each side is 12 ÷ 3 = 4.","The first square runs from x = 0 to x = 4 and is 4 tall, so its top right corner is (4, 4).","The second and third squares meet at x = 8, and the tops are at height 4, so the point is (8, 4)."],"diagram":{"type":"axes","square":true,"x":{"min":-1,"max":14},"y":{"min":-2,"max":6},"polygons":[{"pts":[[0,0],[4,0],[4,4],[0,4]],"fill":false},{"pts":[[4,0],[8,0],[8,4],[4,4]],"fill":false},{"pts":[[8,0],[12,0],[12,4],[8,4]],"fill":false}],"points":[{"at":[12,0],"style":"dot"},{"at":[4,4],"label":"(4, 4)","answer":true},{"at":[8,4],"label":"(8, 4)","answer":true}],"texts":[{"at":[12,-1.4],"text":"(12, 0)"}]},"draw":false,"revision":"f520bdd074ef"},{"id":"g1-coordinates-q10","topicId":"g1-coordinates","topic":"Coordinates","grade":"1","topicGrade":"1","strand":"A","difficulty":10,"marks":5,"tier":"FH","calc":true,"text":"

ABCD is a square.

A is the point with coordinates (2, 1).

C is the point with coordinates (8, 7).

A and C are opposite vertices of the square.

  1. Work out the coordinates of the centre of the square.
  2. Work out the coordinates of the other two vertices, B and D.
","answerType":"multi","answer":"a) (5, 4) b) (2, 7) and (8, 1)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(5, 4)"},{"label":"b","answer":"(2, 7) and (8, 1) in either order"}],"solution":["The centre of a square is the midpoint of a diagonal.","Midpoint of AC: ((2 + 8) ÷ 2, (1 + 7) ÷ 2) = (5, 4).","The other diagonal BD also passes through (5, 4), is the same length as AC and is perpendicular to it.","From the centre, AC goes 3 along and 3 up to C, so BD goes 3 along and 3 down: the other vertices are (5 - 3, 4 + 3) = (2, 7) and (5 + 3, 4 - 3) = (8, 1).","Check: (2, 1), (2, 7), (8, 7), (8, 1) form a square with side 6."],"diagram":{"type":"axes","square":true,"x":{"min":0,"max":9},"y":{"min":0,"max":8},"points":[{"at":[2,1],"label":"A","labelPos":"below-left"},{"at":[8,7],"label":"C"},{"at":[8,1],"label":"B","labelPos":"below-right","answer":true},{"at":[2,7],"label":"D","labelPos":"above-left","answer":true},{"at":[5,4],"label":"(5, 4)","answer":true}],"polygons":[{"pts":[[2,1],[8,1],[8,7],[2,7]],"fill":false,"dashed":true,"answer":true}],"lines":[{"through":[[2,1],[8,7]],"segment":true,"dashed":true}]},"draw":false,"revision":"f7e92b6400bb"},{"id":"g1-factors-and-multiples-q01","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write down the multiple of 4 that is between 25 and 31.

","answerType":"exact","answer":"28","answerDisplay":null,"accept":[],"parts":[],"solution":["Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, ...","The only one between 25 and 31 is 28."],"diagram":null,"draw":false,"revision":"b0d3dbd951da"},{"id":"g1-factors-and-multiples-q02","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

Here is a list of numbers.

12   15   21   26   33

Write down the number from the list that is a multiple of 13.

","answerType":"exact","answer":"26","answerDisplay":null,"accept":[],"parts":[],"solution":["Multiples of 13: 13, 26, 39, ...","26 = 2 × 13, so 26 is the multiple of 13."],"diagram":null,"draw":false,"revision":"45506bf4d550"},{"id":"g1-factors-and-multiples-q03","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":false,"text":"

Here is a list of numbers.

8   14   19   27   35

Write down the number from the list that is a prime number.

","answerType":"exact","answer":"19","answerDisplay":null,"accept":[],"parts":[],"solution":["8 = 2 × 4, 14 = 2 × 7, 27 = 3 × 9, 35 = 5 × 7, so none of these are prime.","19 has no factors other than 1 and 19, so 19 is prime."],"diagram":null,"draw":false,"revision":"447f9f2e4596"},{"id":"g1-factors-and-multiples-q04","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

Write down all the factors of 15.

","answerType":"exact","answer":"1, 3, 5, 15","answerDisplay":null,"accept":["1,3,5,15","15, 5, 3, 1"],"parts":[],"solution":["Work in factor pairs: 1 × 15 and 3 × 5.","The factors of 15 are 1, 3, 5 and 15."],"diagram":null,"draw":false,"revision":"03e8dc96ecc4"},{"id":"g1-factors-and-multiples-q05","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":1,"tier":"F","calc":true,"text":"

Write down the factor of 42 that is between 10 and 20.

","answerType":"exact","answer":"14","answerDisplay":null,"accept":[],"parts":[],"solution":["The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42.","The only factor between 10 and 20 is 14."],"diagram":null,"draw":false,"revision":"64c62cdec70a"},{"id":"g1-factors-and-multiples-q06","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":1,"tier":"F","calc":false,"text":"

Write down the smallest positive even multiple of 9.

","answerType":"exact","answer":"18","answerDisplay":null,"accept":[],"parts":[],"solution":["Multiples of 9: 9, 18, 27, 36, ...","9 is odd, so the smallest positive even multiple is 18."],"diagram":null,"draw":false,"revision":"33d558151d76"},{"id":"g1-factors-and-multiples-q07","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":1,"tier":"F","calc":true,"text":"

Here is a list of numbers.

5   7   9   13   24

Write down the number from the list that is a factor of 36.

","answerType":"exact","answer":"9","answerDisplay":null,"accept":[],"parts":[],"solution":["36 ÷ 9 = 4 exactly, so 9 is a factor of 36.","36 is not divisible by 5, 7, 13 or 24."],"diagram":null,"draw":false,"revision":"4411ceb90029"},{"id":"g1-factors-and-multiples-q08","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":1,"tier":"F","calc":true,"text":"

The even numbers form the pattern

2,  4,  6,  8,  ...

Write down the 12th number in this pattern.

","answerType":"exact","answer":"24","answerDisplay":null,"accept":[],"parts":[],"solution":["The nth even number is 2 × n.","The 12th even number is 2 × 12 = 24."],"diagram":null,"draw":false,"revision":"f6f7614e3eb1"},{"id":"g1-factors-and-multiples-q09","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":9,"marks":2,"tier":"F","calc":true,"text":"

List all the factors of 28.

","answerType":"exact","answer":"1, 2, 4, 7, 14, 28","answerDisplay":null,"accept":["1,2,4,7,14,28"],"parts":[],"solution":["Work in factor pairs: 1 × 28, 2 × 14, 4 × 7.","The factors of 28 are 1, 2, 4, 7, 14 and 28."],"diagram":null,"draw":false,"revision":"79e800012b36"},{"id":"g1-factors-and-multiples-q10","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":10,"marks":2,"tier":"F","calc":false,"text":"

List all the prime numbers between 20 and 35.

","answerType":"exact","answer":"23, 29, 31","answerDisplay":null,"accept":["23,29,31"],"parts":[],"solution":["Check each number from 21 to 34.","21 = 3 × 7, 25 = 5 × 5, 27 = 3 × 9, 33 = 3 × 11 and all even numbers in this interval are divisible by 2 and greater than 2, so are not prime.","The primes between 20 and 35 are 23, 29 and 31."],"diagram":null,"draw":false,"revision":"c57fd804893b"},{"id":"g1-factors-and-multiples-q11","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":11,"marks":2,"tier":"F","calc":true,"text":"

Here is a list of numbers.

6   18   25   32   39

  1. Write down the number from the list that is a square number.
  2. Write down the number from the list that is a multiple of 9.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"25"},{"label":"b","answer":"18"}],"solution":["(a) 25 = 5 × 5, so 25 is a square number.","(b) 18 = 2 × 9, so 18 is a multiple of 9."],"diagram":null,"draw":false,"revision":"19dd99f69427"},{"id":"g1-factors-and-multiples-q12","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":12,"marks":2,"tier":"F","calc":false,"text":"

Meera says

"Every multiple of 5 ends in the digit 5."

Give an example to show that Meera is wrong.

","answerType":"open","answer":"Any multiple of 5 ending in 0, e.g. 10","answerDisplay":null,"accept":[],"parts":[],"solution":["Multiples of 5 end in 5 or 0.","Any multiple of 5 that ends in 0, such as 10, 20 or 30, shows Meera is wrong."],"diagram":null,"draw":false,"revision":"654b70d5e9c0"},{"id":"g1-factors-and-multiples-q13","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":13,"marks":2,"tier":"F","calc":true,"text":"

Find two prime numbers that add together to make 32.

","answerType":"open","answer":"3 and 29 (or 13 and 19)","answerDisplay":null,"accept":[],"parts":[],"solution":["Try prime pairs with sum 32.","3 + 29 = 32 and both 3 and 29 are prime.","13 + 19 = 32 also works."],"diagram":null,"draw":false,"revision":"bd3968cdab56"},{"id":"g1-factors-and-multiples-q14","topicId":"g1-factors-and-multiples","topic":"Factors and Multiples","grade":"1","topicGrade":"1","strand":"N","difficulty":14,"marks":3,"tier":"F","calc":true,"text":"

Work out how many positive whole numbers less than 100 are multiples of both 4 and 6.

","answerType":"exact","answer":"8","answerDisplay":null,"accept":[],"parts":[],"solution":["We count positive whole numbers only, excluding zero and negative multiples. The least common multiple of 4 and 6 is 12 because it is the first number in both positive multiplication lists.","A number that is a multiple of both 4 and 6 must be a multiple of 12.","Multiples of 12 less than 100: 12, 24, 36, 48, 60, 72, 84, 96.","There are 8 such numbers."],"diagram":null,"draw":false,"revision":"5752141b8726"},{"id":"g1-metric-conversions-q01","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Change 7 cm into mm.

","answerType":"exact","answer":"70","answerDisplay":null,"accept":["70 mm"],"parts":[],"solution":["1 cm = 10 mm.","7 × 10 = 70 mm."],"diagram":null,"draw":false,"revision":"0f8c1aea68dc"},{"id":"g1-metric-conversions-q02","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Change 4.5 m into cm.

","answerType":"exact","answer":"450","answerDisplay":null,"accept":["450 cm"],"parts":[],"solution":["1 m = 100 cm.","4.5 × 100 = 450 cm."],"diagram":null,"draw":false,"revision":"db3d76c2be7b"},{"id":"g1-metric-conversions-q03","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":3,"marks":1,"tier":"F","calc":false,"text":"

Change 82 mm into cm.

","answerType":"exact","answer":"8.2","answerDisplay":null,"accept":["8.2 cm"],"parts":[],"solution":["Centimetres are larger units than millimetres, so fewer are needed for the same length. Group the millimetres into tens by dividing by 10.","10 mm = 1 cm.","82 ÷ 10 = 8.2 cm."],"diagram":null,"draw":false,"revision":"29ea313a272e"},{"id":"g1-metric-conversions-q04","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

Change 3.2 km into metres.

","answerType":"exact","answer":"3200","answerDisplay":null,"accept":["3200 m","3,200"],"parts":[],"solution":["1 km = 1000 m.","3.2 × 1000 = 3200 m."],"diagram":null,"draw":false,"revision":"83c4b136a2cf"},{"id":"g1-metric-conversions-q05","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":5,"marks":1,"tier":"F","calc":true,"text":"

Change 2600 g into kg.

","answerType":"exact","answer":"2.6","answerDisplay":null,"accept":["2.6 kg"],"parts":[],"solution":["1000 g = 1 kg.","2600 ÷ 1000 = 2.6 kg."],"diagram":null,"draw":false,"revision":"6e80c911616e"},{"id":"g1-metric-conversions-q06","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":6,"marks":1,"tier":"F","calc":true,"text":"

Change 0.75 litres into millilitres.

","answerType":"exact","answer":"750","answerDisplay":null,"accept":["750 ml"],"parts":[],"solution":["1 litre = 1000 ml.","0.75 × 1000 = 750 ml."],"diagram":null,"draw":false,"revision":"69b9f0c40fbc"},{"id":"g1-metric-conversions-q07","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":7,"marks":1,"tier":"F","calc":true,"text":"

Change 640 cm into metres.

","answerType":"exact","answer":"6.4","answerDisplay":null,"accept":["6.4 m"],"parts":[],"solution":["100 cm = 1 m.","640 ÷ 100 = 6.4 m."],"diagram":null,"draw":false,"revision":"8b494aba1e8c"},{"id":"g1-metric-conversions-q08","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":8,"marks":1,"tier":"F","calc":true,"text":"

Write down a sensible metric unit for measuring the mass of an apple.

","answerType":"exact","answer":"grams","answerDisplay":null,"accept":["g","gram"],"parts":[],"solution":["An apple has a small mass, much less than a kilogram.","A sensible metric unit is grams."],"diagram":null,"draw":false,"revision":"fddfa44f22bd"},{"id":"g1-metric-conversions-q09","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":9,"marks":2,"tier":"F","calc":true,"text":"

Jack is 1.62 m tall.

Ben is 158 cm tall.

Who is taller, and by how many centimetres?

","answerType":"exact","answer":"Jack, by 4 cm","answerDisplay":null,"accept":["Jack by 4cm","Jack, 4 cm"],"parts":[],"solution":["Convert to the same units: 1.62 m = 162 cm.","162 − 158 = 4.","Jack is taller, by 4 cm."],"diagram":null,"draw":false,"revision":"375ae57d2872"},{"id":"g1-metric-conversions-q10","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":10,"marks":2,"tier":"F","calc":true,"text":"

Complete these statements.

  1. 5.4 kg = ______ g
  2. 230 cm = ______ m
","answerType":"multi","answer":"a) 5400 b) 2.3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5400"},{"label":"b","answer":"2.3"}],"solution":["a) 1 kg = 1000 g, so 5.4 × 1000 = 5400 g.","b) 100 cm = 1 m, so 230 ÷ 100 = 2.3 m."],"diagram":null,"draw":false,"revision":"b76978ea39ea"},{"id":"g1-metric-conversions-q11","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":11,"marks":3,"tier":"F","calc":true,"text":"

A bottle holds 1.5 litres of apple juice.

Mia fills 6 glasses from the bottle.

Each glass holds 180 ml.

Work out how much juice is left in the bottle.

Give your answer in millilitres.

","answerType":"exact","answer":"420","answerDisplay":null,"accept":["420 ml"],"parts":[],"solution":["1.5 litres = 1500 ml.","6 × 180 = 1080 ml poured out.","1500 − 1080 = 420 ml left."],"diagram":null,"draw":false,"revision":"a351731cf521"},{"id":"g1-metric-conversions-q12","topicId":"g1-metric-conversions","topic":"Metric Conversions","grade":"1","topicGrade":"1","strand":"R","difficulty":12,"marks":3,"tier":"F","calc":true,"text":"

A watering can holds 4.5 litres when full.

Rosa pours in 1.2 litres of water, then 600 ml, then 750 ml.

Work out how much more water Rosa needs to pour in to fill the can.

Give your answer in millilitres.

","answerType":"exact","answer":"1950","answerDisplay":null,"accept":["1950 ml"],"parts":[],"solution":["Convert everything to ml: 4.5 litres = 4500 ml and 1.2 litres = 1200 ml.","Water poured in so far: 1200 + 600 + 750 = 2550 ml.","4500 − 2550 = 1950 ml more needed."],"diagram":null,"draw":false,"revision":"3b46ed9951dc"},{"id":"g1-multiplication-and-division-q01","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Work out 34 × 26

","answerType":"exact","answer":"884","answerDisplay":null,"accept":[],"parts":[],"solution":["34 × 20 = 680","34 × 6 = 204","680 + 204 = 884"],"diagram":null,"draw":false,"revision":"ec461ccca560"},{"id":"g1-multiplication-and-division-q02","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

Four friends win a prize of £59.40

They share the prize equally.

Work out how much each friend receives.

","answerType":"exact","answer":"£14.85","answerDisplay":null,"accept":["14.85","£14.85"],"parts":[],"solution":["Divide £59.40 by 4","59.40 ÷ 4: 4 × 14 = 56, remainder 3.40","3.40 ÷ 4 = 0.85, so each friend receives £14.85"],"diagram":null,"draw":false,"revision":"f29d541ad120"},{"id":"g1-multiplication-and-division-q03","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

A sunflower is 6 cm tall.

Its height doubles every week.

Work out how tall the sunflower will be after 3 weeks.

","answerType":"exact","answer":"48 cm","answerDisplay":null,"accept":["48","48cm","48 cm"],"parts":[],"solution":["After 1 week: 6 × 2 = 12 cm","After 2 weeks: 12 × 2 = 24 cm","After 3 weeks: 24 × 2 = 48 cm"],"diagram":null,"draw":false,"revision":"8062180deff4"},{"id":"g1-multiplication-and-division-q04","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

Priya works for 9 hours.

She is paid a total of £102.60

Priya is paid the same amount for each hour she works.

Work out how much Priya is paid per hour.

","answerType":"exact","answer":"£11.40","answerDisplay":null,"accept":["11.40","£11.40","11.4"],"parts":[],"solution":["Divide the total pay by the number of hours","102.60 ÷ 9 = 11.40","Priya is paid £11.40 per hour"],"diagram":null,"draw":false,"revision":"685969fcf2fb"},{"id":"g1-multiplication-and-division-q05","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

Work out 246 × 37

","answerType":"exact","answer":"9102","answerDisplay":null,"accept":[],"parts":[],"solution":["246 × 30 = 7380","246 × 7 = 1722","7380 + 1722 = 9102"],"diagram":null,"draw":false,"revision":"9cc6594945ef"},{"id":"g1-multiplication-and-division-q06","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

460 pupils are going on a school trip.

Each coach can carry 52 pupils.

Work out the smallest number of coaches needed for the trip.

","answerType":"exact","answer":"9","answerDisplay":null,"accept":["9","9 coaches"],"parts":[],"solution":["460 ÷ 52 = 8.84... so 8 coaches are not enough","8 × 52 = 416, which leaves 460 - 416 = 44 pupils without a seat","9 coaches are needed (9 × 52 = 468 seats)"],"diagram":null,"draw":false,"revision":"97052b53e7c2"},{"id":"g1-multiplication-and-division-q07","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":2,"tier":"F","calc":false,"text":"

Work out 4.7 × 0.3

","answerType":"exact","answer":"1.41","answerDisplay":null,"accept":[],"parts":[],"solution":["Work out 47 × 3 = 141","4.7 has 1 decimal place and 0.3 has 1 decimal place, so the answer has 2 decimal places","4.7 × 0.3 = 1.41","Compared with 47 × 3, both factors were divided by 10. The product is therefore divided by 100: 141 ÷ 100 = 1.41."],"diagram":null,"draw":false,"revision":"10ee1bf82825"},{"id":"g1-multiplication-and-division-q08","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":3,"tier":"F","calc":false,"text":"

Work out 2.36 × 1.4

","answerType":"exact","answer":"3.304","answerDisplay":null,"accept":[],"parts":[],"solution":["Work out 236 × 14: 236 × 10 = 2360 and 236 × 4 = 944","2360 + 944 = 3304","2.36 has 2 decimal places and 1.4 has 1 decimal place, so the answer has 3 decimal places: 3.304"],"diagram":null,"draw":false,"revision":"ae6f2d3c0287"},{"id":"g1-multiplication-and-division-q09","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":9,"marks":3,"tier":"FH","calc":false,"text":"

Work out 40.32 ÷ 0.7

","answerType":"exact","answer":"57.6","answerDisplay":null,"accept":[],"parts":[],"solution":["Multiplying both dividend and divisor by the same non-zero number keeps their ratio, and therefore the quotient, unchanged. It makes the divisor a whole number.","Multiply both numbers by 10: 40.32 ÷ 0.7 = 403.2 ÷ 7","7 × 57 = 399, remainder 4.2","4.2 ÷ 7 = 0.6, so the answer is 57.6"],"diagram":null,"draw":false,"revision":"8e60cb26ac6e"},{"id":"g1-multiplication-and-division-q10","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"4","topicGrade":"1","strand":"N","difficulty":10,"marks":3,"tier":"H","calc":false,"text":"

Work out 5.394 ÷ 0.06

","answerType":"exact","answer":"89.9","answerDisplay":null,"accept":[],"parts":[],"solution":["Multiply both numbers by 100: 5.394 ÷ 0.06 = 539.4 ÷ 6","6 × 89 = 534, remainder 5.4","5.4 ÷ 6 = 0.9, so the answer is 89.9"],"diagram":null,"draw":false,"revision":"d4ca21a07049"},{"id":"g1-multiplication-and-division-q11","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"1","topicGrade":"1","strand":"N","difficulty":11,"marks":6,"tier":"F","calc":false,"text":"
  1. Work out 3.24 × 2.6
  2. Work out 58.14 ÷ 0.9
","answerType":"multi","answer":"a) 8.424, b) 64.6","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8.424"},{"label":"b","answer":"64.6"}],"solution":["a) 324 × 26: 324 × 20 = 6480 and 324 × 6 = 1944, total 8424","a) Three decimal places in the question, so 3.24 × 2.6 = 8.424","b) Multiply both numbers by 10: 58.14 ÷ 0.9 = 581.4 ÷ 9","b) 9 × 64 = 576, remainder 5.4, and 5.4 ÷ 9 = 0.6, so the answer is 64.6"],"diagram":null,"draw":false,"revision":"22e0de00ca2f"},{"id":"g1-multiplication-and-division-q12","topicId":"g1-multiplication-and-division","topic":"Multiplication and Division","grade":"4","topicGrade":"1","strand":"N","difficulty":12,"marks":6,"tier":"H","calc":false,"text":"
  1. Work out 4.62 × 3.5
  2. Work out 33.48 ÷ 1.2
","answerType":"multi","answer":"a) 16.17, b) 27.9","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"16.17"},{"label":"b","answer":"27.9"}],"solution":["a) 462 × 35: 462 × 30 = 13860 and 462 × 5 = 2310, total 16170","a) Three decimal places in the question gives 16.170, which is 16.17","b) Multiply both numbers by 10: 33.48 ÷ 1.2 = 334.8 ÷ 12","b) 12 × 27 = 324, remainder 10.8, and 10.8 ÷ 12 = 0.9, so the answer is 27.9"],"diagram":null,"draw":false,"revision":"9714c90e2747"},{"id":"g1-negative-numbers-q01","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":true,"text":"

Write these numbers in order of size. Start with the smallest number.

−4   3   −7   0   5

","answerType":"exact","answer":"-7, -4, 0, 3, 5","answerDisplay":null,"accept":["-7 -4 0 3 5","-7, -4, 0, 3, 5"],"parts":[],"solution":["Negative numbers are smaller than 0, and -7 is further below zero than -4","Order: -7, -4, 0, 3, 5"],"diagram":null,"draw":false,"revision":"16467952fc7e"},{"id":"g1-negative-numbers-q02","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

Here are the midnight temperatures, in °C, in five towns.

−2   4   −6   1   −1

Write these temperatures in order. Start with the coldest temperature.

","answerType":"exact","answer":"-6, -2, -1, 1, 4","answerDisplay":null,"accept":["-6 -2 -1 1 4","-6, -2, -1, 1, 4"],"parts":[],"solution":["The coldest temperature is the lowest number: -6","Then -2, then -1, then the positive temperatures 1 and 4","Order: -6, -2, -1, 1, 4"],"diagram":null,"draw":false,"revision":"da75ae0e51a7"},{"id":"g1-negative-numbers-q03","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":true,"text":"

Write down a number that is less than −5

","answerType":"open","answer":"Any number less than -5, for example -6","answerDisplay":null,"accept":[],"parts":[],"solution":["A number less than -5 is further below zero than -5","For example -6, -10 or -5.5"],"diagram":null,"draw":false,"revision":"132ccb73071f"},{"id":"g1-negative-numbers-q04","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":false,"text":"

Work out −3 + 8

","answerType":"exact","answer":"5","answerDisplay":null,"accept":[],"parts":[],"solution":["Start at -3 on a number line and move 8 to the right","-3 + 8 = 5"],"diagram":null,"draw":false,"revision":"fbf56dc7b5b9"},{"id":"g1-negative-numbers-q05","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":1,"tier":"F","calc":false,"text":"

Work out 4 − 9

","answerType":"exact","answer":"-5","answerDisplay":null,"accept":[],"parts":[],"solution":["Start at 4 on a number line and move 9 to the left","4 - 9 = -5"],"diagram":null,"draw":false,"revision":"39310efbf550"},{"id":"g1-negative-numbers-q06","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":1,"tier":"F","calc":false,"text":"

Work out −6 − 5

","answerType":"exact","answer":"-11","answerDisplay":null,"accept":[],"parts":[],"solution":["Start at -6 on a number line and move 5 to the left","-6 - 5 = -11"],"diagram":null,"draw":false,"revision":"5308732e6ff0"},{"id":"g1-negative-numbers-q07","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":1,"tier":"F","calc":true,"text":"

Write these numbers in order of size. Start with the smallest number.

−1.5   0.3   −0.35   −3   1

","answerType":"exact","answer":"-3, -1.5, -0.35, 0.3, 1","answerDisplay":null,"accept":["-3 -1.5 -0.35 0.3 1","-3, -1.5, -0.35, 0.3, 1"],"parts":[],"solution":["The most negative number is the smallest: -3","-1.5 is below -0.35 because it is further from zero","Order: -3, -1.5, -0.35, 0.3, 1"],"diagram":null,"draw":false,"revision":"3ca53978da57"},{"id":"g1-negative-numbers-q08","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":1,"tier":"F","calc":true,"text":"

Find the number that is exactly halfway between −8 and 2

","answerType":"exact","answer":"-3","answerDisplay":null,"accept":[],"parts":[],"solution":["Add the two numbers: -8 + 2 = -6","Halve the total: -6 ÷ 2 = -3","Check: -3 is 5 above -8 and 5 below 2"],"diagram":null,"draw":false,"revision":"ee4291f18b6a"},{"id":"g1-negative-numbers-q09","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":9,"marks":2,"tier":"F","calc":false,"text":"

At midnight the temperature in York was −4 °C.

By 4 am the temperature had fallen by 5 °C.

By 10 am the temperature had risen by 8 °C from its 4 am value.

Work out the temperature at 10 am.

","answerType":"exact","answer":"-1°C","answerDisplay":null,"accept":["-1","-1°C","-1 °C","-1 degrees"],"parts":[],"solution":["Temperature at 4 am: -4 - 5 = -9 °C","Temperature at 10 am: -9 + 8 = -1 °C"],"diagram":null,"draw":false,"revision":"af29b0842353"},{"id":"g1-negative-numbers-q10","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":10,"marks":2,"tier":"F","calc":true,"text":"

The temperature in Cardiff is 3 °C.

The temperature in Oslo is −8 °C.

Work out how many degrees colder Oslo is than Cardiff.

","answerType":"exact","answer":"11","answerDisplay":null,"accept":["11","11°C","11 °C","11 degrees"],"parts":[],"solution":["Difference = 3 - (-8) = 3 + 8","Oslo is 11 degrees colder than Cardiff"],"diagram":null,"draw":false,"revision":"1385dae2d5e0"},{"id":"g1-negative-numbers-q11","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":11,"marks":3,"tier":"F","calc":false,"text":"
  1. Work out (−4)2
  2. Work out 2 − (−7)
  3. Put a pair of brackets in this statement to make it true.
    3 − 5 × 2 = −4
","answerType":"multi","answer":"a) 16, b) 9, c) (3 - 5) × 2 = -4","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"16"},{"label":"b","answer":"9"},{"label":"c","answer":"(3 - 5) × 2 = -4"}],"solution":["a) (-4)² = -4 × -4 = 16 (negative times negative is positive)","b) Subtracting -7 is the same as adding 7: 2 + 7 = 9","c) (3 - 5) × 2 = -2 × 2 = -4, so the brackets go around 3 - 5"],"diagram":null,"draw":false,"revision":"38a630060311"},{"id":"g1-negative-numbers-q12","topicId":"g1-negative-numbers","topic":"Negative Numbers","grade":"1","topicGrade":"1","strand":"N","difficulty":12,"marks":3,"tier":"F","calc":true,"text":"

The diagram shows a number line with an arrow A.

  1. Write down the number the arrow A points to.
  2. A second number B is 14 more than the number at A.
    Work out the value of B.
","answerType":"multi","answer":"a) -6, b) 8","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-6"},{"label":"b","answer":"8"}],"solution":["a) From −10 to 0 there are five equal divisions, so each division is 2. A is two divisions to the right of −10: −10 + 4 = −6.","b) B = -6 + 14 = 8"],"diagram":{"type":"numberline","min":-10,"max":10,"step":2,"labels":[-10,0,10],"markers":[{"at":-6,"style":"arrow","label":"A"}]},"draw":false,"revision":"c083cf525890"},{"id":"g1-pictograms-q01","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

The pictogram shows the number of emails Priya received on three days.

Key: ● represents 4 emails

Write down the number of emails Priya received on Tuesday.

","answerType":"exact","answer":"10","answerDisplay":null,"accept":["10 emails"],"parts":[],"solution":["Each full circle represents 4 emails, so a half circle represents 2 emails.","Tuesday shows 2 full circles and 1 half circle: 4 + 4 + 2 = 10 emails."],"diagram":{"type":"pictogram","key":{"value":4,"unit":"emails"},"rows":[{"label":"Monday","count":2},{"label":"Tuesday","count":2.5},{"label":"Wednesday","count":1}]},"draw":false,"revision":"bd5d1fd1f7de"},{"id":"g1-pictograms-q02","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

The pictogram shows the number of cars washed at a garage on Friday and Saturday.

Key: ● represents 6 cars

Work out how many more cars were washed on Saturday than on Friday.

","answerType":"exact","answer":"9","answerDisplay":null,"accept":["9 cars","9 more"],"parts":[],"solution":["Friday: 3 full circles and 1 half circle = 18 + 3 = 21 cars.","Saturday: 5 full circles = 30 cars.","Difference: 30 - 21 = 9 cars."],"diagram":{"type":"pictogram","key":{"value":6,"unit":"cars"},"rows":[{"label":"Friday","count":3.5},{"label":"Saturday","count":5}]},"draw":false,"revision":"99af1781de43"},{"id":"g1-pictograms-q03","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

A pictogram shows the number of tickets sold at a cinema each day.

Key: ● represents 8 tickets

On Thursday 20 tickets were sold.

Work out how many circles should be drawn in the pictogram for Thursday.

","answerType":"exact","answer":"2 and a half circles","answerDisplay":null,"accept":["2.5","2.5 circles","two and a half circles","2 1/2 circles"],"parts":[],"solution":["20 divided by 8 = 2.5.","So Thursday needs 2 full circles and 1 half circle, which is 2 and a half circles."],"diagram":null,"draw":false,"revision":"95ff3e0da9cb"},{"id":"g1-pictograms-q04","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"

The pictogram shows the number of plants a garden centre sold on Monday, Tuesday and Wednesday.

Key: ● represents 4 plants

The garden centre sold 46 plants in total from Monday to Thursday.

Work out how many circles should be drawn for Thursday.

","answerType":"exact","answer":"3 circles","answerDisplay":null,"accept":["3","three circles"],"parts":[],"solution":["Monday: 3 × 4 = 12 plants. Tuesday: 4 × 4 = 16 plants. Wednesday: 1.5 × 4 = 6 plants.","Total shown so far: 12 + 16 + 6 = 34 plants.","Thursday: 46 - 34 = 12 plants.","12 divided by 4 = 3, so 3 full circles should be drawn."],"diagram":{"type":"pictogram","key":{"value":4,"unit":"plants"},"rows":[{"label":"Monday","count":3},{"label":"Tuesday","count":4},{"label":"Wednesday","count":1.5},{"label":"Thursday","count":null,"answer":3}]},"draw":false,"revision":"e54bcf605240"},{"id":"g1-pictograms-q05","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

The pictogram shows the number of loaves a bakery sold on three days.

Key: ● represents 10 loaves

  1. Write down the number of loaves sold on Monday.
  2. The bakery sold 35 loaves on Thursday. How many circles should be drawn for Thursday?
  3. Work out the total number of loaves sold from Monday to Thursday.
","answerType":"multi","answer":"a) 25 b) 3 and a half circles c) 135","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"25"},{"label":"b","answer":"3 and a half circles"},{"label":"c","answer":"135"}],"solution":["a) Monday shows 2 full circles and 1 half circle: 20 + 5 = 25 loaves.","b) 35 divided by 10 = 3.5, so 3 full circles and 1 half circle.","c) Tuesday = 30 and Wednesday = 4.5 × 10 = 45.","Total: 25 + 30 + 45 + 35 = 135 loaves."],"diagram":{"type":"pictogram","key":{"value":10,"unit":"loaves"},"rows":[{"label":"Monday","count":2.5},{"label":"Tuesday","count":3},{"label":"Wednesday","count":4.5},{"label":"Thursday","count":null,"answer":3.5}]},"draw":false,"revision":"7bd96bd7eee1"},{"id":"g1-pictograms-q06","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":6,"marks":4,"tier":"F","calc":false,"text":"

The pictogram shows the number of muffins three friends sold at a school fair. The key is missing.

Key: ● represents ..... muffins

Ben sold 18 muffins.

  1. Complete the key.
  2. Write down the number of muffins Cara sold.
  3. Amy sold 27 muffins. How many circles should be drawn for Amy?
","answerType":"multi","answer":"a) 6 b) 15 c) 4 and a half circles","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6"},{"label":"b","answer":"15"},{"label":"c","answer":"4 and a half circles"}],"solution":["a) Ben's 3 circles represent 18 muffins, so each circle represents 18 divided by 3 = 6 muffins.","b) Cara: 2.5 × 6 = 15 muffins.","c) Amy: 27 divided by 6 = 4.5, so 4 full circles and 1 half circle."],"diagram":{"type":"pictogram","key":{"value":".....","unit":"muffins","answer":6},"rows":[{"label":"Ben","count":3},{"label":"Cara","count":2.5},{"label":"Amy","count":null,"answer":4.5}]},"draw":false,"revision":"d04f276530c1"},{"id":"g1-pictograms-q07","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

The pictogram shows the number of cakes a stall sold on Saturday and Sunday.

Key: ● represents 4 cakes

Each cake was sold for £2.40.

Work out the total amount of money the stall took from selling cakes on Saturday and Sunday.

","answerType":"exact","answer":"£67.20","answerDisplay":null,"accept":["67.20","£67.20","67.2"],"parts":[],"solution":["Saturday: 4.5 × 4 = 18 cakes.","Sunday: 2.5 × 4 = 10 cakes.","Total cakes: 18 + 10 = 28.","Total money: 28 × £2.40 = £67.20."],"diagram":{"type":"pictogram","key":{"value":4,"unit":"cakes"},"rows":[{"label":"Saturday","count":4.5},{"label":"Sunday","count":2.5}]},"draw":false,"revision":"7787013b9f49"},{"id":"g1-pictograms-q08","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":8,"marks":5,"tier":"F","calc":false,"text":"

The pictogram shows the number of books borrowed from a school library on three days.

Key: ● represents 8 books, ◑ represents 4 books, ◔ represents 2 books

  1. Write down the number of books borrowed on Monday.
  2. Work out how many more books were borrowed on Wednesday than on Tuesday.
  3. On Thursday 22 books were borrowed. Describe the circles that should be drawn for Thursday.
","answerType":"multi","answer":"a) 18 b) 12 c) 2 full circles, 1 half circle and 1 quarter circle","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"18"},{"label":"b","answer":"12"},{"label":"c","answer":"2 full circles, 1 half circle and 1 quarter circle"}],"solution":["a) Monday: 8 + 8 + 2 = 18 books.","b) Wednesday: 3 × 8 = 24 books. Tuesday: 8 + 4 = 12 books. Difference: 24 - 12 = 12 books.","c) 22 = 16 + 4 + 2, so draw 2 full circles (16), 1 half circle (4) and 1 quarter circle (2).","Two full circles and one three-quarter circle show the same amount as two full circles, a half circle and a quarter circle. Either drawing is valid."],"diagram":{"type":"pictogram","key":{"value":8,"unit":"books"},"rows":[{"label":"Monday","count":2.25},{"label":"Tuesday","count":1.5},{"label":"Wednesday","count":3},{"label":"Thursday","count":null,"answer":2.75}]},"draw":false,"revision":"b5712e53fb78"},{"id":"g1-pictograms-q09","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":9,"marks":5,"tier":"F","calc":true,"text":"

The pictogram shows the number of smoothies a cafe sold on three days.

Key: ● represents 12 smoothies

  1. Write down the number of smoothies sold on Wednesday.
  2. On Thursday 18 smoothies were sold. How many circles should be drawn for Thursday?
  3. Leah says, \"The cafe sold more than 100 smoothies in total over the four days.\" Is Leah correct? You must show your working.
","answerType":"multi","answer":"a) 42 b) 1 and a half circles c) Yes, the total is 114 which is more than 100","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"42"},{"label":"b","answer":"1 and a half circles"},{"label":"c","answer":"Yes, total = 114 > 100"}],"solution":["a) Wednesday: 3.5 × 12 = 42 smoothies.","b) 18 divided by 12 = 1.5, so 1 full circle and 1 half circle.","c) Monday: 2.5 × 12 = 30. Tuesday: 2 × 12 = 24.","Total: 30 + 24 + 42 + 18 = 114. 114 > 100, so Leah is correct."],"diagram":{"type":"pictogram","key":{"value":12,"unit":"smoothies"},"rows":[{"label":"Monday","count":2.5},{"label":"Tuesday","count":2},{"label":"Wednesday","count":3.5},{"label":"Thursday","count":null,"answer":1.5}]},"draw":false,"revision":"7d146e1694a6"},{"id":"g1-pictograms-q10","topicId":"g1-pictograms","topic":"Pictograms","grade":"1","topicGrade":"1","strand":"S","difficulty":10,"marks":6,"tier":"F","calc":true,"text":"

The pictogram shows the number of T-shirts sold by four shops in one week.

Key: ● represents 20 T-shirts

  1. Write down the number of T-shirts Shop B sold.
  2. Shop D sold 70 T-shirts. How many circles should be drawn for Shop D?
  3. The four shops had a combined target of 250 T-shirts for the week. Did they meet the target? You must show your working.
","answerType":"multi","answer":"a) 50 b) 3 and a half circles c) Yes, the total is 270 which is 20 more than the target","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"50"},{"label":"b","answer":"3 and a half circles"},{"label":"c","answer":"Yes, total = 270 >= 250"}],"solution":["a) Shop B: 2.5 × 20 = 50 T-shirts.","b) 70 divided by 20 = 3.5, so 3 full circles and 1 half circle.","c) Shop A: 4.5 × 20 = 90. Shop C: 3 × 20 = 60.","Total: 90 + 50 + 60 + 70 = 270.","270 > 250, so yes, they beat the target by 20 T-shirts."],"diagram":{"type":"pictogram","key":{"value":20,"unit":"T-shirts"},"rows":[{"label":"Shop A","count":4.5},{"label":"Shop B","count":2.5},{"label":"Shop C","count":3},{"label":"Shop D","count":null,"answer":3.5}]},"draw":false,"revision":"f5f4b355cd4e"},{"id":"g1-place-value-q01","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":true,"text":"

Write down the value of the 7 in the number 3746

","answerType":"exact","answer":"700","answerDisplay":null,"accept":["700","7 hundreds","seven hundred"],"parts":[],"solution":["The 7 is in the hundreds column","Its value is 700"],"diagram":null,"draw":false,"revision":"048086cfad69"},{"id":"g1-place-value-q02","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Write the number four thousand and sixty-two in figures.

","answerType":"exact","answer":"4062","answerDisplay":null,"accept":["4062","4,062"],"parts":[],"solution":["Four thousand = 4000","Sixty-two = 62","4000 + 62 = 4062 (a zero holds the hundreds place)"],"diagram":null,"draw":false,"revision":"554b3a14624f"},{"id":"g1-place-value-q03","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":true,"text":"

Write down the value of the 2 in the number 526 913

","answerType":"exact","answer":"20000","answerDisplay":null,"accept":["20000","20 000","20,000","twenty thousand"],"parts":[],"solution":["The 2 is in the ten thousands column","Its value is 20 000"],"diagram":null,"draw":false,"revision":"eff047e6c41f"},{"id":"g1-place-value-q04","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":false,"text":"

Work out 4.36 × 100

","answerType":"exact","answer":"436","answerDisplay":null,"accept":[],"parts":[],"solution":["Multiplying by 100 moves every digit two place value columns to the left","4.36 × 100 = 436"],"diagram":null,"draw":false,"revision":"f5f32e5a588c"},{"id":"g1-place-value-q05","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

Here are five numbers.

3407   3047   3470   3074   3740

Write the numbers in order of size. Start with the smallest number.

","answerType":"exact","answer":"3047, 3074, 3407, 3470, 3740","answerDisplay":null,"accept":["3047 3074 3407 3470 3740","3047, 3074, 3407, 3470, 3740"],"parts":[],"solution":["All five numbers have 3 thousands, so compare the hundreds: 3047 and 3074 have 0 hundreds","3047 has 4 tens and 3074 has 7 tens, so 3047 is smaller","Then 3407 and 3470 (4 hundreds), then 3740 (7 hundreds)","Order: 3047, 3074, 3407, 3470, 3740"],"diagram":null,"draw":false,"revision":"87dfce118674"},{"id":"g1-place-value-q06","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":1,"tier":"F","calc":true,"text":"

Write down the value of the 3 in the number 8.234

","answerType":"exact","answer":"0.03","answerDisplay":null,"accept":["0.03","3/100","3 hundredths","three hundredths"],"parts":[],"solution":["The 3 is in the hundredths column","Its value is 0.03, which is 3 hundredths"],"diagram":null,"draw":false,"revision":"00588e44d098"},{"id":"g1-place-value-q07","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":1,"tier":"F","calc":true,"text":"

Here are four numbers.

0.505   0.55   0.5   0.51

Write down the smallest of these numbers.

","answerType":"exact","answer":"0.5","answerDisplay":null,"accept":["0.5","0.500"],"parts":[],"solution":["Write each number with three decimal places: 0.505, 0.550, 0.500, 0.510","The smallest is 0.500, which is 0.5"],"diagram":null,"draw":false,"revision":"527eea6f41e6"},{"id":"g1-place-value-q08","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":1,"tier":"F","calc":true,"text":"

Write these numbers in order of size. Start with the smallest number.

0.72   0.7   0.702   0.727   0.27

","answerType":"exact","answer":"0.27, 0.7, 0.702, 0.72, 0.727","answerDisplay":null,"accept":["0.27 0.7 0.702 0.72 0.727","0.27, 0.7, 0.702, 0.72, 0.727"],"parts":[],"solution":["Write each number with three decimal places: 0.720, 0.700, 0.702, 0.727, 0.270","Order: 0.270, 0.700, 0.702, 0.720, 0.727","So the order is 0.27, 0.7, 0.702, 0.72, 0.727"],"diagram":null,"draw":false,"revision":"b0aabbf318e8"},{"id":"g1-place-value-q09","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":9,"marks":1,"tier":"F","calc":false,"text":"

Write down a five-digit whole number in which the digit 8 has the value 8000

","answerType":"open","answer":"Any five-digit number with 8 in the thousands column, for example 18 234","answerDisplay":null,"accept":[],"parts":[],"solution":["The digit 8 must be in the thousands column","In a five-digit number that is the second digit from the left","For example 18 234 or 28 651"],"diagram":null,"draw":false,"revision":"16bad612ae47"},{"id":"g1-place-value-q10","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":10,"marks":1,"tier":"F","calc":false,"text":"

Here are four digit cards.

3   5   2   7

Using each card once, write down the smallest even number that can be made.

","answerType":"exact","answer":"3572","answerDisplay":null,"accept":[],"parts":[],"solution":["An even number must end in the digit 2","Arrange the remaining digits 3, 5, 7 in ascending order at the front: 357","Smallest even number: 3572"],"diagram":null,"draw":false,"revision":"a21ddd6c7354"},{"id":"g1-place-value-q11","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":11,"marks":2,"tier":"F","calc":true,"text":"

Write < or > in each box to make each statement true.

  1. 0.34 □ 0.4
  2. 2.5 □ 2.45
","answerType":"multi","answer":"a) <, b) >","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"<"},{"label":"b","answer":">"}],"solution":["a) 0.34 is less than 0.40, so 0.34 < 0.4","b) 2.50 is greater than 2.45, so 2.5 > 2.45"],"diagram":null,"draw":false,"revision":"54fa122d176b"},{"id":"g1-place-value-q12","topicId":"g1-place-value","topic":"Place Value","grade":"1","topicGrade":"1","strand":"N","difficulty":12,"marks":2,"tier":"F","calc":false,"text":"

Here are four digit cards.

4   8   6   3

  1. Using each card once, write down the largest odd number that can be made.
  2. Using each card once, write down the smallest number that can be made.
","answerType":"multi","answer":"a) 8643, b) 3468","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8643"},{"label":"b","answer":"3468"}],"solution":["a) An odd number must end in 3 (the only odd digit)","a) Arrange 8, 6, 4 in descending order at the front: 8643","b) Arrange all the digits in ascending order: 3468"],"diagram":null,"draw":false,"revision":"b9c91ed48e64"},{"id":"g1-powers-and-roots-q01","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write down the value of 62

","answerType":"exact","answer":"36","answerDisplay":null,"accept":[],"parts":[],"solution":["6² means 6 × 6","6 × 6 = 36"],"diagram":null,"draw":false,"revision":"d3a54e84d916"},{"id":"g1-powers-and-roots-q02","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Write down the value of √81

","answerType":"exact","answer":"9","answerDisplay":null,"accept":[],"parts":[],"solution":["The √ symbol means the non-negative square root: the non-negative number that multiplies by itself to give 81.","9 × 9 = 81, so √81 = 9"],"diagram":null,"draw":false,"revision":"c44a6741d9b8"},{"id":"g1-powers-and-roots-q03","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":false,"text":"

Work out the value of 25

","answerType":"exact","answer":"32","answerDisplay":null,"accept":[],"parts":[],"solution":["2⁵ = 2 × 2 × 2 × 2 × 2","2, 4, 8, 16, 32, so 2⁵ = 32"],"diagram":null,"draw":false,"revision":"d66217960308"},{"id":"g1-powers-and-roots-q04","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

Work out the value of 3√64

","answerType":"exact","answer":"4","answerDisplay":null,"accept":[],"parts":[],"solution":["The cube root of 64 is the number that gives 64 when cubed","4 × 4 × 4 = 64, so the cube root of 64 is 4"],"diagram":null,"draw":false,"revision":"ecf2f365f9a0"},{"id":"g1-powers-and-roots-q05","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":1,"tier":"F","calc":true,"text":"

Work out the value of 0.32

","answerType":"exact","answer":"0.09","answerDisplay":null,"accept":[],"parts":[],"solution":["0.3² = 0.3 × 0.3","3 × 3 = 9, and there are two decimal places, so 0.3 × 0.3 = 0.09"],"diagram":null,"draw":false,"revision":"29da61d2035f"},{"id":"g1-powers-and-roots-q06","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":1,"tier":"F","calc":true,"text":"

Find the value of √0.64

","answerType":"exact","answer":"0.8","answerDisplay":null,"accept":[],"parts":[],"solution":["0.8 × 0.8 = 0.64","So √0.64 = 0.8"],"diagram":null,"draw":false,"revision":"f74a221ad611"},{"id":"g1-powers-and-roots-q07","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":1,"tier":"F","calc":true,"text":"

Use your calculator to work out the value of 4.63

","answerType":"exact","answer":"97.336","answerDisplay":null,"accept":[],"parts":[],"solution":["4.6³ = 4.6 × 4.6 × 4.6","4.6 × 4.6 = 21.16","21.16 × 4.6 = 97.336"],"diagram":null,"draw":false,"revision":"21995b05a95d"},{"id":"g1-powers-and-roots-q08","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":1,"tier":"F","calc":true,"text":"

Here are five numbers.

6   9   16   27   40

Write down the two numbers that are powers of 3

","answerType":"exact","answer":"9 and 27","answerDisplay":null,"accept":["9 and 27","9, 27","27 and 9"],"parts":[],"solution":["The powers of 3 are 3, 9, 27, 81, ...","From the list, 9 = 3² and 27 = 3³"],"diagram":null,"draw":false,"revision":"4f2b61ff49a6"},{"id":"g1-powers-and-roots-q09","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":9,"marks":1,"tier":"F","calc":true,"text":"

Write down the square number that is between 30 and 45

","answerType":"exact","answer":"36","answerDisplay":null,"accept":[],"parts":[],"solution":["The square numbers are 1, 4, 9, 16, 25, 36, 49, ...","The only one between 30 and 45 is 36"],"diagram":null,"draw":false,"revision":"6c2375f5e875"},{"id":"g1-powers-and-roots-q10","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":10,"marks":2,"tier":"F","calc":true,"text":"
  1. Work out (−5)3
  2. Use your calculator to work out √2.56
","answerType":"multi","answer":"a) -125, b) 1.6","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-125"},{"label":"b","answer":"1.6"}],"solution":["a) (-5)³ = -5 × -5 × -5 = 25 × -5 = -125","b) 1.6 × 1.6 = 2.56, so √2.56 = 1.6"],"diagram":null,"draw":false,"revision":"be54cbccfd7f"},{"id":"g1-powers-and-roots-q11","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":11,"marks":2,"tier":"F","calc":true,"text":"

Liam says,

"When you square a number, the answer is always bigger than the number you started with."

Give an example to show that Liam is wrong.

","answerType":"open","answer":"Any counter-example, e.g. 0.5² = 0.25, which is smaller than 0.5 (or 1² = 1, or 0² = 0)","answerDisplay":null,"accept":[],"parts":[],"solution":["Try a number between 0 and 1, for example 0.5","0.5² = 0.25, and 0.25 is smaller than 0.5, so Liam is wrong","1² = 1 or 0² = 0 also work as counter-examples"],"diagram":null,"draw":false,"revision":"bdbfa104874c"},{"id":"g1-powers-and-roots-q12","topicId":"g1-powers-and-roots","topic":"Powers and Roots","grade":"1","topicGrade":"1","strand":"N","difficulty":12,"marks":3,"tier":"F","calc":true,"text":"

Aisha says,

"32 can be written as the sum of two square numbers."

Is Aisha correct?

You must show your working.

","answerType":"open","answer":"Yes, Aisha is correct: 32 = 16 + 16 = 4² + 4²","answerDisplay":null,"accept":[],"parts":[],"solution":["List the square numbers up to 32: 1, 4, 9, 16, 25","Check pairs that total 32: 16 + 16 = 32","16 = 4², so 32 = 4² + 4² and Aisha is correct"],"diagram":null,"draw":false,"revision":"694f08a80cff"},{"id":"g1-rounding-q01","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write the number 62 to the nearest 10

","answerType":"exact","answer":"60","answerDisplay":null,"accept":[],"parts":[],"solution":["62 is between 60 and 70","The units digit is 2, which is less than 5, so round down to 60"],"diagram":null,"draw":false,"revision":"8db00f8b2c48"},{"id":"g1-rounding-q02","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

Write the number 385 to the nearest 10

","answerType":"exact","answer":"390","answerDisplay":null,"accept":[],"parts":[],"solution":["385 is exactly halfway between 380 and 390","The units digit is 5, so round up to 390"],"diagram":null,"draw":false,"revision":"53175f9bdaf3"},{"id":"g1-rounding-q03","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":false,"text":"

Write the number 4278 to the nearest 100

","answerType":"exact","answer":"4300","answerDisplay":null,"accept":[],"parts":[],"solution":["4278 is between 4200 and 4300","The tens digit is 7, which is 5 or more, so round up to 4300"],"diagram":null,"draw":false,"revision":"637e79019530"},{"id":"g1-rounding-q04","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

Write the number 7849 to the nearest 100

","answerType":"exact","answer":"7800","answerDisplay":null,"accept":[],"parts":[],"solution":["7849 is between 7800 and 7900","The tens digit is 4, which is less than 5, so round down to 7800"],"diagram":null,"draw":false,"revision":"d959b9d090f2"},{"id":"g1-rounding-q05","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":1,"tier":"F","calc":false,"text":"

Write the number 6499 to the nearest 1000

","answerType":"exact","answer":"6000","answerDisplay":null,"accept":[],"parts":[],"solution":["6499 is between 6000 and 7000","The hundreds digit is 4, which is less than 5, so round down to 6000"],"diagram":null,"draw":false,"revision":"032f125ef22b"},{"id":"g1-rounding-q06","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":1,"tier":"F","calc":true,"text":"

Write the number 27 618 to the nearest 1000

","answerType":"exact","answer":"28000","answerDisplay":null,"accept":["28000","28 000","28,000"],"parts":[],"solution":["27 618 is between 27 000 and 28 000","The hundreds digit is 6, which is 5 or more, so round up to 28 000"],"diagram":null,"draw":false,"revision":"3b5cb1e3b998"},{"id":"g1-rounding-q07","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":1,"tier":"F","calc":false,"text":"

Write the number 8.6 to the nearest whole number.

","answerType":"exact","answer":"9","answerDisplay":null,"accept":[],"parts":[],"solution":["8.6 is between 8 and 9","The tenths digit is 6, which is 5 or more, so round up to 9"],"diagram":null,"draw":false,"revision":"a673b3b71998"},{"id":"g1-rounding-q08","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":1,"tier":"F","calc":true,"text":"

Write the number 23.48 to the nearest whole number.

","answerType":"exact","answer":"23","answerDisplay":null,"accept":[],"parts":[],"solution":["23.48 is between 23 and 24","The tenths digit is 4, which is less than 5, so round down to 23"],"diagram":null,"draw":false,"revision":"af7b71457bbb"},{"id":"g1-rounding-q09","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":9,"marks":1,"tier":"F","calc":false,"text":"

Write the number 5.74 correct to 1 decimal place.

","answerType":"exact","answer":"5.7","answerDisplay":null,"accept":[],"parts":[],"solution":["The digit after the first decimal place is 4","4 is less than 5, so round down: 5.7"],"diagram":null,"draw":false,"revision":"00520620e2d3"},{"id":"g1-rounding-q10","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":10,"marks":1,"tier":"F","calc":true,"text":"

Write the number 12.86 correct to 1 decimal place.

","answerType":"exact","answer":"12.9","answerDisplay":null,"accept":[],"parts":[],"solution":["The digit after the first decimal place is 6","6 is 5 or more, so round up: 12.9"],"diagram":null,"draw":false,"revision":"e2a9fd77607e"},{"id":"g1-rounding-q11","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":11,"marks":1,"tier":"F","calc":true,"text":"

Write the number 3.472 correct to 2 decimal places.

","answerType":"exact","answer":"3.47","answerDisplay":null,"accept":[],"parts":[],"solution":["The digit after the second decimal place is 2","2 is less than 5, so round down: 3.47"],"diagram":null,"draw":false,"revision":"943485554e60"},{"id":"g1-rounding-q12","topicId":"g1-rounding","topic":"Rounding","grade":"1","topicGrade":"1","strand":"N","difficulty":12,"marks":2,"tier":"F","calc":true,"text":"
  1. Write the number 4738 correct to 2 significant figures.
  2. Write the number 0.05372 correct to 2 significant figures.
","answerType":"multi","answer":"a) 4700, b) 0.054","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"4700"},{"label":"b","answer":"0.054"}],"solution":["Significant figures start at the first non-zero digit. Keep the requested number of digits and inspect the next digit to decide whether to round up. Leading zeros locate the decimal point; they do not count as significant figures.","a) The first two significant figures are 4 and 7; the next digit is 3, so round down: 4700","b) The first two significant figures are 5 and 3; the next digit is 7, so round up: 0.054"],"diagram":null,"draw":false,"revision":"7a994a47ed1d"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q01","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":1,"marks":1,"tier":"F","calc":true,"text":"

Write down the mathematical name of a polygon with exactly five sides.

","answerType":"exact","answer":"pentagon","answerDisplay":null,"accept":["Pentagon","a pentagon"],"parts":[],"solution":["A polygon with five sides is called a pentagon."],"diagram":null,"draw":false,"revision":"ae41d2a26813"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q02","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"
  1. Write down the number of lines of symmetry of a square.
  2. Write down the number of lines of symmetry of a rectangle that is not a square.
","answerType":"multi","answer":"a) 4 b) 2","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"4"},{"label":"b","answer":"2"}],"solution":["A line of symmetry is a fold line that makes the two halves match exactly. Turning a shape does not change how many such lines it has.","A square has 4 lines of symmetry: one horizontal, one vertical and two diagonal.","A rectangle that is not a square has only 2 lines of symmetry: one horizontal and one vertical. Its diagonals are not lines of symmetry."],"diagram":null,"draw":false,"revision":"8f9fc0656460"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q03","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"
  1. Write down the order of rotational symmetry of a regular hexagon.
  2. Write down the number of lines of symmetry of a regular hexagon.
","answerType":"multi","answer":"a) 6 b) 6","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6"},{"label":"b","answer":"6"}],"solution":["A regular hexagon fits onto itself 6 times during a full turn, so its order of rotational symmetry is 6.","A regular polygon has the same number of lines of symmetry as sides, so a regular hexagon has 6 lines of symmetry."],"diagram":null,"draw":false,"revision":"31692f654bb5"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q04","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"
  1. A quadrilateral has exactly one pair of parallel sides. Write down the mathematical name of this quadrilateral.
  2. A different quadrilateral has two pairs of parallel sides. All four of its sides are equal in length but none of its angles are right angles. Write down the mathematical name of this quadrilateral.
","answerType":"multi","answer":"a) trapezium b) rhombus","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"trapezium"},{"label":"b","answer":"rhombus"}],"solution":["A quadrilateral with exactly one pair of parallel sides is a trapezium.","A quadrilateral with two pairs of parallel sides and four equal sides is a rhombus (it would only be a square if it also had right angles)."],"diagram":null,"draw":false,"revision":"b7b48ea94e35"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q05","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"
  1. A quadrilateral has two pairs of equal adjacent sides and exactly one line of symmetry. Write down the mathematical name of this quadrilateral.
  2. A 3D solid has six faces. Every face is a square. Write down the mathematical name of this solid.
","answerType":"multi","answer":"a) kite b) cube","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"kite"},{"label":"b","answer":"cube"}],"solution":["Two pairs of equal adjacent sides with one line of symmetry describes a kite.","A solid with six square faces is a cube."],"diagram":null,"draw":false,"revision":"7fedf7190907"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q06","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":6,"marks":2,"tier":"F","calc":true,"text":"

On the centimetre grid, draw a quadrilateral that has exactly two lines of symmetry and no right angles.

","answerType":"open","answer":"Any rhombus that is not a square, e.g. vertices (2,0), (4,3), (2,6), (0,3)","answerDisplay":null,"accept":["any non-square rhombus drawn accurately on the grid"],"parts":[],"solution":["A quadrilateral with exactly two lines of symmetry and no right angles must be a rhombus that is not a square.","Draw a rhombus on the grid, for example with vertices at (2,0), (4,3), (2,6) and (0,3). Its diagonals are its two lines of symmetry."],"diagram":{"type":"axes","square":true,"grid":true,"ticks":false,"xy":false,"x":{"min":0,"max":8,"step":1},"y":{"min":0,"max":8,"step":1},"polygons":[{"pts":[[2,0],[4,3],[2,6],[0,3]],"fill":false,"answer":true}]},"draw":true,"revision":"052a03c4f41a"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q07","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":7,"marks":2,"tier":"F","calc":true,"text":"

The diagram shows part of a shape drawn on a centimetre grid, together with a dashed vertical line.

Complete the shape so that the dashed line is a line of symmetry of the completed shape.

","answerType":"open","answer":"The mirror image of the given half drawn correctly on the other side of the dashed line","answerDisplay":null,"accept":["each vertex reflected to the correct square on the opposite side of the line"],"parts":[],"solution":["Reflect every vertex of the given half in the dashed line, keeping each point the same distance from the line on the opposite side.","Join the reflected vertices to complete the shape so that the two halves match exactly."],"diagram":{"type":"axes","square":true,"grid":true,"ticks":false,"xy":false,"x":{"min":0,"max":10,"step":1},"y":{"min":0,"max":8,"step":1},"lines":[{"x":5,"dashed":true}],"curves":[{"pts":[[5,1],[3,1],[3,4],[1,4],[5,7]]},{"pts":[[5,1],[7,1],[7,4],[9,4],[5,7]],"answer":true}]},"draw":true,"revision":"5c5e8e524bdb"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q08","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":8,"marks":2,"tier":"F","calc":false,"text":"

The diagram shows five quadrilaterals, labelled A, B, C, D and E, drawn on a centimetre grid.

  1. Write down the letters of the pair of congruent quadrilaterals.
  2. Write down the mathematical name of quadrilateral A.
","answerType":"multi","answer":"a) A and C b) parallelogram","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"A and C"},{"label":"b","answer":"parallelogram"}],"solution":["Congruent shapes are exactly the same shape and size. A and C are identical parallelograms, one rotated, so A and C are the congruent pair.","Quadrilateral A has two pairs of parallel sides with no right angles, so it is a parallelogram."],"diagram":{"type":"axes","square":true,"grid":true,"ticks":false,"xy":false,"x":{"min":0,"max":15,"step":1},"y":{"min":0,"max":9,"step":1},"polygons":[{"pts":[[1,6],[5,6],[6,8],[2,8]],"label":"A"},{"pts":[[7,6],[12,6],[12,8],[7,8]],"label":"B"},{"pts":[[3,0],[3,4],[1,5],[1,1]],"label":"C"},{"pts":[[5,1],[10,1],[9,4],[6,4]],"label":"D"},{"pts":[[11,1],[14,1],[14,4],[11,4]],"label":"E"}]},"draw":false,"revision":"c6989099c7b6"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q09","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":9,"marks":3,"tier":"F","calc":true,"text":"
  1. Write down the mathematical name of a polygon with exactly eight sides.
  2. A quadrilateral has four right angles. Its opposite sides are equal in length, but not all four sides are equal. Write down the mathematical name of this quadrilateral.
  3. Write down the number of lines of symmetry of the quadrilateral in part (b).
","answerType":"multi","answer":"a) octagon b) rectangle c) 2","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"octagon"},{"label":"b","answer":"rectangle"},{"label":"c","answer":"2"}],"solution":["An eight-sided polygon is an octagon.","Four right angles with equal opposite sides, but not all sides equal, describes a rectangle.","A rectangle that is not a square has 2 lines of symmetry, through the midpoints of opposite sides."],"diagram":null,"draw":false,"revision":"a581e218dfae"},{"id":"g1-symmetry-and-properties-of-2d-shapes-q10","topicId":"g1-symmetry-and-properties-of-2d-shapes","topic":"Symmetry and Properties of 2D Shapes","grade":"1","topicGrade":"1","strand":"G","difficulty":10,"marks":2,"tier":"F","calc":true,"text":"

Amira says,

\"Every quadrilateral with four equal sides must be a square.\"

Is Amira correct? You must give a reason for your answer.

","answerType":"open","answer":"No. A rhombus has four equal sides but its angles do not have to be right angles, so it is not always a square.","answerDisplay":null,"accept":["No, it could be a rhombus","No, with a correct counter-example"],"parts":[],"solution":["A square needs four equal sides AND four right angles.","A rhombus has four equal sides but does not need right angles, so a quadrilateral with four equal sides is not always a square. Amira is not correct."],"diagram":null,"draw":false,"revision":"7bd23e7c0f43"},{"id":"g1-time-q01","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Work out the number of days in 3 weeks.

","answerType":"exact","answer":"21","answerDisplay":null,"accept":["21 days"],"parts":[],"solution":["There are 7 days in 1 week.","3 × 7 = 21 days."],"diagram":null,"draw":false,"revision":"4fb6b9899c5b"},{"id":"g1-time-q02","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Change 240 minutes into hours.

","answerType":"exact","answer":"4","answerDisplay":null,"accept":["4 hours","4 hrs"],"parts":[],"solution":["There are 60 minutes in 1 hour.","240 ÷ 60 = 4 hours."],"diagram":null,"draw":false,"revision":"ff29cd939cd1"},{"id":"g1-time-q03","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":3,"marks":1,"tier":"F","calc":false,"text":"

Change 234 hours into minutes.

","answerType":"exact","answer":"165","answerDisplay":null,"accept":["165 minutes","165 mins"],"parts":[],"solution":["2 hours = 2 × 60 = 120 minutes.","34 of an hour = 45 minutes.","120 + 45 = 165 minutes."],"diagram":null,"draw":false,"revision":"da9c123a0b8c"},{"id":"g1-time-q04","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

A film starts at 19:35 and lasts 50 minutes.

Write down the time the film finishes.

","answerType":"exact","answer":"20:25","answerDisplay":null,"accept":["20 25","8:25 pm","8.25 pm","20.25"],"parts":[],"solution":["19:35 + 25 minutes = 20:00.","20:00 + 25 minutes = 20:25.","The film finishes at 20:25."],"diagram":null,"draw":false,"revision":"c6f9b4f84143"},{"id":"g1-time-q05","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

Change 380 minutes into hours and minutes.

","answerType":"exact","answer":"6 hours 20 minutes","answerDisplay":null,"accept":["6 h 20 min","6 hrs 20 mins","6:20"],"parts":[],"solution":["6 hours = 6 × 60 = 360 minutes.","380 − 360 = 20 minutes.","380 minutes = 6 hours 20 minutes."],"diagram":null,"draw":false,"revision":"3add6ed04a56"},{"id":"g1-time-q06","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":6,"marks":2,"tier":"F","calc":true,"text":"

A delivery van leaves a depot at 10:47 and arrives at a warehouse at 13:12 on the same day.

Work out the length of the journey in minutes.

","answerType":"exact","answer":"145","answerDisplay":null,"accept":["145 minutes","145 mins"],"parts":[],"solution":["From 10:47 to 11:00 is 13 minutes.","From 11:00 to 13:00 is 2 hours = 120 minutes.","From 13:00 to 13:12 is 12 minutes.","13 + 120 + 12 = 145 minutes."],"diagram":null,"draw":false,"revision":"4e8eca6d52e4"},{"id":"g1-time-q07","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":7,"marks":2,"tier":"F","calc":true,"text":"

Aisha finished a swimming session at 16:10.

The session lasted 1 hour 35 minutes.

Work out the time the session started.

","answerType":"exact","answer":"14:35","answerDisplay":null,"accept":["14 35","2:35 pm","2.35 pm","14.35"],"parts":[],"solution":["Count back 1 hour from 16:10 to get 15:10.","Count back 35 minutes from 15:10: 15:10 − 10 minutes = 15:00, then 15:00 − 25 minutes = 14:35.","Check: 14:35 + 1 hour 35 minutes = 16:10."],"diagram":null,"draw":false,"revision":"c515a4ee3b07"},{"id":"g1-time-q08","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

Priya leaves her office at 17:26.

It takes her 18 minutes to walk to the bus station.

Her bus leaves the bus station at 17:40.

Does Priya catch the bus? You must show your working.

","answerType":"open","answer":"No - she arrives at 17:44, which is 4 minutes after the bus leaves","answerDisplay":null,"accept":[],"parts":[],"solution":["17:26 + 18 minutes = 17:44.","The bus leaves at 17:40.","17:44 is after 17:40, so Priya misses the bus by 4 minutes."],"diagram":null,"draw":false,"revision":"f84f11f13f33"},{"id":"g1-time-q09","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":9,"marks":3,"tier":"F","calc":true,"text":"

Marco practises the guitar for 45 minutes on each of Monday, Tuesday, Wednesday, Thursday and Friday.

Work out the total time Marco practises in the week.

Give your answer in hours.

","answerType":"exact","answer":"3.75","answerDisplay":null,"accept":["3.75 hours","3 3/4 hours"],"parts":[],"solution":["5 × 45 = 225 minutes.","225 ÷ 60 = 3.75.","Total practice time is 3.75 hours (3 hours 45 minutes)."],"diagram":null,"draw":false,"revision":"a9e4a10c7d82"},{"id":"g1-time-q10","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

Lena drives from her home to her sister's house.

She leaves home at 08:50.

She drives for 1 hour 48 minutes, stops for a 25 minute break, then drives for another 2 hours 37 minutes.

  1. Work out the time Lena arrives at her sister's house.
  2. Lena wants to arrive before 14:00. Does she succeed? Give a reason.
","answerType":"multi","answer":"a) 13:40 b) Yes, with 20 minutes to spare","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"13:40"},{"label":"b","answer":"Yes - 13:40 is 20 minutes before 14:00"}],"solution":["Total journey time: 1 h 48 min + 25 min = 2 h 13 min, then 2 h 13 min + 2 h 37 min = 4 h 50 min.","08:50 + 4 hours = 12:50.","12:50 + 50 minutes = 13:40, so she arrives at 13:40.","13:40 is before 14:00, so yes - she arrives with 20 minutes to spare."],"diagram":null,"draw":false,"revision":"cec3a4ce9409"},{"id":"g1-time-q11","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":11,"marks":4,"tier":"F","calc":true,"text":"

A lorry driver drives for 2.5 hours at an average speed of 48 miles per hour.

She then drives for 1 hour 15 minutes at an average speed of 40 miles per hour.

Work out the total distance she drives.

","answerType":"exact","answer":"170 miles","answerDisplay":null,"accept":["170"],"parts":[],"solution":["First part: distance = speed × time = 48 × 2.5 = 120 miles.","1 hour 15 minutes = 1.25 hours.","Second part: 40 × 1.25 = 50 miles.","Total distance = 120 + 50 = 170 miles."],"diagram":null,"draw":false,"revision":"9836b3f13c49"},{"id":"g1-time-q12","topicId":"g1-time","topic":"Time","grade":"1","topicGrade":"1","strand":"R","difficulty":12,"marks":5,"tier":"F","calc":true,"text":"

Here is part of a train timetable from Ashville to Redmoor.

Train ATrain BTrain C
Ashville (depart)08:3509:0509:40
Redmoor (arrive)09:4210:0810:51

Maya has a meeting in Redmoor at 10:15.

It takes her 12 minutes to walk from Redmoor station to the meeting.

  1. Which is the latest train Maya can catch from Ashville to be at the meeting on time?
  2. How many minutes early for the meeting will she be?
","answerType":"multi","answer":"a) the 08:35 train (Train A) b) 21 minutes early","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Train A (08:35)"},{"label":"b","answer":"21 minutes"}],"solution":["Train B arrives at 10:08. 10:08 + 12 minutes = 10:20, which is after 10:15, so Train B is too late.","Train A arrives at 09:42. 09:42 + 12 minutes = 09:54, which is before 10:15, so Train A works.","The latest suitable train is Train A, departing 08:35.","From 09:54 to 10:15 is 21 minutes, so she is 21 minutes early."],"diagram":null,"draw":false,"revision":"191db57c941e"},{"id":"g1-writing-simplifying-and-ordering-fractions-q01","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":true,"text":"

The shape is divided into 8 equal parts.

3 of the parts are shaded.

Write down the fraction of the shape that is shaded.

","answerType":"exact","answer":"3/8","answerDisplay":"38","accept":["3⁄8"],"parts":[],"solution":["3 of the 8 equal parts are shaded.","The fraction shaded is 38."],"diagram":{"type":"shape","width":360,"polygons":[{"pts":[[0,0],[1,0],[1,3],[0,3]],"fill":true},{"pts":[[1,0],[2,0],[2,3],[1,3]],"fill":true},{"pts":[[2,0],[3,0],[3,3],[2,3]],"fill":true},{"pts":[[3,0],[4,0],[4,3],[3,3]],"fill":false},{"pts":[[4,0],[5,0],[5,3],[4,3]],"fill":false},{"pts":[[5,0],[6,0],[6,3],[5,3]],"fill":false},{"pts":[[6,0],[7,0],[7,3],[6,3]],"fill":false},{"pts":[[7,0],[8,0],[8,3],[7,3]],"fill":false}]},"draw":false,"revision":"6742dab36f14"},{"id":"g1-writing-simplifying-and-ordering-fractions-q02","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

Write 7 as a fraction of 20.

","answerType":"exact","answer":"7/20","answerDisplay":"720","accept":[],"parts":[],"solution":["7 out of 20 is written as the fraction 720."],"diagram":null,"draw":false,"revision":"80c874b1087c"},{"id":"g1-writing-simplifying-and-ordering-fractions-q03","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":true,"text":"

Write the fraction 1218 in its simplest form.

","answerType":"exact","answer":"2/3","answerDisplay":"23","accept":[],"parts":[],"solution":["The highest common factor of 12 and 18 is 6.","12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 1218 = 23.","Dividing numerator and denominator by the same non-zero number preserves the fraction’s value. Using the highest common factor leaves no further whole-number factor to cancel."],"diagram":null,"draw":false,"revision":"f5c2410ae971"},{"id":"g1-writing-simplifying-and-ordering-fractions-q04","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":false,"text":"

Three of these fractions are equivalent to 34.

68   912   1216   1524

Write down the fraction that is not equivalent to 34.

","answerType":"exact","answer":"15/24","answerDisplay":"1524","accept":[],"parts":[],"solution":["68 = 34, 912 = 34 and 1216 = 34.","1524 simplifies to 58, so 1524 is not equivalent to 34."],"diagram":null,"draw":false,"revision":"ff468c772984"},{"id":"g1-writing-simplifying-and-ordering-fractions-q05","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":5,"marks":1,"tier":"F","calc":true,"text":"

Write these fractions in order of size, starting with the smallest.

12   15   13   110

","answerType":"exact","answer":"1/10, 1/5, 1/3, 1/2","answerDisplay":"110, 15, 13, 12","accept":[],"parts":[],"solution":["For unit fractions, the larger the denominator, the smaller the fraction.","So the order is 110, 15, 13, 12.","Each unit fraction takes one equal piece of the same whole. Dividing the whole into more pieces makes each piece smaller."],"diagram":null,"draw":false,"revision":"732adf9285f0"},{"id":"g1-writing-simplifying-and-ordering-fractions-q06","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":6,"marks":2,"tier":"F","calc":true,"text":"

The grid is made of 12 equal squares.

8 of the squares are shaded.

Write the fraction of the grid that is shaded. Give your fraction in its simplest form.

","answerType":"exact","answer":"2/3","answerDisplay":"23","accept":[],"parts":[],"solution":["8 out of 12 squares are shaded, giving 812.","Divide top and bottom by 4: 812 = 23."],"diagram":{"type":"shape","width":300,"polygons":[{"pts":[[0,0],[1,0],[1,1],[0,1]],"fill":true},{"pts":[[1,0],[2,0],[2,1],[1,1]],"fill":true},{"pts":[[2,0],[3,0],[3,1],[2,1]],"fill":true},{"pts":[[3,0],[4,0],[4,1],[3,1]],"fill":true},{"pts":[[0,1],[1,1],[1,2],[0,2]],"fill":true},{"pts":[[1,1],[2,1],[2,2],[1,2]],"fill":true},{"pts":[[2,1],[3,1],[3,2],[2,2]],"fill":true},{"pts":[[3,1],[4,1],[4,2],[3,2]],"fill":true},{"pts":[[0,2],[1,2],[1,3],[0,3]],"fill":false},{"pts":[[1,2],[2,2],[2,3],[1,3]],"fill":false},{"pts":[[2,2],[3,2],[3,3],[2,3]],"fill":false},{"pts":[[3,2],[4,2],[4,3],[3,3]],"fill":false}]},"draw":false,"revision":"1c9671ac0673"},{"id":"g1-writing-simplifying-and-ordering-fractions-q07","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":7,"marks":2,"tier":"F","calc":true,"text":"

There are 32 students in a class.

20 of the students walk to school.

Write the fraction of the students who walk to school. Give your fraction in its simplest form.

","answerType":"exact","answer":"5/8","answerDisplay":"58","accept":[],"parts":[],"solution":["20 out of 32 students walk, giving 2032.","Divide top and bottom by 4: 2032 = 58."],"diagram":null,"draw":false,"revision":"1fd165b739ee"},{"id":"g1-writing-simplifying-and-ordering-fractions-q08","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":8,"marks":2,"tier":"F","calc":false,"text":"

Dev has 30 pens.

He gives away 12 of the pens.

Write the fraction of the pens that Dev has left. Give your fraction in its simplest form.

","answerType":"exact","answer":"3/5","answerDisplay":"35","accept":[],"parts":[],"solution":["Pens left = 30 − 12 = 18.","Fraction left = 1830.","Divide top and bottom by 6: 1830 = 35."],"diagram":null,"draw":false,"revision":"64f8dc3aa4da"},{"id":"g1-writing-simplifying-and-ordering-fractions-q09","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":9,"marks":2,"tier":"F","calc":true,"text":"

Choose the correct symbol, <, > or =, to make each statement true.

  1. 35  ___  710
  2. 46  ___  23
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"<"},{"label":"b","answer":"="}],"solution":["(a) 35 = 610, and 610 < 710, so the symbol is <.","(b) 4/6 simplifies to 2/3, so the symbol is =."],"diagram":null,"draw":false,"revision":"07332f00f65c"},{"id":"g1-writing-simplifying-and-ordering-fractions-q10","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":10,"marks":2,"tier":"F","calc":true,"text":"

Write these fractions in order of size, starting with the smallest.

23   34   712   56

","answerType":"exact","answer":"7/12, 2/3, 3/4, 5/6","answerDisplay":"712, 23, 34, 56","accept":[],"parts":[],"solution":["Use a common denominator of 12.","23 = 812, 34 = 912, 712 = 712, 56 = 1012.","In order: 712, 23, 34, 56."],"diagram":null,"draw":false,"revision":"84e1a4ebd0ad"},{"id":"g1-writing-simplifying-and-ordering-fractions-q11","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":11,"marks":2,"tier":"F","calc":false,"text":"

A sunflower was 40 cm tall.

It is now 55 cm tall.

Write the increase in height as a fraction of the original height. Give your fraction in its simplest form.

","answerType":"exact","answer":"3/8","answerDisplay":"38","accept":[],"parts":[],"solution":["Increase = 55 − 40 = 15 cm.","Fraction of original height = 1540.","Divide top and bottom by 5: 1540 = 38."],"diagram":null,"draw":false,"revision":"806ed59b22e2"},{"id":"g1-writing-simplifying-and-ordering-fractions-q12","topicId":"g1-writing-simplifying-and-ordering-fractions","topic":"Writing, Simplifying and Ordering Fractions","grade":"1","topicGrade":"1","strand":"N","difficulty":12,"marks":3,"tier":"F","calc":true,"text":"

Freya has 45 marbles.

She gives 9 marbles to her brother.

She then loses 6 marbles.

Work out the fraction of her marbles that Freya has left. Give your fraction in its simplest form.

","answerType":"exact","answer":"2/3","answerDisplay":"23","accept":[],"parts":[],"solution":["Marbles left = 45 − 9 − 6 = 30.","Fraction left = 3045.","Divide top and bottom by 15: 3045 = 23."],"diagram":null,"draw":false,"revision":"723a79c96951"},{"id":"g2-angles-q01","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write down the mathematical name for an angle that is greater than 90° but less than 180°.

","answerType":"exact","answer":"obtuse","answerDisplay":null,"accept":["obtuse angle","Obtuse"],"parts":[],"solution":["An angle between 90° and 180° is called an obtuse angle."],"diagram":null,"draw":false,"revision":"f6644847112a"},{"id":"g2-angles-q02","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

An angle has size 210°.

Write down the mathematical name for this type of angle.

","answerType":"exact","answer":"reflex","answerDisplay":null,"accept":["reflex angle","Reflex"],"parts":[],"solution":["An angle greater than 180° but less than 360° is called a reflex angle, and 210° is between 180° and 360°."],"diagram":null,"draw":false,"revision":"efa579646eb9"},{"id":"g2-angles-q03","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

Two angles fit together on a straight line. One of the angles is 64° and the other angle is y°.

  1. Work out the value of y.
  2. Give a reason for your answer.
","answerType":"multi","answer":"a) y = 116 b) angles on a straight line add up to 180 degrees","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"116"},{"label":"b","answer":"angles on a straight line add up to 180 degrees"}],"solution":["Angles on a straight line add up to 180°.","y = 180 − 64 = 116."],"diagram":null,"draw":false,"revision":"bd9c3e5d55e7"},{"id":"g2-angles-q04","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

Three angles meet at a point. Two of the angles are 145° and 118°. The third angle is x°.

Work out the value of x. Give a reason for your answer.

","answerType":"exact","answer":"x = 97 (angles around a point add up to 360 degrees)","answerDisplay":null,"accept":["97"],"parts":[],"solution":["Angles around a point add up to 360°.","145 + 118 = 263.","x = 360 − 263 = 97."],"diagram":null,"draw":false,"revision":"6a1e558a799b"},{"id":"g2-angles-q05","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

AOB is a straight line. The point C lies above the line.

Angle AOC = 5x° and angle COB = (4x − 9)°.

Work out the value of x. Give a reason for each stage of your working.

","answerType":"exact","answer":"x = 21","answerDisplay":null,"accept":["21"],"parts":[],"solution":["Angles on a straight line add up to 180°, so 5x + 4x − 9 = 180.","9x − 9 = 180, so 9x = 189.","x = 189 ÷ 9 = 21.","Check: 5 × 21 = 105 and 4 × 21 − 9 = 75, and 105 + 75 = 180."],"diagram":{"type":"shape","notAccurate":true,"segments":[{"from":[-5,0],"to":[5,0]},{"from":[0,0],"to":[1.035,3.864]}],"points":[{"at":[-5,0],"label":"A","labelPos":"below-left","style":"none"},{"at":[5,0],"label":"B","labelPos":"below-right","style":"none"},{"at":[0,0],"label":"O","labelPos":"below","style":"none"},{"at":[1.035,3.864],"label":"C","labelPos":"above","style":"none"}],"angles":[{"at":[0,0],"from":[-5,0],"to":[1.035,3.864],"r":0.9},{"at":[0,0],"from":[1.035,3.864],"to":[5,0],"r":0.9}],"texts":[{"at":[-1.157,1.507],"text":"5x°","italic":true},{"at":[1.745,1.339],"text":"(4x − 9)°","italic":true}]},"draw":false,"revision":"7f0aef429a09"},{"id":"g2-angles-q06","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

In triangle ABC,

angle A = 2x°, angle B = (3x + 10)° and angle C = (90 − x)°.

Work out the size of the largest angle in the triangle.

","answerType":"exact","answer":"70°","answerDisplay":null,"accept":["70","70 degrees"],"parts":[],"solution":["Angles in a triangle add up to 180°: 2x + 3x + 10 + 90 − x = 180.","4x + 100 = 180, so 4x = 80 and x = 20.","Angle A = 40°, angle B = 3 × 20 + 10 = 70°, angle C = 90 − 20 = 70°.","The largest angle is 70°."],"diagram":null,"draw":false,"revision":"dc563e4772e2"},{"id":"g2-angles-q07","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":7,"marks":3,"tier":"F","calc":false,"text":"

PQRS is a parallelogram.

Angle P = (3x + 15)° and angle Q = (2x − 10)°. Angles P and Q are next to each other on the same side of the parallelogram.

Work out the value of x. Give a reason for each stage of your working.

","answerType":"exact","answer":"x = 35","answerDisplay":null,"accept":["35"],"parts":[],"solution":["Adjacent angles in a parallelogram add up to 180° (co-interior angles between parallel sides).","3x + 15 + 2x − 10 = 180.","5x + 5 = 180, so 5x = 175 and x = 35.","Check: angle P = 120° and angle Q = 60°, and 120 + 60 = 180."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[6,0],[4.5,2.6],[-1.5,2.6]],"labels":["P","Q","R","S"],"fill":false}],"segments":[{"from":[0,0],"to":[6,0],"parallel":1},{"from":[-1.5,2.6],"to":[4.5,2.6],"parallel":1},{"from":[0,0],"to":[-1.5,2.6],"parallel":2},{"from":[6,0],"to":[4.5,2.6],"parallel":2}],"angles":[{"at":[0,0],"from":[6,0],"to":[-1.5,2.6],"r":0.6},{"at":[6,0],"from":[4.5,2.6],"to":[0,0],"r":0.6}],"texts":[{"at":[1,1.732],"text":"(3x + 15)°","italic":true},{"at":[4.268,1],"text":"(2x − 10)°","italic":true}]},"draw":false,"revision":"c76bd1e6fb86"},{"id":"g2-angles-q08","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

The four angles of a quadrilateral are 74°, 101°, 2t° and (t + 5)°.

Work out the size of the largest angle of the quadrilateral.

","answerType":"exact","answer":"120°","answerDisplay":null,"accept":["120","120 degrees"],"parts":[],"solution":["Angles in a quadrilateral add up to 360°: 74 + 101 + 2t + t + 5 = 360.","3t + 180 = 360, so 3t = 180 and t = 60.","The four angles are 74°, 101°, 120° and 65°.","Check: 74 + 101 + 120 + 65 = 360. The largest angle is 120°."],"diagram":null,"draw":false,"revision":"78584bc60c43"},{"id":"g2-angles-q09","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":9,"marks":3,"tier":"F","calc":true,"text":"

The diagram shows two parallel lines crossed by a straight line.

One of the marked angles is 71°. The angle marked x° is the co-interior angle between the parallel lines on the same side of the crossing line.

Work out the value of x. Give a reason for your answer.

","answerType":"exact","answer":"x = 109 (co-interior angles between parallel lines add up to 180 degrees)","answerDisplay":null,"accept":["109"],"parts":[],"solution":["Co-interior (allied) angles between parallel lines add up to 180°.","x = 180 − 71 = 109."],"diagram":{"type":"shape","notAccurate":true,"segments":[{"from":[0,0],"to":[8,0],"parallel":1},{"from":[0,3],"to":[8,3],"parallel":1},{"from":[1.609,-1.135],"to":[3.424,4.135]}],"angles":[{"at":[3.033,3],"from":[0,3],"to":[2,0],"label":"71°"},{"at":[2,0],"from":[0,0],"to":[3.033,3],"label":"x°","italic":true}]},"draw":false,"revision":"89faa4c2506e"},{"id":"g2-angles-q10","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":10,"marks":4,"tier":"F","calc":false,"text":"

ABC is a straight line. The point D lies above the line.

Triangle BCD is isosceles with BD = CD.

Angle DBA = 134°.

Work out the size of angle BDC. Give a reason for each stage of your working.

","answerType":"exact","answer":"88°","answerDisplay":null,"accept":["88","88 degrees"],"parts":[],"solution":["Angle DBC = 180 − 134 = 46° (angles on a straight line add up to 180°).","Angle DCB = angle DBC = 46° (base angles of an isosceles triangle are equal, since BD = CD).","Angle BDC = 180 − 46 − 46 = 88° (angles in a triangle add up to 180°)."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[4,0],[2,2.071]],"labels":["B","C","D"],"fill":false}],"segments":[{"from":[-3,0],"to":[0,0]},{"from":[0,0],"to":[2,2.071],"ticks":1},{"from":[4,0],"to":[2,2.071],"ticks":1}],"points":[{"at":[-3,0],"label":"A","labelPos":"below-left","style":"none"}],"angles":[{"at":[0,0],"from":[2,2.071],"to":[-3,0],"label":"134°"}]},"draw":false,"revision":"070ee4bfccba"},{"id":"g2-angles-q11","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":11,"marks":4,"tier":"F","calc":true,"text":"

The angles of a triangle are in the ratio 2 : 3 : 7.

Show that the triangle contains an obtuse angle.

","answerType":"open","answer":"The angles are 30, 45 and 105 degrees; 105 is greater than 90 so it is obtuse.","answerDisplay":null,"accept":["angles 30, 45, 105 with 105 identified as obtuse"],"parts":[],"solution":["The ratio has 2 + 3 + 7 = 12 parts.","Angles in a triangle add up to 180°, so one part = 180 ÷ 12 = 15°.","The angles are 2 × 15 = 30°, 3 × 15 = 45° and 7 × 15 = 105°.","105° is greater than 90° and less than 180°, so it is an obtuse angle."],"diagram":null,"draw":false,"revision":"38b7a64e506c"},{"id":"g2-angles-q12","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":12,"marks":4,"tier":"F","calc":false,"text":"

PQR is a triangle. The side PQ is extended to the point S.

Angle RQS = 112° and angle QPR = 44°.

Show that triangle PQR is isosceles. Give a reason for each stage of your working.

","answerType":"open","answer":"Angle PQR = 68 (straight line), angle PRQ = 180 - 44 - 68 = 68 (angles in a triangle), so two angles are equal and the triangle is isosceles.","answerDisplay":null,"accept":["a full chain of reasoning ending with two equal angles of 68 degrees"],"parts":[],"solution":["Angle PQR = 180 − 112 = 68° (angles on a straight line add up to 180°).","Angle PRQ = 180 − 44 − 68 = 68° (angles in a triangle add up to 180°).","Angle PQR = angle PRQ = 68°, so triangle PQR has two equal angles.","A triangle with two equal angles is isosceles, so triangle PQR is isosceles."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[4,0],[2.877,2.779]],"labels":["P","Q","R"],"fill":false}],"segments":[{"from":[4,0],"to":[6,0]}],"points":[{"at":[6,0],"label":"S","labelPos":"below-right","style":"none"}],"angles":[{"at":[4,0],"from":[6,0],"to":[2.877,2.779],"label":"112°"},{"at":[0,0],"from":[4,0],"to":[2.877,2.779],"label":"44°"}]},"draw":false,"revision":"8072313eccd7"},{"id":"g2-angles-q13","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":13,"marks":4,"tier":"FH","calc":true,"text":"

Triangle ABC is isosceles with AB = AC.

angle A : angle B = 2 : 5

Work out the size of angle A.

","answerType":"exact","answer":"30°","answerDisplay":null,"accept":["30","30 degrees"],"parts":[],"solution":["Since AB = AC, the base angles are equal: angle B = angle C (base angles of an isosceles triangle).","Let angle A = 2k and angle B = 5k, so angle C = 5k as well.","Angles in a triangle add up to 180°: 2k + 5k + 5k = 180, so 12k = 180 and k = 15.","Angle A = 2 × 15 = 30°. Check: 30 + 75 + 75 = 180."],"diagram":null,"draw":false,"revision":"cc7fcf2dfe0f"},{"id":"g2-angles-q14","topicId":"g2-angles","topic":"Angles","grade":"2","topicGrade":"2","strand":"G","difficulty":14,"marks":5,"tier":"FH","calc":false,"text":"

One angle of an isosceles triangle is 40°.

Work out the possible sizes of the other two angles of the triangle.

You must show that you have considered every possible case.

","answerType":"open","answer":"Either 70 and 70 degrees (40 is the apex angle) or 40 and 100 degrees (40 is a base angle).","answerDisplay":null,"accept":["70 and 70, or 40 and 100"],"parts":[],"solution":["Case 1: 40° is the angle between the two equal sides. The other two angles are equal, so each is (180 − 40) ÷ 2 = 70°. The angles are 70° and 70°.","Case 2: 40° is one of the two equal base angles. The other base angle is also 40°, and the third angle is 180 − 40 − 40 = 100°. The angles are 40° and 100°.","So the other two angles are either 70° and 70°, or 40° and 100°."],"diagram":null,"draw":false,"revision":"53cd774a63ea"},{"id":"g2-area-and-perimeter-q01","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

A rectangle has length 9 cm and width 4 cm.

  1. Work out the area of the rectangle.
  2. Work out the perimeter of the rectangle.
","answerType":"multi","answer":"a) 36 cm² b) 26 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"36 cm^2"},{"label":"b","answer":"26 cm"}],"solution":["Perimeter is the total distance around the boundary. Area counts the unit squares inside the shape, so its units are squared.","Area = length × width = 9 × 4 = 36 cm².","Perimeter = 2 × (9 + 4) = 2 × 13 = 26 cm."],"diagram":null,"draw":false,"revision":"3be632f77a12"},{"id":"g2-area-and-perimeter-q02","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

A triangle has base 12 cm and perpendicular height 5 cm.

Work out the area of the triangle.

","answerType":"exact","answer":"30 cm²","answerDisplay":null,"accept":["30","30 cm^2","30cm2"],"parts":[],"solution":["Two identical copies of a triangle make a parallelogram with the same base and perpendicular height. The triangle has half that area.","Area of a triangle = 12 × base × perpendicular height.","Area = 12 × 12 × 5 = 30 cm²."],"diagram":null,"draw":false,"revision":"ec82bb113ccc"},{"id":"g2-area-and-perimeter-q03","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Ben wants to work out the perimeter of a square with sides of length 6 cm.

Ben writes:

perimeter = 6 × 6 = 36 cm

Explain the mistake Ben has made.

","answerType":"open","answer":"6 x 6 gives the area of the square, not the perimeter. The perimeter is 4 x 6 = 24 cm.","answerDisplay":null,"accept":["he has worked out the area; perimeter should be 4 x 6 = 24 cm"],"parts":[],"solution":["Multiplying 6 × 6 gives the area of the square (in cm²), not the perimeter.","The perimeter is the total distance around the square: 4 × 6 = 24 cm."],"diagram":null,"draw":false,"revision":"e599c90a209c"},{"id":"g2-area-and-perimeter-q04","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"

An L-shape is made by removing a rectangle 3 cm by 2 cm from one corner of a rectangle 8 cm by 6 cm.

Work out the perimeter of the L-shape.

","answerType":"exact","answer":"28 cm","answerDisplay":null,"accept":["28","28cm"],"parts":[],"solution":["The two cut edges of the removed corner are 3 cm and 2 cm long.","The missing parts of the original sides are also 3 cm and 2 cm, so the total distance around the L-shape is the same as the perimeter of the full 8 cm by 6 cm rectangle.","Perimeter = 8 + 6 + 3 + 2 + (8 − 3) + (6 − 2) = 8 + 6 + 3 + 2 + 5 + 4 = 28 cm.","Check: 2 × (8 + 6) = 28 cm."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[8,0],[8,4],[5,4],[5,6],[0,6]],"sides":["8 cm",null,"3 cm","2 cm",null,"6 cm"],"fill":false}],"angles":[{"at":[0,0],"from":[8,0],"to":[0,6],"right":true},{"at":[8,0],"from":[8,4],"to":[0,0],"right":true},{"at":[8,4],"from":[5,4],"to":[8,0],"right":true},{"at":[5,6],"from":[0,6],"to":[5,4],"right":true},{"at":[0,6],"from":[0,0],"to":[5,6],"right":true}]},"draw":false,"revision":"3b59dd1632ff"},{"id":"g2-area-and-perimeter-q05","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

A rectangle is made from three identical squares placed side by side in a row.

The perimeter of the rectangle is 48 cm.

Work out the area of the rectangle.

","answerType":"exact","answer":"108 cm²","answerDisplay":null,"accept":["108","108 cm^2"],"parts":[],"solution":["Let each square have side s cm. The rectangle is 3s cm long and s cm wide.","Perimeter = 2 × (3s + s) = 8s, so 8s = 48 and s = 6.","The rectangle is 18 cm by 6 cm, so its area = 18 × 6 = 108 cm²."],"diagram":null,"draw":false,"revision":"bbcfd8a39c9a"},{"id":"g2-area-and-perimeter-q06","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":6,"marks":4,"tier":"F","calc":true,"text":"

Rectangle A has length 12 cm and width 5 cm.

Rectangle B has width 4 cm. The perimeter of rectangle B is equal to the perimeter of rectangle A.

Work out the area of rectangle B.

","answerType":"exact","answer":"52 cm²","answerDisplay":null,"accept":["52","52 cm^2"],"parts":[],"solution":["Perimeter of A = 2 × (12 + 5) = 34 cm.","For rectangle B: 2 × (length + 4) = 34, so length + 4 = 17 and length = 13 cm.","Area of B = 13 × 4 = 52 cm²."],"diagram":null,"draw":false,"revision":"3b2825968cde"},{"id":"g2-area-and-perimeter-q07","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

A rectangular patio is 4.5 m long and 3.6 m wide.

The patio is to be covered completely with square paving slabs. Each slab has sides of length 0.9 m and costs £7.25.

Work out the total cost of the slabs needed. You may assume no slabs are cut.

","answerType":"exact","answer":"£145","answerDisplay":null,"accept":["145","£145","£145.00"],"parts":[],"solution":["Slabs along the length: 4.5 ÷ 0.9 = 5.","Slabs along the width: 3.6 ÷ 0.9 = 4.","Number of slabs = 5 × 4 = 20.","Total cost = 20 × £7.25 = £145."],"diagram":null,"draw":false,"revision":"00503a5463ed"},{"id":"g2-area-and-perimeter-q08","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":8,"marks":4,"tier":"F","calc":false,"text":"

A triangle has base (x + 4) cm and perpendicular height 6 cm.

The area of the triangle is 42 cm².

  1. Show that 3x + 12 = 42.
  2. Work out the value of x.
","answerType":"multi","answer":"a) shown b) x = 10","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"area = 1/2 x 6 x (x + 4) = 3(x + 4) = 3x + 12, and this equals 42","answerDisplay":"area = 12 x 6 x (x + 4) = 3(x + 4) = 3x + 12, and this equals 42"},{"label":"b","answer":"10"}],"solution":["Area = 12 × base × height = 12 × (x + 4) × 6 = 3(x + 4) = 3x + 12.","The area is 42 cm², so 3x + 12 = 42.","3x = 30, so x = 10.","Check: base = 14 cm, area = 12 × 14 × 6 = 42 cm²."],"diagram":null,"draw":false,"revision":"5acec94fd4ff"},{"id":"g2-area-and-perimeter-q09","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":9,"marks":5,"tier":"F","calc":true,"text":"

The end wall of a barn is a trapezium. The parallel sides of the trapezium are 6 m and 8 m long, and the perpendicular distance between them is 3 m.

Priya is going to paint the wall. One tin of paint covers 5 m² and costs £13.50.

Work out the total cost of the tins of paint Priya needs to buy.

","answerType":"exact","answer":"£67.50","answerDisplay":null,"accept":["67.50","£67.50","67.5"],"parts":[],"solution":["Area of the trapezium = 12 × (6 + 8) × 3 = 12 × 14 × 3 = 21 m².","Tins needed: 21 ÷ 5 = 4.2, so Priya must buy 5 tins (she cannot buy part of a tin).","Total cost = 5 × £13.50 = £67.50."],"diagram":null,"draw":false,"revision":"7aea05774c41"},{"id":"g2-area-and-perimeter-q10","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":10,"marks":5,"tier":"FH","calc":true,"text":"

Rectangle P has length 20 cm and width x cm.

Square Q has sides of length 8 cm.

The area of rectangle P is twice the area of square Q.

Work out the perimeter of rectangle P.

","answerType":"exact","answer":"52.8 cm","answerDisplay":null,"accept":["52.8","52.8cm"],"parts":[],"solution":["Area of Q = 8 × 8 = 64 cm².","Area of P = 2 × 64 = 128 cm².","20x = 128, so x = 6.4 cm.","Perimeter of P = 2 × (20 + 6.4) = 2 × 26.4 = 52.8 cm."],"diagram":null,"draw":false,"revision":"434439ea29f0"},{"id":"g2-area-and-perimeter-q11","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":11,"marks":5,"tier":"FH","calc":true,"text":"

An isosceles trapezium has parallel sides of length 6 cm and b cm, where b > 6.

The perpendicular distance between the parallel sides is 4 cm, and each of the two sloping sides is 5 cm long.

The area of the trapezium is 36 cm².

  1. Work out the value of b.
  2. Work out the perimeter of the trapezium.
","answerType":"multi","answer":"a) b = 12 b) 28 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"12"},{"label":"b","answer":"28 cm"}],"solution":["Area of a trapezium = 12 × (sum of parallel sides) × height.","12 × (6 + b) × 4 = 36, so 2 × (6 + b) = 36 and 6 + b = 18, giving b = 12.","Perimeter = 6 + 12 + 5 + 5 = 28 cm."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[12,0],[9,4],[3,4]],"sides":["b cm","5 cm","6 cm","5 cm"],"fill":false}],"segments":[{"from":[9,4],"to":[9,0],"dashed":true,"thin":true,"label":"4 cm","labelPos":"left"}],"angles":[{"at":[9,0],"from":[12,0],"to":[9,4],"right":true}]},"draw":false,"revision":"1deb9e6c1b0f"},{"id":"g2-area-and-perimeter-q12","topicId":"g2-area-and-perimeter","topic":"Area and Perimeter","grade":"2","topicGrade":"2","strand":"G","difficulty":12,"marks":7,"tier":"FH","calc":true,"text":"

Jade is tiling a wall in the shape of a trapezium. The parallel sides of the wall are 3 m and 5 m long, and the perpendicular distance between them is 2.4 m.

Each tile is a square with sides of length 30 cm. Tiles are sold in packs of 12. Each pack costs £18.

  1. Work out the total cost of the packs of tiles Jade needs to buy. You may assume no tiles are wasted.
  2. Jade decides to allow 10% extra tiles for cutting and breakage. Does this change the number of packs she needs to buy? You must show your working.
","answerType":"multi","answer":"a) £162 (9 packs) b) Yes - she now needs 118 tiles, which is 10 packs","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£162"},{"label":"b","answer":"yes, 10 packs are now needed instead of 9"}],"solution":["Area of the wall = 12 × (3 + 5) × 2.4 = 12 × 8 × 2.4 = 9.6 m².","Each tile is 0.3 m × 0.3 m = 0.09 m².","Tiles needed: 9.6 ÷ 0.09 = 106.66..., so 107 tiles.","Packs needed: 107 ÷ 12 = 8.91..., so 9 packs. Cost = 9 × £18 = £162.","Allowing 10% extra: 107 × 1.1 = 117.7, so 118 tiles are needed.","9 packs give only 108 tiles, so Jade now needs 10 packs. Yes, the number of packs changes."],"diagram":null,"draw":false,"revision":"8119e6defc17"},{"id":"g2-averages-q01","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Here are seven numbers.

3  7  7  4  9  7  2

Write down the mode of these numbers.

","answerType":"exact","answer":"7","answerDisplay":null,"accept":[],"parts":[],"solution":["The mode is the value that appears most often.","7 appears three times, more than any other number, so the mode is 7."],"diagram":null,"draw":false,"revision":"e01893d71953"},{"id":"g2-averages-q02","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Here are five numbers.

12  5  19  8  3

Work out the range of these numbers.

","answerType":"exact","answer":"16","answerDisplay":null,"accept":[],"parts":[],"solution":["Range = highest value minus lowest value.","19 - 3 = 16."],"diagram":null,"draw":false,"revision":"65962c6645a6"},{"id":"g2-averages-q03","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

Here are four numbers.

6  11  9  14

Work out the mean of these numbers.

","answerType":"exact","answer":"10","answerDisplay":null,"accept":[],"parts":[],"solution":["The mean is the amount each value would have if the total were shared equally. Add all the values, then divide by how many values there are.","Total: 6 + 11 + 9 + 14 = 40.","Mean: 40 divided by 4 = 10."],"diagram":null,"draw":false,"revision":"58261285b3f8"},{"id":"g2-averages-q04","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"

A shoe shop records the sizes of the last eight pairs of trainers it sold.

5  7  6  5  8  5  9  6

  1. Write down the modal shoe size.
  2. Work out the range of the shoe sizes.
","answerType":"multi","answer":"a) 5 b) 4","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5"},{"label":"b","answer":"4"}],"solution":["a) Size 5 appears three times, more than any other size, so the mode is 5.","b) Range: 9 - 5 = 4."],"diagram":null,"draw":false,"revision":"76b7cc2aabad"},{"id":"g2-averages-q05","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

Owen watches five videos. Here are the lengths of the videos, in seconds.

210  185  240  195  230

Work out the mean length of the videos. Give your answer in minutes and seconds.

","answerType":"exact","answer":"3 minutes 32 seconds","answerDisplay":null,"accept":["3 min 32 s","3m 32s","212 seconds = 3 minutes 32 seconds"],"parts":[],"solution":["Total: 210 + 185 + 240 + 195 + 230 = 1060 seconds.","Mean: 1060 divided by 5 = 212 seconds.","212 seconds = 180 seconds + 32 seconds = 3 minutes 32 seconds."],"diagram":null,"draw":false,"revision":"fc874a6ceef6"},{"id":"g2-averages-q06","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

There are 10 boys and 15 girls in a class.

The mean score of the 10 boys in a test was 6.2.

The mean score of the 15 girls in the same test was 5.8.

Mr Field says, \"The mean score for the whole class was 6.\"

Show that Mr Field is wrong.

","answerType":"open","answer":"Whole-class mean = 5.96, not 6","answerDisplay":null,"accept":["5.96"],"parts":[],"solution":["The groups have different sizes, so their means cannot be given equal weight. Recover each group’s total first, then divide the combined total by the combined number of people.","Total for boys: 10 × 6.2 = 62.","Total for girls: 15 × 5.8 = 87.","Whole-class total: 62 + 87 = 149, and 10 + 15 = 25 students.","Mean: 149 divided by 25 = 5.96, which is not 6, so Mr Field is wrong."],"diagram":null,"draw":false,"revision":"b236eb96d0b0"},{"id":"g2-averages-q07","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":7,"marks":3,"tier":"FH","calc":true,"text":"

The mean of 8 numbers is 12.

The mean of 5 of these numbers is 9.

Work out the mean of the other 3 numbers.

","answerType":"exact","answer":"17","answerDisplay":null,"accept":[],"parts":[],"solution":["Total of all 8 numbers: 8 × 12 = 96.","Total of the 5 numbers: 5 × 9 = 45.","Total of the other 3 numbers: 96 - 45 = 51.","Mean: 51 divided by 3 = 17."],"diagram":null,"draw":false,"revision":"a17a4b3c81dd"},{"id":"g2-averages-q08","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

Freya plays six rounds of a quiz. Her mean score for the six rounds is 15.

Here are her scores for the first five rounds.

12  18  9  21  14

Work out her score in the sixth round.

","answerType":"exact","answer":"16","answerDisplay":null,"accept":[],"parts":[],"solution":["Total of all six scores: 6 × 15 = 90.","Total of the first five scores: 12 + 18 + 9 + 21 + 14 = 74.","Sixth score: 90 - 74 = 16."],"diagram":null,"draw":false,"revision":"9e482c72f461"},{"id":"g2-averages-q09","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

The mean of 7 numbers is 14.

One of the numbers is removed.

The mean of the remaining 6 numbers is 12.5.

Work out the number that was removed.

","answerType":"exact","answer":"23","answerDisplay":null,"accept":[],"parts":[],"solution":["Total of the 7 numbers: 7 × 14 = 98.","Total of the remaining 6 numbers: 6 × 12.5 = 75.","Number removed: 98 - 75 = 23."],"diagram":null,"draw":false,"revision":"099bc0f8f428"},{"id":"g2-averages-q10","topicId":"g2-averages","topic":"Averages","grade":"2","topicGrade":"2","strand":"S","difficulty":10,"marks":4,"tier":"FH","calc":true,"text":"

Nine students in Year 7 and nine students in Year 11 ran a race. Here are their times, in seconds.

Year 7:  12  15  15  18  20  21  24  26  28

Year 11:  8  10  13  14  16  19  22  25  27

Compare the times of the Year 7 students with the times of the Year 11 students.

","answerType":"open","answer":"Year 11 median (16 s) is lower than Year 7 median (20 s), so Year 11 were faster on average; Year 7 range (16 s) is smaller than Year 11 range (19 s), so Year 7 times were more consistent","answerDisplay":null,"accept":[],"parts":[],"solution":["Year 7 median: the 5th value = 20 seconds. Year 11 median: the 5th value = 16 seconds.","Year 7 range: 28 - 12 = 16 seconds. Year 11 range: 27 - 8 = 19 seconds.","The Year 11 median is lower, so on average the Year 11 students ran faster.","The Year 7 range is smaller, so the Year 7 times were more consistent."],"diagram":null,"draw":false,"revision":"676c8e6952fc"},{"id":"g2-averages-q11","topicId":"g2-averages","topic":"Averages","grade":"4","topicGrade":"2","strand":"S","difficulty":11,"marks":5,"tier":"H","calc":true,"text":"

A running club has m men and w women.

The mean age of the men is 38.

The mean age of the women is 26.

The mean age of all the members of the club is 30.

Find the ratio m : w. Give your answer in its simplest form.

","answerType":"exact","answer":"1 : 2","answerDisplay":null,"accept":["1:2","m:w = 1:2"],"parts":[],"solution":["Total age of the men: 38m. Total age of the women: 26w.","Mean of everyone: (38m + 26w) divided by (m + w) = 30.","So 38m + 26w = 30m + 30w.","8m = 4w, so 2m = w.","Therefore m : w = 1 : 2."],"diagram":null,"draw":false,"revision":"5e6f440c6131"},{"id":"g2-averages-q12","topicId":"g2-averages","topic":"Averages","grade":"4","topicGrade":"2","strand":"S","difficulty":12,"marks":6,"tier":"H","calc":true,"text":"

Five cards each have a positive whole number written on them.

The mean of the five numbers is 6.

The mode of the five numbers is 4.

The median of the five numbers is 5.

The range of the five numbers is 7.

Find the five numbers.

","answerType":"exact","answer":"4, 4, 5, 6, 11","answerDisplay":null,"accept":["4 4 5 6 11","11, 6, 5, 4, 4"],"parts":[],"solution":["Mean 6 means the total is 5 × 6 = 30.","Write the numbers in order: a, b, c, d, e. The median gives c = 5.","The mode is 4, so 4 must appear at least twice. Both 4s must be below the median, so a = 4 and b = 4.","Range 7 gives e - a = 7, so e = 4 + 7 = 11.","Total: 4 + 4 + 5 + d + 11 = 30, so d = 6.","Check: numbers 4, 4, 5, 6, 11 have mean 6, mode 4, median 5, range 7."],"diagram":null,"draw":false,"revision":"cc4b30e6883f"},{"id":"g2-bar-charts-q01","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

A bar chart shows the favourite fruit of a group of students. The table gives the height of each bar.

FruitNumber of students
Apple9
Banana12
Grapes7
Orange5

Write down the modal fruit.

","answerType":"exact","answer":"Banana","answerDisplay":null,"accept":["banana"],"parts":[],"solution":["The mode is the category with the tallest bar.","Banana has the highest frequency, 12, so the modal fruit is banana."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Apple","Banana","Grapes","Orange"],"label":"Fruit"},"y":{"min":0,"max":14,"step":2,"minor":2,"label":"Number of students"},"bars":[{"cat":0,"height":9},{"cat":1,"height":12},{"cat":2,"height":7},{"cat":3,"height":5}]},"draw":false,"revision":"bc59da165b33"},{"id":"g2-bar-charts-q02","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

A bar chart shows the number of pets sold by a pet shop on four days. The table gives the height of each bar.

DayPets sold
Monday14
Tuesday8
Wednesday11
Thursday17
  1. Write down the day on which the fewest pets were sold.
  2. Work out how many more pets were sold on Thursday than on Tuesday.
","answerType":"multi","answer":"a) Tuesday b) 9","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Tuesday"},{"label":"b","answer":"9"}],"solution":["a) The shortest bar is Tuesday, with 8 pets.","b) 17 - 8 = 9 more pets."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Mon","Tue","Wed","Thu"],"label":"Day"},"y":{"min":0,"max":20,"step":2,"minor":2,"label":"Pets sold"},"bars":[{"cat":0,"height":14},{"cat":1,"height":8},{"cat":2,"height":11},{"cat":3,"height":17}]},"draw":false,"revision":"05460a25f071"},{"id":"g2-bar-charts-q03","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

Harry draws a bar chart to show how students travel to school.

Write down two things that are wrong with Harry's bar chart.

","answerType":"open","answer":"The frequency axis has no label; the bars are not all the same width","answerDisplay":null,"accept":[],"parts":[],"solution":["Error 1: the vertical axis has no label, so you cannot tell what the heights represent.","Error 2: the bar for one category is wider than the others - all bars should be the same width."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Walk","Car","Bus","Cycle"],"label":"Method of travel"},"y":{"min":0,"max":14,"step":2,"minor":2},"bars":[{"cat":0,"height":12,"width":0.5},{"cat":1,"height":9,"width":0.9},{"cat":2,"height":6,"width":0.5},{"cat":3,"height":3,"width":0.5}]},"draw":false,"revision":"553e51cb6ed8"},{"id":"g2-bar-charts-q04","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"

The table shows the favourite colours of a group of children.

ColourFrequency
Red6
Blue9
Green4
Yellow7

Draw a suitable bar chart for this information.

","answerType":"open","answer":"Bar chart with bars of heights 6, 9, 4, 7 for Red, Blue, Green, Yellow, equal widths, gaps between bars, both axes labelled","answerDisplay":null,"accept":[],"parts":[],"solution":["Label the horizontal axis with the four colours and the vertical axis Frequency.","Use a sensible scale, for example 1 square = 1 child.","Draw bars of equal width with gaps between them: Red 6, Blue 9, Green 4, Yellow 7."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Red","Blue","Green","Yellow"],"label":"Colour"},"y":{"min":0,"max":10,"step":1,"label":"Frequency"},"bars":[{"cat":0,"height":6,"answer":true},{"cat":1,"height":9,"answer":true},{"cat":2,"height":4,"answer":true},{"cat":3,"height":7,"answer":true}]},"draw":true,"revision":"1c1b219450f0"},{"id":"g2-bar-charts-q05","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

A bar chart shows how the students in class 8R travel to school. The table gives the height of each bar.

MethodNumber of students
Walk12
Car18
Bus6
  1. Write down the number of students who travel by bus.
  2. Write down the ratio of the number of students who walk to the number who travel by car. Give your ratio in its simplest form.
","answerType":"multi","answer":"a) 6 b) 2 : 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6"},{"label":"b","answer":"2 : 3"}],"solution":["a) The bus bar has height 6.","b) Walk : Car = 12 : 18. Divide both sides by 6 to get 2 : 3."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Walk","Car","Bus"],"label":"Method"},"y":{"min":0,"max":20,"step":2,"minor":2,"label":"Number of students"},"bars":[{"cat":0,"height":12},{"cat":1,"height":18},{"cat":2,"height":6}]},"draw":false,"revision":"25900a5d29e9"},{"id":"g2-bar-charts-q06","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

40 students each joined exactly one after-school club. A bar chart shows the numbers for three of the clubs. The table gives the height of each of these bars.

ClubNumber of students
Chess7
Art13
Drama9
Football?
  1. Work out the number of students who joined the football club.
  2. Write down the modal club.
","answerType":"multi","answer":"a) 11 b) Art","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"11"},{"label":"b","answer":"Art"}],"solution":["a) Chess + Art + Drama = 7 + 13 + 9 = 29 students.","Football: 40 - 29 = 11 students.","b) Art has the highest frequency, 13, so the modal club is art."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Chess","Art","Drama","Football"],"label":"Club"},"y":{"min":0,"max":14,"step":2,"minor":2,"label":"Number of students"},"bars":[{"cat":0,"height":7},{"cat":1,"height":13},{"cat":2,"height":9},{"cat":3,"height":11,"answer":true}]},"draw":false,"revision":"2bdc54a34120"},{"id":"g2-bar-charts-q07","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

A bar chart shows the number of raffle tickets a school sold each day for a week. The table gives the height of each bar.

DayTickets sold
Monday35
Tuesday42
Wednesday28
Thursday51
Friday44

Each ticket costs 50p.

Work out the total amount of money the school took from ticket sales. Give your answer in pounds.

","answerType":"exact","answer":"£100","answerDisplay":null,"accept":["100","£100","£100.00"],"parts":[],"solution":["Total tickets: 35 + 42 + 28 + 51 + 44 = 200.","Total money: 200 × 50p = 10000p = £100."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Mon","Tue","Wed","Thu","Fri"],"label":"Day"},"y":{"min":0,"max":60,"step":10,"minor":2,"label":"Tickets sold"},"bars":[{"cat":0,"height":35},{"cat":1,"height":42},{"cat":2,"height":28},{"cat":3,"height":51},{"cat":4,"height":44}]},"draw":false,"revision":"cf54ec182df1"},{"id":"g2-bar-charts-q08","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":8,"marks":4,"tier":"F","calc":false,"text":"

The table shows the number of boys and the number of girls in three year groups who take part in school sport.

Year groupBoysGirls
Year 71612
Year 81014
Year 9811

Draw a suitable diagram or chart to show this information.

","answerType":"open","answer":"Dual bar chart: for each year group a pair of bars (Boys and Girls) at the correct heights, with a key, labelled axes and a consistent scale","answerDisplay":null,"accept":[],"parts":[],"solution":["A dual (side by side) bar chart is suitable because there are two groups for each year.","For each year group draw a pair of bars: Year 7 boys 16 and girls 12, Year 8 boys 10 and girls 14, Year 9 boys 8 and girls 11.","Include a key showing which bar is boys and which is girls, label both axes and use an even scale."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Year 7","Year 8","Year 9"],"label":"Year group"},"y":{"min":0,"max":16,"step":2,"minor":2,"label":"Number of students"},"key":[{"label":"Boys","fill":"rgba(47,107,224,0.16)"},{"label":"Girls","fill":"rgba(232,163,61,0.35)"}],"bars":[{"cat":0,"height":16,"width":0.32,"offset":-0.17,"answer":true},{"cat":0,"height":12,"width":0.32,"offset":0.17,"fill":"rgba(232,163,61,0.35)","answer":true},{"cat":1,"height":10,"width":0.32,"offset":-0.17,"answer":true},{"cat":1,"height":14,"width":0.32,"offset":0.17,"fill":"rgba(232,163,61,0.35)","answer":true},{"cat":2,"height":8,"width":0.32,"offset":-0.17,"answer":true},{"cat":2,"height":11,"width":0.32,"offset":0.17,"fill":"rgba(232,163,61,0.35)","answer":true}]},"draw":true,"revision":"8e9db202e9d5"},{"id":"g2-bar-charts-q09","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

The composite bar chart shows the number of adults and the number of children who visited a museum on Saturday and on Sunday.

  1. Work out the number of children who visited on Saturday.
  2. Write down the ratio of the number of adults on Saturday to the number of adults on Sunday. Give your ratio in its simplest form.
","answerType":"multi","answer":"a) 20 b) 6 : 5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"20"},{"label":"b","answer":"6 : 5"}],"solution":["a) The Saturday bar reaches 50 in total and the adults section reaches 30, so children = 50 - 30 = 20.","b) Adults on Saturday = 30, adults on Sunday = 25.","30 : 25 = 6 : 5 after dividing both sides by 5."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Sat","Sun"],"label":"Day"},"y":{"min":0,"max":50,"step":5,"label":"Number of visitors"},"key":[{"label":"Adults","fill":"rgba(47,107,224,0.16)"},{"label":"Children","fill":"rgba(232,163,61,0.35)"}],"bars":[{"cat":0,"height":30,"stack":20,"width":0.5,"fill":"rgba(47,107,224,0.16)"},{"cat":1,"height":25,"stack":15,"width":0.5,"fill":"rgba(47,107,224,0.16)"}]},"draw":false,"revision":"3bf789b01096"},{"id":"g2-bar-charts-q10","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":10,"marks":5,"tier":"F","calc":true,"text":"

A cafe sold 60 hot drinks one morning. The table shows information about the drinks sold, but one frequency is missing.

DrinkFrequencyPrice
Coffee24£2.50
Tea18£2.00
Hot chocolate?£2.80
  1. Work out the number of hot chocolates sold.
  2. Work out the total amount of money the cafe took from selling the 60 hot drinks.
","answerType":"multi","answer":"a) 18 b) £146.40","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"18"},{"label":"b","answer":"£146.40"}],"solution":["a) Hot chocolates: 60 - 24 - 18 = 18.","b) Coffee: 24 × £2.50 = £60.","Tea: 18 × £2.00 = £36.","Hot chocolate: 18 × £2.80 = £50.40.","Total: 60 + 36 + 50.40 = £146.40."],"diagram":null,"draw":false,"revision":"49f98903b691"},{"id":"g2-bar-charts-q11","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":11,"marks":6,"tier":"F","calc":false,"text":"

20 students were each asked to choose their favourite sport. Here are the results.

F  T  S  F  B  F  T  S  F  B  F  T  F  S  B  F  T  S  F  B

F = Football, T = Tennis, S = Swimming, B = Basketball

  1. Complete a frequency table for these results.
  2. Draw a bar chart for this information.
  3. Write down the modal sport.
","answerType":"multi","answer":"a) Football 8, Tennis 4, Swimming 4, Basketball 4 b) bar chart with those heights c) Football","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Football 8, Tennis 4, Swimming 4, Basketball 4"},{"label":"b","answer":"Bar chart with bars 8, 4, 4, 4, equal widths and gaps, labelled axes"},{"label":"c","answer":"Football"}],"solution":["a) Counting the letters: F appears 8 times, T appears 4 times, S appears 4 times, B appears 4 times. Check: 8 + 4 + 4 + 4 = 20.","b) Draw bars of equal width with gaps: Football 8, Tennis 4, Swimming 4, Basketball 4. Label both axes.","c) Football has the highest frequency, so the modal sport is football."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Football","Tennis","Swimming","Basketball"],"label":"Sport"},"y":{"min":0,"max":8,"step":1,"label":"Frequency"},"bars":[{"cat":0,"height":8,"answer":true},{"cat":1,"height":4,"answer":true},{"cat":2,"height":4,"answer":true},{"cat":3,"height":4,"answer":true}]},"draw":true,"revision":"dd7415014425"},{"id":"g2-bar-charts-q12","topicId":"g2-bar-charts","topic":"Bar Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":12,"marks":7,"tier":"F","calc":true,"text":"

The bar chart shows the number of visitors to a farm park on four of the five days it was open one week. The bar for Thursday is missing.

There were 180 visitors in total during the week.

  1. Work out the number of visitors on Thursday and complete the bar chart.
  2. Write down the fraction of the week's visitors who came on Monday. Give your fraction in its simplest form.
  3. Write down the ratio of the number of visitors on Wednesday to the number of visitors on Friday. Give your ratio in its simplest form.
","answerType":"multi","answer":"a) 45 b) 1/6 c) 5 : 7","answerDisplay":"a) 45 b) 16 c) 5 : 7","accept":[],"parts":[{"label":"a","answer":"45"},{"label":"b","answer":"1/6","answerDisplay":"16"},{"label":"c","answer":"5 : 7"}],"solution":["a) Monday 30 + Tuesday 45 + Wednesday 25 + Friday 35 = 135 visitors.","Thursday: 180 - 135 = 45 visitors. Draw the Thursday bar to a height of 45.","b) Monday fraction: 30 out of 180 = 30180 = 16.","c) Wednesday : Friday = 25 : 35 = 5 : 7 after dividing both sides by 5."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Mon","Tue","Wed","Thu","Fri"],"label":"Day"},"y":{"min":0,"max":50,"step":5,"label":"Number of visitors"},"bars":[{"cat":0,"height":30},{"cat":1,"height":45},{"cat":2,"height":25},{"cat":3,"height":45,"answer":true},{"cat":4,"height":35}]},"draw":true,"revision":"2bb8cd6b3783"},{"id":"g2-calculation-problems-q01","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

Potatoes cost £1.20 per kilogram.

Work out the cost of 3.5 kg of potatoes.

","answerType":"exact","answer":"£4.20","answerDisplay":null,"accept":["4.20","£4.2","4.2"],"parts":[],"solution":["Cost = 3.5 × £1.20.","3.5 × 1.20 = 4.20, so the cost is £4.20."],"diagram":null,"draw":false,"revision":"36126edb72fc"},{"id":"g2-calculation-problems-q02","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

Lily buys 3 notebooks and a pen.

Each notebook costs £1.35.

The pen costs 85p.

Lily pays with a £10 note.

Work out how much change Lily should get.

","answerType":"exact","answer":"£5.10","answerDisplay":null,"accept":["5.10","£5.1","5.1"],"parts":[],"solution":["3 notebooks: 3 × £1.35 = £4.05.","Total spent: £4.05 + £0.85 = £4.90.","Change: £10.00 − £4.90 = £5.10."],"diagram":null,"draw":false,"revision":"0bc33628265e"},{"id":"g2-calculation-problems-q03","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

A ribbon is 6 metres long.

Pieces of ribbon 35 cm long are cut from it.

  1. Work out the greatest number of 35 cm pieces that can be cut from the ribbon.
  2. Work out the length of ribbon left over.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"17"},{"label":"b","answer":"5 cm"}],"solution":["6 m = 600 cm.","600 ÷ 35 = 17 remainder 5, since 17 × 35 = 595.","(a) 17 pieces can be cut.","(b) 600 − 595 = 5 cm left over."],"diagram":null,"draw":false,"revision":"ff78d4abf356"},{"id":"g2-calculation-problems-q04","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

A sofa costs £624.

Dan pays a deposit of £150.

He pays the rest of the cost in 6 equal monthly payments.

Work out the amount of each monthly payment.

","answerType":"exact","answer":"£79","answerDisplay":null,"accept":["79","£79.00","79.00"],"parts":[],"solution":["Amount left to pay: £624 − £150 = £474.","Each payment: £474 ÷ 6 = £79."],"diagram":null,"draw":false,"revision":"1de02419de87"},{"id":"g2-calculation-problems-q05","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

Amy works in a shop.

Last week she worked 28 hours at her normal rate of £9.50 per hour.

She also worked 6 hours at a higher rate of £14.25 per hour.

Work out the total amount Amy earned last week.

","answerType":"exact","answer":"£351.50","answerDisplay":null,"accept":["351.50","351.5","£351.5"],"parts":[],"solution":["Normal pay: 28 × £9.50 = £266.","Higher-rate pay: 6 × £14.25 = £85.50.","Total: £266 + £85.50 = £351.50."],"diagram":null,"draw":false,"revision":"f5f2000b0ed7"},{"id":"g2-calculation-problems-q06","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

A till contains only £10 notes and £20 notes.

There are 14 £10 notes in the till.

The total value of the notes is £480.

Work out how many £20 notes are in the till.

","answerType":"exact","answer":"17","answerDisplay":null,"accept":[],"parts":[],"solution":["Value of the £10 notes: 14 × £10 = £140.","Value of the £20 notes: £480 − £140 = £340.","Number of £20 notes: 340 ÷ 20 = 17."],"diagram":null,"draw":false,"revision":"ee21224f6a51"},{"id":"g2-calculation-problems-q07","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

Theatre tickets cost £24.75 for an adult and £16.50 for a child.

The Khan family buy tickets for 2 adults and 3 children.

Show that the total cost of the tickets is more than £95.

","answerType":"open","answer":"2 × £24.75 + 3 × £16.50 = £49.50 + £49.50 = £99.00, and £99.00 > £95","answerDisplay":null,"accept":[],"parts":[],"solution":["Adult tickets: 2 × £24.75 = £49.50.","Child tickets: 3 × £16.50 = £49.50.","Total: £49.50 + £49.50 = £99.00.","£99.00 is more than £95, as required."],"diagram":null,"draw":false,"revision":"919fbf1e7a8b"},{"id":"g2-calculation-problems-q08","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

Pens cost £2.35 each.

Priya has £30 to spend on pens.

Work out the greatest number of pens Priya can buy.

","answerType":"exact","answer":"12","answerDisplay":null,"accept":[],"parts":[],"solution":["£30 ÷ £2.35 = 12.76... so try 12 pens.","12 × £2.35 = £28.20, which is within £30.","13 × £2.35 = £30.55, which is too much.","The greatest number of pens is 12."],"diagram":null,"draw":false,"revision":"4069aaf15840"},{"id":"g2-calculation-problems-q09","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

Here are the readings on Jamal's electricity meter at the start and end of March.

Start of March: 4672 units
End of March: 4918 units

Each unit of electricity costs 24.5p.

Work out how much Jamal pays for the electricity he used in March. Give your answer in pounds.

","answerType":"exact","answer":"£60.27","answerDisplay":null,"accept":["60.27"],"parts":[],"solution":["Units used: 4918 − 4672 = 246.","Cost in pence: 246 × 24.5 = 6027p.","Cost in pounds: 6027 ÷ 100 = £60.27."],"diagram":null,"draw":false,"revision":"b394a826cf25"},{"id":"g2-calculation-problems-q10","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

In a cafe, 3 teas and 2 coffees cost £10.70 in total.

One tea costs £1.90.

Work out the cost of one coffee.

","answerType":"exact","answer":"£2.50","answerDisplay":null,"accept":["2.50","2.5","£2.5"],"parts":[],"solution":["Cost of 3 teas: 3 × £1.90 = £5.70.","Cost of 2 coffees: £10.70 − £5.70 = £5.00.","Cost of one coffee: £5.00 ÷ 2 = £2.50."],"diagram":null,"draw":false,"revision":"efea126b5b6b"},{"id":"g2-calculation-problems-q11","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":11,"marks":4,"tier":"F","calc":false,"text":"

Ben buys 4 identical mugs and a jug.

The jug costs £6.50.

Ben pays with a £20 note and gets £3.10 change.

Work out the cost of one mug.

","answerType":"exact","answer":"£2.60","answerDisplay":null,"accept":["2.60","2.6","£2.6"],"parts":[],"solution":["Total spent: £20 − £3.10 = £16.90.","Cost of the 4 mugs: £16.90 − £6.50 = £10.40.","Cost of one mug: £10.40 ÷ 4 = £2.60."],"diagram":null,"draw":false,"revision":"29648ccc206c"},{"id":"g2-calculation-problems-q12","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":12,"marks":4,"tier":"F","calc":true,"text":"

Jade is paid £11.20 per hour for hours worked on weekdays.

For hours worked at the weekend she is paid 1.5 times her weekday rate.

Last week Jade worked 30 hours on weekdays and 4 hours at the weekend.

Work out Jade's total pay for last week.

","answerType":"exact","answer":"£403.20","answerDisplay":null,"accept":["403.20","403.2"],"parts":[],"solution":["Weekend rate: 1.5 × £11.20 = £16.80 per hour.","Weekday pay: 30 × £11.20 = £336.","Weekend pay: 4 × £16.80 = £67.20.","Total pay: £336 + £67.20 = £403.20."],"diagram":null,"draw":false,"revision":"e90fd009511a"},{"id":"g2-calculation-problems-q13","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":13,"marks":5,"tier":"F","calc":true,"text":"

A shop sells rice in two sizes of bag.

Small bag: 500 g for £1.35
Large bag: 2 kg for £4.80

Meena needs exactly 6 kg of rice.

Work out the cheapest way for Meena to buy exactly 6 kg of rice. You must show that your way is the cheapest.

","answerType":"open","answer":"3 large bags for £14.40","answerDisplay":null,"accept":[],"parts":[],"solution":["Small bags: 500 g each, so 6 kg needs 12 small bags, costing 12 × £1.35 = £16.20.","Large bags: 2 kg each, so 6 kg needs 3 large bags, costing 3 × £4.80 = £14.40.","Cost per kg: small bag £2.70 per kg, large bag £2.40 per kg, so replacing any large bag with small bags always costs more.","The cheapest way is 3 large bags, costing £14.40."],"diagram":null,"draw":false,"revision":"1b21b17d6c6a"},{"id":"g2-calculation-problems-q14","topicId":"g2-calculation-problems","topic":"Calculation Problems","grade":"2","topicGrade":"2","strand":"N","difficulty":14,"marks":6,"tier":"F","calc":false,"text":"

Tom has £86.50 in his bank account.

He uses the account to buy 4 books costing £7.95 each and a bag costing £12.60.

His grandmother then pays £25 birthday money into the account.

Work out how much money is in Tom's account now.

","answerType":"exact","answer":"£67.10","answerDisplay":null,"accept":["67.10","67.1"],"parts":[],"solution":["Cost of the books: 4 × £7.95 = £31.80.","Total spent: £31.80 + £12.60 = £44.40.","Balance after shopping: £86.50 − £44.40 = £42.10.","Balance after birthday money: £42.10 + £25 = £67.10."],"diagram":null,"draw":false,"revision":"4172cc5fbdea"},{"id":"g2-fractions-decimals-and-percentages-q01","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write 0.7 as a fraction.

","answerType":"exact","answer":"7/10","answerDisplay":"710","accept":[],"parts":[],"solution":["0.7 means 7 tenths, which is 710."],"diagram":null,"draw":false,"revision":"c1d6eab19a38"},{"id":"g2-fractions-decimals-and-percentages-q02","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Write 30% as a decimal.

","answerType":"exact","answer":"0.3","answerDisplay":null,"accept":["0.30"],"parts":[],"solution":["30% = 30 ÷ 100 = 0.3."],"diagram":null,"draw":false,"revision":"821253a29969"},{"id":"g2-fractions-decimals-and-percentages-q03","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":true,"text":"

Write 0.45 as a percentage.

","answerType":"exact","answer":"45%","answerDisplay":null,"accept":["45"],"parts":[],"solution":["0.45 × 100 = 45, so 0.45 = 45%."],"diagram":null,"draw":false,"revision":"df501a548af2"},{"id":"g2-fractions-decimals-and-percentages-q04","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

Write 60% as a fraction.

","answerType":"exact","answer":"60/100","answerDisplay":"60100","accept":["6/10","3/5"],"parts":[],"solution":["60% means 60 out of 100, which is 60100.","This simplifies to 35."],"diagram":null,"draw":false,"revision":"20e7df679a60"},{"id":"g2-fractions-decimals-and-percentages-q05","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":5,"marks":1,"tier":"F","calc":false,"text":"

Write 9100 as a decimal.

","answerType":"exact","answer":"0.09","answerDisplay":null,"accept":[],"parts":[],"solution":["9100 means 9 hundredths, which is 0.09."],"diagram":null,"draw":false,"revision":"a03652f2a498"},{"id":"g2-fractions-decimals-and-percentages-q06","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":6,"marks":1,"tier":"F","calc":true,"text":"

Write 0.04 as a percentage.

","answerType":"exact","answer":"4%","answerDisplay":null,"accept":["4"],"parts":[],"solution":["0.04 × 100 = 4, so 0.04 = 4%."],"diagram":null,"draw":false,"revision":"5481deffba1d"},{"id":"g2-fractions-decimals-and-percentages-q07","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":7,"marks":1,"tier":"F","calc":true,"text":"

Write 320 as a percentage.

","answerType":"exact","answer":"15%","answerDisplay":null,"accept":["15"],"parts":[],"solution":["320 = 15100, multiplying top and bottom by 5.","15100 = 15%."],"diagram":null,"draw":false,"revision":"032ff64aa795"},{"id":"g2-fractions-decimals-and-percentages-q08","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":8,"marks":1,"tier":"F","calc":false,"text":"

Write 7% as a decimal.

","answerType":"exact","answer":"0.07","answerDisplay":null,"accept":[],"parts":[],"solution":["7% = 7 ÷ 100 = 0.07."],"diagram":null,"draw":false,"revision":"3ff5e6eaf3a6"},{"id":"g2-fractions-decimals-and-percentages-q09","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":9,"marks":1,"tier":"F","calc":true,"text":"

Write 58 as a decimal.

","answerType":"exact","answer":"0.625","answerDisplay":null,"accept":[],"parts":[],"solution":["5 ÷ 8 = 0.625."],"diagram":null,"draw":false,"revision":"1c844f575537"},{"id":"g2-fractions-decimals-and-percentages-q10","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":10,"marks":1,"tier":"F","calc":true,"text":"

Write 0.35 as a fraction. Give your fraction in its simplest form.

","answerType":"exact","answer":"7/20","answerDisplay":"720","accept":[],"parts":[],"solution":["0.35 = 35100.","Divide top and bottom by 5: 35100 = 720."],"diagram":null,"draw":false,"revision":"44bc709de4f1"},{"id":"g2-fractions-decimals-and-percentages-q11","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":11,"marks":1,"tier":"F","calc":true,"text":"

Write down the reciprocal of 0.2

","answerType":"exact","answer":"5","answerDisplay":null,"accept":[],"parts":[],"solution":["The reciprocal of a non-zero number multiplies with it to make 1. It is found by dividing 1 by the original number.","The reciprocal of 0.2 is 1 ÷ 0.2 = 5."],"diagram":null,"draw":false,"revision":"3d4a7103c222"},{"id":"g2-fractions-decimals-and-percentages-q12","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":12,"marks":1,"tier":"F","calc":false,"text":"

Write these numbers in order of size, starting with the smallest.

0.6   55%   12

","answerType":"exact","answer":"1/2, 55%, 0.6","answerDisplay":"12, 55%, 0.6","accept":[],"parts":[],"solution":["Convert to decimals: 0.6 = 0.6, 55% = 0.55, 12 = 0.5.","In order: 12, 55%, 0.6."],"diagram":null,"draw":false,"revision":"378560dd502e"},{"id":"g2-fractions-decimals-and-percentages-q13","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":13,"marks":2,"tier":"F","calc":false,"text":"

Write these numbers in order of size, starting with the smallest.

25   0.44   39%   14

","answerType":"exact","answer":"1/4, 39%, 2/5, 0.44","answerDisplay":"14, 39%, 25, 0.44","accept":[],"parts":[],"solution":["Convert to decimals: 25 = 0.4, 0.44 = 0.44, 39% = 0.39, 14 = 0.25.","In order: 0.25, 0.39, 0.4, 0.44.","So the order is 14, 39%, 25, 0.44."],"diagram":null,"draw":false,"revision":"8cba22599d4e"},{"id":"g2-fractions-decimals-and-percentages-q14","topicId":"g2-fractions-decimals-and-percentages","topic":"Fractions, Decimals and Percentages","grade":"2","topicGrade":"2","strand":"N","difficulty":14,"marks":4,"tier":"F","calc":true,"text":"

The same jacket is sold in two shops.

Shop A: normal price £80, with 15 off.

Shop B: normal price £84, with 25% off.

In which shop is the jacket cheaper, and by how much? You must show your working.

","answerType":"open","answer":"Shop B, by £1 (Shop A: £64, Shop B: £63)","answerDisplay":null,"accept":[],"parts":[],"solution":["Shop A discount: 15 of £80 = £16, so the price is £80 − £16 = £64.","Shop B discount: 25% of £84 = £21, so the price is £84 − £21 = £63.","£63 < £64, so the jacket is cheaper in shop B by £1."],"diagram":null,"draw":false,"revision":"ed4061c15429"},{"id":"g2-fractions-of-an-amount-q01","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Work out 15 of 45

","answerType":"exact","answer":"9","answerDisplay":null,"accept":[],"parts":[],"solution":["The denominator tells you how many equal groups make the whole. Divide by it to find one group, then multiply by the numerator to take the required number of groups.","45 ÷ 5 = 9."],"diagram":null,"draw":false,"revision":"daaa6bd596ab"},{"id":"g2-fractions-of-an-amount-q02","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Work out 18 of 56

","answerType":"exact","answer":"7","answerDisplay":null,"accept":[],"parts":[],"solution":["56 ÷ 8 = 7."],"diagram":null,"draw":false,"revision":"4debf26c2156"},{"id":"g2-fractions-of-an-amount-q03","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":3,"marks":1,"tier":"F","calc":true,"text":"

Work out 16 of 138

","answerType":"exact","answer":"23","answerDisplay":null,"accept":[],"parts":[],"solution":["138 ÷ 6 = 23."],"diagram":null,"draw":false,"revision":"cecf88857a8c"},{"id":"g2-fractions-of-an-amount-q04","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

Work out 34 of 92

","answerType":"exact","answer":"69","answerDisplay":null,"accept":[],"parts":[],"solution":["92 ÷ 4 = 23.","23 × 3 = 69."],"diagram":null,"draw":false,"revision":"fe3c342b513b"},{"id":"g2-fractions-of-an-amount-q05","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

14 of the people at a concert are children.

There are 65 children at the concert.

Work out the total number of people at the concert.

","answerType":"exact","answer":"260","answerDisplay":null,"accept":[],"parts":[],"solution":["14 of the total is 65.","Total = 65 × 4 = 260."],"diagram":null,"draw":false,"revision":"b73111a8fbb3"},{"id":"g2-fractions-of-an-amount-q06","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

Maya has £64.

She spends 38 of her money.

Work out how much money Maya has left.

","answerType":"exact","answer":"£40","answerDisplay":null,"accept":["40","£40.00"],"parts":[],"solution":["£64 ÷ 8 = £8, so 38 of £64 = 3 × £8 = £24.","Money left = £64 − £24 = £40."],"diagram":null,"draw":false,"revision":"10e261cb7f3e"},{"id":"g2-fractions-of-an-amount-q07","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

A crate holds 36 boxes.

59 of the boxes each weigh 2 kg.

The rest of the boxes each weigh 3 kg.

Work out the total weight of the 36 boxes.

","answerType":"exact","answer":"88 kg","answerDisplay":null,"accept":["88"],"parts":[],"solution":["59 of 36 = 36 ÷ 9 × 5 = 20 boxes weighing 2 kg each.","The other 36 − 20 = 16 boxes weigh 3 kg each.","Total weight = 20 × 2 + 16 × 3 = 40 + 48 = 88 kg."],"diagram":null,"draw":false,"revision":"f19b14a0bd23"},{"id":"g2-fractions-of-an-amount-q08","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

Leo has 84 stickers.

He gives 27 of the stickers to Ria.

He then gives half of the stickers he has left to Sam.

Work out how many stickers Leo gives to Sam.

","answerType":"exact","answer":"30","answerDisplay":null,"accept":[],"parts":[],"solution":["27 of 84 = 84 ÷ 7 × 2 = 24 stickers to Ria.","Stickers left = 84 − 24 = 60.","Sam gets half of 60 = 30 stickers."],"diagram":null,"draw":false,"revision":"d042db582cd6"},{"id":"g2-fractions-of-an-amount-q09","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":9,"marks":3,"tier":"F","calc":false,"text":"

A water butt holds 240 litres of water. It is full on Sunday night.

Nia uses 13 of the water on Monday.

She uses 14 of the original 240 litres on Tuesday.

Nia needs 100 litres of water on Wednesday.

Does she have enough water left? You must show your working.

","answerType":"open","answer":"Yes - 80 + 60 = 140 litres used, 240 - 140 = 100 litres left, which is exactly enough","answerDisplay":null,"accept":[],"parts":[],"solution":["Monday: 13 of 240 = 80 litres.","Tuesday: 14 of 240 = 60 litres.","Total used = 80 + 60 = 140 litres, so 240 − 140 = 100 litres are left.","100 litres is exactly the amount Nia needs, so yes, she has enough."],"diagram":null,"draw":false,"revision":"6fe8d9e688d1"},{"id":"g2-fractions-of-an-amount-q10","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

Amir and Beth share a prize of £360.

Amir gets 25 of the prize. Beth gets the rest.

Beth gives 14 of her share to charity.

Work out how much money Beth keeps.

","answerType":"exact","answer":"£162","answerDisplay":null,"accept":["162","£162.00"],"parts":[],"solution":["Amir's share: 25 of £360 = £144.","Beth's share: £360 − £144 = £216.","Beth gives away 14 of £216 = £54.","Beth keeps £216 − £54 = £162."],"diagram":null,"draw":false,"revision":"a4f811dcfc73"},{"id":"g2-fractions-of-an-amount-q11","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":11,"marks":4,"tier":"F","calc":true,"text":"

A school orders 120 calculators.

38 of the calculators are scientific calculators costing £8.50 each.

The rest are basic calculators costing £4 each.

Work out the total cost of the order.

","answerType":"exact","answer":"£682.50","answerDisplay":null,"accept":["682.50","682.5"],"parts":[],"solution":["38 of 120 = 45 scientific calculators.","Basic calculators: 120 − 45 = 75.","Scientific cost: 45 × £8.50 = £382.50.","Basic cost: 75 × £4 = £300.","Total cost: £382.50 + £300 = £682.50."],"diagram":null,"draw":false,"revision":"1ab301155f81"},{"id":"g2-fractions-of-an-amount-q12","topicId":"g2-fractions-of-an-amount","topic":"Fractions of an Amount","grade":"2","topicGrade":"2","strand":"N","difficulty":12,"marks":5,"tier":"F","calc":true,"text":"

Priya, Quinn and Raj share £480.

Priya gets 13 of the money.

Quinn gets 38 of the money.

Raj gets the rest.

Work out how much more money Quinn gets than Raj.

","answerType":"exact","answer":"£40","answerDisplay":null,"accept":["40","£40.00"],"parts":[],"solution":["Priya's share: 13 of £480 = £160.","Quinn's share: 38 of £480 = £180.","Raj's share: £480 − £160 − £180 = £140.","Quinn gets £180 − £140 = £40 more than Raj."],"diagram":null,"draw":false,"revision":"db5257f6da8c"},{"id":"g2-frequency-polygons-q01","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"2","topicGrade":"2","strand":"S","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

The table shows the times, in minutes, that 30 customers waited at a post office.

Time (t minutes)Frequency
0 < t ≤ 105
10 < t ≤ 209
20 < t ≤ 3012
30 < t ≤ 404
  1. Write down the modal class interval.
  2. Write down the class interval that contains the median.
","answerType":"multi","answer":"a) 20 < t ≤ 30 b) 20 < t ≤ 30","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"20 < t <= 30"},{"label":"b","answer":"20 < t <= 30"}],"solution":["a) The modal class is the one with the highest frequency: 20 < t ≤ 30, with frequency 12.","b) There are 30 values, so the median lies between the 15th and 16th values.","Cumulative frequencies: 5, then 14, then 26. The 15th and 16th values are both in 20 < t ≤ 30."],"diagram":null,"draw":false,"revision":"b3ac96c8f943"},{"id":"g2-frequency-polygons-q02","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"2","topicGrade":"2","strand":"S","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

The table shows the masses, in kilograms, of 25 parcels.

Mass (m kg)Frequency
0 < m ≤ 47
4 < m ≤ 810
8 < m ≤ 126
12 < m ≤ 162

Rosie is going to draw a frequency polygon for this information.

Write down the coordinates of the four points she should plot.

","answerType":"exact","answer":"(2, 7), (6, 10), (10, 6), (14, 2)","answerDisplay":null,"accept":["(2,7) (6,10) (10,6) (14,2)"],"parts":[],"solution":["For each class, calculate (lower boundary + upper boundary) ÷ 2 to find its midpoint. This is the horizontal coordinate; the frequency is the vertical coordinate.","Each point is plotted at the midpoint of its class interval.","Midpoints: 2, 6, 10 and 14.","So the points are (2, 7), (6, 10), (10, 6) and (14, 2)."],"diagram":null,"draw":false,"revision":"ddc828499d79"},{"id":"g2-frequency-polygons-q03","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"2","topicGrade":"2","strand":"S","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

The table shows the heights, in centimetres, of 30 students.

Height (h cm)Frequency
140 < h ≤ 1506
150 < h ≤ 16011
160 < h ≤ 1709
170 < h ≤ 1804

Draw a frequency polygon for this information.

","answerType":"open","answer":"Points at (145, 6), (155, 11), (165, 9), (175, 4) joined with straight lines","answerDisplay":null,"accept":[],"parts":[],"solution":["Plot each frequency at the midpoint of its class: (145, 6), (155, 11), (165, 9) and (175, 4).","Join the four points, in order, with straight lines.","Do not join the last point back to the first."],"diagram":{"type":"axes","xy":false,"x":{"min":140,"max":180,"step":10,"minor":2,"label":"Height (cm)"},"y":{"min":0,"max":12,"step":2,"minor":2,"label":"Frequency"},"curves":[{"pts":[[145,6],[155,11],[165,9],[175,4]],"marks":true,"answer":true}]},"draw":true,"revision":"3b3f71938ccd"},{"id":"g2-frequency-polygons-q04","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"2","topicGrade":"2","strand":"S","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

The table shows the ages, in years, of 40 members of a chess club.

Age (a years)Frequency
10 < a ≤ 208
20 < a ≤ 3014
30 < a ≤ 4010
40 < a ≤ 505
50 < a ≤ 603

Draw a frequency polygon for this information.

","answerType":"open","answer":"Points at (15, 8), (25, 14), (35, 10), (45, 5), (55, 3) joined with straight lines","answerDisplay":null,"accept":[],"parts":[],"solution":["Plot each frequency at the midpoint of its class: (15, 8), (25, 14), (35, 10), (45, 5) and (55, 3).","Join the five points, in order, with straight lines."],"diagram":{"type":"axes","xy":false,"x":{"min":10,"max":60,"step":10,"minor":2,"label":"Age (years)"},"y":{"min":0,"max":15,"step":5,"minor":5,"label":"Frequency"},"curves":[{"pts":[[15,8],[25,14],[35,10],[45,5],[55,3]],"marks":true,"answer":true}]},"draw":true,"revision":"0d1bd3d7280b"},{"id":"g2-frequency-polygons-q05","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"2","topicGrade":"2","strand":"S","difficulty":5,"marks":2,"tier":"F","calc":false,"text":"

The table shows the distances, in kilometres, that 25 people cycled one weekend.

Distance (d km)Frequency
0 < d ≤ 24
2 < d ≤ 49
4 < d ≤ 67
6 < d ≤ 85

Meg draws a frequency polygon for this information. She plots the points (2, 4), (4, 8), (6, 7) and (8, 5) and joins them with straight lines.

Write down two mistakes Meg has made.

","answerType":"open","answer":"She plotted at the ends of the intervals instead of the midpoints (should be 1, 3, 5, 7); she plotted a frequency of 8 for the class 2 < d <= 4 instead of 9","answerDisplay":null,"accept":[],"parts":[],"solution":["Mistake 1: the points should be plotted at the midpoints of the class intervals (1, 3, 5 and 7), not at the upper ends.","Mistake 2: the frequency for 2 < d ≤ 4 is 9, but Meg plotted 8."],"diagram":null,"draw":false,"revision":"5cb9b44707cb"},{"id":"g2-frequency-polygons-q06","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"4","topicGrade":"2","strand":"S","difficulty":6,"marks":2,"tier":"H","calc":false,"text":"

The table shows the weights, in kilograms, of 30 parcels handled by a courier.

Weight (w kg)Frequency
0 < w ≤ 13
1 < w ≤ 28
2 < w ≤ 312
3 < w ≤ 46
4 < w ≤ 51

Draw a frequency polygon for this information.

","answerType":"open","answer":"Points at (0.5, 3), (1.5, 8), (2.5, 12), (3.5, 6), (4.5, 1) joined with straight lines","answerDisplay":null,"accept":[],"parts":[],"solution":["Plot each frequency at the midpoint of its class: (0.5, 3), (1.5, 8), (2.5, 12), (3.5, 6) and (4.5, 1).","Join the five points, in order, with straight lines."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":5,"step":1,"minor":2,"label":"Weight (kg)"},"y":{"min":0,"max":14,"step":2,"minor":2,"label":"Frequency"},"curves":[{"pts":[[0.5,3],[1.5,8],[2.5,12],[3.5,6],[4.5,1]],"marks":true,"answer":true}]},"draw":true,"revision":"2a6d0c42cc08"},{"id":"g2-frequency-polygons-q07","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"4","topicGrade":"2","strand":"S","difficulty":7,"marks":2,"tier":"H","calc":true,"text":"

The table shows the lengths, in metres, of 32 boats in a marina.

Length (x metres)Frequency
0 < x ≤ 205
20 < x ≤ 4011
40 < x ≤ 609
60 < x ≤ 807

Amir draws a frequency polygon. He plots the points (10, 5), (30, 11), (50, 9) and (70, 7). He joins the points with a smooth curve, and then joins the last point back to the first point to close the shape.

Write down two things that are wrong with Amir's frequency polygon.

","answerType":"open","answer":"The points should be joined with straight lines, not a smooth curve; the last point should not be joined back to the first","answerDisplay":null,"accept":[],"parts":[],"solution":["Amir's plotted points are correct - they are at the midpoints with the correct frequencies.","Mistake 1: the points in a frequency polygon must be joined with straight lines, not a curve.","Mistake 2: the polygon should not be closed - the last point must not be joined back to the first."],"diagram":null,"draw":false,"revision":"cda1d2d839b6"},{"id":"g2-frequency-polygons-q08","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"4","topicGrade":"2","strand":"S","difficulty":8,"marks":2,"tier":"H","calc":true,"text":"

The table shows the speeds, in miles per hour, of 40 cars passing a speed camera.

Speed (s mph)Frequency
20 < s ≤ 304
30 < s ≤ 409
40 < s ≤ 5015
50 < s ≤ 608
60 < s ≤ 704

Draw a frequency polygon for this information.

","answerType":"open","answer":"Points at (25, 4), (35, 9), (45, 15), (55, 8), (65, 4) joined with straight lines","answerDisplay":null,"accept":[],"parts":[],"solution":["Plot each frequency at the midpoint of its class: (25, 4), (35, 9), (45, 15), (55, 8) and (65, 4).","Join the five points, in order, with straight lines."],"diagram":{"type":"axes","xy":false,"x":{"min":20,"max":70,"step":10,"minor":2,"label":"Speed (mph)"},"y":{"min":0,"max":16,"step":2,"minor":2,"label":"Frequency"},"curves":[{"pts":[[25,4],[35,9],[45,15],[55,8],[65,4]],"marks":true,"answer":true}]},"draw":true,"revision":"b99074af081b"},{"id":"g2-frequency-polygons-q09","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"2","topicGrade":"2","strand":"S","difficulty":9,"marks":3,"tier":"F","calc":true,"text":"

The table shows the waiting times, in minutes, of 30 patients at a clinic.

Time (t minutes)Frequency
0 < t ≤ 56
5 < t ≤ 108
10 < t ≤ 1511
15 < t ≤ 205
  1. Write down the class interval that contains the median waiting time.
  2. Draw a frequency polygon for this information.
","answerType":"multi","answer":"a) 10 < t ≤ 15 b) points at (2.5, 6), (7.5, 8), (12.5, 11), (17.5, 5) joined with straight lines","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"10 < t <= 15"},{"label":"b","answer":"Points at (2.5, 6), (7.5, 8), (12.5, 11), (17.5, 5) joined with straight lines"}],"solution":["a) There are 30 values, so the median lies between the 15th and 16th values.","Cumulative frequencies: 6, then 14, then 25. The 15th and 16th values are both in 10 < t ≤ 15.","b) Plot (2.5, 6), (7.5, 8), (12.5, 11) and (17.5, 5) at the class midpoints and join them, in order, with straight lines."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":20,"step":5,"minor":2,"label":"Time (minutes)"},"y":{"min":0,"max":12,"step":2,"minor":2,"label":"Frequency"},"curves":[{"pts":[[2.5,6],[7.5,8],[12.5,11],[17.5,5]],"marks":true,"answer":true}]},"draw":true,"revision":"21b17b32a0c1"},{"id":"g2-frequency-polygons-q10","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"4","topicGrade":"2","strand":"S","difficulty":10,"marks":3,"tier":"H","calc":true,"text":"

The table shows the masses, in grams, of 30 apples.

Mass (m grams)Frequency
50 < m ≤ 607
60 < m ≤ 7012
70 < m ≤ 809
80 < m ≤ 902
  1. Write down the modal class interval.
  2. Priya draws a frequency polygon for this information. She plots the points (55, 7), (65, 12) and (75, 9) correctly, then plots her final point at (90, 2) and joins all four points with a smooth curve. Write down two mistakes Priya has made.
","answerType":"multi","answer":"a) 60 < m ≤ 70 b) The final point should be (85, 2), not (90, 2); join the points with straight lines, not a smooth curve.","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"60 < m <= 70"},{"label":"b","answer":"Final point should be (85, 2); join the points with straight lines, not a smooth curve."}],"solution":["a) The modal class has the highest frequency, 12, so it is 60 < m ≤ 70.","b) Mistake 1: the final point must be plotted at the midpoint of 80 < m ≤ 90, which is (85, 2), not at the upper end (90, 2).","Mistake 2: a frequency polygon joins consecutive points with straight line segments, not a smooth curve."],"diagram":null,"draw":false,"revision":"366b76153da7"},{"id":"g2-frequency-polygons-q11","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"4","topicGrade":"2","strand":"S","difficulty":11,"marks":3,"tier":"H","calc":true,"text":"

The table shows the journey times, in minutes, of the people who work at an office.

Time (t minutes)Frequency
0 < t ≤ 159
15 < t ≤ 3016
30 < t ≤ 4511
45 < t ≤ 604
  1. Write down the modal class interval.
  2. Work out the number of people who work at the office.
  3. Explain why it is not possible to find the exact range of the journey times from the table.
","answerType":"multi","answer":"a) 15 < t ≤ 30 b) 40 c) The data is grouped, so the exact shortest and longest times are not known","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"15 < t <= 30"},{"label":"b","answer":"40"},{"label":"c","answer":"The data is grouped, so the exact smallest and largest values are unknown"}],"solution":["a) The highest frequency is 16, so the modal class is 15 < t ≤ 30.","b) 9 + 16 + 11 + 4 = 40 people.","c) The table only gives class intervals, not exact values, so the exact shortest and longest journey times are unknown and the exact range cannot be found."],"diagram":null,"draw":false,"revision":"6326212c9356"},{"id":"g2-frequency-polygons-q12","topicId":"g2-frequency-polygons","topic":"Frequency Polygons","grade":"4","topicGrade":"2","strand":"S","difficulty":12,"marks":3,"tier":"H","calc":true,"text":"

The table shows the lengths, in millimetres, of 36 leaves collected for a biology project.

Length (l mm)Frequency
0 < l ≤ 102
10 < l ≤ 207
20 < l ≤ 3013
30 < l ≤ 409
40 < l ≤ 505
  1. Draw a frequency polygon for this information.
  2. Write down the modal class interval.
","answerType":"multi","answer":"a) points at (5, 2), (15, 7), (25, 13), (35, 9), (45, 5) joined with straight lines b) 20 < l ≤ 30","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Points at (5, 2), (15, 7), (25, 13), (35, 9), (45, 5) joined with straight lines"},{"label":"b","answer":"20 < l <= 30"}],"solution":["a) Plot each frequency at the midpoint of its class: (5, 2), (15, 7), (25, 13), (35, 9) and (45, 5). Join the points, in order, with straight lines.","b) The highest frequency is 13, so the modal class is 20 < l ≤ 30."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":50,"step":10,"minor":2,"label":"Length (mm)"},"y":{"min":0,"max":14,"step":2,"minor":2,"label":"Frequency"},"curves":[{"pts":[[5,2],[15,7],[25,13],[35,9],[45,5]],"marks":true,"answer":true}]},"draw":true,"revision":"2bced2e89dd3"},{"id":"g2-function-machines-q01","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Here is a number machine.

  1. Work out the output when the input is 5.
  2. Work out the output when the input is 10.
","answerType":"multi","answer":"a) 17 b) 37","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"17"},{"label":"b","answer":"37"}],"solution":["a) 5 × 4 = 20, then 20 - 3 = 17.","b) 10 × 4 = 40, then 40 - 3 = 37."],"diagram":{"type":"funcmachine","steps":["× 4","− 3"],"input":"5","output":null,"answerOutput":"17"},"draw":false,"revision":"7adcbd9eb724"},{"id":"g2-function-machines-q02","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Here is a number machine.

  1. Work out the output when the input is 6.
  2. Work out the output when the input is 3.5
","answerType":"multi","answer":"a) 26 b) 21","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"26"},{"label":"b","answer":"21"}],"solution":["a) 6 + 7 = 13, then 13 × 2 = 26.","b) 3.5 + 7 = 10.5, then 10.5 × 2 = 21."],"diagram":{"type":"funcmachine","steps":["+ 7","× 2"],"input":"6","output":null,"answerOutput":"26"},"draw":false,"revision":"474f33cd530e"},{"id":"g2-function-machines-q03","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

Here is a number machine.

  1. Work out the output when the input is 4.
  2. Work out the input when the output is 37.
","answerType":"multi","answer":"a) 22 b) 7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"22"},{"label":"b","answer":"7"}],"solution":["To recover an input, undo the last operation first and work backwards. Addition is undone by subtraction; multiplication is undone by division.","a) 4 × 5 = 20, then 20 + 2 = 22.","b) Work backwards using inverse operations: 37 - 2 = 35, then 35 ÷ 5 = 7.","Check: 7 × 5 + 2 = 37."],"diagram":{"type":"funcmachine","steps":["× 5","+ 2"],"input":"4","output":null,"answerOutput":"22"},"draw":false,"revision":"10d646407048"},{"id":"g2-function-machines-q04","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"

Here is a number machine.

  1. Work out the output when the input is 9.
  2. Work out the input when the output is 24.
","answerType":"multi","answer":"a) 15 b) 12","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"15"},{"label":"b","answer":"12"}],"solution":["a) 9 - 4 = 5, then 5 × 3 = 15.","b) Work backwards: 24 ÷ 3 = 8, then 8 + 4 = 12.","Check: (12 - 4) × 3 = 24."],"diagram":{"type":"funcmachine","steps":["− 4","× 3"],"input":"9","output":null,"answerOutput":"15"},"draw":false,"revision":"be534308d1b1"},{"id":"g2-function-machines-q05","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

Here is a number machine.

  1. Work out the output when the input is 2.5
  2. Work out the input when the output is 31.
","answerType":"multi","answer":"a) 10 b) 6","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"10"},{"label":"b","answer":"6"}],"solution":["a) 2.5 × 6 = 15, then 15 - 5 = 10.","b) Work backwards: 31 + 5 = 36, then 36 ÷ 6 = 6.","Check: 6 × 6 - 5 = 31."],"diagram":{"type":"funcmachine","steps":["× 6","− 5"],"input":"2.5","output":null,"answerOutput":"10"},"draw":false,"revision":"0da73dc1f079"},{"id":"g2-function-machines-q06","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

Here is a number machine with a missing number.

When the input is 4, the output is 14.

  1. Work out the missing number in the machine.
  2. Use the completed machine to work out the output when the input is 7.
","answerType":"multi","answer":"a) 3 b) 23","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3"},{"label":"b","answer":"23"}],"solution":["a) Before the + 2 step the number must be 14 - 2 = 12.","a) The input 4 is multiplied to give 12, so the missing number is 12 ÷ 4 = 3.","b) 7 × 3 = 21, then 21 + 2 = 23."],"diagram":{"type":"funcmachine","steps":["× ?","+ 2"],"input":"4","output":"14"},"draw":false,"revision":"4f08aa690dea"},{"id":"g2-function-machines-q07","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

Here is a number machine.

  1. Work out the output when the input is 6.
  2. Find the input when the output is 30.
","answerType":"multi","answer":"a) 32 b) 5.5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"32"},{"label":"b","answer":"5.5"}],"solution":["a) 6 + 2 = 8, then 8 × 4 = 32.","b) Work backwards: 30 ÷ 4 = 7.5, then 7.5 - 2 = 5.5","Check: (5.5 + 2) × 4 = 30."],"diagram":{"type":"funcmachine","steps":["+ 2","× 4"],"input":"6","output":null,"answerOutput":"32"},"draw":false,"revision":"3c4f2b0c44e4"},{"id":"g2-function-machines-q08","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":8,"marks":4,"tier":"F","calc":false,"text":"

Here are two number machines.

  1. Work out the output of machine A when the input is 4.
  2. Work out the input of machine A when the output is 35.
  3. When the input is 4, machine B gives the same output as machine A.
    Work out the value of k.
","answerType":"multi","answer":"a) 17 b) 13 c) 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"17"},{"label":"b","answer":"13"},{"label":"c","answer":"3"}],"solution":["a) 4 × 2 = 8, then 8 + 9 = 17.","b) Work backwards: 35 - 9 = 26, then 26 ÷ 2 = 13.","c) Machine B with input 4: 4 × 5 = 20, then 20 - k.","c) This must equal 17, so 20 - k = 17 and k = 3."],"diagram":{"type":"panels","cols":1,"panelWidth":400,"panelHeight":84,"items":[{"label":"Machine A","diagram":{"type":"funcmachine","steps":["× 2","+ 9"],"input":"4","output":null,"answerOutput":"17"}},{"label":"Machine B","diagram":{"type":"funcmachine","steps":["× 5","− k"],"input":"4","output":null}}]},"draw":false,"revision":"cbd29f9b7149"},{"id":"g2-function-machines-q09","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

Here is a number machine.

  1. Work out the output when the input is 7.
  2. Write down an expression for the output when the input is n.
  3. Find the input when the output is 55.
","answerType":"multi","answer":"a) 27 b) 4n - 1 c) 14","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"27"},{"label":"b","answer":"4n - 1 (accept -1 + 4n)"},{"label":"c","answer":"14"}],"solution":["a) 7 × 4 = 28, then 28 - 1 = 27.","b) The input n is multiplied by 4 then 1 is subtracted, giving 4n - 1.","c) Work backwards: 55 + 1 = 56, then 56 ÷ 4 = 14.","Check: 4 × 14 - 1 = 55."],"diagram":{"type":"funcmachine","steps":["× 4","− 1"],"input":"7","output":null,"answerOutput":"27"},"draw":false,"revision":"6af84b31851d"},{"id":"g2-function-machines-q10","topicId":"g2-function-machines","topic":"Function Machines","grade":"2","topicGrade":"2","strand":"A","difficulty":10,"marks":5,"tier":"F","calc":false,"text":"

Here is a number machine.

  1. Work out the output when the input is 6.
  2. Work out the input when the output is 4.
  3. For one value of the input, the output of the machine is equal to the input.
    Work out this value.
","answerType":"multi","answer":"a) 28 b) -2 c) -5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"28"},{"label":"b","answer":"-2"},{"label":"c","answer":"-5"}],"solution":["a) 6 × 3 = 18, then 18 + 10 = 28.","b) Work backwards: 4 - 10 = -6, then -6 ÷ 3 = -2.","c) Call the input x. The output is 3x + 10, so solve 3x + 10 = x.","c) 2x = -10, so x = -5.","Check: -5 × 3 + 10 = -5, which equals the input."],"diagram":{"type":"funcmachine","steps":["× 3","+ 10"],"input":"6","output":null,"answerOutput":"28"},"draw":false,"revision":"9315df846e47"},{"id":"g2-pie-charts-q01","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

60 people were asked to name their favourite season. The results are shown in a pie chart.

The angle for Summer is 90°.

  1. Write down the fraction of the people who chose Summer. Give your fraction in its simplest form.
  2. Work out the number of people who chose Summer.
","answerType":"multi","answer":"a) 1/4 b) 15","answerDisplay":"a) 14 b) 15","accept":[],"parts":[{"label":"a","answer":"1/4","answerDisplay":"14"},{"label":"b","answer":"15"}],"solution":["The full circle represents the whole group and has angle 360°. A sector’s fraction of 360° is also its fraction of the group.","a) 90 out of 360 degrees = 90360 = 14.","b) 14 of 60 = 15 people."],"diagram":null,"draw":false,"revision":"a47c16efaa95"},{"id":"g2-pie-charts-q02","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

40 students each chose one sport. 15 chose football and 6 chose tennis. Ellie is going to draw a pie chart of the results.

  1. Work out the angle Ellie should use for football.
  2. Work out the angle Ellie should use for tennis.
","answerType":"multi","answer":"a) 135° b) 54°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"135"},{"label":"b","answer":"54"}],"solution":["Each student is worth 360 divided by 40 = 9 degrees.","a) Football: 15 × 9 = 135 degrees.","b) Tennis: 6 × 9 = 54 degrees."],"diagram":null,"draw":false,"revision":"4b6e4936cdb8"},{"id":"g2-pie-charts-q03","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

36 people were asked to name their favourite type of film. The table shows the results.

Type of filmFrequency
Comedy10
Drama12
Action6
Horror8

Draw an accurate pie chart for this information.

","answerType":"multi","answer":"Comedy 100°, Drama 120°, Action 60°, Horror 80°","answerDisplay":null,"accept":[],"parts":[{"label":"Comedy","answer":"100"},{"label":"Drama","answer":"120"},{"label":"Action","answer":"60"},{"label":"Horror","answer":"80"}],"solution":["Each person is worth 360 divided by 36 = 10 degrees.","Comedy: 10 × 10 = 100 degrees. Drama: 12 × 10 = 120 degrees. Action: 6 × 10 = 60 degrees. Horror: 8 × 10 = 80 degrees.","Check: 100 + 120 + 60 + 80 = 360 degrees.","Draw each sector accurately with a protractor and label the sectors."],"diagram":{"type":"pie","blank":true,"slices":[{"label":"Comedy","angle":100,"value":10},{"label":"Drama","angle":120,"value":12},{"label":"Action","angle":60,"value":6},{"label":"Horror","angle":80,"value":8}]},"draw":true,"revision":"cbaee72dcd31"},{"id":"g2-pie-charts-q04","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

45 students were asked how they usually travel to college. The table shows the results.

MethodFrequency
Bus12
Walk15
Car8
Cycle10

Draw an accurate pie chart for this information.

","answerType":"multi","answer":"Bus 96°, Walk 120°, Car 64°, Cycle 80°","answerDisplay":null,"accept":[],"parts":[{"label":"Bus","answer":"96"},{"label":"Walk","answer":"120"},{"label":"Car","answer":"64"},{"label":"Cycle","answer":"80"}],"solution":["Each student is worth 360 divided by 45 = 8 degrees.","Bus: 12 × 8 = 96 degrees. Walk: 15 × 8 = 120 degrees. Car: 8 × 8 = 64 degrees. Cycle: 10 × 8 = 80 degrees.","Check: 96 + 120 + 64 + 80 = 360 degrees.","Draw each sector accurately with a protractor and label the sectors."],"diagram":{"type":"pie","blank":true,"slices":[{"label":"Bus","angle":96,"value":12},{"label":"Walk","angle":120,"value":15},{"label":"Car","angle":64,"value":8},{"label":"Cycle","angle":80,"value":10}]},"draw":true,"revision":"f5fa8d92d109"},{"id":"g2-pie-charts-q05","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

A pie chart shows information about the members of a leisure centre.

The sector for swimming has an angle of 72° and represents 18 members.

Work out the total number of members of the leisure centre.

","answerType":"exact","answer":"90","answerDisplay":null,"accept":["90 members"],"parts":[],"solution":["72 degrees represents 18 members, so 1 degree represents 18 divided by 72 = 0.25 members.","The whole pie chart is 360 degrees: 360 × 0.25 = 90 members.","Alternatively, 360 divided by 72 = 5, and 18 × 5 = 90."],"diagram":null,"draw":false,"revision":"4dbd51b0c329"},{"id":"g2-pie-charts-q06","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

120 people were asked to name their favourite holiday destination. The table shows the angles in a pie chart of the results.

DestinationAngle
Italy150°
Spain90°
France72°
Greece48°
  1. Write down the modal destination.
  2. Work out the number of people who chose France.
","answerType":"multi","answer":"a) Italy b) 24","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Italy"},{"label":"b","answer":"24"}],"solution":["a) Italy has the largest angle, 150 degrees, so it is the mode.","b) France: 72360 of the people = 72360 × 120 = 24 people."],"diagram":null,"draw":false,"revision":"df2df6a5afb9"},{"id":"g2-pie-charts-q07","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

Two schools each drew a pie chart to show their students' favourite lunch.

SchoolNumber of studentsAngle for pizza
School A8045°
School B24030°

Sam says, \"More students chose pizza at School A than at School B, because the angle for pizza is bigger at School A.\"

Is Sam right? You must show your working.

","answerType":"open","answer":"No. School A: 45/360 of 80 = 10 students. School B: 30/360 of 240 = 20 students. More students chose pizza at School B","answerDisplay":"No. School A: 45360 of 80 = 10 students. School B: 30360 of 240 = 20 students. More students chose pizza at School B","accept":[],"parts":[],"solution":["School A: 45360 × 80 = 10 students chose pizza.","School B: 30360 × 240 = 20 students chose pizza.","20 > 10, so Sam is wrong. The pie charts represent different numbers of students, so a bigger angle does not mean a bigger number."],"diagram":null,"draw":false,"revision":"b0fe6e1b26c6"},{"id":"g2-pie-charts-q08","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

The table shows the activities chosen by the 60 members of Club A.

ActivityFrequency
Swimming25
Gym20
Yoga15
  1. Draw an accurate pie chart for this information.
  2. Club B has 120 members. In the pie chart for Club B, the angle for yoga is 60°. Which club has more members who chose yoga? You must show your working.
","answerType":"multi","answer":"a) Swimming 150°, Gym 120°, Yoga 90° b) Club B (20 members compared with 15)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Swimming 150, Gym 120, Yoga 90"},{"label":"b","answer":"Club B, because 60/360 of 120 = 20, which is more than 15","answerDisplay":"Club B, because 60360 of 120 = 20, which is more than 15"}],"solution":["a) Each member is worth 360 divided by 60 = 6 degrees.","Swimming: 25 × 6 = 150 degrees. Gym: 20 × 6 = 120 degrees. Yoga: 15 × 6 = 90 degrees. Check: 150 + 120 + 90 = 360.","b) Club B yoga members: 60360 × 120 = 20.","Club A has 15 yoga members, so Club B has more."],"diagram":{"type":"pie","blank":true,"slices":[{"label":"Swimming","angle":150,"value":25},{"label":"Gym","angle":120,"value":20},{"label":"Yoga","angle":90,"value":15}]},"draw":true,"revision":"3e0e94f561db"},{"id":"g2-pie-charts-q09","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

200 people entered a raffle. A pie chart shows the colours of the tickets they bought.

The angle for red tickets is 126°.

  1. Work out the number of people who bought a red ticket.
  2. One of the 200 people is chosen at random. Write down the probability that this person bought a red ticket. Give your answer as a fraction in its simplest form.

Each person bought exactly one ticket.

","answerType":"multi","answer":"a) 70 b) 7/20","answerDisplay":"a) 70 b) 720","accept":[],"parts":[{"label":"a","answer":"70"},{"label":"b","answer":"7/20","answerDisplay":"720"}],"solution":["a) Red tickets: 126360 × 200 = 70 people.","b) Probability: 70200 = 720. This is the same as 126360 simplified."],"diagram":null,"draw":false,"revision":"41d919fb9ae0"},{"id":"g2-pie-charts-q10","topicId":"g2-pie-charts","topic":"Pie Charts","grade":"2","topicGrade":"2","strand":"S","difficulty":10,"marks":5,"tier":"F","calc":true,"text":"

300 people voted for the name of a new park. A pie chart shows the results.

The angle for Alder Park is 96° and the angle for Birch Park is 144°. The only other choice was Cedar Park.

  1. Work out the angle for Cedar Park.
  2. Work out the number of people who voted for Cedar Park.
  3. Work out how many more people voted for Birch Park than for Alder Park.
","answerType":"multi","answer":"a) 120° b) 100 c) 40","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"120"},{"label":"b","answer":"100"},{"label":"c","answer":"40"}],"solution":["a) Cedar Park: 360 - 96 - 144 = 120 degrees.","b) Cedar Park votes: 120360 × 300 = 100 people.","c) Birch Park: 144360 × 300 = 120 votes. Alder Park: 96360 × 300 = 80 votes.","Difference: 120 - 80 = 40 people."],"diagram":null,"draw":false,"revision":"62444260306a"},{"id":"g2-probability-q01","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

A fair six-sided dice is rolled once.

Write down the probability that the dice lands on a number less than 3.

","answerType":"exact","answer":"1/3","answerDisplay":"13","accept":["2/6","0.33...","0.3 recurring"],"parts":[],"solution":["The numbers less than 3 are 1 and 2, so there are 2 successful outcomes out of 6 equally likely outcomes.","P(less than 3) = 26 = 13"],"diagram":null,"draw":false,"revision":"1a2e0568aafc"},{"id":"g2-probability-q02","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

Here is a probability scale, labelled from 0 to 1.

  1. Mark with a cross the probability that a fair coin lands on tails.
  2. Mark with a cross the probability that an ordinary fair dice lands on the number 8.
","answerType":"multi","answer":"a) cross at 1/2; b) cross at 0","answerDisplay":"a) cross at 12; b) cross at 0","accept":[],"parts":[{"label":"a","answer":"cross at 1/2 (halfway along the scale)","answerDisplay":"cross at 12 (halfway along the scale)"},{"label":"b","answer":"cross at 0"}],"solution":["A fair coin lands on tails with probability 12, so the cross goes exactly halfway between 0 and 1.","An ordinary dice only shows 1 to 6, so landing on 8 is impossible and the cross goes at 0."],"diagram":{"type":"numberline","min":0,"max":1,"step":0.5,"format":"frac","markers":[{"at":0.5,"style":"cross","label":"a","answer":true},{"at":0,"style":"cross","label":"b","answer":true}]},"draw":true,"revision":"861661fe38ef"},{"id":"g2-probability-q03","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Priya buys 12 tickets in a school raffle.

A total of 400 tickets are sold.

One ticket is picked at random to win the prize.

Work out the probability that Priya does not win the prize.

","answerType":"exact","answer":"97/100","answerDisplay":"97100","accept":["388/400","0.97","97%"],"parts":[],"solution":["The number of tickets that are not Priya's is 400 - 12 = 388.","P(Priya does not win) = 388400 = 97100"],"diagram":null,"draw":false,"revision":"cfd6887222a7"},{"id":"g2-probability-q04","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":4,"marks":2,"tier":"F","calc":false,"text":"

A bag contains 4 green counters, 7 yellow counters and 9 white counters.

A counter is taken from the bag at random.

Find the probability that the counter is not green.

","answerType":"exact","answer":"4/5","answerDisplay":"45","accept":["16/20","0.8","80%"],"parts":[],"solution":["Total number of counters = 4 + 7 + 9 = 20.","Counters that are not green = 7 + 9 = 16.","P(not green) = 1620 = 45"],"diagram":null,"draw":false,"revision":"3079fc93f110"},{"id":"g2-probability-q05","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

A fair eight-sided spinner has sections numbered 1 to 8.

The spinner is spun once.

  1. Write down the word that best describes the probability that the spinner lands on 9.
  2. Find the probability that the spinner lands on a multiple of 3.
","answerType":"multi","answer":"a) impossible; b) 1/4","answerDisplay":"a) impossible; b) 14","accept":[],"parts":[{"label":"a","answer":"impossible"},{"label":"b","answer":"1/4","answerDisplay":"14"}],"solution":["The spinner only shows 1 to 8, so landing on 9 cannot happen: it is impossible.","The multiples of 3 from 1 to 8 are 3 and 6, so P(multiple of 3) = 28 = 14."],"diagram":null,"draw":false,"revision":"e9b1e0ce9232"},{"id":"g2-probability-q06","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

Ten cards each have a number written on them:

2  3  3  5  5  5  7  8  8  9

  1. Write down the mode of the numbers.
  2. One card is picked at random. Find the probability that the number on the card is greater than 7.
","answerType":"multi","answer":"a) 5; b) 3/10","answerDisplay":"a) 5; b) 310","accept":[],"parts":[{"label":"a","answer":"5"},{"label":"b","answer":"3/10","answerDisplay":"310"}],"solution":["The number 5 appears three times, more than any other number, so the mode is 5.","The numbers greater than 7 are 8, 8 and 9, which is 3 cards out of 10.","P(greater than 7) = 310"],"diagram":null,"draw":false,"revision":"13ce8a8aaf28"},{"id":"g2-probability-q07","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":7,"marks":3,"tier":"F","calc":false,"text":"

Ben has these notes and coins in a tin:

two £10 notes, three £5 notes, four £2 coins and one £1 coin.

Ben takes one note or coin from the tin at random.

Work out the probability that its value is less than £5.

","answerType":"exact","answer":"1/2","answerDisplay":"12","accept":["5/10","0.5","50%"],"parts":[],"solution":["Total number of notes and coins = 2 + 3 + 4 + 1 = 10.","The items worth less than £5 are the four £2 coins and the one £1 coin, which is 5 items.","P(value less than £5) = 510 = 12"],"diagram":null,"draw":false,"revision":"c417ae9d754a"},{"id":"g2-probability-q08","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

A spinner can land on red, blue or green only.

P(red) = 0.3 and P(blue) = 0.25.

  1. Work out the probability that the spinner lands on red or blue.
  2. Work out the probability that the spinner lands on green.
","answerType":"multi","answer":"a) 0.55; b) 0.45","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"0.55"},{"label":"b","answer":"0.45"}],"solution":["P(red or blue) = 0.3 + 0.25 = 0.55.","The probabilities of red, blue and green must add up to 1.","P(green) = 1 - 0.55 = 0.45"],"diagram":null,"draw":false,"revision":"b489144a5884"},{"id":"g2-probability-q09","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":9,"marks":3,"tier":"F","calc":true,"text":"

Box A contains 12 balls. 3 of the balls are red.

Box B contains 18 balls. 5 of the balls are red.

Leah takes one ball at random from each box.

From which box is she more likely to take a red ball?

You must show your working.

","answerType":"exact","answer":"Box B","answerDisplay":null,"accept":["B","Box B because 5/18 > 3/12"],"parts":[],"solution":["P(red from Box A) = 312 = 0.25.","P(red from Box B) = 518 = 0.277... (to 3 decimal places 0.278).","0.278 > 0.25, so she is more likely to take a red ball from Box B."],"diagram":null,"draw":false,"revision":"55cb11187413"},{"id":"g2-probability-q10","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

A four-sided spinner can land on 1, 2, 3 or 4.

The table shows some of the probabilities.

Number1234
Probability0.20.350.15
  1. Work out the probability that the spinner lands on 4.
  2. The spinner is spun 200 times. Work out an estimate for the number of times it lands on 4.
","answerType":"multi","answer":"a) 0.3; b) 60","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"0.3"},{"label":"b","answer":"60"}],"solution":["The probabilities must add up to 1, and 0.2 + 0.35 + 0.15 = 0.7.","P(4) = 1 - 0.7 = 0.3.","Estimated number of 4s in 200 spins = 0.3 × 200 = 60"],"diagram":null,"draw":false,"revision":"4265d0f0462b"},{"id":"g2-probability-q11","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":11,"marks":4,"tier":"F","calc":false,"text":"

A bag contains only red, blue and green counters.

P(green) = 0.4

P(red) = P(blue)

There are 20 green counters in the bag.

Work out the number of red counters in the bag.

","answerType":"exact","answer":"15","answerDisplay":null,"accept":["15 red counters"],"parts":[],"solution":["P(green) = 0.4 and there are 20 green counters, so the total number of counters is 20 ÷ 0.4 = 50.","P(red) + P(blue) = 1 - 0.4 = 0.6, and P(red) = P(blue), so P(red) = 0.3.","Number of red counters = 0.3 × 50 = 15"],"diagram":null,"draw":false,"revision":"e5c29cac7713"},{"id":"g2-probability-q12","topicId":"g2-probability","topic":"Probability","grade":"2","topicGrade":"2","strand":"P","difficulty":12,"marks":5,"tier":"F","calc":true,"text":"

Sam and Tia both play the same fairground game many times.

Sam says the probability of winning a game is 0.35.

Tia says the probability of winning a game is 38.

  1. Whose probability of winning is larger? You must show your working.
  2. Ravi plays the game 300 times. Using Sam's probability, work out an estimate for the number of games Ravi wins.
","answerType":"multi","answer":"a) Tia, because 3/8 = 0.375 > 0.35; b) 105","answerDisplay":"a) Tia, because 38 = 0.375 > 0.35; b) 105","accept":[],"parts":[{"label":"a","answer":"Tia (3/8 = 0.375, which is larger than 0.35)","answerDisplay":"Tia (38 = 0.375, which is larger than 0.35)"},{"label":"b","answer":"105"}],"solution":["Convert Tia's fraction to a decimal: 38 = 0.375.","0.375 > 0.35, so Tia's probability is larger.","Using Sam's probability, estimated wins = 0.35 × 300 = 105"],"diagram":null,"draw":false,"revision":"0289aa2ac94e"},{"id":"g2-simplifying-algebra-q01","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":1,"marks":1,"tier":"F","calc":true,"text":"

Simplify  a + a + a + a

","answerType":"exact","answer":"4a","answerDisplay":null,"accept":["4 a"],"parts":[],"solution":["There are four lots of a added together.","a + a + a + a = 4a."],"diagram":null,"draw":false,"revision":"e301d1f0ff17"},{"id":"g2-simplifying-algebra-q02","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

Simplify  3 × g × 5

","answerType":"exact","answer":"15g","answerDisplay":null,"accept":["15 g"],"parts":[],"solution":["Multiply the numbers together: 3 × 5 = 15.","3 × g × 5 = 15g."],"diagram":null,"draw":false,"revision":"e344290fb980"},{"id":"g2-simplifying-algebra-q03","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":3,"marks":1,"tier":"F","calc":false,"text":"

Simplify  7c − 2c + 4c

","answerType":"exact","answer":"9c","answerDisplay":null,"accept":["9 c"],"parts":[],"solution":["The coefficient is the number multiplying a letter. Add the coefficients of matching letter terms, just as 4 lots of an object plus 5 lots make 9 lots.","All the terms are in c, so combine the coefficients: 7 - 2 + 4 = 9.","7c - 2c + 4c = 9c."],"diagram":null,"draw":false,"revision":"daeddb6fd94b"},{"id":"g2-simplifying-algebra-q04","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":4,"marks":1,"tier":"F","calc":true,"text":"

Simplify  d × d × d

","answerType":"exact","answer":"d^3","answerDisplay":null,"accept":["d cubed","d³"],"parts":[],"solution":["d multiplied by itself three times is d to the power 3.","d × d × d = d3."],"diagram":null,"draw":false,"revision":"40c19388f8df"},{"id":"g2-simplifying-algebra-q05","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":5,"marks":1,"tier":"F","calc":true,"text":"

Simplify  10e ÷ 2

","answerType":"exact","answer":"5e","answerDisplay":null,"accept":["5 e"],"parts":[],"solution":["Divide the coefficient by 2: 10 ÷ 2 = 5.","10e ÷ 2 = 5e."],"diagram":null,"draw":false,"revision":"64b9068ad7b0"},{"id":"g2-simplifying-algebra-q06","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":6,"marks":2,"tier":"F","calc":false,"text":"

Simplify  6x + 3y − 2x + 5y

","answerType":"exact","answer":"4x + 8y","answerDisplay":null,"accept":["8y + 4x","4x+8y","8y+4x"],"parts":[],"solution":["Like terms have exactly the same letters and powers. Combine x terms with x terms and y terms with y terms; an x and a y need not have the same value.","Collect the x terms: 6x - 2x = 4x.","Collect the y terms: 3y + 5y = 8y.","The simplified expression is 4x + 8y."],"diagram":null,"draw":false,"revision":"46d1cbea3386"},{"id":"g2-simplifying-algebra-q07","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":7,"marks":2,"tier":"F","calc":true,"text":"
  1. Simplify  4 × 3h
  2. Simplify  9a + 2b − 3a + b
","answerType":"multi","answer":"a) 12h b) 6a + 3b","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"12h"},{"label":"b","answer":"6a + 3b (accept 3b + 6a)"}],"solution":["a) 4 × 3h = 12h.","b) Collect the a terms: 9a - 3a = 6a. Collect the b terms: 2b + b = 3b.","b) The answer is 6a + 3b."],"diagram":null,"draw":false,"revision":"eb34a828a54e"},{"id":"g2-simplifying-algebra-q08","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":8,"marks":2,"tier":"F","calc":true,"text":"

Simplify  7pq − 2qp + 3pq

","answerType":"exact","answer":"8pq","answerDisplay":null,"accept":["8qp","8 pq"],"parts":[],"solution":["qp means the same as pq, so all three terms are like terms.","7pq - 2pq + 3pq = 8pq."],"diagram":null,"draw":false,"revision":"cb613d17adcd"},{"id":"g2-simplifying-algebra-q09","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":9,"marks":3,"tier":"F","calc":false,"text":"
  1. Simplify  5 × 2m
  2. Simplify  n × n × n × n
  3. Simplify  3c × 4d
","answerType":"multi","answer":"a) 10m b) n^4 c) 12cd","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"10m"},{"label":"b","answer":"n^4"},{"label":"c","answer":"12cd"}],"solution":["a) 5 × 2m = 10m.","b) n multiplied by itself four times is n4.","c) Multiply the numbers: 3 × 4 = 12, then attach the letters: 12cd."],"diagram":null,"draw":false,"revision":"eefcc2b22ca2"},{"id":"g2-simplifying-algebra-q10","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":10,"marks":3,"tier":"F","calc":true,"text":"
  1. Simplify  8x + 5 − 3x + 2
  2. Simplify  2g × 5g
","answerType":"multi","answer":"a) 5x + 7 b) 10g^2","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5x + 7 (accept 7 + 5x)"},{"label":"b","answer":"10g^2"}],"solution":["a) Collect the x terms: 8x - 3x = 5x. Collect the numbers: 5 + 2 = 7. Answer 5x + 7.","b) Multiply the numbers: 2 × 5 = 10. Multiply the letters: g × g = g2. Answer 10g2."],"diagram":null,"draw":false,"revision":"f3e1721afac6"},{"id":"g2-simplifying-algebra-q11","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":11,"marks":4,"tier":"F","calc":true,"text":"
  1. Simplify  t3 × t4
  2. Simplify  12h5 ÷ 3h2
  3. Simplify  5u + 3v − 2u + 7v
","answerType":"multi","answer":"a) t^7 b) 4h^3 c) 3u + 10v","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"t^7"},{"label":"b","answer":"4h^3"},{"label":"c","answer":"3u + 10v (accept 10v + 3u)"}],"solution":["a) When multiplying powers of the same letter, add the indices: 3 + 4 = 7, so t7.","b) Divide the numbers: 12 ÷ 3 = 4. Subtract the indices: 5 - 2 = 3. Answer 4h3.","c) Collect the u terms: 5u - 2u = 3u. Collect the v terms: 3v + 7v = 10v. Answer 3u + 10v.","The original fraction has h in its denominator, so h must not be zero."],"diagram":null,"draw":false,"revision":"f5b8b7841949"},{"id":"g2-simplifying-algebra-q12","topicId":"g2-simplifying-algebra","topic":"Simplifying Algebra","grade":"2","topicGrade":"2","strand":"A","difficulty":12,"marks":4,"tier":"F","calc":false,"text":"
  1. Simplify  9w − 4w + w
  2. Simplify  6e × 2f
  3. Factorise fully  8x + 12
","answerType":"multi","answer":"a) 6w b) 12ef c) 4(2x + 3)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6w"},{"label":"b","answer":"12ef"},{"label":"c","answer":"4(2x + 3)"}],"solution":["a) 9w - 4w + w = 6w.","b) 6 × 2 = 12 and the letters are e and f, so 12ef.","c) The highest common factor of 8x and 12 is 4.","c) 8x + 12 = 4(2x + 3). Check by expanding: 4 × 2x = 8x and 4 × 3 = 12."],"diagram":null,"draw":false,"revision":"1f5336d51ea0"},{"id":"g2-solving-one-step-equations-q01","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Solve  x + 7 = 12

","answerType":"exact","answer":"x = 5","answerDisplay":null,"accept":["5","x=5"],"parts":[],"solution":["Subtract 7 from both sides.","x = 12 - 7 = 5."],"diagram":null,"draw":false,"revision":"8f6064b0a281"},{"id":"g2-solving-one-step-equations-q02","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

Solve  y − 9 = 14

","answerType":"exact","answer":"y = 23","answerDisplay":null,"accept":["23","y=23"],"parts":[],"solution":["Add 9 to both sides.","y = 14 + 9 = 23."],"diagram":null,"draw":false,"revision":"8472dedf657d"},{"id":"g2-solving-one-step-equations-q03","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"
  1. Solve  4m = 28
  2. Solve  t ÷ 5 = 6
","answerType":"multi","answer":"a) m = 7 b) t = 30","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"m = 7"},{"label":"b","answer":"t = 30"}],"solution":["a) Divide both sides by 4: m = 28 ÷ 4 = 7.","b) Multiply both sides by 5: t = 6 × 5 = 30."],"diagram":null,"draw":false,"revision":"323e02d6aeae"},{"id":"g2-solving-one-step-equations-q04","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"
  1. Solve  w + 13 = 40
  2. Solve  8k = 60
","answerType":"multi","answer":"a) w = 27 b) k = 7.5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"w = 27"},{"label":"b","answer":"k = 7.5"}],"solution":["a) Subtract 13 from both sides: w = 40 - 13 = 27.","b) Divide both sides by 8: k = 60 ÷ 8 = 7.5"],"diagram":null,"draw":false,"revision":"9c9a80bf6381"},{"id":"g2-solving-one-step-equations-q05","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"
  1. Solve  c − 8 = 11
  2. Solve  6d = 54
  3. Solve  e ÷ 4 = 7
","answerType":"multi","answer":"a) c = 19 b) d = 9 c) e = 28","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"c = 19"},{"label":"b","answer":"d = 9"},{"label":"c","answer":"e = 28"}],"solution":["a) Add 8 to both sides: c = 11 + 8 = 19.","b) Divide both sides by 6: d = 54 ÷ 6 = 9.","c) Multiply both sides by 4: e = 7 × 4 = 28."],"diagram":null,"draw":false,"revision":"12db0565fa0f"},{"id":"g2-solving-one-step-equations-q06","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"
  1. Solve  3f = 19.5
  2. Solve  g + 4.7 = 9.2
  3. Solve  h ÷ 6 = 2.5
","answerType":"multi","answer":"a) f = 6.5 b) g = 4.5 c) h = 15","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"f = 6.5"},{"label":"b","answer":"g = 4.5"},{"label":"c","answer":"h = 15"}],"solution":["a) Divide both sides by 3: f = 19.5 ÷ 3 = 6.5","b) Subtract 4.7 from both sides: g = 9.2 - 4.7 = 4.5","c) Multiply both sides by 6: h = 2.5 × 6 = 15."],"diagram":null,"draw":false,"revision":"56362db68ac4"},{"id":"g2-solving-one-step-equations-q07","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":false,"text":"
  1. Solve  p + 9 = 4
  2. Solve  5q = −30
  3. Solve  r − 3 = −10
","answerType":"multi","answer":"a) p = -5 b) q = -6 c) r = -7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"p = -5"},{"label":"b","answer":"q = -6"},{"label":"c","answer":"r = -7"}],"solution":["a) Subtract 9 from both sides: p = 4 - 9 = -5.","b) Divide both sides by 5: q = -30 ÷ 5 = -6.","c) Add 3 to both sides: r = -10 + 3 = -7."],"diagram":null,"draw":false,"revision":"220d73d7ce21"},{"id":"g2-solving-one-step-equations-q08","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"
  1. Solve  7s = 42
  2. Solve  2t + 5 = 17
","answerType":"multi","answer":"a) s = 6 b) t = 6","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"s = 6"},{"label":"b","answer":"t = 6"}],"solution":["a) Divide both sides by 7: s = 42 ÷ 7 = 6.","b) Subtract 5 from both sides: 2t = 12.","b) Divide both sides by 2: t = 6."],"diagram":null,"draw":false,"revision":"3c67165a7977"},{"id":"g2-solving-one-step-equations-q09","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":9,"marks":3,"tier":"F","calc":false,"text":"
  1. Solve  15 = v + 8
  2. Solve  4w − 3 = 25
","answerType":"multi","answer":"a) v = 7 b) w = 7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"v = 7"},{"label":"b","answer":"w = 7"}],"solution":["a) The equation is the same as v + 8 = 15, so v = 15 - 8 = 7.","b) Add 3 to both sides: 4w = 28.","b) Divide both sides by 4: w = 7."],"diagram":null,"draw":false,"revision":"5720eb8f8970"},{"id":"g2-solving-one-step-equations-q10","topicId":"g2-solving-one-step-equations","topic":"Solving One Step Equations","grade":"2","topicGrade":"2","strand":"A","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"
  1. Solve  x − 17 = 26
  2. Solve  5y + 8 = 43
  3. Solve  20 − z = 12
","answerType":"multi","answer":"a) x = 43 b) y = 7 c) z = 8","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x = 43"},{"label":"b","answer":"y = 7"},{"label":"c","answer":"z = 8"}],"solution":["a) Add 17 to both sides: x = 26 + 17 = 43.","b) Subtract 8 from both sides: 5y = 35, then divide by 5: y = 7.","c) 20 - z = 12 means z is the difference between 20 and 12, so z = 8.","Check: 20 - 8 = 12."],"diagram":null,"draw":false,"revision":"8932a82603c6"},{"id":"g2-stem-and-leaf-q01","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

The stem and leaf diagram shows the masses, in grams, of some tomatoes.

StemLeaves
21  4  6
30  2  5  8
43  7

Key: 2 | 1 = 21 grams

Find the range of the masses.

","answerType":"exact","answer":"26 grams","answerDisplay":null,"accept":["26","26 g"],"parts":[],"solution":["The smallest mass is 21 grams and the largest mass is 47 grams.","Range: 47 - 21 = 26 grams."],"diagram":null,"draw":false,"revision":"806a2b245837"},{"id":"g2-stem-and-leaf-q02","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

Here are the ages, in years, of 15 people at a family party.

23  31  8  17  25  36  12  29  34  15  21  9  27  33  18

Draw an ordered stem and leaf diagram for these ages. Include a key.

","answerType":"multi","answer":"0 | 8 9 ; 1 | 2 5 7 8 ; 2 | 1 3 5 7 9 ; 3 | 1 3 4 6, with key 1 | 2 = 12 years","answerDisplay":null,"accept":[],"parts":[{"label":"stem 0","answer":"8 9"},{"label":"stem 1","answer":"2 5 7 8"},{"label":"stem 2","answer":"1 3 5 7 9"},{"label":"stem 3","answer":"1 3 4 6"}],"solution":["Use the tens digit as the stem and the units digit as the leaf.","Order the leaves in each row: 0 | 8 9, then 1 | 2 5 7 8, then 2 | 1 3 5 7 9, then 3 | 1 3 4 6.","Check the count: 2 + 4 + 5 + 4 = 15 values.","Include a key such as 1 | 2 = 12 years."],"diagram":{"type":"shape","notAccurate":false,"bounds":[[0,-1],[10,6]],"segments":[{"from":[2,1],"to":[2,5]}],"texts":[{"at":[1,5.5],"text":"Stem"},{"at":[6,5.5],"text":"Leaf"},{"at":[1,4.5],"text":"0","answer":true},{"at":[6,4.5],"text":"8 9","answer":true},{"at":[1,3.5],"text":"1","answer":true},{"at":[6,3.5],"text":"2 5 7 8","answer":true},{"at":[1,2.5],"text":"2","answer":true},{"at":[6,2.5],"text":"1 3 5 7 9","answer":true},{"at":[1,1.5],"text":"3","answer":true},{"at":[6,1.5],"text":"1 3 4 6","answer":true},{"at":[5,0],"text":"Key: 1 | 2 = 12 years","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"91e83f79f3e7"},{"id":"g2-stem-and-leaf-q03","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

The stem and leaf diagram shows the number of customers in a shop on each of 13 days.

StemLeaves
12  3  7
20  4  5  8  9
31  4  6
40  2

Key: 1 | 2 = 12 customers

Find the median number of customers.

","answerType":"exact","answer":"28","answerDisplay":null,"accept":["28 customers"],"parts":[],"solution":["There are 13 values, so the median is the 7th value when the data is in order.","Counting through the leaves in order: 12, 13, 17, 20, 24, 25, 28, ...","The 7th value is 28, so the median is 28 customers."],"diagram":null,"draw":false,"revision":"ee72a6eff4e3"},{"id":"g2-stem-and-leaf-q04","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"

Here are the times, in seconds, that 14 swimmers took to swim one length of a pool.

42  57  61  45  53  68  44  50  62  59  47  55  66  51

Draw an ordered stem and leaf diagram for these times. Include a key.

","answerType":"multi","answer":"4 | 2 4 5 7 ; 5 | 0 1 3 5 7 9 ; 6 | 1 2 6 8, with key 4 | 2 = 42 seconds","answerDisplay":null,"accept":[],"parts":[{"label":"stem 4","answer":"2 4 5 7"},{"label":"stem 5","answer":"0 1 3 5 7 9"},{"label":"stem 6","answer":"1 2 6 8"}],"solution":["Use the tens digit as the stem and the units digit as the leaf.","Order the leaves in each row: 4 | 2 4 5 7, then 5 | 0 1 3 5 7 9, then 6 | 1 2 6 8.","Check the count: 4 + 6 + 4 = 14 values.","Include a key such as 4 | 2 = 42 seconds."],"diagram":{"type":"shape","notAccurate":false,"bounds":[[0,-1],[10,6]],"segments":[{"from":[2,1],"to":[2,5]}],"texts":[{"at":[1,5.5],"text":"Stem"},{"at":[6,5.5],"text":"Leaf"},{"at":[1,4.5],"text":"4","answer":true},{"at":[6,4.5],"text":"2 4 5 7","answer":true},{"at":[1,3.5],"text":"5","answer":true},{"at":[6,3.5],"text":"0 1 3 5 7 9","answer":true},{"at":[1,2.5],"text":"6","answer":true},{"at":[6,2.5],"text":"1 2 6 8","answer":true},{"at":[5,0],"text":"Key: 4 | 2 = 42 seconds","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"445c0fa17114"},{"id":"g2-stem-and-leaf-q05","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

The stem and leaf diagram shows the number of press-ups 11 students did in one minute.

StemLeaves
06  8
11  3  5  7  9
22  4  6  8

Key: 1 | 1 = 11 press-ups

  1. Find the median number of press-ups.
  2. Find the range of the number of press-ups.
","answerType":"multi","answer":"a) 17 b) 22","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"17"},{"label":"b","answer":"22"}],"solution":["a) There are 11 values, so the median is the 6th value: 6, 8, 11, 13, 15, 17, ... The median is 17.","b) Range: 28 - 6 = 22."],"diagram":null,"draw":false,"revision":"bfd74142a1e8"},{"id":"g2-stem-and-leaf-q06","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

Nine boys and nine girls sat the same quiz. Their scores are shown in two stem and leaf diagrams.

Boys

StemLeaves
05
12  8
23  7
31  5  8
44

Girls

StemLeaves
21  4  6  8
30  3  5  7
40

Key: 2 | 1 = 21 points

Compare the boys' scores with the girls' scores.

","answerType":"open","answer":"Girls' median (30) is higher than boys' median (27), so the girls scored more on average; girls' range (19) is smaller than boys' range (39), so the girls' scores were more consistent","answerDisplay":null,"accept":[],"parts":[],"solution":["Boys' median: the 5th of 9 values = 27. Girls' median: the 5th of 9 values = 30.","Boys' range: 44 - 5 = 39. Girls' range: 40 - 21 = 19.","The girls' median is higher, so on average the girls scored more points.","The girls' range is smaller, so the girls' scores were more consistent."],"diagram":null,"draw":false,"revision":"e1a48d7a8f40"},{"id":"g2-stem-and-leaf-q07","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":7,"marks":4,"tier":"FH","calc":true,"text":"

The stem and leaf diagram shows the times, in minutes, that 15 students in Class A took to finish a puzzle.

StemLeaves
21  4  8
30  2  3  5  7  9
42  4  6  8
50  3

Key: 2 | 1 = 21 minutes

For Class B, the median time was 38 minutes and the range of the times was 15 minutes.

  1. Find the median time for Class A.
  2. Find the range of the times for Class A.
  3. Compare the times of Class A with the times of Class B.
","answerType":"multi","answer":"a) 37 minutes b) 32 minutes c) Class A median (37) is lower than Class B median (38), so Class A were slightly faster on average; Class A range (32) is larger than Class B range (15), so Class B times were more consistent","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"37 minutes"},{"label":"b","answer":"32 minutes"},{"label":"c","answer":"Class A slightly faster on average (lower median); Class B more consistent (smaller range)"}],"solution":["a) There are 15 values, so the median is the 8th value: 21, 24, 28, 30, 32, 33, 35, 37, ... The median is 37 minutes.","b) Range: 53 - 21 = 32 minutes.","c) Class A's median (37) is lower than Class B's (38), so on average Class A finished slightly faster.","Class A's range (32) is larger than Class B's (15), so Class B's times were more consistent."],"diagram":null,"draw":false,"revision":"0f8898c9b335"},{"id":"g2-stem-and-leaf-q08","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

Here are the heights, in centimetres, of 15 sunflower plants.

132  145  128  150  141  136  129  147  153  138  144  131  149  140  135

  1. Draw an ordered stem and leaf diagram for these heights. Include a key.
  2. Find the median height.
","answerType":"multi","answer":"a) 12 | 8 9 ; 13 | 1 2 5 6 8 ; 14 | 0 1 4 5 7 9 ; 15 | 0 3, key 12 | 8 = 128 cm b) 140 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"12 | 8 9 ; 13 | 1 2 5 6 8 ; 14 | 0 1 4 5 7 9 ; 15 | 0 3"},{"label":"b","answer":"140 cm"}],"solution":["a) Use stems 12, 13, 14, 15 (the hundreds and tens digits) with the units digit as the leaf.","Ordered rows: 12 | 8 9, then 13 | 1 2 5 6 8, then 14 | 0 1 4 5 7 9, then 15 | 0 3. Check: 2 + 5 + 6 + 2 = 15 values. Key: 12 | 8 = 128 cm.","b) The median is the 8th of 15 values: 128, 129, 131, 132, 135, 136, 138, 140, ...","The median height is 140 cm."],"diagram":{"type":"shape","notAccurate":false,"bounds":[[0,-1],[10,6]],"segments":[{"from":[2,1],"to":[2,5]}],"texts":[{"at":[1,5.5],"text":"Stem"},{"at":[6,5.5],"text":"Leaf"},{"at":[1,4.5],"text":"12","answer":true},{"at":[6,4.5],"text":"8 9","answer":true},{"at":[1,3.5],"text":"13","answer":true},{"at":[6,3.5],"text":"1 2 5 6 8","answer":true},{"at":[1,2.5],"text":"14","answer":true},{"at":[6,2.5],"text":"0 1 4 5 7 9","answer":true},{"at":[1,1.5],"text":"15","answer":true},{"at":[6,1.5],"text":"0 3","answer":true},{"at":[5,0],"text":"Key: 12 | 8 = 128 cm","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"301815ba8227"},{"id":"g2-stem-and-leaf-q09","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"2","topicGrade":"2","strand":"S","difficulty":9,"marks":5,"tier":"F","calc":true,"text":"

20 players each scored points in a darts game. Here are their scores.

12  45  23  38  41  17  29  34  47  21  36  43  15  27  32  49  24  39  44  30

  1. Draw an ordered stem and leaf diagram for these scores. Include a key.
  2. One of the 20 players is chosen at random. Find the probability that the player scored 40 points or more. Give your answer as a fraction in its simplest form.
","answerType":"multi","answer":"a) 1 | 2 5 7 ; 2 | 1 3 4 7 9 ; 3 | 0 2 4 6 8 9 ; 4 | 1 3 4 5 7 9, key 1 | 2 = 12 points b) 3/10","answerDisplay":"a) 1 | 2 5 7 ; 2 | 1 3 4 7 9 ; 3 | 0 2 4 6 8 9 ; 4 | 1 3 4 5 7 9, key 1 | 2 = 12 points b) 310","accept":[],"parts":[{"label":"a","answer":"1 | 2 5 7 ; 2 | 1 3 4 7 9 ; 3 | 0 2 4 6 8 9 ; 4 | 1 3 4 5 7 9"},{"label":"b","answer":"3/10","answerDisplay":"310"}],"solution":["a) Ordered rows: 1 | 2 5 7, then 2 | 1 3 4 7 9, then 3 | 0 2 4 6 8 9, then 4 | 1 3 4 5 7 9. Check: 3 + 5 + 6 + 6 = 20 values. Key: 1 | 2 = 12 points.","b) Scores of 40 or more are the stem-4 row: 41, 43, 44, 45, 47, 49, which is 6 players.","Probability: 620 = 310."],"diagram":{"type":"shape","notAccurate":false,"bounds":[[0,-1],[10,6]],"segments":[{"from":[2,1],"to":[2,5]}],"texts":[{"at":[1,5.5],"text":"Stem"},{"at":[6,5.5],"text":"Leaf"},{"at":[1,4.5],"text":"1","answer":true},{"at":[6,4.5],"text":"2 5 7","answer":true},{"at":[1,3.5],"text":"2","answer":true},{"at":[6,3.5],"text":"1 3 4 7 9","answer":true},{"at":[1,2.5],"text":"3","answer":true},{"at":[6,2.5],"text":"0 2 4 6 8 9","answer":true},{"at":[1,1.5],"text":"4","answer":true},{"at":[6,1.5],"text":"1 3 4 5 7 9","answer":true},{"at":[5,0],"text":"Key: 1 | 2 = 12 points","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"67d48ad9b433"},{"id":"g2-stem-and-leaf-q10","topicId":"g2-stem-and-leaf","topic":"Stem and Leaf","grade":"4","topicGrade":"2","strand":"S","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

A gardener grew 12 bean plants using Fertiliser A and 12 bean plants using Fertiliser B. The stem and leaf diagrams show the heights of the plants, in centimetres, after six weeks.

Fertiliser A

StemLeaves
14  7
21  3  6  8
30  2  5  7
41  3

Fertiliser B

StemLeaves
25  7  9
31  2  4  6  8  9
40  2  5

Key: 2 | 5 = 25 cm

The gardener wants to use the fertiliser that grows taller, more consistent plants. Compare the two distributions and state which fertiliser the gardener should choose.

","answerType":"open","answer":"Fertiliser B: its median (35 cm) is higher than A's (29 cm), so B plants are taller on average, and its range (20 cm) is smaller than A's (29 cm), so B plants are more consistent","answerDisplay":null,"accept":[],"parts":[],"solution":["Fertiliser A median: mean of the 6th and 7th of 12 values = (28 + 30) divided by 2 = 29 cm.","Fertiliser B median: (34 + 36) divided by 2 = 35 cm.","Fertiliser A range: 43 - 14 = 29 cm. Fertiliser B range: 45 - 25 = 20 cm.","B has the higher median, so its plants are taller on average, and B has the smaller range, so its heights are more consistent.","The gardener should choose Fertiliser B."],"diagram":null,"draw":false,"revision":"7cd140b534f0"},{"id":"g2-systematic-listing-q01","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Ola is choosing an outfit.

She has 3 shirts: red (R), blue (B) and green (G).

She has 2 pairs of shorts: black (K) and white (W).

List all the possible outfits Ola can choose. One outfit, RK, has been done for you.

","answerType":"exact","answer":"RK, RW, BK, BW, GK, GW","answerDisplay":null,"accept":["RK RW BK BW GK GW"],"parts":[],"solution":["Pair each shirt with each pair of shorts in turn.","Red: RK, RW. Blue: BK, BW. Green: GK, GW.","There are 6 outfits in total."],"diagram":null,"draw":false,"revision":"19eb3210f510"},{"id":"g2-systematic-listing-q02","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

A cafe sells 3 hot drinks: tea (T), coffee (C) and hot chocolate (H).

It sells 2 cakes: flapjack (F) and brownie (B).

Noah chooses one hot drink and one cake.

List all the possible combinations Noah can choose.

","answerType":"exact","answer":"TF, TB, CF, CB, HF, HB","answerDisplay":null,"accept":["TF TB CF CB HF HB"],"parts":[],"solution":["Pair each drink with each cake in turn.","Tea: TF, TB. Coffee: CF, CB. Hot chocolate: HF, HB.","There are 6 combinations in total."],"diagram":null,"draw":false,"revision":"f77c7015ba7c"},{"id":"g2-systematic-listing-q03","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Bag A contains three cards numbered 1, 2 and 3.

Bag B contains two cards numbered 5 and 7.

Erin takes one card from bag A and one card from bag B.

Write down all the possible pairs of cards Erin can take.

","answerType":"exact","answer":"(1, 5), (1, 7), (2, 5), (2, 7), (3, 5), (3, 7)","answerDisplay":null,"accept":["1 and 5, 1 and 7, 2 and 5, 2 and 7, 3 and 5, 3 and 7"],"parts":[],"solution":["Pair each card from bag A with each card from bag B.","1 with 5, 1 with 7, 2 with 5, 2 with 7, 3 with 5, 3 with 7.","There are 6 possible pairs."],"diagram":null,"draw":false,"revision":"af39b86464d9"},{"id":"g2-systematic-listing-q04","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":4,"marks":2,"tier":"F","calc":false,"text":"

Ali throws a fair coin 3 times.

Each throw lands on heads (H) or tails (T).

List all the possible outcomes of the 3 throws. One outcome, HHH, has been done for you.

","answerType":"exact","answer":"HHH, HHT, HTH, HTT, THH, THT, TTH, TTT","answerDisplay":null,"accept":["HHH HHT HTH HTT THH THT TTH TTT"],"parts":[],"solution":["Work systematically, starting with outcomes beginning H.","HHH, HHT, HTH, HTT, then THH, THT, TTH, TTT.","There are 8 outcomes in total."],"diagram":null,"draw":false,"revision":"fd5d4408ac91"},{"id":"g2-systematic-listing-q05","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

A locker code is a letter followed by a prime number.

The letter is chosen from P, Q and R.

The prime number is chosen from 3 and 7.

List all the possible locker codes.

","answerType":"exact","answer":"P3, P7, Q3, Q7, R3, R7","answerDisplay":null,"accept":["P3 P7 Q3 Q7 R3 R7"],"parts":[],"solution":["Pair each letter with each prime in turn.","P: P3, P7. Q: Q3, Q7. R: R3, R7.","There are 6 possible codes."],"diagram":null,"draw":false,"revision":"5b44ef35aa1a"},{"id":"g2-systematic-listing-q06","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":6,"marks":2,"tier":"F","calc":true,"text":"

Here are three digit cards.

2   5   8

Ravi uses two different cards each time to make two-digit numbers.

List all the two-digit numbers Ravi can make.

","answerType":"exact","answer":"25, 28, 52, 58, 82, 85","answerDisplay":null,"accept":["25 28 52 58 82 85"],"parts":[],"solution":["Work systematically by tens digit.","Tens digit 2: 25, 28. Tens digit 5: 52, 58. Tens digit 8: 82, 85.","There are 6 numbers in total."],"diagram":null,"draw":false,"revision":"a8d208ca76b6"},{"id":"g2-systematic-listing-q07","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":7,"marks":2,"tier":"F","calc":true,"text":"

A pizza shop offers 4 toppings: mushroom (M), pepper (P), olive (O) and sweetcorn (S).

Jo chooses 2 different toppings for her pizza.

List all the possible pairs of toppings Jo can choose.

","answerType":"exact","answer":"MP, MO, MS, PO, PS, OS","answerDisplay":null,"accept":["MP MO MS PO PS OS"],"parts":[],"solution":["Work systematically. The order of the two toppings does not matter.","With M: MP, MO, MS. With P (not already listed): PO, PS. With O: OS.","There are 6 possible pairs."],"diagram":null,"draw":false,"revision":"c31447435508"},{"id":"g2-systematic-listing-q08","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":8,"marks":2,"tier":"F","calc":true,"text":"

Spinner A has three equal sections numbered 1, 2 and 3.

Spinner B has two equal sections numbered 4 and 5.

Both spinners are spun and the two numbers are multiplied together.

List all the possible products.

","answerType":"exact","answer":"4, 5, 8, 10, 12, 15","answerDisplay":null,"accept":["4 5 8 10 12 15"],"parts":[],"solution":["Multiply each number on spinner A by each number on spinner B.","1 × 4 = 4, 1 × 5 = 5, 2 × 4 = 8, 2 × 5 = 10, 3 × 4 = 12, 3 × 5 = 15.","The possible products are 4, 5, 8, 10, 12 and 15."],"diagram":null,"draw":false,"revision":"6ed13cc76e46"},{"id":"g2-systematic-listing-q09","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":9,"marks":2,"tier":"F","calc":false,"text":"

Ivy has three coins: a 5p coin, a 10p coin and a 20p coin.

She picks two different coins and adds their values.

List all the possible totals.

","answerType":"exact","answer":"15p, 25p, 30p","answerDisplay":null,"accept":["15p 25p 30p","15, 25, 30"],"parts":[],"solution":["The possible pairs are 5p and 10p, 5p and 20p, 10p and 20p.","The totals are 15p, 25p and 30p."],"diagram":null,"draw":false,"revision":"4f34fac1da95"},{"id":"g2-systematic-listing-q10","topicId":"g2-systematic-listing","topic":"Systematic Listing","grade":"2","topicGrade":"2","strand":"N","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

Here are four digit cards.

2   5   6   9

  1. Using each card once, write down the largest four-digit number that can be made.
  2. Using each card once, write down the smallest four-digit even number that can be made.
  3. Using two different cards each time, list all the two-digit numbers greater than 60 that can be made.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"9652"},{"label":"b","answer":"2596"},{"label":"c","answer":"62, 65, 69, 92, 95, 96"}],"solution":["(a) Put the digits in decreasing order: 9652.","(b) The number must end in 2 or 6. Ending in 6 allows the start 259, giving 2596. Ending in 2 the smallest is 5692, which is bigger. So the answer is 2596.","(c) Tens digit 6: 62, 65, 69. Tens digit 9: 92, 95, 96. Numbers starting 2 or 5 are less than 60.","There are 6 numbers: 62, 65, 69, 92, 95, 96."],"diagram":null,"draw":false,"revision":"e8d7ecb23c09"},{"id":"g2-time-series-and-line-graphs-q01","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"2","topicGrade":"2","strand":"S","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

A line graph shows the midday temperature in a town on the first day of each month. The table gives the plotted values.

MonthTemperature (°C)
January6
February7
March10
April13
May17
June21
  1. Write down the month in which the temperature was 13°C.
  2. Work out the difference between the temperature in June and the temperature in January.
","answerType":"multi","answer":"a) April b) 15°C","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"April"},{"label":"b","answer":"15"}],"solution":["a) The value 13°C is plotted for April.","b) 21 - 6 = 15°C."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Jan","Feb","Mar","Apr","May","Jun"],"label":"Month"},"y":{"min":0,"max":25,"step":5,"minor":5,"label":"Temperature (°C)"},"curves":[{"pts":[[0,6],[1,7],[2,10],[3,13],[4,17],[5,21]],"marks":true}]},"draw":false,"revision":"117fd450709d"},{"id":"g2-time-series-and-line-graphs-q02","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"2","topicGrade":"2","strand":"S","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

The time series graph shows the number of ice creams a kiosk sold each month from January to June.

  1. Write down the number of ice creams sold in April.
  2. Between which two consecutive months was the biggest increase in sales?
","answerType":"multi","answer":"a) 260 b) April and May","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"260"},{"label":"b","answer":"April and May"}],"solution":["a) The point for April is at 260 ice creams.","b) The increases are: Jan to Feb 30, Feb to Mar 60, Mar to Apr 50, Apr to May 70, May to Jun 60.","The biggest increase, 70, is between April and May - the steepest line segment on the graph."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Jan","Feb","Mar","Apr","May","Jun"],"label":"Month"},"y":{"min":0,"max":420,"step":30,"label":"Number of ice creams sold"},"curves":[{"pts":[[0,120],[1,150],[2,210],[3,260],[4,330],[5,390]],"marks":true}]},"draw":false,"revision":"20f9c7ff2def"},{"id":"g2-time-series-and-line-graphs-q03","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"2","topicGrade":"2","strand":"S","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

A shop sells woolly hats. The table shows the number of hats sold in each quarter of 2024 and 2025.

Quarter20242025
Q1 (Jan to Mar)6258
Q2 (Apr to Jun)3026
Q3 (Jul to Sep)1815
Q4 (Oct to Dec)5551
  1. Describe the pattern in sales within each year.
  2. Describe the trend in sales from 2024 to 2025.
","answerType":"open","answer":"a) Sales are highest in the colder quarters (Q1 and Q4) and lowest in summer (Q3) - a seasonal pattern b) Every quarter is lower in 2025 than in 2024, so the trend is downward","answerDisplay":null,"accept":[],"parts":[],"solution":["a) In both years sales are high in Q1 and Q4 (winter) and low in Q2 and Q3 (summer). This is a seasonal pattern - woolly hats sell more in cold months.","b) Comparing like quarters: 62 to 58, 30 to 26, 18 to 15, 55 to 51. Every quarter fell, so the overall trend is downward."],"diagram":null,"draw":false,"revision":"6a2e1c3ed608"},{"id":"g2-time-series-and-line-graphs-q04","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"2","topicGrade":"2","strand":"S","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

The table shows the number of visitors to a website each day for one week.

DayVisitors
Monday85
Tuesday70
Wednesday90
Thursday105
Friday120
Saturday150
Sunday140

Draw a time series graph for this information.

","answerType":"open","answer":"Points plotted at (Mon, 85), (Tue, 70), (Wed, 90), (Thu, 105), (Fri, 120), (Sat, 150), (Sun, 140), joined in order with straight lines","answerDisplay":null,"accept":[],"parts":[],"solution":["Put the days in order along the horizontal axis and Visitors on the vertical axis.","Plot each value: 85, 70, 90, 105, 120, 150, 140.","Join the points, in day order, with straight lines."],"diagram":{"type":"axes","xy":false,"x":{"cats":["Mon","Tue","Wed","Thu","Fri","Sat","Sun"],"label":"Day"},"y":{"min":0,"max":160,"step":20,"minor":2,"label":"Visitors"},"curves":[{"pts":[[0,85],[1,70],[2,90],[3,105],[4,120],[5,150],[6,140]],"marks":true,"answer":true}]},"draw":true,"revision":"8094c8cdc26d"},{"id":"g2-time-series-and-line-graphs-q05","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"2","topicGrade":"2","strand":"S","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

A company's sales rose from £192000 in January to £198000 in June.

The company prints a line graph of its monthly sales. The vertical axis of the graph starts at £190000 and ends at £200000. The headline above the graph says \"Sales have rocketed!\"

Explain why the graph could be misleading.

","answerType":"open","answer":"The vertical axis does not start at zero, so the small rise looks very steep; the actual increase is only £6000, about 3%, which is not a rocketing rise","answerDisplay":null,"accept":[],"parts":[],"solution":["The vertical axis starts at £190000, not zero, so the line uses the whole height of the graph and the rise looks much bigger than it is.","The actual increase is 198000 - 192000 = £6000, which is only about 3% - a small rise, not a rocketing one."],"diagram":null,"draw":false,"revision":"30125bf0367c"},{"id":"g2-time-series-and-line-graphs-q06","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"4","topicGrade":"2","strand":"S","difficulty":6,"marks":2,"tier":"H","calc":true,"text":"

An estate agent draws a time series graph to show how average house prices in a town have changed.

Write down two reasons why the graph is misleading.

","answerType":"open","answer":"The vertical axis starts at 240000 rather than zero, exaggerating the rise; the years on the horizontal axis are not equally spaced in time but are plotted at equal gaps, distorting the rate of change","answerDisplay":null,"accept":[],"parts":[],"solution":["Reason 1: the vertical axis starts at £240000, not zero, so the increase looks much larger than it really is.","Reason 2: the years 2018, 2021, 2022 and 2023 are plotted at equal gaps even though the time between them is not equal, so the steepness of the line does not show the true rate of change."],"diagram":{"type":"axes","xy":false,"x":{"cats":["2018","2021","2022","2023"],"label":"Year"},"y":{"min":240000,"max":300000,"step":10000,"label":"Average house price (£)"},"curves":[{"pts":[[0,245000],[1,262000],[2,280000],[3,295000]],"marks":true}]},"draw":false,"revision":"775f982aea20"},{"id":"g2-time-series-and-line-graphs-q07","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"2","topicGrade":"2","strand":"S","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

A bookshop recorded the number of books it sold each month from January to June.

MonthBooks sold
January240
February255
March230
April265
May250
June248

The shop's target was a mean of 250 books sold per month.

Did the shop meet its target? You must show your working.

","answerType":"open","answer":"No. Total = 1488, mean = 1488 divided by 6 = 248, which is below the target of 250","answerDisplay":null,"accept":[],"parts":[],"solution":["Total: 240 + 255 + 230 + 265 + 250 + 248 = 1488 books.","Mean: 1488 divided by 6 = 248 books per month.","248 < 250, so the shop did not meet its target."],"diagram":null,"draw":false,"revision":"45b80c4cb87f"},{"id":"g2-time-series-and-line-graphs-q08","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"2","topicGrade":"2","strand":"S","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

The table shows the amount of electricity, in kilowatt hours, a household used in each quarter of two years.

QuarterkWh
2024 Q11150
2024 Q2820
2024 Q3700
2024 Q41080
2025 Q11100
2025 Q2780
2025 Q3650
2025 Q41020

The first six points have been plotted on the time series graph.

  1. Complete the time series graph by plotting the last two points and joining them.
  2. Describe the trend in electricity use over the two years.
","answerType":"multi","answer":"a) points at (2025 Q3, 650) and (2025 Q4, 1020) plotted and joined b) Use follows a seasonal pattern (high in Q1 and Q4, low in Q3) and comparing like quarters it is slowly falling","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Plot (2025 Q3, 650) and (2025 Q4, 1020) and join with straight lines"},{"label":"b","answer":"Seasonal pattern, with a slight downward trend year on year"}],"solution":["a) Plot 650 for 2025 Q3 and 1020 for 2025 Q4, joining the points with straight lines in order.","b) Within each year use is highest in Q1 and Q4 and lowest in Q3 - a seasonal pattern.","Comparing the same quarter in each year (1150 to 1100, 820 to 780, 700 to 650, 1080 to 1020), every quarter fell, so the overall trend is slightly downward."],"diagram":{"type":"axes","xy":false,"width":560,"x":{"cats":["24 Q1","24 Q2","24 Q3","24 Q4","25 Q1","25 Q2","25 Q3","25 Q4"],"label":"Quarter"},"y":{"min":600,"max":1200,"step":100,"minor":2,"label":"Electricity used (kWh)"},"curves":[{"pts":[[0,1150],[1,820],[2,700],[3,1080],[4,1100],[5,780]],"marks":true},{"pts":[[5,780],[6,650],[7,1020]],"marks":true,"answer":true}]},"draw":true,"revision":"25accab6f6a6"},{"id":"g2-time-series-and-line-graphs-q09","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"4","topicGrade":"2","strand":"S","difficulty":9,"marks":3,"tier":"H","calc":true,"text":"

A factory records the number of bicycles it makes each month.

MonthBicycles
January410
February380
March395
April420
May405
June402
  1. The factory manager says, \"The mean monthly output for these six months is more than 400.\" Show that the manager is correct.
  2. Work out how much greater the mean output for April to June is than the mean output for January to March.
","answerType":"multi","answer":"a) Total = 2412, mean = 402, and 402 > 400 b) 14","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Mean = 2412 / 6 = 402 > 400","answerDisplay":"Mean = 24126 = 402 > 400"},{"label":"b","answer":"14"}],"solution":["a) Total: 410 + 380 + 395 + 420 + 405 + 402 = 2412. Mean: 2412 divided by 6 = 402, which is more than 400, so the manager is correct.","b) January to March: 410 + 380 + 395 = 1185, mean 395.","April to June: 420 + 405 + 402 = 1227, mean 409.","Difference: 409 - 395 = 14 bicycles."],"diagram":null,"draw":false,"revision":"4e31bfb4a248"},{"id":"g2-time-series-and-line-graphs-q10","topicId":"g2-time-series-and-line-graphs","topic":"Time Series and Line Graphs","grade":"4","topicGrade":"2","strand":"S","difficulty":10,"marks":3,"tier":"H","calc":true,"text":"

The table shows the number of members of a gym at the end of each year.

YearMembers
20212000
20222200
20232420
  1. Show that the number of members increased by the same percentage each year.
  2. Assuming this trend continues, predict the number of members at the end of 2024.
  3. Give one reason why your prediction may not be reliable.
","answerType":"multi","answer":"a) 2200/2000 = 1.1 and 2420/2200 = 1.1, a 10% increase each year b) 2662 c) The trend may not continue in the future","answerDisplay":"a) 22002000 = 1.1 and 24202200 = 1.1, a 10% increase each year b) 2662 c) The trend may not continue in the future","accept":[],"parts":[{"label":"a","answer":"2200 / 2000 = 1.1 and 2420 / 2200 = 1.1, so a 10% increase each year","answerDisplay":"22002000 = 1.1 and 24202200 = 1.1, so a 10% increase each year"},{"label":"b","answer":"2662"},{"label":"c","answer":"The trend may not continue - membership could change for other reasons"}],"solution":["a) 2200 divided by 2000 = 1.1 and 2420 divided by 2200 = 1.1, so members grew by 10% in each year.","b) 2420 × 1.1 = 2662 members.","c) The prediction assumes the 10% growth continues, but the trend may not continue - for example a new competitor could open or prices could change."],"diagram":null,"draw":false,"revision":"5d48e77b4caa"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q01","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

Here are the first four terms of a sequence.

3    6    12    24

  1. Write down the next term of the sequence.
  2. Write down the term-to-term rule of the sequence.
","answerType":"multi","answer":"a) 48 b) multiply by 2","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"48"},{"label":"b","answer":"multiply by 2 (accept double the previous term)"}],"solution":["Each term is double the term before: 3 × 2 = 6, 6 × 2 = 12, 12 × 2 = 24.","The next term is 24 × 2 = 48.","The term-to-term rule is multiply by 2."],"diagram":null,"draw":false,"revision":"a9a2cfb392a1"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q02","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Here are the first five square numbers.

1    4    9    16    25

  1. Write down the next square number after 25.
  2. Write down the 8th square number.
","answerType":"multi","answer":"a) 36 b) 64","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"36"},{"label":"b","answer":"64"}],"solution":["The square numbers are 1 × 1, 2 × 2, 3 × 3 and so on.","The next one after 25 is 6 × 6 = 36.","The 8th square number is 8 × 8 = 64."],"diagram":null,"draw":false,"revision":"04ebe0eac3a1"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q03","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Here are the first five terms of a Fibonacci sequence.

2    3    5    8    13

  1. Write down the next term of the sequence.
  2. Describe the rule for finding the next term of the sequence.
","answerType":"multi","answer":"a) 21 b) add the two previous terms","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"21"},{"label":"b","answer":"open: add the two previous terms together"}],"solution":["In a Fibonacci sequence each term is the sum of the two terms before it.","Check: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13.","The next term is 8 + 13 = 21."],"diagram":null,"draw":false,"revision":"cd061994f50f"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q04","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

Here are the first four cube numbers.

1    8    27    64

  1. Write down the next cube number after 64.
  2. Write down the 10th cube number.
  3. Explain why 200 is not a cube number.
","answerType":"multi","answer":"a) 125 b) 1000 c) 5^3 = 125 and 6^3 = 216, and 200 is between them, so 200 is not a cube number","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"125"},{"label":"b","answer":"1000"},{"label":"c","answer":"open: 5 cubed = 125 and 6 cubed = 216, and 200 lies between them"}],"solution":["The cube numbers are 13, 23, 33 and so on.","The next one after 64 is 53 = 125.","The 10th cube number is 103 = 1000.","53 = 125 and 63 = 216, so 200 falls between two consecutive cube numbers and cannot be one."],"diagram":null,"draw":false,"revision":"49e5c607ed1b"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q05","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

Here are the first four terms of a sequence.

14    11    8    5

  1. Work out the next two terms of the sequence.
  2. Write down the term-to-term rule of the sequence.
","answerType":"multi","answer":"a) 2 and -1 b) subtract 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2 and -1"},{"label":"b","answer":"subtract 3 (accept take away 3)"}],"solution":["The terms go down by 3 each time: 14, 11, 8, 5.","Next term: 5 - 3 = 2.","The term after that: 2 - 3 = -1.","The term-to-term rule is subtract 3."],"diagram":null,"draw":false,"revision":"596d1e8ddf30"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q06","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

Here are the first four terms of a sequence.

96    48    24    12

  1. Write down the next term of the sequence.
  2. Work out the 8th term of the sequence.
","answerType":"multi","answer":"a) 6 b) 0.75","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6"},{"label":"b","answer":"0.75 (accept 3/4)","answerDisplay":"0.75 (accept 34)"}],"solution":["Each term is half the term before, so the 5th term is 12 ÷ 2 = 6.","Keep halving: 6th term = 3, 7th term = 1.5, 8th term = 0.75"],"diagram":null,"draw":false,"revision":"da0d031806cd"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q07","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":false,"text":"

Here are the first five triangular numbers.

1    3    6    10    15

  1. Describe how the sequence continues from one term to the next.
  2. Work out the next two terms of the sequence.
","answerType":"multi","answer":"a) the differences increase by 1 each time (add 2, then 3, then 4, ...) b) 21 and 28","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"open: the amount added increases by 1 each time (add 2, add 3, add 4, ...)"},{"label":"b","answer":"21 and 28"}],"solution":["The differences between terms are 2, 3, 4, 5, so the amount added goes up by 1 each time.","Next term: 15 + 6 = 21.","The term after that: 21 + 7 = 28."],"diagram":null,"draw":false,"revision":"b5454a9346fb"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q08","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

Here are three sequences.

Sequence A:  7    11    15    19

Sequence B:  2    6    18    54

Sequence C:  1    4    5    9    14

  1. Write down the letter of the geometric sequence.
  2. Work out the next term of sequence C.
  3. Work out the 10th term of sequence A.
","answerType":"multi","answer":"a) B b) 23 c) 43","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"B"},{"label":"b","answer":"23"},{"label":"c","answer":"43"}],"solution":["Sequence B multiplies by 3 each time, so it is geometric.","Sequence C is Fibonacci-type: each term is the sum of the two before it (1 + 4 = 5, 4 + 5 = 9, 5 + 9 = 14), so the next term is 9 + 14 = 23.","Sequence A goes up by 4 each time. The 10th term is 7 + 9 × 4 = 7 + 36 = 43."],"diagram":null,"draw":false,"revision":"869ed88f21ee"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q09","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"2","topicGrade":"2","strand":"A","difficulty":9,"marks":3,"tier":"FH","calc":false,"text":"

In a Fibonacci sequence, each term after the second is the sum of the two previous terms.

The first term of a Fibonacci sequence is p.

The second term of the sequence is 5.

The fourth term of the sequence is 22.

Work out the value of p.

","answerType":"exact","answer":"p = 12","answerDisplay":null,"accept":["12","p=12"],"parts":[],"solution":["Third term = p + 5.","Fourth term = (p + 5) + 5 = p + 10.","So p + 10 = 22, giving p = 12.","Check: the sequence is 12, 5, 17, 22 and 5 + 17 = 22."],"diagram":null,"draw":false,"revision":"b2b7c9e0b04d"},{"id":"g2-types-of-sequences-fibonacci-geometric-square-and-cube-q10","topicId":"g2-types-of-sequences-fibonacci-geometric-square-and-cube","topic":"Types of Sequences (Fibonacci, Geometric, Square and Cube)","grade":"6","topicGrade":"2","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

The first two terms of a geometric sequence are

5    5√2

  1. Work out the 4th term of the sequence.
    Give your answer in the form a√2 where a is an integer.
  2. Find the first term of the sequence that is greater than 100.
","answerType":"multi","answer":"a) 10√2 b) 80√2","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"10√2"},{"label":"b","answer":"80√2 (the 10th term, approximately 113.1)"}],"solution":["The common ratio is 5√2 ÷ 5 = √2.","a) 3rd term = 5√2 × √2 = 10. 4th term = 10 × √2 = 10√2.","b) The terms are 5, 5√2, 10, 10√2, 20, 20√2, 40, 40√2, 80, 80√2, ...","b) 80 < 100 but 80√2 ≈ 113.1 > 100.","b) The first term greater than 100 is 80√2."],"diagram":null,"draw":false,"revision":"63b4403f1b1d"},{"id":"g2-using-a-calculator-q01","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"2","topicGrade":"2","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":true,"text":"

Use your calculator to work out √1296

","answerType":"exact","answer":"36","answerDisplay":null,"accept":[],"parts":[],"solution":["√1296 = 36, since 36 × 36 = 1296."],"diagram":null,"draw":false,"revision":"786e94232f1f"},{"id":"g2-using-a-calculator-q02","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"2","topicGrade":"2","strand":"N","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

Use your calculator to work out 2.54

","answerType":"exact","answer":"39.0625","answerDisplay":null,"accept":[],"parts":[],"solution":["2.52 = 6.25.","6.252 = 39.0625, so 2.54 = 39.0625."],"diagram":null,"draw":false,"revision":"55ab4fc8fb91"},{"id":"g2-using-a-calculator-q03","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"2","topicGrade":"2","strand":"N","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Use your calculator to work out

(2.42 + 1.8) ÷ 0.53

Write down all the figures on your calculator display.

","answerType":"exact","answer":"60.48","answerDisplay":null,"accept":[],"parts":[],"solution":["2.42 = 5.76, so the numerator is 5.76 + 1.8 = 7.56.","0.53 = 0.125.","7.56 ÷ 0.125 = 60.48."],"diagram":null,"draw":false,"revision":"41708a37bc8c"},{"id":"g2-using-a-calculator-q04","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"2","topicGrade":"2","strand":"N","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

Use your calculator to work out

√29.16 ÷ 1.22

Write down all the figures on your calculator display.

","answerType":"exact","answer":"3.75","answerDisplay":null,"accept":[],"parts":[],"solution":["√29.16 = 5.4, since 5.4 × 5.4 = 29.16.","1.22 = 1.44.","5.4 ÷ 1.44 = 3.75."],"diagram":null,"draw":false,"revision":"55dfe29c2941"},{"id":"g2-using-a-calculator-q05","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"4","topicGrade":"2","strand":"N","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

Use your calculator to work out

(√8.41 + 2.5) ÷ tan 45°

","answerType":"exact","answer":"5.4","answerDisplay":null,"accept":["5.40"],"parts":[],"solution":["Use degree mode because the angle is given in degrees.","√8.41 = 2.9, since 2.9 × 2.9 = 8.41.","2.9 + 2.5 = 5.4.","tan 45° = 1, so the answer is 5.4 ÷ 1 = 5.4."],"diagram":null,"draw":false,"revision":"cf86e0b79543"},{"id":"g2-using-a-calculator-q06","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"2","topicGrade":"2","strand":"N","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

Use your calculator to work out

(13.7 + 9.28) ÷ 0.64

  1. Write down all the figures on your calculator display.
  2. Write your answer to part (a) correct to 2 decimal places.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"35.90625"},{"label":"b","answer":"35.91"}],"solution":["13.7 + 9.28 = 22.98.","22.98 ÷ 0.64 = 35.90625.","(b) The third decimal place is 6, so round up: 35.91."],"diagram":null,"draw":false,"revision":"f878b5278a44"},{"id":"g2-using-a-calculator-q07","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"2","topicGrade":"2","strand":"N","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"
  1. Use your calculator to work out 4.12 − 3.6
  2. Write down the reciprocal of 0.25
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"13.21"},{"label":"b","answer":"4"}],"solution":["(a) 4.12 = 16.81, so 16.81 − 3.6 = 13.21.","(b) The reciprocal of 0.25 is 1 ÷ 0.25 = 4."],"diagram":null,"draw":false,"revision":"bc39e77ae569"},{"id":"g2-using-a-calculator-q08","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"2","topicGrade":"2","strand":"N","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

Use your calculator to work out

(8.2 + 3.55) ÷ 3.25

  1. Give this answer to 6 decimal places.
  2. Write your answer to part (a) correct to 4 significant figures.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3.615385"},{"label":"b","answer":"3.615"}],"solution":["8.2 + 3.55 = 11.75.","11.75 ÷ 3.25 = 47 ÷ 13 = 3.615384615...","(b) The fifth significant figure is 3, so round down: 3.615.","To 6 decimal places this is 3.615385. Keep the original calculator value for any later rounding."],"diagram":null,"draw":false,"revision":"7c28c0258069"},{"id":"g2-using-a-calculator-q09","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"2","topicGrade":"2","strand":"N","difficulty":9,"marks":3,"tier":"F","calc":true,"text":"

Use your calculator to work out

(2.82 − 1.9) ÷ (0.3 + 0.45)

  1. Write down all the figures on your calculator display.
  2. Write your answer to part (a) correct to 1 significant figure.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"7.92"},{"label":"b","answer":"8"}],"solution":["2.82 = 7.84, so the numerator is 7.84 − 1.9 = 5.94.","The denominator is 0.3 + 0.45 = 0.75.","5.94 ÷ 0.75 = 7.92.","(b) 7.92 to 1 significant figure is 8."],"diagram":null,"draw":false,"revision":"371f57837ccf"},{"id":"g2-using-a-calculator-q10","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"4","topicGrade":"2","strand":"N","difficulty":10,"marks":3,"tier":"H","calc":true,"text":"
  1. Use your calculator to work out the cube root of 9.261
  2. Work out the reciprocal of 0.35
    Give this answer to 6 decimal places.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2.1"},{"label":"b","answer":"2.857143"}],"solution":["(a) 2.13 = 2.1 × 2.1 × 2.1 = 9.261, so the cube root of 9.261 is 2.1.","(b) The reciprocal of 0.35 is 1 ÷ 0.35 = 207 = 2.857142857...","To 6 decimal places this is 2.857143. Keep the original calculator value for any later rounding."],"diagram":null,"draw":false,"revision":"3f0934f394a5"},{"id":"g2-using-a-calculator-q11","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"4","topicGrade":"2","strand":"N","difficulty":11,"marks":3,"tier":"H","calc":true,"text":"

Use your calculator to work out

√6.25 + tan 60°

  1. Give this answer to 6 decimal places.
  2. Write your answer to part (a) correct to 3 significant figures.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"4.232051"},{"label":"b","answer":"4.23"}],"solution":["Use degree mode because the angle is given in degrees.","√6.25 = 2.5.","tan 60° = √3 ≈ 1.732050808.","2.5 + √3 ≈ 4.232050808.","(b) 4.232050808 to 3 significant figures is 4.23.","To 6 decimal places this is 4.232051. Keep the original calculator value for any later rounding."],"diagram":null,"draw":false,"revision":"5c82bcad83ae"},{"id":"g2-using-a-calculator-q12","topicId":"g2-using-a-calculator","topic":"Using a Calculator","grade":"4","topicGrade":"2","strand":"N","difficulty":12,"marks":3,"tier":"H","calc":true,"text":"

Use your calculator to work out

(5.42 − √53.29) ÷ sin 30°

  1. Write down all the figures on your calculator display.
  2. Write your answer to part (a) correct to 3 significant figures.
","answerType":"multi","answer":null,"answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"43.72"},{"label":"b","answer":"43.7"}],"solution":["Use degree mode because the angle is given in degrees.","5.42 = 29.16.","√53.29 = 7.3, since 7.3 × 7.3 = 53.29.","Numerator: 29.16 − 7.3 = 21.86.","sin 30° = 0.5, so 21.86 ÷ 0.5 = 43.72.","(b) 43.72 to 3 significant figures is 43.7."],"diagram":null,"draw":false,"revision":"098dc324b7f9"},{"id":"g2-writing-an-expression-q01","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Pencils cost c pence each.

Write down an expression, in terms of c, for the total cost, in pence, of 7 pencils.

","answerType":"exact","answer":"7c","answerDisplay":null,"accept":["7 c","c × 7","7 x c"],"parts":[],"solution":["Each pencil costs c pence and there are 7 pencils.","Total cost = 7 × c = 7c pence."],"diagram":null,"draw":false,"revision":"9361ac3a84bf"},{"id":"g2-writing-an-expression-q02","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

A ribbon is k metres long.

Write an expression, in terms of k, for the length of the ribbon in centimetres.

","answerType":"exact","answer":"100k","answerDisplay":null,"accept":["100 k","k × 100","100 x k"],"parts":[],"solution":["There are 100 centimetres in 1 metre.","k metres = 100 × k = 100k centimetres."],"diagram":null,"draw":false,"revision":"afbf1017cdc7"},{"id":"g2-writing-an-expression-q03","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

Amy is x years old.

Ben is 5 years older than Amy.

Chloe is twice as old as Amy.

  1. Write down an expression, in terms of x, for Ben's age.
  2. Write down an expression, in terms of x, for Chloe's age.
","answerType":"multi","answer":"a) x + 5 b) 2x","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x + 5 (accept 5 + x)"},{"label":"b","answer":"2x"}],"solution":["Ben is 5 more than Amy, so Ben's age is x + 5.","Chloe is twice Amy's age, so Chloe's age is 2 × x = 2x."],"diagram":null,"draw":false,"revision":"69b63d261b92"},{"id":"g2-writing-an-expression-q04","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

A bag contains n sweets.

  1. Priya takes 4 sweets out of the bag.
    Write an expression, in terms of n, for the number of sweets left in the bag.
  2. Sam has 3 full bags. Each full bag contains n sweets.
    Write an expression, in terms of n, for the total number of sweets Sam has.
","answerType":"multi","answer":"a) n - 4 b) 3n","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"n - 4"},{"label":"b","answer":"3n"}],"solution":["a) Start with n sweets and take away 4, leaving n - 4.","b) 3 bags of n sweets is 3 × n = 3n."],"diagram":null,"draw":false,"revision":"ec7ad4841024"},{"id":"g2-writing-an-expression-q05","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

At a cinema, an adult ticket costs £a and a child ticket costs £c.

The Khan family buy tickets for 2 adults and 3 children.

  1. Write down an expression, in terms of a and c, for the total cost, in pounds, of the tickets.
  2. The family pay with a £50 note.
    Write down an expression, in terms of a and c, for the change, in pounds, they should receive.
","answerType":"multi","answer":"a) 2a + 3c b) 50 - 2a - 3c","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2a + 3c (accept 3c + 2a)"},{"label":"b","answer":"50 - 2a - 3c (accept 50 - (2a + 3c))"}],"solution":["a) 2 adult tickets cost 2 × a = 2a pounds and 3 child tickets cost 3 × c = 3c pounds.","a) Total cost = 2a + 3c.","b) Change = amount paid minus total cost = 50 - 2a - 3c."],"diagram":null,"draw":false,"revision":"14fc7c0ba7da"},{"id":"g2-writing-an-expression-q06","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

A plumber charges a fixed call-out fee of £40 plus £28 for each hour of work.

The plumber works for h hours on a job.

The total charge for the job is £C.

Write a formula for C in terms of h.

","answerType":"exact","answer":"C = 40 + 28h","answerDisplay":null,"accept":["C = 28h + 40","C=40+28h","C=28h+40"],"parts":[],"solution":["The hourly part of the charge is 28 × h = 28h pounds.","Add the fixed £40 call-out fee.","C = 40 + 28h."],"diagram":null,"draw":false,"revision":"bdb3d523ebb2"},{"id":"g2-writing-an-expression-q07","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":false,"text":"

Tom is y years old.

His sister Freya is 7 years older than Tom.

  1. Write down an expression, in terms of y, for Freya's age.
  2. The sum of Tom's age and Freya's age is 23 years.
    Work out Tom's age.
","answerType":"multi","answer":"a) y + 7 b) 8","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"y + 7 (accept 7 + y)"},{"label":"b","answer":"8"}],"solution":["a) Freya is 7 more than Tom, so her age is y + 7.","b) Sum of ages: y + (y + 7) = 23, so 2y + 7 = 23.","b) 2y = 16, so y = 8. Tom is 8 years old.","Check: Tom 8, Freya 15, and 8 + 15 = 23."],"diagram":null,"draw":false,"revision":"8e74d48c1668"},{"id":"g2-writing-an-expression-q08","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"
  1. Write  p × p × p  using index notation.
  2. In a quiz, a team scores 5 points for each correct answer and loses 2 points for each wrong answer.
    A team gets c correct answers and w wrong answers.
    Write an expression, in terms of c and w, for the team's total number of points.
","answerType":"multi","answer":"a) p^3 b) 5c - 2w","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"p^3"},{"label":"b","answer":"5c - 2w (accept -2w + 5c)"}],"solution":["a) p multiplied by itself three times is p3.","b) Correct answers score 5 × c = 5c points.","b) Wrong answers lose 2 × w = 2w points, so the total is 5c - 2w."],"diagram":null,"draw":false,"revision":"8efa272545e1"},{"id":"g2-writing-an-expression-q09","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

A taxi journey costs a fixed charge of £3 plus £2 for each mile travelled.

A journey of m miles costs £C in total.

  1. Write a formula for C in terms of m.
  2. A journey costs £25 in total.
    Work out the number of miles travelled.
","answerType":"multi","answer":"a) C = 3 + 2m b) 11","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"C = 3 + 2m (accept C = 2m + 3)"},{"label":"b","answer":"11"}],"solution":["a) The mileage part costs 2 × m = 2m pounds, plus the fixed £3, so C = 3 + 2m.","b) Substitute C = 25: 25 = 3 + 2m.","b) 2m = 22, so m = 11. The journey was 11 miles.","Check: 3 + 2 × 11 = 3 + 22 = 25."],"diagram":null,"draw":false,"revision":"1ce64c293122"},{"id":"g2-writing-an-expression-q10","topicId":"g2-writing-an-expression","topic":"Writing an Expression","grade":"2","topicGrade":"2","strand":"A","difficulty":10,"marks":5,"tier":"F","calc":false,"text":"

Here are the first three patterns in a sequence made from grey square tiles.

Pattern 1 uses 5 tiles. Pattern 2 uses 8 tiles. Pattern 3 uses 11 tiles.

The number of tiles goes up by the same amount from each pattern to the next.

  1. Work out the number of tiles in pattern 5.
  2. Write down an expression, in terms of n, for the number of tiles in pattern n.
  3. Explain why there is no pattern in this sequence that uses exactly 100 tiles.
","answerType":"multi","answer":"a) 17 b) 3n + 2 c) 3n + 2 = 100 gives 3n = 98, and 98 is not a multiple of 3, so n is not a whole number","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"17"},{"label":"b","answer":"3n + 2 (accept 2 + 3n)"},{"label":"c","answer":"open: 3n + 2 = 100 gives 3n = 98, which has no whole number solution (98 is not a multiple of 3)"}],"solution":["a) The pattern goes up by 3 tiles each time: pattern 4 uses 11 + 3 = 14 tiles, pattern 5 uses 14 + 3 = 17 tiles.","b) The common difference is 3, so the expression starts 3n. For n = 1, 3n = 3, but pattern 1 has 5 tiles, so add 2: 3n + 2.","b) Check: n = 2 gives 8 and n = 3 gives 11, both correct.","c) If 3n + 2 = 100 then 3n = 98. 98 ÷ 3 is not a whole number, so no pattern uses exactly 100 tiles."],"diagram":{"type":"shape","width":440,"polygons":[{"pts":[[0,0],[1,0],[1,1],[0,1]],"fill":true},{"pts":[[1,0],[2,0],[2,1],[1,1]],"fill":true},{"pts":[[2,0],[3,0],[3,1],[2,1]],"fill":true},{"pts":[[2,1],[3,1],[3,2],[2,2]],"fill":true},{"pts":[[2,2],[3,2],[3,3],[2,3]],"fill":true},{"pts":[[5,0],[6,0],[6,1],[5,1]],"fill":true},{"pts":[[6,0],[7,0],[7,1],[6,1]],"fill":true},{"pts":[[7,0],[8,0],[8,1],[7,1]],"fill":true},{"pts":[[7,1],[8,1],[8,2],[7,2]],"fill":true},{"pts":[[7,2],[8,2],[8,3],[7,3]],"fill":true},{"pts":[[8,0],[9,0],[9,1],[8,1]],"fill":true},{"pts":[[8,1],[9,1],[9,2],[8,2]],"fill":true},{"pts":[[8,2],[9,2],[9,3],[8,3]],"fill":true},{"pts":[[11,0],[12,0],[12,1],[11,1]],"fill":true},{"pts":[[12,0],[13,0],[13,1],[12,1]],"fill":true},{"pts":[[13,0],[14,0],[14,1],[13,1]],"fill":true},{"pts":[[13,1],[14,1],[14,2],[13,2]],"fill":true},{"pts":[[13,2],[14,2],[14,3],[13,3]],"fill":true},{"pts":[[14,0],[15,0],[15,1],[14,1]],"fill":true},{"pts":[[14,1],[15,1],[15,2],[14,2]],"fill":true},{"pts":[[14,2],[15,2],[15,3],[14,3]],"fill":true},{"pts":[[15,0],[16,0],[16,1],[15,1]],"fill":true},{"pts":[[15,1],[16,1],[16,2],[15,2]],"fill":true},{"pts":[[15,2],[16,2],[16,3],[15,3]],"fill":true}],"texts":[{"at":[1.5,-0.9],"text":"Pattern 1","size":12},{"at":[7,-0.9],"text":"Pattern 2","size":12},{"at":[13.5,-0.9],"text":"Pattern 3","size":12}]},"draw":false,"revision":"660d7dadbf8f"},{"id":"g3-area-and-circumference-of-circles-q01","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":1,"marks":3,"tier":"F","calc":true,"text":"

A circle has radius 6 cm.

  1. Work out the circumference of the circle. Give your answer correct to 1 decimal place.
  2. Work out the area of the circle. Give your answer correct to 1 decimal place.
","answerType":"multi","answer":"a) 37.7 cm b) 113.1 cm²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"37.7 cm"},{"label":"b","answer":"113.1 cm^2"}],"solution":["Circumference is the distance around the circle; area is the space inside it. The radius runs from the centre to the edge, and the diameter is twice the radius. Use the calculator’s π key and round only the final answers.","Circumference = 2πr = 2 × π × 6 = 37.699... = 37.7 cm (1 dp).","Area = πr² = π × 36 = 113.097... = 113.1 cm² (1 dp)."],"diagram":null,"draw":false,"revision":"53b3f8060576"},{"id":"g3-area-and-circumference-of-circles-q02","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

A circle has diameter 18 cm.

  1. Work out the circumference of the circle. Give your answer in terms of π.
  2. Work out the area of the circle. Give your answer in terms of π.
","answerType":"multi","answer":"a) 18π cm b) 81π cm²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"18pi cm"},{"label":"b","answer":"81pi cm^2"}],"solution":["Circumference = πd = 18π cm.","Radius = 18 ÷ 2 = 9 cm, so area = πr² = π × 81 = 81π cm²."],"diagram":null,"draw":false,"revision":"29bc25d10e5e"},{"id":"g3-area-and-circumference-of-circles-q03","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":3,"marks":4,"tier":"F","calc":false,"text":"

The circumference of a circle is 14π cm.

Work out the area of the circle. Give your answer in terms of π.

","answerType":"exact","answer":"49π cm²","answerDisplay":null,"accept":["49pi","49pi cm^2","49π cm²"],"parts":[],"solution":["Circumference = 2πr, so 2πr = 14π and r = 7 cm.","Area = πr² = π × 7² = 49π cm²."],"diagram":null,"draw":false,"revision":"4177362fd86b"},{"id":"g3-area-and-circumference-of-circles-q04","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":4,"marks":4,"tier":"F","calc":true,"text":"

A circle of radius 5 cm is drawn inside a square with sides of length 10 cm, so that the circle touches all four sides of the square.

The region inside the square but outside the circle is shaded.

Work out the shaded area. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"21.5 cm²","answerDisplay":null,"accept":["21.5","21.5 cm^2"],"parts":[],"solution":["Area of the square = 10 × 10 = 100 cm².","Area of the circle = π × 5² = 25π = 78.539... cm².","Shaded area = 100 − 78.539... = 21.460... = 21.5 cm² (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[10,0],[10,10],[0,10]],"sides":["10 cm",null,null,null],"fill":true}],"circles":[{"c":[5,5],"r":5,"fill":"#ffffff","centre":true}],"segments":[{"from":[5,5],"to":[10,5],"thin":true,"label":"5 cm","labelPos":"above"}]},"draw":false,"revision":"f78eec5abf45"},{"id":"g3-area-and-circumference-of-circles-q05","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":5,"marks":4,"tier":"FH","calc":true,"text":"

A rectangle is 15 cm long and 8 cm wide. A circle of radius 3 cm is drawn completely inside the rectangle and is cut out.

The rest of the rectangle is shaded.

Work out the shaded area as a percentage of the area of the rectangle. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"76.4%","answerDisplay":null,"accept":["76.4","76.4 %"],"parts":[],"solution":["Area of the rectangle = 15 × 8 = 120 cm².","Area of the circle = π × 3² = 9π = 28.274... cm².","Shaded area = 120 − 28.274... = 91.725... cm².","Percentage = 91.725... ÷ 120 × 100 = 76.438... = 76.4% (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[15,0],[15,8],[0,8]],"sides":["15 cm","8 cm",null,null],"fill":true}],"circles":[{"c":[7.5,4],"r":3,"fill":"#ffffff","centre":true}],"segments":[{"from":[7.5,4],"to":[10.5,4],"thin":true,"label":"3 cm","labelPos":"above"}]},"draw":false,"revision":"0f6f47e905f1"},{"id":"g3-area-and-circumference-of-circles-q06","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":6,"marks":4,"tier":"FH","calc":true,"text":"

The diagram shows a square OABC with sides of length 6 cm. A quarter circle, centre O and radius 6 cm, is drawn inside the square from A to C.

The region between the quarter circle and the corner B is shaded.

Show that the shaded area is more than 21% of the area of the square.

","answerType":"open","answer":"Shaded area = 36 - 9pi = 7.725... cm^2, which is 21.46...% of 36 cm^2, and 21.46% > 21%.","answerDisplay":null,"accept":["a complete calculation reaching approximately 21.5% with a comparison to 21%"],"parts":[],"solution":["Area of the square = 6 × 6 = 36 cm².","Area of the quarter circle = 14 × π × 6² = 9π = 28.274... cm².","Shaded area = 36 − 28.274... = 7.725... cm².","Percentage = 7.725... ÷ 36 × 100 = 21.46...%, which is more than 21%."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[6,0],[6,6],[0,6]],"labels":["O","A","B","C"],"sides":["6 cm",null,null,null],"fill":true}],"arcs":[{"c":[0,0],"r":6,"from":0,"to":90,"sector":true,"fill":"#ffffff"}]},"draw":false,"revision":"3e60004e1c28"},{"id":"g3-area-and-circumference-of-circles-q07","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

A circular pond has radius 3 m.

The pond is surrounded by a path of constant width 1 m.

Calculate the area of the path. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"22.0 m²","answerDisplay":null,"accept":["22.0"],"parts":[],"solution":["The outer edge of the path is a circle of radius 3 + 1 = 4 m.","Area of the path = π × 4² − π × 3² = 16π − 9π = 7π m².","7π = 21.991... = 22.0 m² (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"circles":[{"c":[0,0],"r":4,"fill":true},{"c":[0,0],"r":3,"fill":"#ffffff","centre":true}],"segments":[{"from":[0,0],"to":[-3,0],"thin":true,"label":"3 m","labelPos":"above"},{"from":[0,3],"to":[0,4],"thin":true,"arrow":"both","label":"1 m","labelPos":"right"}],"texts":[{"at":[0,-1.2],"text":"pond"},{"at":[0,-3.5],"text":"path"}]},"draw":false,"revision":"b0a7a623478c"},{"id":"g3-area-and-circumference-of-circles-q08","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

A square has sides of length 5 cm.

A circle is drawn through all four vertices of the square.

Calculate the circumference of the circle. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"22.2 cm","answerDisplay":null,"accept":["22.2","22.2cm"],"parts":[],"solution":["The diameter of the circle is the diagonal of the square.","By Pythagoras, diagonal² = 5² + 5² = 50, so diagonal = √50 = 7.0710... cm.","Circumference = πd = π × 7.0710... = 22.214... = 22.2 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[-2.5,-2.5],[2.5,-2.5],[2.5,2.5],[-2.5,2.5]],"sides":["5 cm",null,null,null],"fill":false}],"circles":[{"c":[0,0],"r":3.536}],"segments":[{"from":[-2.5,-2.5],"to":[2.5,2.5],"dashed":true,"answer":true}]},"draw":false,"revision":"5c63bfe4878a"},{"id":"g3-area-and-circumference-of-circles-q09","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":9,"marks":4,"tier":"H","calc":false,"text":"

OAB is a quarter circle with centre O. OA and OB are radii and angle AOB = 90°.

The straight line AB has length 10 cm.

Work out the area of the quarter circle. Give your answer in terms of π.

","answerType":"exact","answer":"12.5π cm²","answerDisplay":null,"accept":["12.5pi","25pi/2","12.5pi cm^2"],"parts":[],"solution":["Let the radius be r. Triangle OAB is right-angled at O with OA = OB = r.","By Pythagoras, r² + r² = 10², so 2r² = 100 and r² = 50.","Area of the quarter circle = 14πr² = 14 × π × 50 = 12.5π cm²."],"diagram":{"type":"shape","notAccurate":true,"arcs":[{"c":[0,0],"r":7.071,"from":0,"to":90}],"segments":[{"from":[0,0],"to":[7.071,0]},{"from":[0,0],"to":[0,7.071]},{"from":[7.071,0],"to":[0,7.071],"label":"10 cm","labelPos":"above-right"}],"points":[{"at":[0,0],"label":"O","labelPos":"below-left","style":"none"},{"at":[7.071,0],"label":"A","labelPos":"below-right","style":"none"},{"at":[0,7.071],"label":"B","labelPos":"above-left","style":"none"}],"angles":[{"at":[0,0],"from":[7.071,0],"to":[0,7.071],"right":true}]},"draw":false,"revision":"62688dfc098f"},{"id":"g3-area-and-circumference-of-circles-q10","topicId":"g3-area-and-circumference-of-circles","topic":"Area and Circumference of Circles","grade":"3","topicGrade":"3","strand":"G","difficulty":10,"marks":5,"tier":"F","calc":true,"text":"

A flower bed is a semicircle with diameter 20 m.

Fencing is to be put around the whole perimeter of the flower bed, including the straight side.

Fencing costs £4.20 per metre and is only sold in whole metres.

Work out the total cost of the fencing.

","answerType":"exact","answer":"£218.40","answerDisplay":null,"accept":["218.40","£218.40","218.4"],"parts":[],"solution":["Curved edge = 12 × π × 20 = 10π = 31.415... m.","Total perimeter = 31.415... + 20 = 51.415... m.","Fencing is sold in whole metres, so 52 m must be bought.","Total cost = 52 × £4.20 = £218.40."],"diagram":null,"draw":false,"revision":"a421a1fd7eda"},{"id":"g3-area-of-compound-shapes-q01","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

An L-shape is made by joining a rectangle 8 cm by 5 cm to a rectangle 4 cm by 3 cm. The two rectangles do not overlap.

Work out the total area of the L-shape.

","answerType":"exact","answer":"52 cm²","answerDisplay":null,"accept":["52","52 cm^2"],"parts":[],"solution":["Area of the first rectangle = 8 × 5 = 40 cm².","Area of the second rectangle = 4 × 3 = 12 cm².","Total area = 40 + 12 = 52 cm²."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[8,0],[8,5],[4,5],[4,8],[0,8]],"sides":["8 cm","5 cm",null,"3 cm","4 cm",null],"fill":false}],"segments":[{"from":[0,5],"to":[4,5],"dashed":true,"thin":true}]},"draw":false,"revision":"a252aa68f94e"},{"id":"g3-area-of-compound-shapes-q02","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

A rectangle is 12 cm long and 9 cm wide.

A rectangle 5 cm by 4 cm is removed from one corner.

Work out the area of the remaining shape.

","answerType":"exact","answer":"88 cm²","answerDisplay":null,"accept":["88","88 cm^2"],"parts":[],"solution":["Area of the full rectangle = 12 × 9 = 108 cm².","Area removed = 5 × 4 = 20 cm².","Remaining area = 108 − 20 = 88 cm²."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[12,0],[12,5],[7,5],[7,9],[0,9]],"sides":["12 cm",null,"5 cm","4 cm",null,"9 cm"],"fill":false}]},"draw":false,"revision":"471b015cf3a8"},{"id":"g3-area-of-compound-shapes-q03","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":3,"marks":4,"tier":"F","calc":true,"text":"

The front of a shed is made from a rectangle with a triangle on top.

The rectangle is 10 cm wide and 6 cm high in a scale drawing. The triangle sits exactly on top of the rectangle, with base 10 cm and perpendicular height 4 cm.

Work out the total area of the scale drawing.

","answerType":"exact","answer":"80 cm²","answerDisplay":null,"accept":["80","80 cm^2"],"parts":[],"solution":["Area of the rectangle = 10 × 6 = 60 cm².","Area of the triangle = 12 × 10 × 4 = 20 cm².","Total area = 60 + 20 = 80 cm²."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[10,0],[10,6],[5,10],[0,6]],"sides":["10 cm","6 cm",null,null,null],"fill":false}],"segments":[{"from":[0,6],"to":[10,6],"dashed":true,"thin":true},{"from":[5,10],"to":[5,6],"dashed":true,"thin":true,"label":"4 cm","labelPos":"right"}],"angles":[{"at":[5,6],"from":[10,6],"to":[5,10],"right":true}]},"draw":false,"revision":"852e1d16d1ee"},{"id":"g3-area-of-compound-shapes-q04","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":4,"marks":4,"tier":"F","calc":false,"text":"

Two identical squares, each with sides of length 6 cm, overlap.

The overlapping region is a square with sides of length 2 cm.

Work out the total area covered by the two squares.

","answerType":"exact","answer":"68 cm²","answerDisplay":null,"accept":["68","68 cm^2"],"parts":[],"solution":["Area of each square = 6 × 6 = 36 cm².","The two squares together, counting the overlap twice, cover 36 + 36 = 72 cm².","The overlap, area 2 × 2 = 4 cm², has been counted twice, so subtract it once.","Total area covered = 72 − 4 = 68 cm²."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[6,0],[6,6],[0,6]],"sides":["6 cm",null,null,null],"fill":false},{"pts":[[4,4],[10,4],[10,10],[4,10]],"sides":[null,null,"6 cm",null],"fill":false},{"pts":[[4,4],[6,4],[6,6],[4,6]],"sides":["2 cm",null,null,null],"fill":true}]},"draw":false,"revision":"6c318df3cd2a"},{"id":"g3-area-of-compound-shapes-q05","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":5,"marks":5,"tier":"F","calc":true,"text":"

The diagram shows the floor of a workshop. The floor is an L-shape made from a rectangle 7 m by 4 m joined to a rectangle 3 m by 2 m.

Marco is going to paint the whole floor. One tin of floor paint covers 8 m² and costs £22.

Work out the total cost of the tins of paint Marco needs to buy.

","answerType":"exact","answer":"£110","answerDisplay":null,"accept":["110","£110","£110.00"],"parts":[],"solution":["Area of the floor = 7 × 4 + 3 × 2 = 28 + 6 = 34 m².","Tins needed: 34 ÷ 8 = 4.25, so Marco must buy 5 tins.","Total cost = 5 × £22 = £110."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[10,0],[10,2],[7,2],[7,4],[0,4]],"sides":[null,"2 m","3 m","2 m","7 m","4 m"],"fill":false}],"segments":[{"from":[7,0],"to":[7,2],"dashed":true,"thin":true}],"texts":[{"at":[3.5,-0.55],"text":"7 m"},{"at":[8.5,-0.55],"text":"3 m"}]},"draw":false,"revision":"9ca99ffcb0bb"},{"id":"g3-area-of-compound-shapes-q06","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":6,"marks":5,"tier":"F","calc":true,"text":"

A rectangular garden is 9 m long and 7 m wide.

A rectangular patio 4 m by 3 m is laid inside the garden. The rest of the garden is to be covered with grass seed.

One bag of grass seed covers 15 m² and costs £9.99.

Work out the total cost of the bags of grass seed needed.

","answerType":"exact","answer":"£39.96","answerDisplay":null,"accept":["39.96","£39.96"],"parts":[],"solution":["Area of the garden = 9 × 7 = 63 m².","Area of the patio = 4 × 3 = 12 m².","Area to be seeded = 63 − 12 = 51 m².","Bags needed: 51 ÷ 15 = 3.4, so 4 bags must be bought.","Total cost = 4 × £9.99 = £39.96."],"diagram":null,"draw":false,"revision":"d0649e3e3e23"},{"id":"g3-area-of-compound-shapes-q07","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":7,"marks":5,"tier":"FH","calc":true,"text":"

A window is made from a rectangle with a semicircle on top.

The rectangle is 1.2 m high and 0.8 m wide. The semicircle sits exactly on top of the rectangle, so its diameter is 0.8 m.

Glass for the window costs £95 per square metre.

Work out the cost of the glass for the window. Give your answer correct to the nearest penny.

","answerType":"exact","answer":"£115.08","answerDisplay":null,"accept":["115.08","£115.08"],"parts":[],"solution":["Area of the rectangle = 1.2 × 0.8 = 0.96 m².","Radius of the semicircle = 0.8 ÷ 2 = 0.4 m.","Area of the semicircle = 12 × π × 0.4² = 12 × π × 0.16 = 0.25132... m².","Total area = 0.96 + 0.25132... = 1.21132... m².","Cost = 1.21132... × £95 = £115.076... = £115.08 (nearest penny)."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[8,0],[8,12],[0,12]],"sides":["0.8 m","1.2 m",null,null],"fill":false}],"arcs":[{"c":[4,12],"r":4,"from":0,"to":180}]},"draw":false,"revision":"b083fe7255c3"},{"id":"g3-area-of-compound-shapes-q08","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":8,"marks":5,"tier":"FH","calc":false,"text":"

A field is made from a rectangle 60 m by 40 m joined along one 40 m side to a right-angled triangle. The triangle has base 40 m and perpendicular height 30 m.

Fertiliser for the field costs 5p per square metre.

Work out the total cost of the fertiliser. Give your answer in pounds.

","answerType":"exact","answer":"£150","answerDisplay":null,"accept":["150","£150","£150.00"],"parts":[],"solution":["Area of the rectangle = 60 × 40 = 2400 m².","Area of the triangle = 12 × 40 × 30 = 600 m².","Total area = 2400 + 600 = 3000 m².","Cost = 3000 × 5p = 15000p = £150."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[90,0],[60,40],[0,40]],"sides":[null,null,"60 m","40 m"],"fill":false}],"segments":[{"from":[60,0],"to":[60,40],"dashed":true,"thin":true,"label":"40 m","labelPos":"left"}],"angles":[{"at":[60,0],"from":[90,0],"to":[60,40],"right":true}],"texts":[{"at":[30,-4.5],"text":"60 m"},{"at":[75,-4.5],"text":"30 m"}]},"draw":false,"revision":"04020bb35d1e"},{"id":"g3-area-of-compound-shapes-q09","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"

Two identical squares are cut from a rectangular sheet of card 14 cm long and 10 cm wide. The squares do not overlap.

The area of the card remaining is 90 cm².

Work out the side length of each square.

","answerType":"exact","answer":"5 cm","answerDisplay":null,"accept":["5","5cm"],"parts":[],"solution":["Area of the sheet = 14 × 10 = 140 cm².","Area removed = 140 − 90 = 50 cm².","Each square has area 50 ÷ 2 = 25 cm².","Side length = √25 = 5 cm."],"diagram":null,"draw":false,"revision":"e21a324ca5a0"},{"id":"g3-area-of-compound-shapes-q10","topicId":"g3-area-of-compound-shapes","topic":"Area of Compound Shapes","grade":"3","topicGrade":"3","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

The diagram shows a shape made from a square and a quarter circle.

The square OABC has sides of length 10 cm. The quarter circle has centre O and radius 10 cm, and is joined to the square along the side OC so that the two parts do not overlap.

Work out the total area of the shape. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"178.5 cm²","answerDisplay":null,"accept":["178.5","178.5 cm^2"],"parts":[],"solution":["Area of the square = 10 × 10 = 100 cm².","Area of the quarter circle = 14 × π × 10² = 25π = 78.539... cm².","Total area = 100 + 78.539... = 178.539... = 178.5 cm² (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[10,0],[10,10],[0,10]],"labels":["O","A","B","C"],"sides":["10 cm",null,null,null],"fill":false}],"arcs":[{"c":[0,0],"r":10,"from":90,"to":180}],"segments":[{"from":[0,0],"to":[-10,0],"label":"10 cm","labelPos":"below"}]},"draw":false,"revision":"de020a0351cc"},{"id":"g3-best-buy-questions-q01","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":1,"marks":3,"tier":"F","calc":false,"text":"

A shop sells two sizes of box of teabags.

Small box: 80 teabags for £2.40

Large box: 240 teabags for £6.60

Which box is the better value for money? Show your working.

","answerType":"open","answer":"Large box - 2.75p per teabag compared with 3p per teabag","answerDisplay":null,"accept":["Large box","240 for £6.60"],"parts":[],"solution":["Compare the cost of the same quantity in each box. Dividing price by the number of teabags gives the cost of one teabag, so the lower unit price is better value.","Small box: 240p ÷ 80 = 3p per teabag","Large box: 660p ÷ 240 = 2.75p per teabag","2.75p is less than 3p, so the large box is the better value"],"diagram":null,"draw":false,"revision":"4b216026cdb1"},{"id":"g3-best-buy-questions-q02","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

Washing-up liquid is sold in two sizes of bottle.

500 ml bottle for £1.20

750 ml bottle for £1.65

Work out which bottle is the better value for money. You must show your working.

","answerType":"open","answer":"750 ml bottle - 0.22p per ml compared with 0.24p per ml","answerDisplay":null,"accept":["750 ml bottle","750ml"],"parts":[],"solution":["500 ml bottle: 120p ÷ 500 = 0.24p per ml","750 ml bottle: 165p ÷ 750 = 0.22p per ml","0.22p per ml is cheaper, so the 750 ml bottle is the better value"],"diagram":null,"draw":false,"revision":"f7fbcd8843c9"},{"id":"g3-best-buy-questions-q03","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

Two shops sell the same tins of soup.

Shop A: 3 tins for £2.10

Shop B: 4 tins for £2.72

Decide which shop gives the better value for money. Show your working.

","answerType":"open","answer":"Shop B - 68p per tin compared with 70p per tin","answerDisplay":null,"accept":["Shop B","B"],"parts":[],"solution":["Shop A: £2.10 ÷ 3 = 70p per tin","Shop B: £2.72 ÷ 4 = 68p per tin","68p is less than 70p, so Shop B gives the better value"],"diagram":null,"draw":false,"revision":"d0f2781ec4aa"},{"id":"g3-best-buy-questions-q04","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":4,"marks":4,"tier":"F","calc":true,"text":"

Two shops sell the same orange juice.

Shop A sells a 2 litre carton for £1.85.

Shop B sells a 1.5 litre carton for £1.35.

Show which carton is the better value for money.

","answerType":"open","answer":"Shop B's carton - 90p per litre compared with 92.5p per litre","answerDisplay":null,"accept":["Shop B","1.5 litre carton"],"parts":[],"solution":["Shop A: 185p ÷ 2 = 92.5p per litre","Shop B: 135p ÷ 1.5 = 90p per litre","90p per litre is less than 92.5p per litre, so Shop B's carton is the better value"],"diagram":null,"draw":false,"revision":"6115736be919"},{"id":"g3-best-buy-questions-q05","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

A supermarket sells two special offer boxes of cereal.

Box A: 500 g for £3.20, with 25% extra free

Box B: 750 g for £3.60, with 10% off the price

Decide which box is the better value for money. You must show your working.

","answerType":"open","answer":"Box B - 0.432p per gram compared with 0.512p per gram","answerDisplay":null,"accept":["Box B","B"],"parts":[],"solution":["Box A actually contains 500 × 1.25 = 625 g for £3.20","Box A: 320p ÷ 625 = 0.512p per gram","Box B actually costs £3.60 × 0.9 = £3.24 for 750 g","Box B: 324p ÷ 750 = 0.432p per gram, so Box B is the better value"],"diagram":null,"draw":false,"revision":"6a174949cb94"},{"id":"g3-best-buy-questions-q06","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":6,"marks":4,"tier":"F","calc":true,"text":"

Peanut butter is sold in three sizes of jar.

Small: 250 g for £1.10

Medium: 400 g for £1.68

Large: 600 g for £2.58

Work out which jar is the best value for money. You must show your working.

","answerType":"open","answer":"Medium jar - 0.42p per gram (small is 0.44p per gram, large is 0.43p per gram)","answerDisplay":null,"accept":["Medium","medium jar","400 g"],"parts":[],"solution":["Small: 110p ÷ 250 = 0.44p per gram","Medium: 168p ÷ 400 = 0.42p per gram","Large: 258p ÷ 600 = 0.43p per gram","The medium jar has the lowest price per gram, so it is the best value"],"diagram":null,"draw":false,"revision":"27054f0fc12d"},{"id":"g3-best-buy-questions-q07","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

Layla needs 18 identical notebooks for a club.

Shop A sells the notebooks at £1.20 each, with an offer of buy 2 get 1 free.

Shop B sells the notebooks at £1.05 each, with 10% off the total cost when you buy 15 or more.

Work out which shop is cheaper for 18 notebooks, and by how much.

","answerType":"open","answer":"Shop A by £2.61 (Shop A £14.40, Shop B £17.01)","answerDisplay":null,"accept":["Shop A, £2.61","A by £2.61"],"parts":[],"solution":["Shop A: with buy 2 get 1 free, every 3 notebooks cost 2 × £1.20 = £2.40","18 notebooks = 6 groups of 3, so cost = 6 × £2.40 = £14.40","Shop B: 18 × £1.05 = £18.90, then 10% off gives £18.90 × 0.9 = £17.01","Shop A is cheaper by £17.01 − £14.40 = £2.61"],"diagram":null,"draw":false,"revision":"4f40193f2cb5"},{"id":"g3-best-buy-questions-q08","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":8,"marks":4,"tier":"FH","calc":true,"text":"

The same perfume is sold in London and in Paris.

In London it costs £42 for a 50 ml bottle.

In Paris it costs 52.80 euros for a 60 ml bottle.

£1 = 1.20 euros

Work out in which city the perfume is better value for money. You must show your working.

","answerType":"open","answer":"Paris - about 73.3p per ml compared with 84p per ml in London","answerDisplay":null,"accept":["Paris"],"parts":[],"solution":["Change the Paris price to pounds: 52.80 ÷ 1.20 = £44","London: £42 ÷ 50 = £0.84 per ml","Paris: £44 ÷ 60 = £0.733... per ml","The price per ml is lower in Paris, so the perfume is better value in Paris"],"diagram":null,"draw":false,"revision":"ef9be3149454"},{"id":"g3-best-buy-questions-q09","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":9,"marks":4,"tier":"FH","calc":true,"text":"

In England, petrol costs £1.44 per litre.

In America, petrol costs 3.80 dollars per US gallon.

1 US gallon = 3.8 litres

£1 = 1.25 dollars

Show that petrol is cheaper in America than in England.

","answerType":"open","answer":"America: $1 per litre = £0.80 per litre, which is less than £1.44 per litre","answerDisplay":null,"accept":["80p per litre < £1.44 per litre"],"parts":[],"solution":["Price per litre in America = 3.80 ÷ 3.8 = 1 dollar per litre","Change to pounds: 1 ÷ 1.25 = £0.80 per litre","£0.80 is less than £1.44, so petrol is cheaper in America"],"diagram":null,"draw":false,"revision":"e930e5ab89a8"},{"id":"g3-best-buy-questions-q10","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

A teacher needs 48 cartons of juice for a school trip.

Shop A sells packs of 8 cartons for £2.16 per pack.

Shop B sells packs of 6 cartons for £1.68 per pack.

Work out which shop is cheaper for exactly 48 cartons, and by how much.

","answerType":"open","answer":"Shop A by 48p (Shop A £12.96, Shop B £13.44)","answerDisplay":null,"accept":["Shop A, 48p","A by £0.48"],"parts":[],"solution":["Shop A: 48 ÷ 8 = 6 packs, so cost = 6 × £2.16 = £12.96","Shop B: 48 ÷ 6 = 8 packs, so cost = 8 × £1.68 = £13.44","Shop A is cheaper by £13.44 − £12.96 = £0.48"],"diagram":null,"draw":false,"revision":"b463bfc8bdcc"},{"id":"g3-best-buy-questions-q11","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":11,"marks":5,"tier":"F","calc":true,"text":"

Sam is organising a barbecue and needs at least 30 bread rolls.

Bread rolls are sold in packs of 6 for £1.10 and packs of 15 for £2.55.

Work out the cheapest way for Sam to buy at least 30 rolls, and the cost.

","answerType":"open","answer":"Two packs of 15 for £5.10","answerDisplay":null,"accept":["2 x 15 packs, £5.10","£5.10"],"parts":[],"solution":["Two packs of 15 gives exactly 30 rolls: 2 × £2.55 = £5.10","Five packs of 6 gives exactly 30 rolls: 5 × £1.10 = £5.50","One pack of 15 and three packs of 6 gives 33 rolls: £2.55 + 3 × £1.10 = £5.85","The cheapest way is two packs of 15, costing £5.10"],"diagram":null,"draw":false,"revision":"bb71160a9bf4"},{"id":"g3-best-buy-questions-q12","topicId":"g3-best-buy-questions","topic":"Best Buy Questions","grade":"3","topicGrade":"3","strand":"R","difficulty":12,"marks":5,"tier":"F","calc":true,"text":"

Mia needs at least 40 paper cups and at least 40 paper plates for a party.

Cups: pack of 25 for £1.80, or pack of 10 for £0.85

Plates: pack of 20 for £2.40, or pack of 8 for £1.10

Work out the cheapest total cost for Mia. You must show your working.

","answerType":"exact","answer":"£8.20","answerDisplay":null,"accept":["8.20","£8.20"],"parts":[],"solution":["Cups: four packs of 10 gives exactly 40 cups for 4 × £0.85 = £3.40","Compare: two packs of 25 gives 50 cups for £3.60, and 25 + 10 + 10 gives 45 cups for £3.50, so £3.40 is cheapest","Plates: two packs of 20 gives exactly 40 plates for 2 × £2.40 = £4.80","Compare: five packs of 8 gives 40 plates for £5.50, and 20 + 8 + 8 + 8 gives 44 plates for £5.70, so £4.80 is cheapest","Cheapest total = £3.40 + £4.80 = £8.20"],"diagram":null,"draw":false,"revision":"b04c4916e22b"},{"id":"g3-conversions-and-units-q01","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":1,"marks":1,"tier":"F","calc":true,"text":"

Change 3.5 m2 to cm2.

","answerType":"exact","answer":"35000 cm^2","answerDisplay":null,"accept":["35000","35 000","35,000"],"parts":[],"solution":["1 m2 = 100 cm × 100 cm = 10 000 cm2","3.5 × 10 000 = 35 000 cm2"],"diagram":null,"draw":false,"revision":"ddce457a7ff8"},{"id":"g3-conversions-and-units-q02","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Change 6 cm3 to mm3.

","answerType":"exact","answer":"6000 mm^3","answerDisplay":null,"accept":["6000","6 000","6,000"],"parts":[],"solution":["1 cm3 = 10 mm × 10 mm × 10 mm = 1000 mm3","6 × 1000 = 6000 mm3"],"diagram":null,"draw":false,"revision":"773e78063ada"},{"id":"g3-conversions-and-units-q03","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

A gull flies at a speed of 15 metres per second.

Change 15 metres per second to kilometres per hour.

","answerType":"exact","answer":"54 km/h","answerDisplay":null,"accept":["54","54 km per hour","54km/h"],"parts":[],"solution":["In 1 hour the gull flies 15 × 3600 = 54 000 metres","54 000 m = 54 km, so the speed is 54 km/h"],"diagram":null,"draw":false,"revision":"6199ded04e3e"},{"id":"g3-conversions-and-units-q04","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

The conversion graph can be used to change between litres and pints.

  1. Use the graph to change 6 litres to pints.
  2. Use the graph to change 14 pints to litres.
","answerType":"multi","answer":"a) 10.5 pints, b) 8 litres","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"10.5 pints"},{"label":"b","answer":"8 litres"}],"solution":["The line passes through (10 litres, 17.5 pints), so 1 litre = 1.75 pints","a) 6 litres = 6 × 1.75 = 10.5 pints","b) 14 pints = 14 ÷ 1.75 = 8 litres"],"diagram":{"type":"axes","height":300,"x":{"min":0,"max":10,"step":1,"label":"Litres","minor":2},"y":{"min":0,"max":20,"step":2,"minor":4,"label":"Pints"},"lines":[{"through":[[0,0],[10,17.5]],"from":0,"to":10}]},"draw":false,"revision":"aea7e0bcb27a"},{"id":"g3-conversions-and-units-q05","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

Nadia wants to put a shelf that is 32 inches long into an alcove that is 80 cm wide.

1 inch = 2.54 cm

Show that the shelf will not fit in the alcove.

","answerType":"open","answer":"32 x 2.54 = 81.28 cm, which is more than 80 cm, so the shelf will not fit","answerDisplay":null,"accept":["81.28 cm > 80 cm so it does not fit"],"parts":[],"solution":["Change the shelf length to centimetres: 32 × 2.54 = 81.28 cm","81.28 cm is greater than 80 cm, so the shelf is too long for the alcove"],"diagram":null,"draw":false,"revision":"4778c0a694c1"},{"id":"g3-conversions-and-units-q06","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

The fuel tank of Dev's van holds 12 gallons when it is full.

1 gallon = 4.55 litres

Diesel costs £1.42 per litre.

The tank is empty. Work out the cost of filling the tank. Give your answer to the nearest penny.

","answerType":"exact","answer":"£77.53","answerDisplay":null,"accept":["77.53","£77.53"],"parts":[],"solution":["Change gallons to litres: 12 × 4.55 = 54.6 litres","Cost = 54.6 × £1.42 = £77.532","To the nearest penny this is £77.53"],"diagram":null,"draw":false,"revision":"758d0cdcbdf7"},{"id":"g3-conversions-and-units-q07","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

Amira drives 231 miles to visit her sister.

Her car travels 44 miles per gallon.

1 gallon = 4.55 litres

Petrol costs £1.48 per litre.

Work out the cost of the petrol for the journey.

Give your answer to the nearest penny.

","answerType":"exact","answer":"£35.35","answerDisplay":null,"accept":["35.35","£35.35"],"parts":[],"solution":["Gallons used = 231 ÷ 44 = 5.25 gallons","Litres used = 5.25 × 4.55 = 23.8875 litres","Cost = 23.8875 × £1.48 = £35.3535","The petrol costs £35.35 to the nearest penny"],"diagram":null,"draw":false,"revision":"4f6524c37110"},{"id":"g3-conversions-and-units-q08","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

Two empty water tanks are filled at the same time.

Tank A holds 180 litres and is filled at a rate of 12 litres per minute.

Tank B holds 220 litres and is filled at a rate of 16 litres per minute.

Which tank is full first? Work out how many seconds earlier it is full.

","answerType":"open","answer":"Tank B is full first, 75 seconds earlier","answerDisplay":null,"accept":["Tank B, 75 seconds","B by 1.25 minutes"],"parts":[],"solution":["Tank A: 180 ÷ 12 = 15 minutes","Tank B: 220 ÷ 16 = 13.75 minutes","Tank B is full first","Difference = 15 − 13.75 = 1.25 minutes = 75 seconds"],"diagram":null,"draw":false,"revision":"f0aea7396793"},{"id":"g3-conversions-and-units-q09","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":9,"marks":4,"tier":"FH","calc":true,"text":"

Water flows out of a pipe at a rate of 0.9 m3 per minute.

1 m3 = 1000 litres

Work out the rate of flow in litres per second.

","answerType":"exact","answer":"15 litres per second","answerDisplay":null,"accept":["15","15 l/s","15 litres/second"],"parts":[],"solution":["Change to litres: 0.9 m3 = 0.9 × 1000 = 900 litres per minute","Change to seconds: 900 ÷ 60 = 15 litres per second"],"diagram":null,"draw":false,"revision":"4ff519f6834a"},{"id":"g3-conversions-and-units-q10","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":10,"marks":4,"tier":"FH","calc":true,"text":"

A cyclist travels 5.4 km in 12 minutes at a steady speed.

A sprinter runs at 8 metres per second.

Show that the sprinter is faster than the cyclist.

","answerType":"open","answer":"Cyclist: 27 km/h = 7.5 m/s, which is less than 8 m/s, so the sprinter is faster","answerDisplay":null,"accept":["7.5 m/s < 8 m/s","sprinter 28.8 km/h > 27 km/h"],"parts":[],"solution":["Cyclist's speed = 5.4 ÷ 12 × 60 = 27 km/h","27 km/h = 27 000 ÷ 3600 = 7.5 m/s","7.5 m/s is less than 8 m/s, so the sprinter is faster","Alternative: sprinter = 8 × 3.6 = 28.8 km/h, which is more than 27 km/h"],"diagram":null,"draw":false,"revision":"9b540274c4ed"},{"id":"g3-conversions-and-units-q11","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":11,"marks":4,"tier":"FH","calc":true,"text":"

Elin uses the rule "5 miles = 8 kilometres" to change miles to kilometres.

A more accurate conversion is 1 mile = 1.609 kilometres.

Elin's journey is 150 miles.

Work out the difference, in kilometres, between the two conversions of 150 miles.

","answerType":"exact","answer":"1.35 km","answerDisplay":null,"accept":["1.35","1.35 kilometres"],"parts":[],"solution":["Using the rule: 150 ÷ 5 × 8 = 240 km","Using the accurate conversion: 150 × 1.609 = 241.35 km","Difference = 241.35 − 240 = 1.35 km"],"diagram":null,"draw":false,"revision":"74113418a153"},{"id":"g3-conversions-and-units-q12","topicId":"g3-conversions-and-units","topic":"Conversions and Units","grade":"3","topicGrade":"3","strand":"R","difficulty":12,"marks":5,"tier":"FH","calc":true,"text":"

Room A has a floor area of 84 m2 and is shared by 24 people.

Room B has a floor area of 540 000 cm2 and is shared by 15 people.

  1. Which room gives more floor area per person? You must show your working.
  2. Cal says "540 000 cm2 = 5400 m2 because 100 cm = 1 m". Explain the mistake Cal has made.
","answerType":"multi","answer":"a) Room B (3.6 m^2 per person compared with 3.5 m^2), b) 1 m^2 = 10 000 cm^2, not 100, so he should divide by 10 000","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Room B; 3.6 m^2 per person compared with 3.5 m^2 per person"},{"label":"b","answer":"1 m^2 = 100 x 100 = 10 000 cm^2, so he should divide by 10 000, not 100"}],"solution":["a) Room A: 84 ÷ 24 = 3.5 m2 per person","Change Room B to m2: 540 000 ÷ 10 000 = 54 m2","Room B: 54 ÷ 15 = 3.6 m2 per person, so Room B gives more area per person","b) 1 m2 = 100 cm × 100 cm = 10 000 cm2, so Cal should have divided by 10 000, not by 100"],"diagram":null,"draw":false,"revision":"c7d625fce7a6"},{"id":"g3-drawing-linear-graphs-q01","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

A straight line has equation y = 3x − 2

  1. Write down the gradient of the line.
  2. Write down the coordinates of the point where the line crosses the y-axis.
","answerType":"multi","answer":"a) 3; b) (0, -2)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3"},{"label":"b","answer":"(0, -2)"}],"solution":["In y = mx + c, m is the change in y for a 1-unit increase in x, called the gradient. c is the value of y when x = 0, called the y-intercept.","The equation is in the form y = mx + c","a) The gradient is the coefficient of x, which is 3","b) The line crosses the y-axis where x = 0, so at (0, -2)"],"diagram":null,"draw":false,"revision":"df0e6e34d33e"},{"id":"g3-drawing-linear-graphs-q02","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

Here are the equations of three straight lines.

Line A: y = 2x + 1

Line B: y = 3x − 2

Line C: y = 2x − 5

  1. Which two lines are parallel?
  2. Which line passes through the point (0, −2)?
","answerType":"multi","answer":"a) A and C; b) B","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"A and C"},{"label":"b","answer":"B"}],"solution":["a) Parallel lines have equal gradients. Lines A and C both have gradient 2","b) At (0, -2) the y-intercept is -2. Line B has y-intercept -2"],"diagram":null,"draw":false,"revision":"85e99dc1eeb6"},{"id":"g3-drawing-linear-graphs-q03","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

Here is a table of values for y = 2x + 1

x: −2, −1, 0, 1, 2, 3

y: −3, ?, 1, 3, ?, 7

  1. Work out the missing value of y when x = −1
  2. Work out the missing value of y when x = 2
  3. Write down the coordinates of the point where the line crosses the y-axis.
","answerType":"multi","answer":"a) -1; b) 5; c) (0, 1)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-1"},{"label":"b","answer":"5"},{"label":"c","answer":"(0, 1)"}],"solution":["a) When x = -1, y = 2(-1) + 1 = -1","b) When x = 2, y = 2(2) + 1 = 5","c) When x = 0, y = 1, so the line crosses the y-axis at (0, 1)"],"diagram":null,"draw":false,"revision":"57250f3b3cfd"},{"id":"g3-drawing-linear-graphs-q04","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

Here is a table of values for y = 3x − 4

x: −1, 0, 1, 2, 3

y: ?, −4, −1, 2, ?

  1. Work out the two missing values of y.
  2. Does the point (10, 26) lie on the line y = 3x − 4? You must show how you get your answer.
","answerType":"multi","answer":"a) -7 and 5; b) Yes, because 3(10) - 4 = 26","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-7 and 5"},{"label":"b","answer":"Yes, 3(10) - 4 = 26"}],"solution":["a) When x = -1, y = 3(-1) - 4 = -7","When x = 3, y = 3(3) - 4 = 5","b) Substitute x = 10: y = 30 - 4 = 26, so (10, 26) does lie on the line"],"diagram":null,"draw":false,"revision":"8d444dcc8016"},{"id":"g3-drawing-linear-graphs-q05","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

On the grid, draw the graph of y = 2x − 3 for values of x from −2 to 3

","answerType":"open","answer":"Straight line through (-2, -7), (0, -3) and (3, 3)","answerDisplay":null,"accept":[],"parts":[],"solution":["Make a table of values: x = -2 gives y = -7, x = 0 gives y = -3, x = 1 gives y = -1, x = 3 gives y = 3","Plot the points and join them with a single straight line with a ruler"],"diagram":{"type":"axes","square":true,"width":250,"x":{"min":-2,"max":3},"y":{"min":-8,"max":4},"lines":[{"m":2,"c":-3,"from":-2,"to":3,"answer":true,"label":"y = 2x − 3","labelAt":[2,1]}]},"draw":true,"revision":"df9a5553aa38"},{"id":"g3-drawing-linear-graphs-q06","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

Here is a table of values for y = 12x + 2

x: −4, −2, 0, 2, 4

y: 0, ?, 2, ?, 4

  1. Work out the two missing values of y.
  2. Write down the gradient of the line.
","answerType":"multi","answer":"a) 1 and 3; b) 1/2","answerDisplay":"a) 1 and 3; b) 12","accept":[],"parts":[{"label":"a","answer":"1 and 3"},{"label":"b","answer":"1/2","answerDisplay":"12"}],"solution":["a) When x = -2, y = -1 + 2 = 1","When x = 2, y = 1 + 2 = 3","b) The gradient is the coefficient of x, which is 12"],"diagram":null,"draw":false,"revision":"529ca2952f46"},{"id":"g3-drawing-linear-graphs-q07","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

On the grid, draw the graph of x + y = 6 for values of x from 0 to 6

","answerType":"open","answer":"Straight line through (0, 6) and (6, 0)","answerDisplay":null,"accept":[],"parts":[],"solution":["Rearrange to y = 6 - x","When x = 0, y = 6. When x = 3, y = 3. When x = 6, y = 0","Plot the points and join them with a single straight line with a ruler"],"diagram":{"type":"axes","square":true,"width":360,"x":{"min":0,"max":6},"y":{"min":0,"max":6},"lines":[{"through":[[0,6],[6,0]],"from":0,"to":6,"answer":true,"label":"x + y = 6","labelAt":[4,2]}]},"draw":true,"revision":"1ac820163ac1"},{"id":"g3-drawing-linear-graphs-q08","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

A straight line passes through the points (0, −1) and (4, 7).

  1. Work out the gradient of the line.
  2. Write down the equation of the line in the form y = mx + c
","answerType":"multi","answer":"a) 2; b) y = 2x - 1","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2"},{"label":"b","answer":"y = 2x - 1"}],"solution":["a) Gradient = change in y over change in x = 7 - (-1)4 - 0 = 84 = 2","b) The line crosses the y-axis at (0, -1), so c = -1","The equation is y = 2x - 1"],"diagram":null,"draw":false,"revision":"3fdd285f75ff"},{"id":"g3-drawing-linear-graphs-q09","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":9,"marks":3,"tier":"FH","calc":false,"text":"

The straight line y = 4x + c passes through the point (3, 10).

  1. Find the value of c.
  2. Write down the coordinates of the point where the line crosses the y-axis.
","answerType":"multi","answer":"a) c = -2; b) (0, -2)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"c = -2"},{"label":"b","answer":"(0, -2)"}],"solution":["a) Substitute x = 3 and y = 10: 10 = 4(3) + c","10 = 12 + c, so c = -2","b) The y-intercept is c, so the line crosses the y-axis at (0, -2)"],"diagram":null,"draw":false,"revision":"46fb92b930de"},{"id":"g3-drawing-linear-graphs-q10","topicId":"g3-drawing-linear-graphs","topic":"Drawing Linear Graphs","grade":"3","topicGrade":"3","strand":"A","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"
  1. Complete the table of values for y = 2 − 2x
    x: −2, −1, 0, 1, 2, 3
    y: 6, ?, 2, 0, ?, −4
  2. On the grid, draw the graph of y = 2 − 2x for values of x from −2 to 3
","answerType":"multi","answer":"a) 4 and -2; b) straight line through (-2, 6) and (3, -4)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"4 and -2"},{"label":"b","answer":"Straight line through (-2, 6), (0, 2) and (3, -4)"}],"solution":["a) When x = -1, y = 2 - 2(-1) = 4","When x = 2, y = 2 - 2(2) = -2","b) Plot all six points and join them with a single straight line with a ruler. The line slopes downwards from left to right"],"diagram":{"type":"axes","square":true,"width":250,"x":{"min":-2,"max":3},"y":{"min":-5,"max":7},"lines":[{"m":-2,"c":2,"from":-2,"to":3,"answer":true,"label":"y = 2 − 2x","labelAt":[-1.5,5]}]},"draw":true,"revision":"ee346b0e478d"},{"id":"g3-error-intervals-q01","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

The mass of a parcel is 340 g, correct to the nearest 10 g.

Complete the error interval for the mass, m grams, of the parcel.

....... ≤ m < .......

","answerType":"exact","answer":"335 ≤ m < 345","answerDisplay":null,"accept":["335 ≤ m < 345","335 <= m < 345","335, 345"],"parts":[],"solution":["Rounding to the nearest 10 g means the true mass is within 5 g of 340 g.","Lower bound: 340 - 5 = 335. Upper bound: 340 + 5 = 345.","So 335 ≤ m < 345.","Include the lower boundary because a value exactly halfway rounds up to the stated result. Exclude the upper boundary because that halfway value rounds up to the next result instead."],"diagram":null,"draw":false,"revision":"a5f7bdfe5a59"},{"id":"g3-error-intervals-q02","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

The length of a pencil is 27 cm, correct to the nearest centimetre.

Complete the error interval for the length, L cm, of the pencil.

....... ≤ L < .......

","answerType":"exact","answer":"26.5 ≤ L < 27.5","answerDisplay":null,"accept":["26.5 ≤ L < 27.5","26.5 <= L < 27.5","26.5, 27.5"],"parts":[],"solution":["Rounding to the nearest centimetre means the true length is within 0.5 cm of 27 cm.","Lower bound: 27 - 0.5 = 26.5. Upper bound: 27 + 0.5 = 27.5.","So 26.5 ≤ L < 27.5.","Include the lower boundary because a value exactly halfway rounds up to the stated result. Exclude the upper boundary because that halfway value rounds up to the next result instead."],"diagram":null,"draw":false,"revision":"c6c9d5ab232f"},{"id":"g3-error-intervals-q03","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

The attendance at a football match is 5600, correct to the nearest 100

Complete the error interval for the attendance, n.

....... ≤ n < .......

","answerType":"exact","answer":"5550 ≤ n < 5650","answerDisplay":null,"accept":["5550 ≤ n < 5650","5550 <= n < 5650","5550, 5650"],"parts":[],"solution":["Rounding to the nearest 100 means the true value is within 50 of 5600.","Lower bound: 5600 - 50 = 5550. Upper bound: 5600 + 50 = 5650.","So 5550 ≤ n < 5650.","Include the lower boundary because a value exactly halfway rounds up to the stated result. Exclude the upper boundary because that halfway value rounds up to the next result instead."],"diagram":null,"draw":false,"revision":"8219440d11fc"},{"id":"g3-error-intervals-q04","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

The mass of a cat is 8.3 kg, correct to 1 decimal place.

Complete the error interval for the mass, w kg, of the cat.

....... ≤ w < .......

","answerType":"exact","answer":"8.25 ≤ w < 8.35","answerDisplay":null,"accept":["8.25 ≤ w < 8.35","8.25 <= w < 8.35","8.25, 8.35"],"parts":[],"solution":["Rounding to 1 decimal place means the true mass is within 0.05 kg of 8.3 kg.","Lower bound: 8.3 - 0.05 = 8.25. Upper bound: 8.3 + 0.05 = 8.35.","So 8.25 ≤ w < 8.35.","Include the lower boundary because a value exactly halfway rounds up to the stated result. Exclude the upper boundary because that halfway value rounds up to the next result instead."],"diagram":null,"draw":false,"revision":"7af96873af40"},{"id":"g3-error-intervals-q05","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":5,"marks":2,"tier":"FH","calc":true,"text":"

The time taken for a ball to roll down a ramp is 4.70 seconds, correct to 2 decimal places.

Complete the error interval for the time, t seconds.

....... ≤ t < .......

","answerType":"exact","answer":"4.695 ≤ t < 4.705","answerDisplay":null,"accept":["4.695 ≤ t < 4.705","4.695 <= t < 4.705","4.695, 4.705"],"parts":[],"solution":["Rounding to 2 decimal places means the true time is within 0.005 seconds of 4.70.","Lower bound: 4.70 - 0.005 = 4.695. Upper bound: 4.70 + 0.005 = 4.705.","So 4.695 ≤ t < 4.705.","Include the lower boundary because a value exactly halfway rounds up to the stated result. Exclude the upper boundary because that halfway value rounds up to the next result instead."],"diagram":null,"draw":false,"revision":"1f7111d4979f"},{"id":"g3-error-intervals-q06","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":6,"marks":2,"tier":"FH","calc":true,"text":"

A number d is truncated to 1 decimal place.

The result is 12.4

Complete the error interval for d.

....... ≤ d < .......

","answerType":"exact","answer":"12.4 ≤ d < 12.5","answerDisplay":null,"accept":["12.4 ≤ d < 12.5","12.4 <= d < 12.5","12.4, 12.5"],"parts":[],"solution":["Truncating to 1 decimal place removes everything after the first decimal digit, so the value has not been rounded up.","The value must be at least 12.4 but strictly less than 12.5. Removing the later digits from any value in this interval gives 12.4.","So 12.4 ≤ d < 12.5."],"diagram":null,"draw":false,"revision":"4a98746d6151"},{"id":"g3-error-intervals-q07","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":7,"marks":2,"tier":"FH","calc":true,"text":"

A digital scale shows the weight of a suitcase as 52 kg.

The scale truncates the weight to a whole number of kilograms.

Complete the error interval for the weight, w kg, of the suitcase.

....... ≤ w < .......

","answerType":"exact","answer":"52 ≤ w < 53","answerDisplay":null,"accept":["52 ≤ w < 53","52 <= w < 53","52, 53"],"parts":[],"solution":["Truncating to a whole number just removes the decimal part, so nothing is rounded up.","The value must be at least 52 but strictly less than 53. Removing the later digits from any value in this interval gives 52.","So 52 ≤ w < 53."],"diagram":null,"draw":false,"revision":"d5b1f8d55c9d"},{"id":"g3-error-intervals-q08","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":8,"marks":2,"tier":"H","calc":true,"text":"

A number x is 470, correct to 2 significant figures.

Complete the error interval for x.

....... ≤ x < .......

","answerType":"exact","answer":"465 ≤ x < 475","answerDisplay":null,"accept":["465 ≤ x < 475","465 <= x < 475","465, 475"],"parts":[],"solution":["To 2 significant figures, 470 has been rounded to the nearest 10.","Lower bound: 470 - 5 = 465. Upper bound: 470 + 5 = 475.","So 465 ≤ x < 475.","Include the lower boundary because a value exactly halfway rounds up to the stated result. Exclude the upper boundary because that halfway value rounds up to the next result instead."],"diagram":null,"draw":false,"revision":"6e363a24c1e1"},{"id":"g3-error-intervals-q09","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":9,"marks":2,"tier":"H","calc":true,"text":"

A number y is 0.08, correct to 1 significant figure.

Complete the error interval for y.

....... ≤ y < .......

","answerType":"exact","answer":"0.075 ≤ y < 0.085","answerDisplay":null,"accept":["0.075 ≤ y < 0.085","0.075 <= y < 0.085","0.075, 0.085"],"parts":[],"solution":["To 1 significant figure, 0.08 has been rounded to the nearest 0.01.","Lower bound: 0.08 - 0.005 = 0.075. Upper bound: 0.08 + 0.005 = 0.085.","So 0.075 ≤ y < 0.085.","Include the lower boundary because a value exactly halfway rounds up to the stated result. Exclude the upper boundary because that halfway value rounds up to the next result instead."],"diagram":null,"draw":false,"revision":"c2aa208bebfb"},{"id":"g3-error-intervals-q10","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":10,"marks":2,"tier":"H","calc":true,"text":"

A number k is 15.0, correct to 3 significant figures.

Complete the error interval for k.

....... ≤ k < .......

","answerType":"exact","answer":"14.95 ≤ k < 15.05","answerDisplay":null,"accept":["14.95 ≤ k < 15.05","14.95 <= k < 15.05","14.95, 15.05"],"parts":[],"solution":["To 3 significant figures, 15.0 has been rounded to the nearest 0.1 - the trailing zero counts as a significant figure.","Lower bound: 15.0 - 0.05 = 14.95. Upper bound: 15.0 + 0.05 = 15.05.","So 14.95 ≤ k < 15.05.","Include the lower boundary because a value exactly halfway rounds up to the stated result. Exclude the upper boundary because that halfway value rounds up to the next result instead."],"diagram":null,"draw":false,"revision":"75a95c049cf6"},{"id":"g3-error-intervals-q11","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":11,"marks":2,"tier":"H","calc":true,"text":"

A number p is truncated to 2 decimal places.

The result is 7.26

Complete the error interval for p.

....... ≤ p < .......

","answerType":"exact","answer":"7.26 ≤ p < 7.27","answerDisplay":null,"accept":["7.26 ≤ p < 7.27","7.26 <= p < 7.27","7.26, 7.27"],"parts":[],"solution":["Truncating to 2 decimal places removes every digit after the second decimal place, with no rounding up.","The value must be at least 7.26 but strictly less than 7.27. Removing the later digits from any value in this interval gives 7.26.","So 7.26 ≤ p < 7.27."],"diagram":null,"draw":false,"revision":"6f1ad4113f0f"},{"id":"g3-error-intervals-q12","topicId":"g3-error-intervals","topic":"Error Intervals","grade":"3","topicGrade":"3","strand":"N","difficulty":12,"marks":3,"tier":"F","calc":true,"text":"
  1. Write down the reciprocal of 0.4
  2. A number q is rounded to 2 significant figures.

    The result is 2.5

    Complete the error interval for q.

    ....... ≤ q < .......

","answerType":"multi","answer":"a) 2.5 b) 2.45 ≤ q < 2.55","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2.5"},{"label":"b","answer":"2.45 ≤ q < 2.55"}],"solution":["a) The reciprocal of 0.4 is 1 / 0.4 = 104 = 2.5.","b) To 2 significant figures, 2.5 has been rounded to the nearest 0.1.","b) Lower bound: 2.5 - 0.05 = 2.45. Upper bound: 2.5 + 0.05 = 2.55, so 2.45 ≤ q < 2.55."],"diagram":null,"draw":false,"revision":"d3ebcb1dc8f2"},{"id":"g3-estimating-q01","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Katie says

"6.2 × 48.9 = 3031.8"

Without working out the exact answer, explain how you can tell that Katie's answer must be wrong.

","answerType":"open","answer":"6.2 x 48.9 is roughly 6 x 50 = 300, so the answer should be close to 300, not close to 3000. Katie's answer is about ten times too big.","answerDisplay":null,"accept":[],"parts":[],"solution":["Round each number to 1 significant figure: 6.2 is about 6 and 48.9 is about 50.","6 x 50 = 300, so the true answer must be close to 300.","3031.8 is about ten times bigger than 300, so Katie's answer must be wrong."],"diagram":null,"draw":false,"revision":"62003f4688c1"},{"id":"g3-estimating-q02","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

Work out an estimate for the value of

48.7 × 6.120.216

You must round each number to 1 significant figure.

","answerType":"exact","answer":"1500","answerDisplay":null,"accept":["1500","1,500"],"parts":[],"solution":["Round each number to 1 significant figure: 48.7 becomes 50, 6.12 becomes 6, 0.216 becomes 0.2.","Numerator: 50 x 6 = 300.","300 / 0.2 = 3000 / 2 = 1500."],"diagram":null,"draw":false,"revision":"7c87d494f2a9"},{"id":"g3-estimating-q03","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

41,872 fans attend a rugby match.

  1. Round 41,872 to the nearest 1000
  2. Each fan pays £5 for a match programme.

    Use your answer to part (a) to work out an estimate for the total amount paid for programmes.

","answerType":"multi","answer":"a) 42,000 b) £210,000","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"42,000"},{"label":"b","answer":"£210,000"}],"solution":["a) 41,872 is between 41,000 and 42,000, and 872 is more than 500, so it rounds up to 42,000.","b) 42,000 x 5 = 210,000, so the estimate is 210,000 pounds."],"diagram":null,"draw":false,"revision":"02c2940f7e31"},{"id":"g3-estimating-q04","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"
  1. Work out an estimate for 62.3 × 4.87
  2. You are given that 623 × 487 = 303,401

    Use this fact to write down the exact value of 62.3 × 4.87

","answerType":"multi","answer":"a) 300 b) 303.401","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"300"},{"label":"b","answer":"303.401"}],"solution":["a) Round to 1 significant figure: 62.3 is about 60 and 4.87 is about 5, so the estimate is 60 x 5 = 300.","b) 62.3 is 623 / 10 and 4.87 is 487 / 100, so the answer is 303,401 / 1000 = 303.401.","b) Check against the estimate: 303.401 is close to 300, so the decimal point is in the right place."],"diagram":null,"draw":false,"revision":"3a139d0f1944"},{"id":"g3-estimating-q05","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":5,"marks":4,"tier":"F","calc":false,"text":"

Marco is building a fence.

He buys 28 fence panels costing £19.75 each.

He also buys a tin of paint costing £41.80

By rounding each number to 1 significant figure, work out an estimate for the total amount Marco pays.

","answerType":"exact","answer":"£640","answerDisplay":null,"accept":["640","£640"],"parts":[],"solution":["Round each number to 1 significant figure: 28 becomes 30, 19.75 becomes 20, 41.80 becomes 40.","Estimate for the panels: 30 x 20 = 600 pounds.","Add the paint: 600 + 40 = 640 pounds."],"diagram":null,"draw":false,"revision":"ec546b6a683c"},{"id":"g3-estimating-q06","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":6,"marks":4,"tier":"FH","calc":false,"text":"

A lorry driver has to drive 289 miles from Leeds to Plymouth.

Her average speed for the journey is 48 miles per hour.

Work out an estimate for the time the journey takes.

Round both numbers to 1 significant figure before calculating.

","answerType":"exact","answer":"6 hours","answerDisplay":null,"accept":["6","6 hours"],"parts":[],"solution":["Round each number to 1 significant figure: 289 becomes 300 and 48 becomes 50.","Time = distance / speed = 300 / 50.","300 / 50 = 6, so the journey takes about 6 hours."],"diagram":null,"draw":false,"revision":"9634a896dd27"},{"id":"g3-estimating-q07","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":7,"marks":4,"tier":"FH","calc":false,"text":"

Amara drives 189 km at an average speed of 62 km/h.

  1. By rounding both numbers to 1 significant figure, work out an estimate for the time her journey takes.

    Give your answer in hours and minutes.

  2. Is your answer to part (a) an underestimate or an overestimate of the actual time?

    Give a reason for your answer.

","answerType":"multi","answer":"a) 3 hours 20 minutes b) Overestimate - the distance was rounded up and the speed was rounded down, both of which make the estimated time bigger than the actual time.","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3 hours 20 minutes"},{"label":"b","answer":"Overestimate, because the distance was rounded up and the speed was rounded down"}],"solution":["a) Round to 1 significant figure: 189 becomes 200 and 62 becomes 60.","a) Time = 200 / 60 = 313 hours = 3 hours 20 minutes.","b) Rounding the distance up gives more distance to cover, and rounding the speed down means covering it more slowly - both make the estimate bigger, so it is an overestimate."],"diagram":null,"draw":false,"revision":"fc7c5d7ecd07"},{"id":"g3-estimating-q08","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":8,"marks":4,"tier":"FH","calc":false,"text":"

Dan is walking a long-distance path that is 990 km long.

He walks at an average speed of 4.8 km/h for 8 hours each day.

By rounding each number to 1 significant figure, work out an estimate for the number of days Dan takes to walk the whole path.

","answerType":"exact","answer":"25 days","answerDisplay":null,"accept":["25","25 days"],"parts":[],"solution":["Round to 1 significant figure: 990 becomes 1000 and 4.8 becomes 5.","Distance walked each day: 5 x 8 = 40 km.","Number of days: 1000 / 40 = 25 days."],"diagram":null,"draw":false,"revision":"36d96553a39f"},{"id":"g3-estimating-q09","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":9,"marks":4,"tier":"FH","calc":false,"text":"

A rectangular floor is 4.2 m long and 3.1 m wide.

Flooring costs £24.60 per square metre.

  1. By rounding each number to 1 significant figure, work out an estimate for the cost of flooring for the whole floor.
  2. Is your answer to part (a) an underestimate or an overestimate of the actual cost?

    Give a reason for your answer.

","answerType":"multi","answer":"a) £240 b) Underestimate - all three numbers were rounded down, so the estimate must be smaller than the actual cost.","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£240"},{"label":"b","answer":"Underestimate, because every number was rounded down"}],"solution":["a) Round to 1 significant figure: 4.2 becomes 4, 3.1 becomes 3, 24.60 becomes 20.","a) Estimated area: 4 x 3 = 12 square metres.","a) Estimated cost: 12 x 20 = 240 pounds.","b) 4.2, 3.1 and 24.60 were all rounded down, so the estimate is smaller than the actual cost - it is an underestimate."],"diagram":null,"draw":false,"revision":"d8789705b68a"},{"id":"g3-estimating-q10","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":10,"marks":4,"tier":"H","calc":false,"text":"

A square field has an area of 7900 m²

Work out an estimate for the length of one side of the field.

You must show your working and explain how you know your estimate is sensible.

","answerType":"open","answer":"About 89 m. The side length is the square root of 7900. Since 80 squared = 6400 and 90 squared = 8100, the side is between 80 m and 90 m, and because 7900 is very close to 8100 the side is close to 90 m - about 89 m (89 squared = 7921).","answerDisplay":null,"accept":[],"parts":[],"solution":["The side length of a square is the square root of its area, so we need the square root of 7900.","80 squared = 6400 and 90 squared = 8100, so the side is between 80 m and 90 m.","7900 is much closer to 8100 than to 6400, so the side is close to 90 m.","Try 89: 89 squared = 7921, which is very close to 7900, so about 89 m is a sensible estimate."],"diagram":null,"draw":false,"revision":"d028b1a4fe88"},{"id":"g3-estimating-q11","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":11,"marks":4,"tier":"H","calc":false,"text":"
  1. Work out an estimate for the value of 9.72 × 5.30.48
  2. Given that 1.84 = 10.4976, write down the value of 184
  3. Write down the value of 5−2
","answerType":"multi","answer":"a) 100 b) 104,976 c) 1/25","answerDisplay":"a) 100 b) 104,976 c) 125","accept":[],"parts":[{"label":"a","answer":"100"},{"label":"b","answer":"104,976"},{"label":"c","answer":"1/25","answerDisplay":"125"}],"solution":["a) Round to 1 significant figure: 9.72 becomes 10, 5.3 becomes 5, 0.48 becomes 0.5. Then (10 x 5) / 0.5 = 50 / 0.5 = 100.","b) 18 = 1.8 x 10, so 18^4 = 1.8^4 x 10^4 = 10.4976 x 10,000 = 104,976.","c) 5^-2 = 1 / 5^2 = 125."],"diagram":null,"draw":false,"revision":"87d99b907fb6"},{"id":"g3-estimating-q12","topicId":"g3-estimating","topic":"Estimating","grade":"3","topicGrade":"3","strand":"N","difficulty":12,"marks":6,"tier":"F","calc":false,"text":"

One evening a theatre sells 412 adult tickets and 243 child tickets.

An adult ticket costs £11.75

A child ticket costs £5.20

  1. By rounding each number to 1 significant figure, work out an estimate for the total amount of money the theatre takes from ticket sales.
  2. Is your answer to part (a) an underestimate or an overestimate of the actual amount?

    Give a reason for your answer.

","answerType":"multi","answer":"a) £5,000 b) Underestimate - every number was rounded down (412 to 400, 11.75 to 10, 243 to 200, 5.20 to 5), so the actual amount must be more than £5,000.","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£5,000"},{"label":"b","answer":"Underestimate, because every number was rounded down"}],"solution":["a) Round to 1 significant figure: 412 becomes 400, 11.75 becomes 10, 243 becomes 200, 5.20 becomes 5.","a) Adult tickets: 400 x 10 = 4000 pounds. Child tickets: 200 x 5 = 1000 pounds.","a) Total estimate: 4000 + 1000 = 5000 pounds.","b) All four numbers were rounded down, so the estimate is smaller than the actual takings - it is an underestimate."],"diagram":null,"draw":false,"revision":"ef9b696bc3e4"},{"id":"g3-exchange-rates-q01","topicId":"g3-exchange-rates","topic":"Exchange Rates","grade":"3","topicGrade":"3","strand":"R","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

The exchange rate is £1 = €1.14.

Nadia changes £250 into euros.

Work out how many euros she gets.

","answerType":"exact","answer":"€285","answerDisplay":null,"accept":["285","285 euros"],"parts":[],"solution":["Each pound is worth €1.14.","250 × 1.14 = 285.","Nadia gets €285."],"diagram":null,"draw":false,"revision":"d17340fdd9ca"},{"id":"g3-exchange-rates-q02","topicId":"g3-exchange-rates","topic":"Exchange Rates","grade":"3","topicGrade":"3","strand":"R","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

On holiday in France, Callum sees a jacket priced at €84.

The exchange rate is £1 = €1.20.

Work out the cost of the jacket in pounds.

","answerType":"exact","answer":"£70","answerDisplay":null,"accept":["70","£70.00","70.00"],"parts":[],"solution":["The rate means each £1 costs €1.20. Dividing €84 by €1.20 per pound finds how many pounds are equivalent.","To change euros into pounds, divide by the exchange rate.","84 ÷ 1.20 = 70.","The jacket costs £70. Check: 70 × 1.20 = 84."],"diagram":null,"draw":false,"revision":"5ec07a40591b"},{"id":"g3-exchange-rates-q03","topicId":"g3-exchange-rates","topic":"Exchange Rates","grade":"3","topicGrade":"3","strand":"R","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

Theo changes £320 into dollars.

The exchange rate is £1 = $1.30.

He is given as many $50 notes as possible, with the rest in smaller notes.

Work out the number of $50 notes Theo is given.

","answerType":"exact","answer":"8","answerDisplay":null,"accept":["8 notes"],"parts":[],"solution":["320 × 1.30 = 416, so Theo gets $416.","416 ÷ 50 = 8.32, so 8 whole $50 notes fit.","8 × $50 = $400, leaving $16 in smaller notes. Theo gets 8 fifty-dollar notes."],"diagram":null,"draw":false,"revision":"ec4329eca071"},{"id":"g3-exchange-rates-q04","topicId":"g3-exchange-rates","topic":"Exchange Rates","grade":"3","topicGrade":"3","strand":"R","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

A games console costs £399 in London.

The same console costs €448.50 in Dublin.

The exchange rate is £1 = €1.15.

In which city is the console cheaper, and by how much? Give your answer in pounds.

","answerType":"exact","answer":"Dublin, by £9","answerDisplay":null,"accept":["Dublin by £9","Dublin, £9 cheaper"],"parts":[],"solution":["Convert the Dublin price to pounds: 448.50 ÷ 1.15 = £390.","London price = £399.","399 − 390 = £9.","The console is cheaper in Dublin, by £9."],"diagram":null,"draw":false,"revision":"b9b59ffdb550"},{"id":"g3-exchange-rates-q05","topicId":"g3-exchange-rates","topic":"Exchange Rates","grade":"3","topicGrade":"3","strand":"R","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

Ana is offered two jobs.

A job in Manchester pays £13.50 per hour.

A job in Berlin pays €16.24 per hour.

The exchange rate is £1 = €1.16.

Which job pays more per hour? You must show your working.

","answerType":"open","answer":"The Berlin job - €16.24 / 1.16 = £14.00 per hour, which is more than £13.50","answerDisplay":"The Berlin job - €16.241.16 = £14.00 per hour, which is more than £13.50","accept":[],"parts":[],"solution":["Convert the Berlin rate to pounds: 16.24 ÷ 1.16 = £14.00 per hour.","Compare: £14.00 > £13.50.","The Berlin job pays more, by 50p per hour."],"diagram":null,"draw":false,"revision":"99a5bdf04fe8"},{"id":"g3-exchange-rates-q06","topicId":"g3-exchange-rates","topic":"Exchange Rates","grade":"3","topicGrade":"3","strand":"R","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

Before a trip to America, Prav changes £400 into dollars at an exchange rate of £1 = $1.25.

In America he spends $380.

When he gets home, he changes his remaining dollars back into pounds at an exchange rate of £1 = $1.20.

Work out how many pounds Prav gets back.

","answerType":"exact","answer":"£100","answerDisplay":null,"accept":["100","£100.00","100.00"],"parts":[],"solution":["400 × 1.25 = 500, so Prav takes $500.","Dollars left = 500 − 380 = $120.","Changing back: 120 ÷ 1.20 = £100."],"diagram":null,"draw":false,"revision":"9710e32d8628"},{"id":"g3-exchange-rates-q07","topicId":"g3-exchange-rates","topic":"Exchange Rates","grade":"3","topicGrade":"3","strand":"R","difficulty":7,"marks":3,"tier":"FH","calc":true,"text":"

A phone costs £520 in a UK shop.

In a New York shop the same phone costs $650 plus sales tax at 8%.

The exchange rate is £1 = $1.35.

Show that the phone costs the same in both shops.

","answerType":"open","answer":"New York total = 650 × 1.08 = $702; 702 / 1.35 = £520, the same as the UK price","answerDisplay":"New York total = 650 × 1.08 = $702; 7021.35 = £520, the same as the UK price","accept":[],"parts":[],"solution":["New York price including tax = 650 × 1.08 = $702.","Convert to pounds: 702 ÷ 1.35 = £520.","This equals the UK price of £520, so the phone costs the same in both shops."],"diagram":null,"draw":false,"revision":"28f6458d9f32"},{"id":"g3-exchange-rates-q08","topicId":"g3-exchange-rates","topic":"Exchange Rates","grade":"3","topicGrade":"3","strand":"R","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

Mia is booking a holiday.

Her flights cost £186.

Her hotel costs €78 per night for 5 nights.

She also buys $260 of spending money.

The exchange rates are £1 = €1.20 and £1 = $1.30.

Work out the total cost of the holiday in pounds.

","answerType":"exact","answer":"£711","answerDisplay":null,"accept":["711","£711.00","711.00"],"parts":[],"solution":["Hotel total = 78 × 5 = €390.","Hotel in pounds = 390 ÷ 1.20 = £325.","Spending money in pounds = 260 ÷ 1.30 = £200.","Total = 186 + 325 + 200 = £711."],"diagram":null,"draw":false,"revision":"2927f07721dc"},{"id":"g3-fractions-q01","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Work out 1 − 38

","answerType":"exact","answer":"5/8","answerDisplay":"58","accept":["5/8","0.625"],"parts":[],"solution":["Write 1 as 88.","88 - 38 = 58."],"diagram":null,"draw":false,"revision":"6972f6c466c3"},{"id":"g3-fractions-q02","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

Work out 49 × 310

Give your answer as a fraction in its simplest form.

","answerType":"exact","answer":"2/15","answerDisplay":"215","accept":["2/15"],"parts":[],"solution":["Multiply the numerators and multiply the denominators: 4 x 3 = 12 and 9 x 10 = 90, so the product is 1290.","Divide top and bottom by 6: 1290 = 215."],"diagram":null,"draw":false,"revision":"959be2a66966"},{"id":"g3-fractions-q03","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

Find the number that is exactly halfway between 14 and 58

","answerType":"exact","answer":"7/16","answerDisplay":"716","accept":["7/16","0.4375"],"parts":[],"solution":["Write both fractions with denominator 8: 14 = 28.","Add them: 28 + 58 = 78.","Halve the total: 78 divided by 2 = 716."],"diagram":null,"draw":false,"revision":"8d14be03c858"},{"id":"g3-fractions-q04","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":4,"marks":2,"tier":"F","calc":false,"text":"

Freya is asked to work out 23 + 16

Here is her working.

23 + 16 = 39 = 13

Describe two ways you can tell that Freya's method is wrong.

","answerType":"open","answer":"She added the numerators and the denominators instead of using a common denominator; also her answer 1/3 is smaller than 2/3, which is impossible when adding a positive fraction. (Correct answer: 2/3 = 4/6, so 4/6 + 1/6 = 5/6.)","answerDisplay":"She added the numerators and the denominators instead of using a common denominator; also her answer 13 is smaller than 23, which is impossible when adding a positive fraction. (Correct answer: 23 = 46, so 46 + 16 = 56.)","accept":[],"parts":[],"solution":["Mistake 1: you cannot add fractions by adding numerators and adding denominators - you need a common denominator.","Mistake 2: her answer 13 is less than the 23 she started with, but adding a positive fraction must give a bigger answer.","The correct answer is 23 + 16 = 46 + 16 = 56.","The denominator names the size of the pieces. Convert thirds into sixths so that both fractions count equal-sized pieces; then add the number of pieces and keep that denominator."],"diagram":null,"draw":false,"revision":"c57e8d4dea50"},{"id":"g3-fractions-q05","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"
  1. One of these fractions is not equivalent to 35

    610    915    1225    1525

    Write down this fraction.

  2. Work out 25 + 13
","answerType":"multi","answer":"a) 12/25 b) 11/15","answerDisplay":"a) 1225 b) 1115","accept":[],"parts":[{"label":"a","answer":"12/25","answerDisplay":"1225"},{"label":"b","answer":"11/15","answerDisplay":"1115"}],"solution":["a) 610, 915 and 1525 all simplify to 35, but 1225 does not (35 = 1525, not 1225).","b) Use denominator 15: 25 = 615 and 13 = 515.","615 + 515 = 1115."],"diagram":null,"draw":false,"revision":"8646eaa547e8"},{"id":"g3-fractions-q06","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":6,"marks":3,"tier":"FH","calc":false,"text":"

Work out 425 − 134

Give your answer as a mixed number in its simplest form.

","answerType":"exact","answer":"2 13/20","answerDisplay":"21320","accept":["2 13/20"],"parts":[],"solution":["Write as improper fractions: 425 = 225 and 134 = 74.","Use denominator 20: 225 = 8820 and 74 = 3520.","8820 - 3520 = 5320 = 21320."],"diagram":null,"draw":false,"revision":"6c12a63d6f71"},{"id":"g3-fractions-q07","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":7,"marks":3,"tier":"FH","calc":false,"text":"

Work out 313 ÷ 212

Give your answer as a mixed number in its simplest form.

","answerType":"exact","answer":"1 1/3","answerDisplay":"113","accept":["1 1/3"],"parts":[],"solution":["Write as improper fractions: 313 = 103 and 212 = 52.","To divide by 5/2, multiply by its reciprocal 2/5: (5/2) × (2/5) = 1, so this undoes multiplication by 5/2.","Dividing by 52 is the same as multiplying by 25: 103 x 25 = 2015.","Simplify: 2015 = 43 = 113."],"diagram":null,"draw":false,"revision":"93567167e289"},{"id":"g3-fractions-q08","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":8,"marks":3,"tier":"FH","calc":false,"text":"

Show that 223 × 178 = 5

","answerType":"open","answer":"2 2/3 = 8/3 and 1 7/8 = 15/8, so the product is (8 x 15)/(3 x 8) = 120/24 = 5","answerDisplay":"223 = 83 and 178 = 158, so the product is (8 x 15)(3 x 8) = 12024 = 5","accept":[],"parts":[],"solution":["Write both numbers as improper fractions: 223 = 83 and 178 = 158.","Multiply: 83 x 158 = 12024.","Simplify: 12024 = 5, as required. (Or cancel the 8s first: 153 = 5.)"],"diagram":null,"draw":false,"revision":"e0eefe499808"},{"id":"g3-fractions-q09","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":9,"marks":3,"tier":"FH","calc":true,"text":"

In a sale, the price of a coat is reduced by 27 of its original price.

The sale price of the coat is £45

Work out the original price of the coat.

","answerType":"exact","answer":"£63","answerDisplay":null,"accept":["63","£63","63.00"],"parts":[],"solution":["A reduction of 27 leaves 57 of the original price.","So 57 of the original price is 45, which means 17 is 45 / 5 = 9.","The original price is 7 x 9 = 63, so the coat originally cost 63 pounds."],"diagram":null,"draw":false,"revision":"a4c34a0dbfa9"},{"id":"g3-fractions-q10","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":10,"marks":4,"tier":"FH","calc":false,"text":"
  1. Work out 134 + 256

    Give your answer as a mixed number in its simplest form.

  2. Work out 212 × 115
","answerType":"multi","answer":"a) 4 7/12 b) 3","answerDisplay":"a) 4712 b) 3","accept":[],"parts":[{"label":"a","answer":"4 7/12","answerDisplay":"4712"},{"label":"b","answer":"3"}],"solution":["a) 134 = 74 and 256 = 176. Using denominator 12: 74 = 2112 and 176 = 3412.","a) 2112 + 3412 = 5512 = 4712.","b) 212 = 52 and 115 = 65, so 52 x 65 = 3010 = 3."],"diagram":null,"draw":false,"revision":"05db6735a9f2"},{"id":"g3-fractions-q11","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":11,"marks":4,"tier":"FH","calc":false,"text":"

Work out (12 + 25) ÷ 310

","answerType":"exact","answer":"3","answerDisplay":null,"accept":["3"],"parts":[],"solution":["First add the fractions in the bracket using denominator 10: 12 = 510 and 25 = 410, so the bracket is 910.","Dividing by 310 is the same as multiplying by 103.","910 x 103 = 9030 = 3."],"diagram":null,"draw":false,"revision":"2e4878e19cee"},{"id":"g3-fractions-q12","topicId":"g3-fractions","topic":"Fractions","grade":"3","topicGrade":"3","strand":"N","difficulty":12,"marks":5,"tier":"FH","calc":false,"text":"
  1. Work out 78 − 25
  2. Show that 123 × 214 = 334
","answerType":"multi","answer":"a) 19/40 b) 5/3 x 9/4 = 45/12 = 15/4 = 3 3/4","answerDisplay":"a) 1940 b) 53 x 94 = 4512 = 154 = 334","accept":[],"parts":[{"label":"a","answer":"19/40","answerDisplay":"1940"},{"label":"b","answer":"5/3 x 9/4 = 45/12 = 15/4 = 3 3/4","answerDisplay":"53 x 94 = 4512 = 154 = 334"}],"solution":["a) Use denominator 40: 78 = 3540 and 25 = 1640.","a) 3540 - 1640 = 1940.","b) Write as improper fractions: 123 = 53 and 214 = 94.","b) 53 x 94 = 4512 = 154 = 334, as required."],"diagram":null,"draw":false,"revision":"5f750079445b"},{"id":"g3-frequency-trees-q01","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":1,"marks":3,"tier":"F","calc":false,"text":"

There are 60 students in Year 10 and Year 11 at a small school.

24 of the students are in Year 10 and the rest are in Year 11.

Of the Year 10 students, 15 study French and 9 do not.

Of the Year 11 students, 21 study French.

  1. Work out the number of students in Year 11.
  2. Work out the number of Year 11 students who do not study French.
","answerType":"multi","answer":"a) 36; b) 15","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"36"},{"label":"b","answer":"15"}],"solution":["A frequency tree counts people, rather than showing probabilities. The two branches leaving each group must add to the number in that group.","Number in Year 11 = 60 - 24 = 36.","Year 11 students not studying French = 36 - 21 = 15.","Check: 15 + 9 = 24 and 21 + 15 = 36, so every branch adds up correctly."],"diagram":{"type":"tree","kind":"freq","root":"60","stage1":{"outcomes":["Year 10","Year 11"],"values":["24",null]},"stage2":{"outcomes":["French","Not French"],"values":[["15","9"],["21",null]]},"answer":{"stage1":["24","36"],"stage2":[["15","9"],["21","15"]]}},"draw":false,"revision":"199448130e41"},{"id":"g3-frequency-trees-q02","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":2,"marks":4,"tier":"F","calc":false,"text":"

80 adults went to a gym on Saturday.

45 of the adults went to a morning class. The rest went to an afternoon class.

Of the adults at the morning class, 30 did yoga.

Of the adults at the afternoon class, 12 did yoga.

  1. Complete the frequency tree.
  2. One of the 80 adults is chosen at random. Find the probability that this adult went to the morning class and did yoga.
","answerType":"multi","answer":"a) afternoon 35, morning not yoga 15, afternoon not yoga 23; b) 3/8","answerDisplay":"a) afternoon 35, morning not yoga 15, afternoon not yoga 23; b) 38","accept":[],"parts":[{"label":"a","answer":"afternoon = 35; morning not yoga = 15; afternoon not yoga = 23"},{"label":"b","answer":"3/8","answerDisplay":"38"}],"solution":["Afternoon class = 80 - 45 = 35.","Morning, not yoga = 45 - 30 = 15.","Afternoon, not yoga = 35 - 12 = 23.","P(morning and yoga) = 3080 = 38"],"diagram":{"type":"tree","kind":"freq","root":"80","stage1":{"outcomes":["Morning","Afternoon"],"values":["45",null]},"stage2":{"outcomes":["Yoga","Not yoga"],"values":[["30",null],["12",null]]},"answer":{"stage1":["45","35"],"stage2":[["30","15"],["12","23"]]}},"draw":false,"revision":"a2a4f7fb16d7"},{"id":"g3-frequency-trees-q03","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":3,"marks":4,"tier":"F","calc":true,"text":"

200 shoppers visited a supermarket one morning.

60% of the shoppers were female. The rest were male.

45 of the female shoppers used a loyalty card.

26 of the male shoppers used a loyalty card.

  1. Work out the number of female shoppers and the number of male shoppers.
  2. Work out the number of female shoppers who did not use a loyalty card.
  3. Work out the number of male shoppers who did not use a loyalty card.
","answerType":"multi","answer":"a) 120 female, 80 male; b) 75; c) 54","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"120 female and 80 male"},{"label":"b","answer":"75"},{"label":"c","answer":"54"}],"solution":["Female shoppers = 60% of 200 = 0.6 × 200 = 120, so male shoppers = 200 - 120 = 80.","Female with no loyalty card = 120 - 45 = 75.","Male with no loyalty card = 80 - 26 = 54."],"diagram":{"type":"tree","kind":"freq","root":"200","stage1":{"outcomes":["Female","Male"],"values":[null,null]},"stage2":{"outcomes":["Loyalty card","No card"],"values":[["45",null],["26",null]]},"answer":{"stage1":["120","80"],"stage2":[["45","75"],["26","54"]]}},"draw":false,"revision":"44cfac010a95"},{"id":"g3-frequency-trees-q04","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":4,"marks":4,"tier":"F","calc":true,"text":"

150 patients had an appointment at a health centre one day.

90 of the appointments were in the morning. The rest were in the afternoon.

81 of the morning appointments started on time.

42 of the afternoon appointments started on time.

  1. Complete the frequency tree.
  2. One of the 150 patients is chosen at random. Work out the probability that their appointment started on time.
","answerType":"multi","answer":"a) afternoon 60, morning late 9, afternoon late 18; b) 41/50","answerDisplay":"a) afternoon 60, morning late 9, afternoon late 18; b) 4150","accept":[],"parts":[{"label":"a","answer":"afternoon = 60; morning not on time = 9; afternoon not on time = 18"},{"label":"b","answer":"41/50","answerDisplay":"4150"}],"solution":["Afternoon appointments = 150 - 90 = 60.","Morning not on time = 90 - 81 = 9; afternoon not on time = 60 - 42 = 18.","Appointments on time altogether = 81 + 42 = 123.","P(on time) = 123150 = 4150 = 0.82"],"diagram":{"type":"tree","kind":"freq","root":"150","stage1":{"outcomes":["Morning","Afternoon"],"values":["90",null]},"stage2":{"outcomes":["On time","Not on time"],"values":[["81",null],["42",null]]},"answer":{"stage1":["90","60"],"stage2":[["81","9"],["42","18"]]}},"draw":false,"revision":"69a92db3d89a"},{"id":"g3-frequency-trees-q05","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":5,"marks":5,"tier":"F","calc":false,"text":"

A running club has 120 members. Each member is an adult or a junior.

70 of the members are adults. The rest are juniors.

25 of the adults ran a marathon last year.

11 of the juniors ran a marathon last year.

  1. Work out the number of juniors in the club.
  2. Work out the number of adults who ran a marathon last year.
  3. One of the members who ran a marathon last year is chosen at random. Work out the probability that this member is a junior.
","answerType":"multi","answer":"a) 50; b) 28; c) 11/39","answerDisplay":"a) 50; b) 28; c) 1139","accept":[],"parts":[{"label":"a","answer":"50"},{"label":"b","answer":"28"},{"label":"c","answer":"11/39","answerDisplay":"1139"}],"solution":["Juniors = 120 - 70 = 50.","Adults who ran a marathon = 25 of 70 = 28.","Total members who ran a marathon = 28 + 11 = 39.","P(junior) = 1139"],"diagram":{"type":"tree","kind":"freq","root":"120","stage1":{"outcomes":["Adult","Junior"],"values":["70",null]},"stage2":{"outcomes":["Marathon","No marathon"],"values":[[null,null],["11",null]]},"answer":{"stage1":["70","50"],"stage2":[["28","42"],["11","39"]]}},"draw":false,"revision":"77b566f12636"},{"id":"g3-frequency-trees-q06","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":6,"marks":5,"tier":"F","calc":true,"text":"

A gardener planted 240 seeds.

150 of the seeds were planted in a greenhouse. The rest were planted outdoors.

80% of the greenhouse seeds germinated.

63 of the outdoor seeds germinated.

  1. Complete the frequency tree.
  2. What percentage of the outdoor seeds germinated?
","answerType":"multi","answer":"a) outdoors 90, greenhouse germinated 120, greenhouse not 30, outdoors not 27; b) 70%","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"outdoors = 90; greenhouse germinated = 120; greenhouse not germinated = 30; outdoors not germinated = 27"},{"label":"b","answer":"70%"}],"solution":["Outdoor seeds = 240 - 150 = 90.","Greenhouse seeds that germinated = 80% of 150 = 120, so 150 - 120 = 30 did not.","Outdoor seeds that did not germinate = 90 - 63 = 27.","Percentage of outdoor seeds that germinated = 6390 × 100 = 70%"],"diagram":{"type":"tree","kind":"freq","root":"240","stage1":{"outcomes":["Greenhouse","Outdoors"],"values":["150",null]},"stage2":{"outcomes":["Germinated","Did not"],"values":[[null,null],["63",null]]},"answer":{"stage1":["150","90"],"stage2":[["120","30"],["63","27"]]}},"draw":false,"revision":"5aefd2e72b74"},{"id":"g3-frequency-trees-q07","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":7,"marks":5,"tier":"F","calc":true,"text":"

300 passengers took a flight.

40% of the passengers were travelling for business. The rest were travelling for leisure.

34 of the business passengers checked in a bag.

117 of the leisure passengers checked in a bag.

  1. Complete the frequency tree.
  2. One of the 300 passengers is chosen at random. Work out the probability that this passenger was travelling for leisure and did not check in a bag.
","answerType":"multi","answer":"a) business 120, leisure 180, business bag 90, business no bag 30, leisure no bag 63; b) 21/100","answerDisplay":"a) business 120, leisure 180, business bag 90, business no bag 30, leisure no bag 63; b) 21100","accept":[],"parts":[{"label":"a","answer":"business = 120; leisure = 180; business with bag = 90; business no bag = 30; leisure no bag = 63"},{"label":"b","answer":"21/100","answerDisplay":"21100"}],"solution":["Business passengers = 40% of 300 = 120, so leisure passengers = 300 - 120 = 180.","Business passengers with a bag = 34 of 120 = 90, so 120 - 90 = 30 had no bag.","Leisure passengers with no bag = 180 - 117 = 63.","P(leisure and no bag) = 63300 = 21100 = 0.21"],"diagram":{"type":"tree","kind":"freq","root":"300","stage1":{"outcomes":["Business","Leisure"],"values":[null,null]},"stage2":{"outcomes":["Bag","No bag"],"values":[[null,null],["117",null]]},"answer":{"stage1":["120","180"],"stage2":[["90","30"],["117","63"]]}},"draw":false,"revision":"b72506e3e7cc"},{"id":"g3-frequency-trees-q08","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":8,"marks":5,"tier":"F","calc":false,"text":"

Every student at a sports academy is a boy or a girl.

45 of the boys attended training on Monday and 15 of the boys did not.

There are 40 girls. 28 of the girls attended training on Monday.

  1. Work out the total number of students at the academy.
  2. One student is chosen at random. Work out the probability that this student attended training on Monday.
","answerType":"multi","answer":"a) 100; b) 73/100","answerDisplay":"a) 100; b) 73100","accept":[],"parts":[{"label":"a","answer":"100"},{"label":"b","answer":"73/100","answerDisplay":"73100"}],"solution":["Number of boys = 45 + 15 = 60.","Total students = 60 + 40 = 100.","Students who attended training = 45 + 28 = 73.","P(attended) = 73100"],"diagram":{"type":"tree","kind":"freq","root":null,"stage1":{"outcomes":["Boys","Girls"],"values":[null,"40"]},"stage2":{"outcomes":["Attended","Did not"],"values":[["45","15"],["28",null]]},"answer":{"stage1":["60","40"],"stage2":[["45","15"],["28","12"]],"root":"100"}},"draw":false,"revision":"72cdd09c2bb0"},{"id":"g3-frequency-trees-q09","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":9,"marks":5,"tier":"F","calc":true,"text":"

160 cars were given a safety test at a garage.

35 of the cars were petrol cars. The rest were diesel cars.

84 of the petrol cars passed the test. The rest of the petrol cars failed.

16 of the diesel cars failed the test. The rest of the diesel cars passed.

  1. Work out the number of diesel cars tested.
  2. Work out the number of petrol cars that failed the test.
  3. What percentage of the 160 cars failed the test?
","answerType":"multi","answer":"a) 64; b) 12; c) 17.5%","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"64"},{"label":"b","answer":"12"},{"label":"c","answer":"17.5%"}],"solution":["Petrol cars = 35 of 160 = 96, so diesel cars = 160 - 96 = 64.","Petrol cars that failed = 96 - 84 = 12.","Total failures = 12 + 16 = 28.","Percentage failed = 28160 × 100 = 17.5%"],"diagram":{"type":"tree","kind":"freq","root":"160","stage1":{"outcomes":["Petrol","Diesel"],"values":[null,null]},"stage2":{"outcomes":["Passed","Failed"],"values":[["84",null],[null,"16"]]},"answer":{"stage1":["96","64"],"stage2":[["84","12"],["48","16"]]}},"draw":false,"revision":"18884f30500f"},{"id":"g3-frequency-trees-q10","topicId":"g3-frequency-trees","topic":"Frequency Trees","grade":"3","topicGrade":"3","strand":"P","difficulty":10,"marks":5,"tier":"F","calc":true,"text":"

180 people were in the audience at a theatre show.

55% of the audience were adults. The rest were children.

23 of the adults bought a programme.

27 of the children bought a programme.

Programmes cost £4 each.

  1. Work out the number of adults and the number of children in the audience.
  2. Work out the number of adults who bought a programme.
  3. Work out the total amount of money the theatre received from programme sales.
","answerType":"multi","answer":"a) 99 adults, 81 children; b) 66; c) £372","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"99 adults and 81 children"},{"label":"b","answer":"66"},{"label":"c","answer":"£372"}],"solution":["Adults = 55% of 180 = 0.55 × 180 = 99, so children = 180 - 99 = 81.","Adults who bought a programme = 23 of 99 = 66.","Total programmes sold = 66 + 27 = 93.","Money received = 93 × £4 = £372"],"diagram":{"type":"tree","kind":"freq","root":"180","stage1":{"outcomes":["Adults","Children"],"values":[null,null]},"stage2":{"outcomes":["Programme","No programme"],"values":[[null,null],["27",null]]},"answer":{"stage1":["99","81"],"stage2":[["66","33"],["27","54"]]}},"draw":false,"revision":"af207ed0d486"},{"id":"g3-percentage-change-q01","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":1,"marks":1,"tier":"H","calc":true,"text":"

The value of a van is multiplied by 0.88 each year.

Write down the percentage decrease in the value of the van each year.

","answerType":"exact","answer":"12%","answerDisplay":null,"accept":["12"],"parts":[],"solution":["A multiplier of 0.88 means the van keeps 88% of its value.","100% − 88% = 12% decrease each year."],"diagram":null,"draw":false,"revision":"bbf3b9ebc03d"},{"id":"g3-percentage-change-q02","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

Last year a rail season ticket cost £640.

This year the same season ticket costs £688.

Work out the percentage increase in the cost of the season ticket.

","answerType":"exact","answer":"7.5%","answerDisplay":null,"accept":["7.5"],"parts":[],"solution":["Increase = 688 − 640 = £48.","Compare the increase with the original £640, which represents 100%. Using the new price as the denominator would answer a different question.","Percentage increase = 48 ÷ 640 = 0.075.","0.075 × 100 = 7.5%."],"diagram":null,"draw":false,"revision":"3115e084b7dc"},{"id":"g3-percentage-change-q03","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

Kane buys a bike for £250.

A year later he sells the bike for £210.

Work out his percentage loss.

","answerType":"exact","answer":"16%","answerDisplay":null,"accept":["16"],"parts":[],"solution":["Loss = 250 − 210 = £40.","Percentage loss = 40 ÷ 250 = 0.16.","0.16 × 100 = 16%."],"diagram":null,"draw":false,"revision":"67a784df1278"},{"id":"g3-percentage-change-q04","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":4,"marks":3,"tier":"FH","calc":false,"text":"

A trader buys a table for £68.

She sells the table for £85.

Work out her percentage profit.

","answerType":"exact","answer":"25%","answerDisplay":null,"accept":["25"],"parts":[],"solution":["Profit = 85 − 68 = £17.","Percentage profit = 17 ÷ 68 = 0.25.","0.25 × 100 = 25%."],"diagram":null,"draw":false,"revision":"97cf0d0fd68d"},{"id":"g3-percentage-change-q05","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":5,"marks":3,"tier":"FH","calc":true,"text":"

In 2019 Femi bought a flat for £180 000.

In 2026 he sold the flat for £207 000.

Work out his percentage profit.

","answerType":"exact","answer":"15%","answerDisplay":null,"accept":["15"],"parts":[],"solution":["Profit = 207 000 − 180 000 = £27 000.","Percentage profit = 27 000 ÷ 180 000 = 0.15.","0.15 × 100 = 15%."],"diagram":null,"draw":false,"revision":"8d9d49d09034"},{"id":"g3-percentage-change-q06","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":6,"marks":3,"tier":"H","calc":false,"text":"

A shop buys phone cases for £3.20 each.

The shop sells each phone case for £4.40.

Show that the shop's percentage profit on each phone case is 37.5%.

","answerType":"open","answer":"Profit = £1.20; 1.20 / 3.20 = 0.375 = 37.5%","answerDisplay":"Profit = £1.20; 1.203.20 = 0.375 = 37.5%","accept":[],"parts":[],"solution":["Profit on each case = 4.40 − 3.20 = £1.20.","Percentage profit = 1.20 ÷ 3.20 = 0.375.","0.375 × 100 = 37.5%, as required."],"diagram":null,"draw":false,"revision":"c5b3f666ca27"},{"id":"g3-percentage-change-q07","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

Gina buys 40 plants for £1.80 each.

She sells 32 of the plants for £3 each.

She sells the rest of the plants for £1.50 each.

Work out Gina's percentage profit.

","answerType":"exact","answer":"50%","answerDisplay":null,"accept":["50"],"parts":[],"solution":["Total cost = 40 × £1.80 = £72.","Sales: 32 × £3 = £96, and the remaining 8 plants: 8 × £1.50 = £12.","Total sales = 96 + 12 = £108.","Profit = 108 − 72 = £36.","Percentage profit = 36 ÷ 72 = 0.5 = 50%."],"diagram":null,"draw":false,"revision":"b74b1dbfff03"},{"id":"g3-percentage-change-q08","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":8,"marks":4,"tier":"FH","calc":true,"text":"

A grocer buys a 25 kg sack of potatoes for £15.

He puts all the potatoes into 2.5 kg bags.

He sells each bag for £2.10.

He sells all the bags.

Work out his percentage profit.

","answerType":"exact","answer":"40%","answerDisplay":null,"accept":["40"],"parts":[],"solution":["Number of bags = 25 ÷ 2.5 = 10.","Total sales = 10 × £2.10 = £21.","Profit = 21 − 15 = £6.","Percentage profit = 6 ÷ 15 = 0.4 = 40%."],"diagram":null,"draw":false,"revision":"9194fca6f29f"},{"id":"g3-percentage-change-q09","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":9,"marks":4,"tier":"FH","calc":true,"text":"

Ben's monthly electricity bill was £96. It went up by 15%.

Ben's monthly broadband bill was £30. It went up by 40%.

Ben says, "My broadband bill went up by more money than my electricity bill."

Is Ben correct? You must show your working.

","answerType":"open","answer":"No - electricity went up by £14.40 but broadband only went up by £12","answerDisplay":null,"accept":[],"parts":[],"solution":["Electricity increase = 15% of £96 = 0.15 × 96 = £14.40.","Broadband increase = 40% of £30 = 0.4 × 30 = £12.","£14.40 > £12, so Ben is not correct."],"diagram":null,"draw":false,"revision":"16a55170d287"},{"id":"g3-percentage-change-q10","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":10,"marks":4,"tier":"H","calc":false,"text":"

The membership of running club A increased from 1500 to 1725.

The membership of running club B increased from 2400 to 2700.

Cam says, "Club B had the greater percentage increase because its membership went up by more people."

Show that Cam is wrong.

","answerType":"open","answer":"Club A: 225/1500 = 15%; club B: 300/2400 = 12.5%; 15% > 12.5% so club A had the greater percentage increase","answerDisplay":"Club A: 2251500 = 15%; club B: 3002400 = 12.5%; 15% > 12.5% so club A had the greater percentage increase","accept":[],"parts":[],"solution":["Club A increase = 1725 − 1500 = 225, and 225 ÷ 1500 = 0.15 = 15%.","Club B increase = 2700 − 2400 = 300, and 300 ÷ 2400 = 0.125 = 12.5%.","15% > 12.5%, so club A had the greater percentage increase and Cam is wrong."],"diagram":null,"draw":false,"revision":"129f182a2c25"},{"id":"g3-percentage-change-q11","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":11,"marks":5,"tier":"FH","calc":true,"text":"

A florist buys 60 roses for a total of £45.

She sells all 60 roses at the same price each.

She wants to make a profit of at least 40%.

Work out the least possible price for each rose.

","answerType":"exact","answer":"£1.05","answerDisplay":null,"accept":["1.05","105p"],"parts":[],"solution":["For a 40% profit the total sales must be at least 45 × 1.4 = £63.","Price per rose = 63 ÷ 60 = £1.05.","So the least possible price is £1.05 per rose."],"diagram":null,"draw":false,"revision":"83a52ae8c6b7"},{"id":"g3-percentage-change-q12","topicId":"g3-percentage-change","topic":"Percentage Change","grade":"3","topicGrade":"3","strand":"R","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

Rob and Sita buy a boat for £6400.

They pay for the boat in the ratio 5 : 3.

Later they sell the boat for £8800.

They share the money from the sale in the ratio 7 : 4, with Rob taking the larger share.

Work out Rob's percentage profit on the money he paid.

","answerType":"exact","answer":"40%","answerDisplay":null,"accept":["40"],"parts":[],"solution":["Buying: 5 + 3 = 8 parts, so 1 part = 6400 ÷ 8 = £800. Rob paid 5 × 800 = £4000.","Selling: 7 + 4 = 11 parts, so 1 part = 8800 ÷ 11 = £800. Rob receives 7 × 800 = £5600.","Rob's profit = 5600 − 4000 = £1600.","Percentage profit = 1600 ÷ 4000 = 0.4 = 40%."],"diagram":null,"draw":false,"revision":"973df86a28a2"},{"id":"g3-percentages-q01","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write down 50% of £86

","answerType":"exact","answer":"£43","answerDisplay":null,"accept":["43","£43","43.00"],"parts":[],"solution":["50% is one half.","Half of 86 is 43, so the answer is 43 pounds."],"diagram":null,"draw":false,"revision":"88957b1b6873"},{"id":"g3-percentages-q02","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

Work out 15% of £60

","answerType":"exact","answer":"£9","answerDisplay":null,"accept":["9","£9","9.00"],"parts":[],"solution":["10% of 60 is 6.","5% is half of 10%, so 5% of 60 is 3.","15% = 10% + 5% = 6 + 3 = 9 pounds."],"diagram":null,"draw":false,"revision":"293455610f2a"},{"id":"g3-percentages-q03","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

A bottle contains 850 ml of juice.

34% of the juice is orange juice.

Work out how many millilitres of orange juice are in the bottle.

","answerType":"exact","answer":"289 ml","answerDisplay":null,"accept":["289","289 ml"],"parts":[],"solution":["Percent means “out of 100”, so 34% = 34/100 = 0.34. Multiplying by 0.34 finds 34 hundredths of the whole amount.","34% as a decimal is 0.34.","0.34 x 850 = 289, so there are 289 ml of orange juice."],"diagram":null,"draw":false,"revision":"e69b81a49948"},{"id":"g3-percentages-q04","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

Hannah puts £2400 into a savings account.

The account pays 3% simple interest each year.

Work out the total amount of interest Hannah gets after 3 years.

","answerType":"exact","answer":"£216","answerDisplay":null,"accept":["216","£216","216.00"],"parts":[],"solution":["Interest for one year: 3% of 2400 = 0.03 x 2400 = 72 pounds.","Simple interest is the same each year, so over 3 years it is 3 x 72 = 216 pounds."],"diagram":null,"draw":false,"revision":"94cf7ffe7f96"},{"id":"g3-percentages-q05","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

Work out 30% of 60% of £250

","answerType":"exact","answer":"£45","answerDisplay":null,"accept":["45","£45","45.00"],"parts":[],"solution":["60% of 250 = 0.6 x 250 = 150.","30% of 150 = 0.3 x 150 = 45, so the answer is 45 pounds."],"diagram":null,"draw":false,"revision":"eb83d334ea4a"},{"id":"g3-percentages-q06","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

Ben's rent is £240 each week.

His rent is increased by 35%.

Work out Ben's new weekly rent.

","answerType":"exact","answer":"£324","answerDisplay":null,"accept":["324","£324","324.00"],"parts":[],"solution":["10% of 240 is 24, so 30% is 3 x 24 = 72.","5% is half of 10%, so 5% of 240 is 12.","The increase is 72 + 12 = 84 pounds, so the new rent is 240 + 84 = 324 pounds."],"diagram":null,"draw":false,"revision":"8ddb5f3219b6"},{"id":"g3-percentages-q07","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

62% of the members of a swimming club are adults.

The rest of the members are children.

There are 133 children in the club.

Work out the total number of members of the club.

","answerType":"exact","answer":"350","answerDisplay":null,"accept":["350","350 members"],"parts":[],"solution":["The children make up 100% - 62% = 38% of the members.","So 38% of the total is 133.","Total = 133 / 0.38 = 350 members."],"diagram":null,"draw":false,"revision":"99ae7540c611"},{"id":"g3-percentages-q08","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

Priya invests £1500 in a savings account for 4 years.

The account pays simple interest at a rate of R% per year.

At the end of 4 years, Priya has received a total of £180 in interest.

Work out the value of R.

","answerType":"exact","answer":"3","answerDisplay":null,"accept":["3","3%","R = 3"],"parts":[],"solution":["Interest received each year: 180 / 4 = 45 pounds.","45 as a percentage of 1500 is 45 / 1500 = 0.03.","0.03 = 3%, so R = 3."],"diagram":null,"draw":false,"revision":"deb66da736f6"},{"id":"g3-percentages-q09","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":9,"marks":3,"tier":"F","calc":true,"text":"

Ravi is choosing between two vouchers for the same online shop.

Voucher A gives 24% off a £75 order.

Voucher B gives 32% off a £55 order.

Which voucher saves more money?

You must show your working.

","answerType":"open","answer":"Voucher A. It saves 24% of £75 = £18, while Voucher B saves 32% of £55 = £17.60, so Voucher A saves £0.40 more.","answerDisplay":null,"accept":[],"parts":[],"solution":["Voucher A saving: 0.24 x 75 = 18 pounds.","Voucher B saving: 0.32 x 55 = 17.60 pounds.","18 is more than 17.60, so Voucher A saves more money (by 40p)."],"diagram":null,"draw":false,"revision":"6ff8e483a799"},{"id":"g3-percentages-q10","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

Kirsty buys a sofa costing £840

She pays a deposit of 35% of the cost.

She pays the rest of the cost in 12 equal monthly payments.

Work out the amount of each monthly payment.

","answerType":"exact","answer":"£45.50","answerDisplay":null,"accept":["45.50","£45.50","45.5"],"parts":[],"solution":["Deposit: 35% of 840 = 0.35 x 840 = 294 pounds.","Amount left to pay: 840 - 294 = 546 pounds.","Each monthly payment: 546 / 12 = 45.50 pounds."],"diagram":null,"draw":false,"revision":"1ec467cf2081"},{"id":"g3-percentages-q11","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":11,"marks":4,"tier":"F","calc":true,"text":"

Jack is given £480

He spends 38 of the money on rent.

He then spends 40% of the money he has left on food.

Work out how much money Jack has left after paying for rent and food.

","answerType":"exact","answer":"£180","answerDisplay":null,"accept":["180","£180","180.00"],"parts":[],"solution":["Rent: 38 of 480 = 3 x 60 = 180 pounds, leaving 480 - 180 = 300 pounds.","Food: 40% of 300 = 0.4 x 300 = 120 pounds.","Money left: 300 - 120 = 180 pounds."],"diagram":null,"draw":false,"revision":"c802573bf687"},{"id":"g3-percentages-q12","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":12,"marks":5,"tier":"F","calc":true,"text":"

A cinema has 350 seats.

One evening, 92% of the seats are filled.

Each person pays £8.50 for a ticket.

The cinema has to give 20% of the ticket money to the film distributor.

Work out how much of the ticket money the cinema keeps for that evening.

","answerType":"exact","answer":"£2189.60","answerDisplay":null,"accept":["2189.60","£2189.60","2,189.60"],"parts":[],"solution":["Number of people: 92% of 350 = 0.92 x 350 = 322.","Ticket money: 322 x 8.50 = 2737 pounds.","The distributor takes 20%, so the cinema keeps 80%.","0.8 x 2737 = 2189.60 pounds."],"diagram":null,"draw":false,"revision":"a0e988f74885"},{"id":"g3-percentages-q13","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":13,"marks":5,"tier":"F","calc":true,"text":"

A company shares a bonus of £2400 between three workers.

Aisha receives £912

Ben receives £768

Cara receives the rest of the bonus.

Cara says

"I received more than 30% of the bonus."

Is Cara correct?

You must show how you get your answer.

","answerType":"open","answer":"No. Cara receives 2400 - 912 - 768 = £720, and 720/2400 = 30% exactly, so she received exactly 30%, not more than 30%.","answerDisplay":"No. Cara receives 2400 - 912 - 768 = £720, and 7202400 = 30% exactly, so she received exactly 30%, not more than 30%.","accept":[],"parts":[],"solution":["Aisha and Ben together receive 912 + 768 = 1680 pounds.","Cara receives 2400 - 1680 = 720 pounds.","720 / 2400 = 0.3 = 30%.","30% is not more than 30%, so Cara is not correct."],"diagram":null,"draw":false,"revision":"9a7f505a5915"},{"id":"g3-percentages-q14","topicId":"g3-percentages","topic":"Percentages","grade":"3","topicGrade":"3","strand":"N","difficulty":14,"marks":6,"tier":"F","calc":true,"text":"

A shop's takings in 2025 were 15% less than its takings in 2024

The shop's takings in 2025 were £61,200

  1. Work out the shop's takings in 2024
  2. In 2024, the manager was paid a bonus of 2.5% of the 2024 takings.

    Work out the manager's bonus for 2024

","answerType":"multi","answer":"a) £72,000 b) £1,800","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£72,000"},{"label":"b","answer":"£1,800"}],"solution":["a) A 15% decrease means the 2025 takings are 85% of the 2024 takings.","a) 2024 takings = 61,200 / 0.85 = 72,000 pounds.","b) Bonus = 2.5% of 72,000 = 0.025 x 72,000 = 1,800 pounds."],"diagram":null,"draw":false,"revision":"f780bc2f7bfa"},{"id":"g3-proportion-q01","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

A recipe for 4 people uses 300 g of pasta.

Work out how much pasta is needed for 10 people.

","answerType":"exact","answer":"750 g","answerDisplay":null,"accept":["750","0.75 kg"],"parts":[],"solution":["Pasta for 1 person = 300 ÷ 4 = 75 g.","Pasta for 10 people = 75 × 10 = 750 g."],"diagram":null,"draw":false,"revision":"8c072f094eec"},{"id":"g3-proportion-q02","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Rosie asked 60 students at her school to name their favourite sport.

24 of the students said football.

There are 900 students at the school.

Work out an estimate for the number of students at the school whose favourite sport is football.

","answerType":"exact","answer":"360","answerDisplay":null,"accept":["360 students"],"parts":[],"solution":["Fraction saying football = 2460 = 25.","Estimate = (25) × 900 = 360 students.","This assumes the sample represents the school’s preferences. The result is an estimate, not a guaranteed total."],"diagram":null,"draw":false,"revision":"d1c34b9302ad"},{"id":"g3-proportion-q03","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

3 kg of potatoes cost £2.64.

Work out the cost of 7 kg of these potatoes.

","answerType":"exact","answer":"£6.16","answerDisplay":null,"accept":["6.16"],"parts":[],"solution":["1 kg costs 2.64 ÷ 3 = £0.88.","7 kg costs 7 × £0.88 = £6.16."],"diagram":null,"draw":false,"revision":"2afa3f222e26"},{"id":"g3-proportion-q04","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

There are 380 calories in 100 g of a breakfast cereal.

There are 64 calories in 100 ml of milk.

Jess has a bowl of 45 g of the cereal with 125 ml of the milk.

Work out the total number of calories in Jess's bowl.

","answerType":"exact","answer":"251","answerDisplay":null,"accept":["251 calories"],"parts":[],"solution":["Calories in the cereal = 380 × 0.45 = 171.","Calories in the milk = 64 × 1.25 = 80.","Total = 171 + 80 = 251 calories."],"diagram":null,"draw":false,"revision":"1a59a315cf06"},{"id":"g3-proportion-q05","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

Here is a recipe for 12 biscuits.

12 biscuits: 180 g butter, 240 g flour, 100 g sugar

Meg wants to make 30 biscuits.

She has 425 g of butter, plenty of flour and plenty of sugar.

Does Meg have enough butter to make 30 biscuits? You must show your working.

","answerType":"open","answer":"No - 30 biscuits need 450 g of butter and she only has 425 g, so she is 25 g short","answerDisplay":null,"accept":[],"parts":[],"solution":["30 ÷ 12 = 2.5, so multiply the recipe by 2.5.","Butter needed = 180 × 2.5 = 450 g.","425 g < 450 g, so Meg does not have enough butter. She is 25 g short."],"diagram":null,"draw":false,"revision":"8f20712861df"},{"id":"g3-proportion-q06","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

A car park charges 6p for each minute of parking.

Dan drives into the car park at 09:40.

When he leaves, he pays £3.90.

Work out the time Dan leaves the car park.

","answerType":"exact","answer":"10:45","answerDisplay":null,"accept":["10 45","10.45","10:45 am"],"parts":[],"solution":["£3.90 = 390p.","Minutes parked = 390 ÷ 6 = 65 minutes.","09:40 + 65 minutes = 09:40 + 1 hour 5 minutes = 10:45."],"diagram":null,"draw":false,"revision":"ad4aa305109c"},{"id":"g3-proportion-q07","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":7,"marks":3,"tier":"F","calc":false,"text":"

It takes 4 identical pumps 9 hours to empty a tank.

Work out how long it would take 6 of these pumps to empty the same tank.

","answerType":"exact","answer":"6 hours","answerDisplay":null,"accept":["6","6 hrs"],"parts":[],"solution":["Total work = 4 × 9 = 36 pump-hours.","With 6 pumps: 36 ÷ 6 = 6 hours.","More pumps means less time, so 6 hours makes sense."],"diagram":null,"draw":false,"revision":"c174b89737ed"},{"id":"g3-proportion-q08","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

Machine A makes 45 boxes each hour.

Machine B makes 64 boxes each hour.

On Monday, machine A runs for 4 hours and machine B runs for 212 hours.

Work out the total number of boxes made on Monday.

","answerType":"exact","answer":"340","answerDisplay":null,"accept":["340 boxes"],"parts":[],"solution":["Machine A: 45 × 4 = 180 boxes.","Machine B: 64 × 2.5 = 160 boxes.","Total = 180 + 160 = 340 boxes."],"diagram":null,"draw":false,"revision":"8aaa830b5c06"},{"id":"g3-proportion-q09","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

A recipe for 8 flapjacks uses 240 g of oats.

Noah wants to make 30 flapjacks.

He has 850 g of oats.

Work out how many more grams of oats Noah needs.

","answerType":"exact","answer":"50 g","answerDisplay":null,"accept":["50"],"parts":[],"solution":["Oats for 1 flapjack = 240 ÷ 8 = 30 g.","Oats for 30 flapjacks = 30 × 30 = 900 g.","Shortfall = 900 − 850 = 50 g."],"diagram":null,"draw":false,"revision":"90cba2351bce"},{"id":"g3-proportion-q10","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":10,"marks":5,"tier":"F","calc":true,"text":"

Grass seed needs to be spread at 35 g per square metre.

Harry's rectangular lawn is 12 m long and 8 m wide.

Grass seed is sold in 1 kg boxes.

Each box costs £6.99.

Work out the total cost of the boxes of grass seed Harry needs to buy.

","answerType":"exact","answer":"£27.96","answerDisplay":null,"accept":["27.96"],"parts":[],"solution":["Area of lawn = 12 × 8 = 96 m².","Seed needed = 96 × 35 = 3360 g = 3.36 kg.","3.36 kg needs 4 boxes (3 boxes = 3 kg is not enough).","Cost = 4 × £6.99 = £27.96."],"diagram":null,"draw":false,"revision":"413c1cfb6b88"},{"id":"g3-proportion-q11","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":11,"marks":5,"tier":"FH","calc":false,"text":"

Here is a recipe for 10 scones.

10 scones: 350 g flour, 80 g butter, 150 ml milk

Tariq has 1.4 kg of flour, 350 g of butter and 700 ml of milk.

Work out the greatest number of scones Tariq can make.

","answerType":"exact","answer":"40","answerDisplay":null,"accept":["40 scones"],"parts":[],"solution":["Flour: 1.4 kg = 1400 g, and 1400 ÷ 350 = 4 batches.","Butter: 350 ÷ 80 = 4.375 batches.","Milk: 700 ÷ 150 = 4.66... batches.","The flour runs out first, allowing exactly 4 batches.","Greatest number of scones = 4 × 10 = 40."],"diagram":null,"draw":false,"revision":"8a1ed8f8f24c"},{"id":"g3-proportion-q12","topicId":"g3-proportion","topic":"Proportion","grade":"3","topicGrade":"3","strand":"R","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

A recipe for 24 brownies uses 300 g of dark chocolate.

A bakery needs to make 180 brownies.

Dark chocolate is sold in 200 g bars.

Each bar costs £1.35.

Work out the total cost of the bars of chocolate the bakery needs to buy.

","answerType":"exact","answer":"£16.20","answerDisplay":null,"accept":["16.20","16.2"],"parts":[],"solution":["180 ÷ 24 = 7.5, so multiply the recipe by 7.5.","Chocolate needed = 300 × 7.5 = 2250 g.","Bars needed = 2250 ÷ 200 = 11.25, so 12 bars must be bought.","Cost = 12 × £1.35 = £16.20."],"diagram":null,"draw":false,"revision":"f7bf489b8668"},{"id":"g3-ratio-q01","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Kemi and Layla share some money in the ratio 3 : 7.

Kemi says, "I get 37 of the money."

Kemi is wrong. Explain why.

","answerType":"open","answer":"There are 3 + 7 = 10 parts in total, so Kemi gets 3/10 of the money, not 3/7","answerDisplay":"There are 3 + 7 = 10 parts in total, so Kemi gets 310 of the money, not 37","accept":[],"parts":[],"solution":["The total number of parts is 3 + 7 = 10.","Kemi gets 3 parts out of 10, which is 310 of the money.","37 compares her share with Layla's share, not with the total."],"diagram":null,"draw":false,"revision":"9c0a04bee086"},{"id":"g3-ratio-q02","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

In a gym class, the ratio of the number of men to the number of women is 4 : 5.

There are 20 men in the class.

Work out the total number of people in the class.

","answerType":"exact","answer":"45","answerDisplay":null,"accept":["45 people"],"parts":[],"solution":["4 parts represent 20 men, so 1 part = 20 ÷ 4 = 5 people.","Women = 5 parts = 5 × 5 = 25.","Total = 20 + 25 = 45 people."],"diagram":null,"draw":false,"revision":"5a090154ef48"},{"id":"g3-ratio-q03","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":3,"marks":3,"tier":"FH","calc":false,"text":"

Amir, Bella and Carl share £132 in the ratio 1 : 4 : 6.

Work out how much money each of them gets.

","answerType":"multi","answer":"Amir £12, Bella £48, Carl £72","answerDisplay":null,"accept":[],"parts":[{"label":"Amir","answer":"£12"},{"label":"Bella","answer":"£48"},{"label":"Carl","answer":"£72"}],"solution":["Total parts = 1 + 4 + 6 = 11.","1 part = £132 ÷ 11 = £12.","Amir gets 1 × £12 = £12, Bella gets 4 × £12 = £48, Carl gets 6 × £12 = £72.","Check: 12 + 48 + 72 = 132."],"diagram":null,"draw":false,"revision":"430ee2d87196"},{"id":"g3-ratio-q04","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":4,"marks":3,"tier":"FH","calc":true,"text":"

Each month Tom is paid £1800.

He saves 20% of the £1800.

He splits the rest between rent and bills in the ratio 5 : 3.

Work out how much Tom spends on rent each month.

","answerType":"exact","answer":"£900","answerDisplay":null,"accept":["900","900.00","£900.00"],"parts":[],"solution":["Savings = 20% of £1800 = £360, leaving 1800 − 360 = £1440.","Total parts = 5 + 3 = 8, so 1 part = 1440 ÷ 8 = £180.","Rent = 5 × £180 = £900."],"diagram":null,"draw":false,"revision":"78f0a2cd024d"},{"id":"g3-ratio-q05","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":5,"marks":3,"tier":"FH","calc":true,"text":"

In a fruit bowl there are apples, bananas and pears.

The ratio of apples to bananas is 3 : 2.

The ratio of bananas to pears is 4 : 3.

There are 24 apples in the bowl.

Work out the number of pears in the bowl.

","answerType":"exact","answer":"12","answerDisplay":null,"accept":["12 pears"],"parts":[],"solution":["Make the banana parts match: apples : bananas = 3 : 2 = 6 : 4.","So apples : bananas : pears = 6 : 4 : 3.","6 parts = 24 apples, so 1 part = 4.","Pears = 3 parts = 3 × 4 = 12."],"diagram":null,"draw":false,"revision":"46256ec5889b"},{"id":"g3-ratio-q06","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":6,"marks":4,"tier":"FH","calc":true,"text":"

A shop orders 240 candles.

The ratio of large candles to small candles ordered is 3 : 5.

Each large candle costs £4.20.

Each small candle costs £2.50.

Work out the total cost of the order.

","answerType":"exact","answer":"£753","answerDisplay":null,"accept":["753","£753.00","753.00"],"parts":[],"solution":["Total parts = 3 + 5 = 8, so 1 part = 240 ÷ 8 = 30 candles.","Large candles = 3 × 30 = 90. Small candles = 5 × 30 = 150.","Cost of large candles = 90 × £4.20 = £378.","Cost of small candles = 150 × £2.50 = £375.","Total cost = 378 + 375 = £753."],"diagram":null,"draw":false,"revision":"c761da21aebc"},{"id":"g3-ratio-q07","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

Ella, Fred and Grace share some money in the ratio 3 : 5 : 8.

Grace gets £35 more than Ella.

Work out the total amount of money they share.

","answerType":"exact","answer":"£112","answerDisplay":null,"accept":["112","£112.00","112.00"],"parts":[],"solution":["The difference between Grace's parts and Ella's parts is 8 − 3 = 5 parts.","5 parts = £35, so 1 part = £7.","Total parts = 3 + 5 + 8 = 16.","Total = 16 × £7 = £112."],"diagram":null,"draw":false,"revision":"24b7e8877aa9"},{"id":"g3-ratio-q08","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":8,"marks":4,"tier":"FH","calc":false,"text":"

The ratio of the amount of money Priya has to the amount of money Quinn has is 7 : 5.

Priya gives Quinn £18.

Now they each have the same amount of money.

Work out how much money Priya had originally.

","answerType":"exact","answer":"£126","answerDisplay":null,"accept":["126","£126.00","126.00"],"parts":[],"solution":["Let Priya have 7x and Quinn have 5x.","After the gift: 7x − 18 = 5x + 18.","2x = 36, so x = 18.","Priya originally had 7 × 18 = £126.","Check: 126 − 18 = 108 and 90 + 18 = 108, which are equal."],"diagram":null,"draw":false,"revision":"e2b7cde8a81f"},{"id":"g3-ratio-q09","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":9,"marks":4,"tier":"FH","calc":true,"text":"

A school buys pens in packs of 12 and pencils in packs of 20.

The ratio of the number of packs of pens to the number of packs of pencils bought is 5 : 2.

Show that the ratio of the number of pens to the number of pencils bought is 3 : 2.

","answerType":"open","answer":"For every 5 packs of pens and 2 packs of pencils there are 60 pens and 40 pencils; 60 : 40 = 3 : 2","answerDisplay":null,"accept":[],"parts":[],"solution":["Take 5 packs of pens and 2 packs of pencils.","Pens = 5 × 12 = 60. Pencils = 2 × 20 = 40.","Pens : pencils = 60 : 40.","Divide both parts by 20 to get 3 : 2, as required."],"diagram":null,"draw":false,"revision":"2d5cc40981be"},{"id":"g3-ratio-q10","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":10,"marks":4,"tier":"FH","calc":false,"text":"

A is the point with coordinates (1, 3).

B is the point with coordinates (13, 11).

The point P lies on the line segment AB so that AP : PB = 3 : 1.

Work out the coordinates of P.

","answerType":"exact","answer":"(10, 9)","answerDisplay":null,"accept":["10, 9","(10,9)"],"parts":[],"solution":["From A to B the x-coordinate increases by 13 − 1 = 12 and the y-coordinate increases by 11 − 3 = 8.","AP : PB = 3 : 1 means P is 34 of the way from A to B.","x-coordinate of P = 1 + (34) × 12 = 1 + 9 = 10.","y-coordinate of P = 3 + (34) × 8 = 3 + 6 = 9.","P is (10, 9)."],"diagram":{"type":"axes","square":true,"x":{"min":0,"max":14,"step":1},"y":{"min":0,"max":12,"step":1},"points":[{"at":[1,3],"label":"A","labelPos":"below-right"},{"at":[13,11],"label":"B"},{"at":[10,9],"label":"P","labelPos":"below-right","answer":true}],"lines":[{"through":[[1,3],[13,11]],"segment":true}]},"draw":false,"revision":"1f7242a89721"},{"id":"g3-ratio-q11","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":11,"marks":4,"tier":"H","calc":false,"text":"

In a box, the ratio of the number of red counters to the number of blue counters is 4 : 7.

12 more red counters are put into the box.

The ratio of the number of red counters to the number of blue counters is now 2 : 3.

Work out the number of blue counters in the box.

","answerType":"exact","answer":"126","answerDisplay":null,"accept":["126 blue counters"],"parts":[],"solution":["Let there be 4x red and 7x blue counters at the start.","After adding 12 red: (4x + 12) : 7x = 2 : 3.","Cross multiply: 3(4x + 12) = 2 × 7x, so 12x + 36 = 14x.","2x = 36, so x = 18.","Blue counters = 7 × 18 = 126.","Check: red = 72 + 12 = 84, and 84 : 126 = 2 : 3."],"diagram":null,"draw":false,"revision":"99703e8b2964"},{"id":"g3-ratio-q12","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":12,"marks":5,"tier":"FH","calc":true,"text":"

The price of a sofa is £480 plus VAT at 20%.

Zara pays for the sofa with a deposit followed by 8 equal monthly payments.

The ratio of the deposit to the total of the 8 monthly payments is 2 : 7.

Work out the amount of each monthly payment.

","answerType":"exact","answer":"£56","answerDisplay":null,"accept":["56","£56.00","56.00"],"parts":[],"solution":["Total price = £480 × 1.2 = £576.","Total parts = 2 + 7 = 9, so 1 part = 576 ÷ 9 = £64.","Deposit = 2 × £64 = £128. Monthly payments total = 7 × £64 = £448.","Each monthly payment = 448 ÷ 8 = £56."],"diagram":null,"draw":false,"revision":"7b6ff6ce86aa"},{"id":"g3-ratio-q13","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":13,"marks":5,"tier":"H","calc":true,"text":"

Leah and Sam share £480 in the ratio 5 : 7.

Sam gives 30% of his share to charity.

Leah gives £60 of her share to charity.

Sam gives more money to charity than Leah does.

Work out how much more.

","answerType":"exact","answer":"£24","answerDisplay":null,"accept":["24","£24.00","24.00"],"parts":[],"solution":["Total parts = 5 + 7 = 12, so 1 part = 480 ÷ 12 = £40.","Leah's share = 5 × £40 = £200. Sam's share = 7 × £40 = £280.","Sam gives 30% of £280 = £84.","Leah gives £60.","Difference = 84 − 60 = £24."],"diagram":null,"draw":false,"revision":"3c5ef613048d"},{"id":"g3-ratio-q14","topicId":"g3-ratio","topic":"Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":14,"marks":6,"tier":"H","calc":false,"text":"

Every student in year 11 studies exactly one language, French or Spanish.

The ratio of the number of boys to the number of girls in year 11 is 4 : 5.

For the boys, the ratio of the number studying French to the number studying Spanish is 1 : 3.

For the girls, the ratio of the number studying French to the number studying Spanish is 2 : 3.

Work out the fraction of the year 11 students who study French.

","answerType":"exact","answer":"1/3","answerDisplay":"13","accept":["3/9","one third"],"parts":[],"solution":["Boys are 49 of the year and girls are 59.","Of the boys, 1 part in 1 + 3 = 4 studies French, so French boys = (14) × (49) = 19 of the year.","Of the girls, 2 parts in 2 + 3 = 5 study French, so French girls = (25) × (59) = 29 of the year.","Fraction studying French = 19 + 29 = 39 = 13."],"diagram":null,"draw":false,"revision":"2ed9059000f1"},{"id":"g3-scale-drawings-q01","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

A map has a scale of 1 : 50 000.

On the map, the distance between two churches is 6 cm.

Work out the real distance between the churches. Give your answer in kilometres.

","answerType":"exact","answer":"3 km","answerDisplay":null,"accept":["3","3 kilometres","3km"],"parts":[],"solution":["A scale of 1 : 50 000 compares lengths in the same units: 1 cm on the map represents 50 000 cm in real life. Convert to kilometres after applying the scale.","Real distance = 6 × 50 000 = 300 000 cm","300 000 cm = 3000 m = 3 km"],"diagram":null,"draw":false,"revision":"020fcb00bc30"},{"id":"g3-scale-drawings-q02","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

A model car is made using a scale of 1 : 18.

The real car is 4.5 m long.

Work out the length of the model car. Give your answer in centimetres.

","answerType":"exact","answer":"25 cm","answerDisplay":null,"accept":["25","25cm"],"parts":[],"solution":["Change the real length to centimetres: 4.5 m = 450 cm","Model length = 450 ÷ 18 = 25 cm"],"diagram":null,"draw":false,"revision":"16dc67d639a9"},{"id":"g3-scale-drawings-q03","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

Priya is making a scale drawing of her school field.

She uses a scale of 1 cm to 4 m.

A path across the field is 24 m long.

Work out the length of the path on the scale drawing.

","answerType":"exact","answer":"6 cm","answerDisplay":null,"accept":["6","6cm"],"parts":[],"solution":["Each centimetre on the drawing represents 4 m","Drawing length = 24 ÷ 4 = 6 cm"],"diagram":null,"draw":false,"revision":"e11bb0eb56d1"},{"id":"g3-scale-drawings-q04","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

A playground is a rectangle 12 m long and 8 m wide.

Draw an accurate scale drawing of the playground. Use a scale of 1 cm to 2 m.

","answerType":"open","answer":"An accurate rectangle 6 cm by 4 cm","answerDisplay":null,"accept":["rectangle 6 cm by 4 cm"],"parts":[],"solution":["Length on drawing = 12 ÷ 2 = 6 cm","Width on drawing = 8 ÷ 2 = 4 cm","Draw an accurate rectangle 6 cm by 4 cm"],"diagram":{"type":"shape","grid":true,"width":420,"bounds":[[0,0],[8,6]],"polygons":[{"pts":[[1,1],[7,1],[7,5],[1,5]],"fill":false,"sides":["6 cm","4 cm",null,null],"answer":true}]},"draw":true,"revision":"8497953016a3"},{"id":"g3-scale-drawings-q05","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

A map of a national park has a scale of 1 : 25 000.

On the map, the distance from a car park to a waterfall is 9.6 cm.

Work out the real distance from the car park to the waterfall. Give your answer in kilometres.

","answerType":"exact","answer":"2.4 km","answerDisplay":null,"accept":["2.4","2.4 kilometres","2.4km"],"parts":[],"solution":["Real distance = 9.6 × 25 000 = 240 000 cm","240 000 cm = 2400 m","2400 m = 2.4 km"],"diagram":null,"draw":false,"revision":"976d4d78e4ed"},{"id":"g3-scale-drawings-q06","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

Two villages are 4.5 km apart.

A map has a scale of 1 : 30 000.

Work out the distance between the two villages on the map. Give your answer in centimetres.

","answerType":"exact","answer":"15 cm","answerDisplay":null,"accept":["15","15cm"],"parts":[],"solution":["Change the real distance to centimetres: 4.5 km = 450 000 cm","Map distance = 450 000 ÷ 30 000 = 15 cm"],"diagram":null,"draw":false,"revision":"aa6ad1218c14"},{"id":"g3-scale-drawings-q07","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

A model aeroplane is made using a scale of 1 : 72.

The wingspan of the model is 25 cm.

Work out the wingspan of the real aeroplane. Give your answer in metres.

","answerType":"exact","answer":"18 m","answerDisplay":null,"accept":["18","18 metres","18m"],"parts":[],"solution":["Real wingspan = 25 × 72 = 1800 cm","1800 cm = 18 m"],"diagram":null,"draw":false,"revision":"afcf45a2a9ec"},{"id":"g3-scale-drawings-q08","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

In a photograph, Marcus is standing next to a lamp post.

Marcus is 1.8 m tall.

In the photograph, Marcus is 2 cm tall and the lamp post is 7 cm tall.

Estimate the real height of the lamp post.

Assume Marcus and the lamp post are the same distance from the camera and the photograph has a uniform scale.

","answerType":"exact","answer":"6.3 m","answerDisplay":null,"accept":["6.3","6.3 metres","6.3m"],"parts":[],"solution":["In the photograph, 2 cm represents 1.8 m, so 1 cm represents 0.9 m","Estimate of the real height = 7 × 0.9 = 6.3 m"],"diagram":null,"draw":false,"revision":"9dd75b0de4b9"},{"id":"g3-scale-drawings-q09","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":9,"marks":3,"tier":"FH","calc":true,"text":"

On the plan of a community hall, the scale is 1 cm to 1.5 m.

On the plan, the main room is a rectangle 6 cm long and 4 cm wide.

Work out the real area of the main room. Give your answer in m2.

","answerType":"exact","answer":"54 m^2","answerDisplay":null,"accept":["54","54 m2","54 square metres"],"parts":[],"solution":["Real length = 6 × 1.5 = 9 m","Real width = 4 × 1.5 = 6 m","Real area = 9 × 6 = 54 m2"],"diagram":null,"draw":false,"revision":"ff6a0a45d29c"},{"id":"g3-scale-drawings-q10","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

A map has a scale of 1 : 200 000.

  1. On the map, the distance between two towns is 4.2 cm. Work out the real distance between the towns, in kilometres.
  2. The real distance between two bridges is 15 km. Work out the distance between the bridges on the map, in centimetres.
","answerType":"multi","answer":"a) 8.4 km, b) 7.5 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8.4 km"},{"label":"b","answer":"7.5 cm"}],"solution":["a) 4.2 × 200 000 = 840 000 cm = 8400 m = 8.4 km","b) 15 km = 1 500 000 cm","1 500 000 ÷ 200 000 = 7.5 cm"],"diagram":null,"draw":false,"revision":"0f7b5d671cfa"},{"id":"g3-scale-drawings-q11","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":11,"marks":4,"tier":"F","calc":true,"text":"

Jade is planning a walk.

On a map with a scale of 1 : 50 000, her route is 14 cm long.

Jade walks at a steady speed of 4 km per hour.

Work out how long the walk will take. Give your answer in hours and minutes.

","answerType":"exact","answer":"1 hour 45 minutes","answerDisplay":null,"accept":["1 h 45 min","1 hour 45 minutes"],"parts":[],"solution":["Real distance = 14 × 50 000 = 700 000 cm = 7 km","Time = 7 ÷ 4 = 1.75 hours","1.75 hours = 1 hour 45 minutes"],"diagram":null,"draw":false,"revision":"1124ad6239b0"},{"id":"g3-scale-drawings-q12","topicId":"g3-scale-drawings","topic":"Scale Drawings","grade":"3","topicGrade":"3","strand":"R","difficulty":12,"marks":5,"tier":"F","calc":true,"text":"

A scale drawing of a rectangular garden uses a scale of 1 cm to 2.5 m.

On the drawing, the garden is 8 cm long and 5 cm wide.

Fencing costs £4.20 per metre.

Harry has £250 to fence the whole perimeter of the garden.

Does Harry have enough money? You must show your working.

","answerType":"open","answer":"No - the perimeter is 65 m, costing £273, which is more than £250","answerDisplay":null,"accept":["No, £273 > £250"],"parts":[],"solution":["Real length = 8 × 2.5 = 20 m and real width = 5 × 2.5 = 12.5 m","Perimeter = 2 × (20 + 12.5) = 65 m","Cost = 65 × £4.20 = £273","£273 is more than £250, so Harry does not have enough money"],"diagram":null,"draw":false,"revision":"ae5330c25056"},{"id":"g3-solving-equations-q01","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

Solve w4 + 2 = 9

","answerType":"exact","answer":"w = 28","answerDisplay":null,"accept":["28","w=28"],"parts":[],"solution":["Keep the two sides equal by applying the same operation to both. Undo the addition first, then undo division by 4.","Subtract 2 from both sides: w4 = 7","Multiply both sides by 4: w = 28"],"diagram":null,"draw":false,"revision":"4e2f02fc2f31"},{"id":"g3-solving-equations-q02","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Solve 5x − 7 = 23

","answerType":"exact","answer":"x = 6","answerDisplay":null,"accept":["6","x=6"],"parts":[],"solution":["Add 7 to both sides: 5x = 30","Divide both sides by 5: x = 6"],"diagram":null,"draw":false,"revision":"99a192b035cf"},{"id":"g3-solving-equations-q03","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Solve 3(x + 4) = 27

","answerType":"exact","answer":"x = 5","answerDisplay":null,"accept":["5","x=5"],"parts":[],"solution":["Divide both sides by 3: x + 4 = 9","Subtract 4 from both sides: x = 5"],"diagram":null,"draw":false,"revision":"e74b031869a8"},{"id":"g3-solving-equations-q04","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

Solve 4(2x − 3) = 36

","answerType":"exact","answer":"x = 6","answerDisplay":null,"accept":["6","x=6"],"parts":[],"solution":["Expand the bracket: 8x - 12 = 36","Add 12 to both sides: 8x = 48","Divide both sides by 8: x = 6"],"diagram":null,"draw":false,"revision":"fb4a9a7236f9"},{"id":"g3-solving-equations-q05","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

Solve 2x + 53 = 7

","answerType":"exact","answer":"x = 8","answerDisplay":null,"accept":["8","x=8"],"parts":[],"solution":["Multiply both sides by 3: 2x + 5 = 21","Subtract 5 from both sides: 2x = 16","Divide both sides by 2: x = 8"],"diagram":null,"draw":false,"revision":"1f343db81893"},{"id":"g3-solving-equations-q06","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

Priya thinks of a number.

She multiplies the number by 4 and then subtracts 9.

Her answer is 35.

Work out the number Priya thought of.

","answerType":"exact","answer":"11","answerDisplay":null,"accept":["n = 11","x = 11"],"parts":[],"solution":["Let n be the number, so 4n - 9 = 35","Add 9 to both sides: 4n = 44","Divide both sides by 4: n = 11","Check: 4 times 11 = 44, 44 - 9 = 35"],"diagram":null,"draw":false,"revision":"aa8a0abbb298"},{"id":"g3-solving-equations-q07","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

Solve 7x − 5 = 3x + 23

","answerType":"exact","answer":"x = 7","answerDisplay":null,"accept":["7","x=7"],"parts":[],"solution":["Subtract 3x from both sides: 4x - 5 = 23","Add 5 to both sides: 4x = 28","Divide both sides by 4: x = 7","Check: 7(7) - 5 = 44 and 3(7) + 23 = 44"],"diagram":null,"draw":false,"revision":"405b32aeb893"},{"id":"g3-solving-equations-q08","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":8,"marks":3,"tier":"FH","calc":true,"text":"

Solve 5(x − 2) = 3x + 8

","answerType":"exact","answer":"x = 9","answerDisplay":null,"accept":["9","x=9"],"parts":[],"solution":["Expand the bracket: 5x - 10 = 3x + 8","Subtract 3x from both sides: 2x - 10 = 8","Add 10 to both sides: 2x = 18","Divide both sides by 2: x = 9","Check: 5(9 - 2) = 35 and 3(9) + 8 = 35"],"diagram":null,"draw":false,"revision":"28afc066f78e"},{"id":"g3-solving-equations-q09","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":9,"marks":3,"tier":"H","calc":false,"text":"

Solve 4x + 13 = x + 3

","answerType":"exact","answer":"x = 8","answerDisplay":null,"accept":["8","x=8"],"parts":[],"solution":["Multiply both sides by 3: 4x + 1 = 3x + 9","Subtract 3x from both sides: x + 1 = 9","Subtract 1 from both sides: x = 8","Check: 32 + 13 = 11 and 8 + 3 = 11"],"diagram":null,"draw":false,"revision":"255650936276"},{"id":"g3-solving-equations-q10","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"
  1. Solve 2(3x + 4) = 5x + 17
  2. Factorise fully 6y2 + 9y
","answerType":"multi","answer":"a) x = 9; b) 3y(2y + 3)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x = 9"},{"label":"b","answer":"3y(2y + 3)"}],"solution":["a) Expand: 6x + 8 = 5x + 17","Subtract 5x from both sides: x + 8 = 17, so x = 9","Check: 2(27 + 4) = 62 and 5(9) + 17 = 62","b) The highest common factor of 6y squared and 9y is 3y","6y^2 + 9y = 3y(2y + 3)"],"diagram":null,"draw":false,"revision":"3bab5097a227"},{"id":"g3-solving-equations-q11","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":11,"marks":4,"tier":"H","calc":true,"text":"
  1. Solve 9 − 2x = 3(x − 7)
  2. Simplify (m5)2 ÷ m3
","answerType":"multi","answer":"a) x = 6; b) m^7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x = 6"},{"label":"b","answer":"m^7"}],"solution":["a) Expand: 9 - 2x = 3x - 21","Add 2x to both sides: 9 = 5x - 21","Add 21 to both sides: 30 = 5x, so x = 6","Check: 9 - 12 = -3 and 3(6 - 7) = -3","b) (m^5)^2 = m^10","m^10 divided by m^3 = m^7","The original fraction requires m ≠ 0; keep this restriction after simplifying."],"diagram":null,"draw":false,"revision":"0a59293f365b"},{"id":"g3-solving-equations-q12","topicId":"g3-solving-equations","topic":"Solving Equations","grade":"3","topicGrade":"3","strand":"A","difficulty":12,"marks":5,"tier":"F","calc":true,"text":"
  1. Simplify 4a + 3b − a + 5b
  2. Solve 5x − 42 = 18
  3. Solve 4(y + 3) = 2y + 22
","answerType":"multi","answer":"a) 3a + 8b; b) x = 8; c) y = 5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3a + 8b"},{"label":"b","answer":"x = 8"},{"label":"c","answer":"y = 5"}],"solution":["a) 4a - a = 3a and 3b + 5b = 8b, so the answer is 3a + 8b","b) Multiply both sides by 2: 5x - 4 = 36","Add 4: 5x = 40, so x = 8","c) Expand: 4y + 12 = 2y + 22","Subtract 2y: 2y + 12 = 22","Subtract 12: 2y = 10, so y = 5","Check: 4(5 + 3) = 32 and 2(5) + 22 = 32"],"diagram":null,"draw":false,"revision":"3bfc7e1932a7"},{"id":"g3-substitution-q01","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

T = 4x + 3y

Work out the value of T when x = 5 and y = 2.

","answerType":"exact","answer":"26","answerDisplay":null,"accept":["T = 26"],"parts":[],"solution":["Substitution means replacing each letter with its given value. Numbers next to letters mean multiplication, so 4x becomes 4 × 5. Multiply before adding.","Substitute the values: T = 4 × 5 + 3 × 2.","T = 20 + 6 = 26."],"diagram":null,"draw":false,"revision":"ca7df8b70422"},{"id":"g3-substitution-q02","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

v = 5t − 8

Work out the value of v when t = 3.4

","answerType":"exact","answer":"9","answerDisplay":null,"accept":["v = 9","9.0"],"parts":[],"solution":["Substitute t = 3.4: v = 5 × 3.4 - 8.","v = 17 - 8 = 9."],"diagram":null,"draw":false,"revision":"09b005ee012c"},{"id":"g3-substitution-q03","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

E = 3a − 2b

Work out the value of E when a = −4 and b = 5.

","answerType":"exact","answer":"-22","answerDisplay":null,"accept":["E = -22"],"parts":[],"solution":["Substitute the values: E = 3 × (-4) - 2 × 5.","E = -12 - 10 = -22."],"diagram":null,"draw":false,"revision":"7dfc0c0d057c"},{"id":"g3-substitution-q04","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

Work out the value of  x2 + 7x  when x = 6.

","answerType":"exact","answer":"78","answerDisplay":null,"accept":[],"parts":[],"solution":["x2 = 6 × 6 = 36.","7x = 7 × 6 = 42.","36 + 42 = 78."],"diagram":null,"draw":false,"revision":"8c5c891fcaef"},{"id":"g3-substitution-q05","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

5y = 35

  1. Work out the value of y.
  2. Work out the value of  2y2
","answerType":"multi","answer":"a) 7 b) 98","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"7"},{"label":"b","answer":"98"}],"solution":["a) y = 35 ÷ 5 = 7.","b) Square first: y2 = 49.","b) Then multiply by 2: 2 × 49 = 98."],"diagram":null,"draw":false,"revision":"d94fa989c0a7"},{"id":"g3-substitution-q06","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"
  1. Simplify  4c × 2d
  2. Work out the value of  4c × 2d  when c = 3 and d = 1.5
","answerType":"multi","answer":"a) 8cd b) 36","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8cd"},{"label":"b","answer":"36"}],"solution":["a) 4 × 2 = 8 and the letters are c and d, so 8cd.","b) Substitute into 8cd: 8 × 3 × 1.5","b) 8 × 3 = 24, then 24 × 1.5 = 36."],"diagram":null,"draw":false,"revision":"d5a6738fb0f9"},{"id":"g3-substitution-q07","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":7,"marks":4,"tier":"F","calc":false,"text":"

F = 5v + 20

  1. Work out the value of F when v = 6.
  2. Work out the value of v when F = 95.
","answerType":"multi","answer":"a) 50 b) 15","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"50"},{"label":"b","answer":"15"}],"solution":["a) F = 5 × 6 + 20 = 30 + 20 = 50.","b) Substitute F = 95: 95 = 5v + 20.","b) Subtract 20: 5v = 75, so v = 15.","Check: 5 × 15 + 20 = 95."],"diagram":null,"draw":false,"revision":"1c5181c2026f"},{"id":"g3-substitution-q08","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

A tool hire company uses this rule to work out the cost of hiring a floor sander.

Total cost = £45 × number of days + £60 deposit

  1. Meera hires the sander for 7 days.
    Work out her total cost.
  2. Liam's total cost is £555.
    Work out the number of days Liam hires the sander for.
","answerType":"multi","answer":"a) £375 b) 11 days","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£375"},{"label":"b","answer":"11"}],"solution":["a) 45 × 7 = 315, then 315 + 60 = 375. Total cost £375.","b) Take off the deposit: 555 - 60 = 495.","b) Divide by the daily rate: 495 ÷ 45 = 11 days.","Check: 45 × 11 + 60 = 555."],"diagram":null,"draw":false,"revision":"bbaf4d699b98"},{"id":"g3-substitution-q09","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":9,"marks":4,"tier":"F","calc":false,"text":"
  1. Work out the value of  2a − 3b  when a = −1 and b = 4.
  2. Work out the value of  4a2 + b  when a = −3 and b = −7.
","answerType":"multi","answer":"a) -14 b) 29","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-14"},{"label":"b","answer":"29"}],"solution":["a) 2 × (-1) = -2 and 3 × 4 = 12, so -2 - 12 = -14.","b) a2 = (-3) × (-3) = 9, so 4a2 = 36.","b) 36 + (-7) = 29."],"diagram":null,"draw":false,"revision":"7d4911b2e6f9"},{"id":"g3-substitution-q10","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":10,"marks":4,"tier":"FH","calc":true,"text":"

v = u + at

  1. Work out the value of v when u = 7, a = 9.8 and t = 5.
  2. Work out the value of t when v = 87, u = 12 and a = 5.
","answerType":"multi","answer":"a) 56 b) 15","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"56"},{"label":"b","answer":"15"}],"solution":["a) v = 7 + 9.8 × 5 = 7 + 49 = 56.","b) Substitute: 87 = 12 + 5t.","b) Subtract 12: 5t = 75, so t = 15.","Check: 12 + 5 × 15 = 87."],"diagram":null,"draw":false,"revision":"0603df3ac337"},{"id":"g3-substitution-q11","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":11,"marks":4,"tier":"FH","calc":false,"text":"

m = 3n − 2p

  1. Work out the value of m when n = −5 and p = 4.
  2. Work out the value of n when m = 19 and p = −2.
","answerType":"multi","answer":"a) -23 b) 5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-23"},{"label":"b","answer":"5"}],"solution":["a) m = 3 × (-5) - 2 × 4 = -15 - 8 = -23.","b) Substitute: 19 = 3n - 2 × (-2) = 3n + 4.","b) Subtract 4: 3n = 15, so n = 5.","Check: 3 × 5 - 2 × (-2) = 15 + 4 = 19."],"diagram":null,"draw":false,"revision":"f62c07c57a7d"},{"id":"g3-substitution-q12","topicId":"g3-substitution","topic":"Substitution","grade":"3","topicGrade":"3","strand":"A","difficulty":12,"marks":6,"tier":"F","calc":true,"text":"
  1. Work out the value of  5x + 2y  when x = 4 and y = −3.
  2. Work out the value of  x2 − 4x  when x = −2.
  3. D = (h + 3k) ÷ 2
    Work out the value of h when D = 10 and k = 4.
","answerType":"multi","answer":"a) 14 b) 12 c) 8","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"14"},{"label":"b","answer":"12"},{"label":"c","answer":"8"}],"solution":["a) 5 × 4 = 20 and 2 × (-3) = -6, so 20 + (-6) = 14.","b) x2 = (-2) × (-2) = 4 and 4x = 4 × (-2) = -8.","b) 4 - (-8) = 4 + 8 = 12.","c) Substitute: 10 = (h + 12) ÷ 2.","c) Multiply both sides by 2: h + 12 = 20, so h = 8.","Check: (8 + 12) ÷ 2 = 10."],"diagram":null,"draw":false,"revision":"c628e548072a"},{"id":"g3-transformations-q01","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

The point P has coordinates (3, 5).

P is reflected in the y-axis to give the point P′.

Write down the coordinates of P′.

","answerType":"exact","answer":"(-3, 5)","answerDisplay":null,"accept":["(-3,5)"],"parts":[],"solution":["Reflecting in the y-axis changes the sign of the x-coordinate and leaves the y-coordinate unchanged.","So (3, 5) maps to (-3, 5)."],"diagram":null,"draw":false,"revision":"2acec338b661"},{"id":"g3-transformations-q02","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

A triangle has vertices at (1, 1), (4, 1) and (1, 3).

The triangle is reflected in the x-axis.

Write down the coordinates of the vertices of the image.

","answerType":"exact","answer":"(1, -1), (4, -1), (1, -3)","answerDisplay":null,"accept":["(1,-1), (4,-1), (1,-3) in any order"],"parts":[],"solution":["Reflecting in the x-axis changes the sign of each y-coordinate.","(1, 1) maps to (1, -1); (4, 1) maps to (4, -1); (1, 3) maps to (1, -3)."],"diagram":null,"draw":false,"revision":"2a8b40ef38ba"},{"id":"g3-transformations-q03","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

A triangle has vertices at (2, 3), (5, 3) and (2, 7).

The triangle is translated by the vector (4, −2), that is 4 units to the right and 2 units down.

Write down the coordinates of the vertices of the image.

","answerType":"exact","answer":"(6, 1), (9, 1), (6, 5)","answerDisplay":null,"accept":["(6,1), (9,1), (6,5) in any order"],"parts":[],"solution":["Add 4 to each x-coordinate and subtract 2 from each y-coordinate.","(2, 3) maps to (6, 1); (5, 3) maps to (9, 1); (2, 7) maps to (6, 5)."],"diagram":null,"draw":false,"revision":"ffc388d852ea"},{"id":"g3-transformations-q04","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":4,"marks":2,"tier":"F","calc":false,"text":"

A triangle has vertices at (1, 2), (4, 2) and (4, 4).

The triangle is rotated 180° about the origin.

Write down the coordinates of the vertices of the image.

","answerType":"exact","answer":"(-1, -2), (-4, -2), (-4, -4)","answerDisplay":null,"accept":["(-1,-2), (-4,-2), (-4,-4) in any order"],"parts":[],"solution":["A rotation of 180° about the origin maps (x, y) to (-x, -y).","(1, 2) maps to (-1, -2); (4, 2) maps to (-4, -2); (4, 4) maps to (-4, -4)."],"diagram":null,"draw":false,"revision":"189484a05d2f"},{"id":"g3-transformations-q05","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

Triangle A has vertices at (1, 1), (3, 1) and (1, 4).

Triangle B has vertices at (5, −1), (7, −1) and (5, 2).

Describe fully the single transformation that maps triangle A onto triangle B.

","answerType":"open","answer":"Translation by the vector (4, -2): 4 right and 2 down.","answerDisplay":null,"accept":["translation by vector (4, -2)"],"parts":[],"solution":["Compare matching vertices: (1, 1) maps to (5, -1). The x-change is 5 - 1 = +4 and the y-change is -1 - 1 = -2.","Check with (3, 1) to (7, -1): +4 and -2. Check with (1, 4) to (5, 2): +4 and -2. Consistent.","The single transformation is a translation by the vector (4, -2)."],"diagram":null,"draw":false,"revision":"0054a494e40b"},{"id":"g3-transformations-q06","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":6,"marks":2,"tier":"F","calc":true,"text":"

A triangle has vertices at (1, 2), (3, 2) and (3, 5).

The triangle is reflected in the line x = 4.

Write down the coordinates of the vertices of the image.

","answerType":"exact","answer":"(7, 2), (5, 2), (5, 5)","answerDisplay":null,"accept":["(7,2), (5,2), (5,5) in any order"],"parts":[],"solution":["Reflecting in the line x = 4 maps (x, y) to (8 - x, y), because the image is the same distance from the line on the other side.","(1, 2) is 3 left of the line, so its image is 3 right of the line: (7, 2).","(3, 2) maps to (5, 2) and (3, 5) maps to (5, 5)."],"diagram":null,"draw":false,"revision":"f4cbf6067764"},{"id":"g3-transformations-q07","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":7,"marks":2,"tier":"F","calc":false,"text":"

Triangle P has vertices at (2, 1), (5, 1) and (5, 3).

Triangle Q has vertices at (2, −1), (5, −1) and (5, −3).

Describe fully the single transformation that maps triangle P onto triangle Q.

","answerType":"open","answer":"Reflection in the x-axis.","answerDisplay":null,"accept":["reflection in the line y = 0"],"parts":[],"solution":["Each vertex of Q has the same x-coordinate as the matching vertex of P, with the y-coordinate's sign changed.","This is a reflection in the x-axis (the line y = 0)."],"diagram":null,"draw":false,"revision":"ca521fd2d1a8"},{"id":"g3-transformations-q08","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":8,"marks":2,"tier":"F","calc":false,"text":"

A triangle has vertices at (1, 1), (4, 1) and (1, 3).

The triangle is rotated 90° clockwise about the origin.

Write down the coordinates of the vertices of the image.

","answerType":"exact","answer":"(1, -1), (1, -4), (3, -1)","answerDisplay":null,"accept":["(1,-1), (1,-4), (3,-1) in any order"],"parts":[],"solution":["For each vertex, draw a line from the origin and turn that line a quarter turn clockwise, keeping its length unchanged. A point to the right of the origin rotates below it.","A rotation of 90° clockwise about the origin maps (x, y) to (y, -x).","(1, 1) maps to (1, -1); (4, 1) maps to (1, -4); (1, 3) maps to (3, -1)."],"diagram":{"type":"axes","square":true,"x":{"min":-2,"max":5,"step":1},"y":{"min":-5,"max":4,"step":1},"polygons":[{"pts":[[1,1],[4,1],[1,3]],"label":"Original"},{"pts":[[1,-1],[1,-4],[3,-1]],"answer":true,"label":"Image"}],"points":[{"at":[0,0],"label":"O","style":"dot"}],"alt":"Original triangle on coordinate axes. Its clockwise rotation is revealed with the solution."},"draw":false,"revision":"283c26d9ea5b"},{"id":"g3-transformations-q09","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":9,"marks":2,"tier":"F","calc":true,"text":"

A triangle has vertices at (1, 2), (2, 2) and (1, 4).

The triangle is enlarged by scale factor 3, centre the origin.

Write down the coordinates of the vertices of the image.

","answerType":"exact","answer":"(3, 6), (6, 6), (3, 12)","answerDisplay":null,"accept":["(3,6), (6,6), (3,12) in any order"],"parts":[],"solution":["An enlargement with centre the origin and scale factor 3 maps (x, y) to (3x, 3y).","(1, 2) maps to (3, 6); (2, 2) maps to (6, 6); (1, 4) maps to (3, 12)."],"diagram":null,"draw":false,"revision":"0c519c3eea52"},{"id":"g3-transformations-q10","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":10,"marks":2,"tier":"FH","calc":true,"text":"

Rectangle A has vertices at (1, 1), (3, 1), (3, 2) and (1, 2).

Rectangle B has vertices at (2, 2), (6, 2), (6, 4) and (2, 4).

Describe fully the single transformation that maps rectangle A onto rectangle B.

","answerType":"open","answer":"Enlargement, scale factor 2, centre (0, 0).","answerDisplay":null,"accept":["enlargement scale factor 2 centre the origin"],"parts":[],"solution":["The sides of B are twice as long as the sides of A, so the scale factor is 2.","Every vertex of B is exactly double the matching vertex of A: (1, 1) maps to (2, 2), (3, 1) maps to (6, 2), and so on.","So the transformation is an enlargement, scale factor 2, centre the origin (0, 0)."],"diagram":null,"draw":false,"revision":"afeddf13f3c6"},{"id":"g3-transformations-q11","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":11,"marks":2,"tier":"H","calc":true,"text":"

A triangle has vertices at (4, 2), (8, 2) and (4, 6).

The triangle is enlarged by scale factor 12, centre the origin.

Write down the coordinates of the vertices of the image.

","answerType":"exact","answer":"(2, 1), (4, 1), (2, 3)","answerDisplay":null,"accept":["(2,1), (4,1), (2,3) in any order"],"parts":[],"solution":["An enlargement with centre the origin and scale factor 12 maps (x, y) to (x2, y2).","(4, 2) maps to (2, 1); (8, 2) maps to (4, 1); (4, 6) maps to (2, 3)."],"diagram":null,"draw":false,"revision":"41cda1dc225e"},{"id":"g3-transformations-q12","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":12,"marks":3,"tier":"H","calc":false,"text":"

Triangle A has vertices at (2, 1), (5, 1) and (2, 3).

Triangle A is reflected in the y-axis to give triangle B.

  1. Write down the coordinates of the vertices of triangle B.
  2. Triangle C has vertices at (−1, −2), (−4, −2) and (−1, 0). Find the vector of the translation that maps triangle B onto triangle C.
","answerType":"multi","answer":"a) (-2, 1), (-5, 1), (-2, 3) b) vector (1, -3)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(-2, 1), (-5, 1), (-2, 3)"},{"label":"b","answer":"(1, -3)"}],"solution":["Reflecting in the y-axis changes the sign of each x-coordinate: (2, 1) maps to (-2, 1), (5, 1) maps to (-5, 1), (2, 3) maps to (-2, 3).","Compare matching vertices of B and C: (-2, 1) maps to (-1, -2), a change of +1 in x and -3 in y.","Check with (-5, 1) to (-4, -2): +1 and -3. Check with (-2, 3) to (-1, 0): +1 and -3.","The translation vector is (1, -3)."],"diagram":null,"draw":false,"revision":"fba1b7d18c9f"},{"id":"g3-transformations-q13","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":13,"marks":3,"tier":"H","calc":false,"text":"

Triangle T has vertices at (1, 1), (3, 1) and (1, 2).

Triangle T is reflected in the y-axis to give triangle T1.

Triangle T1 is then reflected in the x-axis to give triangle T2.

Describe fully the single transformation that maps triangle T onto triangle T2.

","answerType":"open","answer":"Rotation of 180 degrees about the origin.","answerDisplay":null,"accept":["rotation 180 about (0, 0)"],"parts":[],"solution":["Reflecting in the y-axis maps (x, y) to (-x, y), so T1 has vertices (-1, 1), (-3, 1), (-1, 2).","Reflecting T1 in the x-axis maps (x, y) to (x, -y), so T2 has vertices (-1, -1), (-3, -1), (-1, -2).","Overall (x, y) has mapped to (-x, -y), which is a rotation of 180° about the origin.","So the single transformation is a rotation of 180°, centre (0, 0)."],"diagram":null,"draw":false,"revision":"8a3d2b6ccaaa"},{"id":"g3-transformations-q14","topicId":"g3-transformations","topic":"Transformations","grade":"3","topicGrade":"3","strand":"G","difficulty":14,"marks":4,"tier":"H","calc":true,"text":"

Triangle A has vertices at (2, 2), (4, 2) and (2, 6).

Triangle B has vertices at (1, 1), (2, 1) and (1, 3).

Triangle C has vertices at (−1, 1), (−2, 1) and (−1, 3).

  1. Describe fully the single transformation that maps triangle A onto triangle B.
  2. Describe fully the single transformation that maps triangle B onto triangle C.
","answerType":"multi","answer":"a) Enlargement, scale factor 1/2, centre (0, 0) b) Reflection in the y-axis","answerDisplay":"a) Enlargement, scale factor 12, centre (0, 0) b) Reflection in the y-axis","accept":[],"parts":[{"label":"a","answer":"enlargement, scale factor 1/2, centre (0, 0)","answerDisplay":"enlargement, scale factor 12, centre (0, 0)"},{"label":"b","answer":"reflection in the y-axis"}],"solution":["Every vertex of B is exactly half the matching vertex of A: (2, 2) maps to (1, 1), (4, 2) maps to (2, 1), (2, 6) maps to (1, 3).","So A to B is an enlargement, scale factor 12, centre the origin (0, 0).","Every vertex of C has the same y-coordinate as the matching vertex of B with the sign of the x-coordinate changed.","So B to C is a reflection in the y-axis (the line x = 0)."],"diagram":null,"draw":false,"revision":"54437bfb37b3"},{"id":"g3-two-way-tables-q01","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":1,"marks":3,"tier":"F","calc":true,"text":"

80 students each chose one club: Art, Music or Drama.

The two-way table shows some information about their choices.

ArtMusicDramaTotal
Boys14935
Girls1811
Total80

Complete the two-way table.

","answerType":"multi","answer":"Boys Music 12; Girls Art 16; Girls Total 45; Totals: Art 30, Music 30, Drama 20","answerDisplay":null,"accept":[],"parts":[{"label":"Boys Music","answer":"12"},{"label":"Girls Art","answer":"16"},{"label":"Girls Total","answer":"45"},{"label":"Art Total","answer":"30"},{"label":"Music Total","answer":"30"},{"label":"Drama Total","answer":"20"}],"solution":["Boys Music = 35 - 14 - 9 = 12.","Girls Total = 80 - 35 = 45.","Girls Art = 45 - 18 - 11 = 16.","Column totals: Art = 14 + 16 = 30, Music = 12 + 18 = 30, Drama = 9 + 11 = 20.","Check: 30 + 30 + 20 = 80."],"diagram":null,"draw":false,"revision":"d1b2b92a2460"},{"id":"g3-two-way-tables-q02","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

80 people watched a match at a sports club. Each person watched either football or rugby.

45 of the people were adults. The rest were children.

28 of the children watched football.

19 of the adults watched rugby.

Use this information to complete the two-way table.

FootballRugbyTotal
Adults
Children
Total
","answerType":"multi","answer":"Adults: Football 26, Rugby 19, Total 45; Children: Football 28, Rugby 7, Total 35; Totals: Football 54, Rugby 26, 80","answerDisplay":null,"accept":[],"parts":[{"label":"Adults Football","answer":"26"},{"label":"Children Rugby","answer":"7"},{"label":"Children Total","answer":"35"},{"label":"Football Total","answer":"54"},{"label":"Rugby Total","answer":"26"}],"solution":["Children total = 80 - 45 = 35.","Adults football = 45 - 19 = 26.","Children rugby = 35 - 28 = 7.","Football total = 26 + 28 = 54, Rugby total = 19 + 7 = 26.","Check: 54 + 26 = 80."],"diagram":null,"draw":false,"revision":"434ecb247035"},{"id":"g3-two-way-tables-q03","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

120 Year 10 students each study exactly one language, French or Spanish.

66 of the students are girls.

28 boys study French.

39 girls study Spanish.

Work out how many students study French.

","answerType":"exact","answer":"55","answerDisplay":null,"accept":["55 students"],"parts":[],"solution":["Boys = 120 - 66 = 54.","Girls studying French = 66 - 39 = 27.","Students studying French = 28 + 27 = 55."],"diagram":null,"draw":false,"revision":"359ba7dd020b"},{"id":"g3-two-way-tables-q04","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

90 members of a gym each went to one class, yoga or spin.

25 of the members are men.

21 of the men went to spin.

25 of the women went to yoga.

Work out how many members went to yoga in total.

","answerType":"exact","answer":"40","answerDisplay":null,"accept":["40 members"],"parts":[],"solution":["Men = 25 of 90 = 36, so women = 90 - 36 = 54.","Men at yoga = 36 - 21 = 15.","Yoga total = 15 + 25 = 40."],"diagram":null,"draw":false,"revision":"31bb0114e33d"},{"id":"g3-two-way-tables-q05","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

200 people were asked whether they prefer to shop online or in store.

112 of the people were aged under 40. The rest were aged 40 or over.

74 of the people aged under 40 prefer to shop online.

61 of the people aged 40 or over prefer to shop in store.

Work out how many of the 200 people prefer to shop online.

","answerType":"exact","answer":"101","answerDisplay":null,"accept":["101 people"],"parts":[],"solution":["People aged 40 or over = 200 - 112 = 88.","Aged 40 or over preferring online = 88 - 61 = 27.","Online total = 74 + 27 = 101."],"diagram":null,"draw":false,"revision":"cf7983f1b035"},{"id":"g3-two-way-tables-q06","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

150 students travel to school by bus, by car or by walking.

The two-way table shows some information.

BusCarWalkTotal
Year 7243080
Year 818
Total55150

Complete the two-way table.

","answerType":"multi","answer":"Year 7 Car 26; Year 8: Bus 31, Walk 21, Total 70; Totals: Car 44, Walk 51","answerDisplay":null,"accept":[],"parts":[{"label":"Year 7 Car","answer":"26"},{"label":"Year 8 Bus","answer":"31"},{"label":"Year 8 Walk","answer":"21"},{"label":"Year 8 Total","answer":"70"},{"label":"Car Total","answer":"44"},{"label":"Walk Total","answer":"51"}],"solution":["Year 7 car = 80 - 24 - 30 = 26.","Year 8 total = 150 - 80 = 70.","Year 8 bus = 55 - 24 = 31.","Year 8 walk = 70 - 31 - 18 = 21.","Car total = 26 + 18 = 44, Walk total = 30 + 21 = 51.","Check: 55 + 44 + 51 = 150."],"diagram":null,"draw":false,"revision":"4141c76344cb"},{"id":"g3-two-way-tables-q07","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

A sports club has 132 members. Each member plays exactly one sport, badminton or squash.

72 of the members are female.

The number of male members who play squash is twice the number of male members who play badminton.

45 female members play badminton.

Work out the total number of members who play badminton.

","answerType":"exact","answer":"65","answerDisplay":null,"accept":["65 members"],"parts":[],"solution":["Male members = 132 - 72 = 60.","Let male badminton players = b, then male squash players = 2b, so 3b = 60 and b = 20.","Male badminton = 20, male squash = 40.","Badminton total = 45 + 20 = 65."],"diagram":null,"draw":false,"revision":"39958b727dc1"},{"id":"g3-two-way-tables-q08","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

One morning a cafe sold 90 drinks. Each drink was tea, coffee or hot chocolate, and each drink was small or large.

34 of the drinks were tea. 14 of the teas were large.

41 of the drinks were coffee. 26 of the coffees were large.

9 of the hot chocolates were large.

Work out how many small drinks the cafe sold.

","answerType":"exact","answer":"41","answerDisplay":null,"accept":["41 drinks","41 small drinks"],"parts":[],"solution":["Small teas = 34 - 14 = 20.","Small coffees = 41 - 26 = 15.","Hot chocolates = 90 - 34 - 41 = 15, so small hot chocolates = 15 - 9 = 6.","Small drinks = 20 + 15 + 6 = 41."],"diagram":null,"draw":false,"revision":"5c90e2b6367b"},{"id":"g3-two-way-tables-q09","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

80 students sat a mock exam. Each student is in Year 10 or Year 11.

47 of the students are in Year 10.

35 of the Year 10 students passed the exam.

62 students passed the exam in total.

One of the students who passed the exam is chosen at random.

Find the probability that this student is in Year 11.

Give your answer as an exact fraction.

","answerType":"exact","answer":"27/62","answerDisplay":"2762","accept":["27 / 62"],"parts":[],"solution":["Year 11 students who passed = 62 - 35 = 27.","The student is chosen from the 62 who passed.","P(Year 11) = 2762."],"diagram":null,"draw":false,"revision":"3f631ab7b754"},{"id":"g3-two-way-tables-q10","topicId":"g3-two-way-tables","topic":"Two Way Tables","grade":"3","topicGrade":"3","strand":"S","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

120 people at a conference are delegates or speakers. Each person chose one workshop: Coding, Design or Marketing.

There are 30 speakers.

13 of the speakers chose Coding.

8 speakers chose Marketing.

36 delegates chose Design.

Equal numbers of delegates chose Coding and Marketing.

  1. Complete the two-way table to show this information.
    CodingDesignMarketingTotal
    Delegates36
    Speakers830
    Total120
  2. One of the 120 people is chosen at random. Work out the probability that this person chose Coding.
","answerType":"multi","answer":"(a) Delegates: Coding 27, Design 36, Marketing 27, Total 90. Speakers: Coding 10, Design 12, Marketing 8, Total 30. Totals: Coding 37, Design 48, Marketing 35, Grand total 120. (b) 37/120","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Delegates 27, 36, 27, total 90; speakers 10, 12, 8, total 30; totals 37, 48, 35, grand total 120."},{"label":"b","answer":"37/120","answerDisplay":"37120"}],"solution":["Speakers choosing Coding = 13 of 30 = 10, so speakers choosing Design = 30 - 10 - 8 = 12.","Delegates = 120 - 30 = 90.","Delegates choosing Coding and Marketing = 90 - 36 = 54, shared equally, so 27 each.","Complete the totals too: speakers 30, delegates 90; Coding 37, Design 48, Marketing 35. Both the row totals and column totals add to 120.","Total choosing Coding = 10 + 27 = 37.","P(Coding) = 37120."],"diagram":null,"draw":false,"revision":"8f3815ff5b59"},{"id":"g3-writing-and-simplifying-ratio-q01","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write down the ratio 20 : 35 in its simplest form.

","answerType":"exact","answer":"4 : 7","answerDisplay":null,"accept":["4:7"],"parts":[],"solution":["The highest common factor of 20 and 35 is 5.","20 ÷ 5 = 4 and 35 ÷ 5 = 7.","20 : 35 = 4 : 7.","Dividing both quantities by the same number preserves their relative sizes. Using the highest common factor leaves no further whole-number factor to cancel."],"diagram":null,"draw":false,"revision":"7ec5f42e3191"},{"id":"g3-writing-and-simplifying-ratio-q02","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Write down the ratio 42 : 6 in the form n : 1.

","answerType":"exact","answer":"7 : 1","answerDisplay":null,"accept":["7:1"],"parts":[],"solution":["Divide both parts by 6.","42 ÷ 6 = 7 and 6 ÷ 6 = 1.","42 : 6 = 7 : 1."],"diagram":null,"draw":false,"revision":"cedd72c703e9"},{"id":"g3-writing-and-simplifying-ratio-q03","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

In a choir there are 16 sopranos and 24 altos.

Write down the ratio of the number of sopranos to the number of altos.

Give your ratio in its simplest form.

","answerType":"exact","answer":"2 : 3","answerDisplay":null,"accept":["2:3"],"parts":[],"solution":["The ratio of sopranos to altos is 16 : 24.","The highest common factor of 16 and 24 is 8.","16 ÷ 8 = 2 and 24 ÷ 8 = 3, so the ratio is 2 : 3."],"diagram":null,"draw":false,"revision":"217b323dccc8"},{"id":"g3-writing-and-simplifying-ratio-q04","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":4,"marks":2,"tier":"F","calc":false,"text":"

Write down the ratio 45 cm : 1.5 m in its simplest form.

","answerType":"exact","answer":"3 : 10","answerDisplay":null,"accept":["3:10"],"parts":[],"solution":["Convert to the same units: 1.5 m = 150 cm.","The ratio is 45 : 150.","The highest common factor of 45 and 150 is 15.","45 ÷ 15 = 3 and 150 ÷ 15 = 10, so the ratio is 3 : 10."],"diagram":null,"draw":false,"revision":"84a9ed9159d7"},{"id":"g3-writing-and-simplifying-ratio-q05","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

A box contains red pens, green pens and blue pens.

There are twice as many green pens as red pens.

There are three times as many blue pens as red pens.

Write down the ratio of red pens to green pens to blue pens.

","answerType":"exact","answer":"1 : 2 : 3","answerDisplay":null,"accept":["1:2:3"],"parts":[],"solution":["Let the number of red pens be 1 part.","Green = twice red = 2 parts. Blue = three times red = 3 parts.","The ratio red : green : blue = 1 : 2 : 3."],"diagram":null,"draw":false,"revision":"2e7d939b88d9"},{"id":"g3-writing-and-simplifying-ratio-q06","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":6,"marks":2,"tier":"F","calc":true,"text":"

At a cinema screening there are 84 adults and 63 children.

Write down the ratio of the number of adults to the number of children.

Give your ratio in its simplest form.

","answerType":"exact","answer":"4 : 3","answerDisplay":null,"accept":["4:3"],"parts":[],"solution":["The ratio of adults to children is 84 : 63.","The highest common factor of 84 and 63 is 21.","84 ÷ 21 = 4 and 63 ÷ 21 = 3, so the ratio is 4 : 3."],"diagram":null,"draw":false,"revision":"1c3a81177549"},{"id":"g3-writing-and-simplifying-ratio-q07","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":7,"marks":2,"tier":"F","calc":true,"text":"

In a tennis club, 35% of the members are juniors.

The rest of the members are seniors.

Write down the ratio of juniors to seniors in its simplest form.

","answerType":"exact","answer":"7 : 13","answerDisplay":null,"accept":["7:13"],"parts":[],"solution":["Seniors make up 100% − 35% = 65% of the members.","The ratio of juniors to seniors is 35 : 65.","The highest common factor of 35 and 65 is 5.","35 ÷ 5 = 7 and 65 ÷ 5 = 13, so the ratio is 7 : 13."],"diagram":null,"draw":false,"revision":"6b7998d8793c"},{"id":"g3-writing-and-simplifying-ratio-q08","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":8,"marks":2,"tier":"F","calc":true,"text":"

Sam has 12 stamps.

Tia has half as many stamps as Sam.

Leo has 3 times as many stamps as Tia.

Write down the ratio of Sam's stamps to Tia's stamps to Leo's stamps.

Give your ratio in its simplest form.

","answerType":"exact","answer":"2 : 1 : 3","answerDisplay":null,"accept":["2:1:3"],"parts":[],"solution":["Tia has 12 ÷ 2 = 6 stamps.","Leo has 3 × 6 = 18 stamps.","The ratio is 12 : 6 : 18.","Divide each part by 6 to get 2 : 1 : 3."],"diagram":null,"draw":false,"revision":"5a65c9587d15"},{"id":"g3-writing-and-simplifying-ratio-q09","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":9,"marks":2,"tier":"F","calc":true,"text":"

In a bag of counters, 38 of the counters are yellow.

The rest of the counters are white.

Write down the ratio of yellow counters to white counters.

","answerType":"exact","answer":"3 : 5","answerDisplay":null,"accept":["3:5"],"parts":[],"solution":["If 3 out of every 8 counters are yellow, then 8 − 3 = 5 out of every 8 are white.","The ratio of yellow to white is 3 : 5."],"diagram":null,"draw":false,"revision":"86f64251c92e"},{"id":"g3-writing-and-simplifying-ratio-q10","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":10,"marks":3,"tier":"F","calc":false,"text":"

At an animal shelter, the ratio of cats to dogs is 8 : 20.

There are no other animals at the shelter.

  1. Write down the fraction of the animals that are cats. Give your fraction in its simplest form.
  2. Write the ratio 8 : 20 in the form 1 : n.
","answerType":"multi","answer":"a) 2/7 b) 1 : 2.5","answerDisplay":"a) 27 b) 1 : 2.5","accept":[],"parts":[{"label":"a","answer":"2/7","answerDisplay":"27"},{"label":"b","answer":"1 : 2.5"}],"solution":["a) Total parts = 8 + 20 = 28, so cats are 828 of the animals.","a) 828 simplifies to 27.","b) Divide both parts of 8 : 20 by 8.","b) 8 ÷ 8 = 1 and 20 ÷ 8 = 2.5, so the ratio is 1 : 2.5."],"diagram":null,"draw":false,"revision":"d315438835da"},{"id":"g3-writing-and-simplifying-ratio-q11","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":11,"marks":3,"tier":"F","calc":true,"text":"

Here are the first five terms of an arithmetic sequence.

4   10   16   22   28

  1. Write down the next term of the sequence.
  2. Write down the ratio of the 2nd term to the 5th term. Give your ratio in its simplest form.
","answerType":"multi","answer":"a) 34 b) 5 : 14","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"34"},{"label":"b","answer":"5 : 14"}],"solution":["a) The sequence goes up by 6 each time, so the next term is 28 + 6 = 34.","b) The 2nd term is 10 and the 5th term is 28, so the ratio is 10 : 28.","b) The highest common factor of 10 and 28 is 2, so the ratio simplifies to 5 : 14."],"diagram":null,"draw":false,"revision":"18ac1522a269"},{"id":"g3-writing-and-simplifying-ratio-q12","topicId":"g3-writing-and-simplifying-ratio","topic":"Writing and Simplifying Ratio","grade":"3","topicGrade":"3","strand":"R","difficulty":12,"marks":3,"tier":"F","calc":true,"text":"

In class A, the ratio of girls to boys is 5 : 4.

In class B, 47 of the students are girls.

Show that class B has the greater proportion of girls.

","answerType":"open","answer":"In class A girls are 5/9 of the class (about 55.6%); in class B girls are 4/7 (about 57.1%); 4/7 > 5/9 so class B has the greater proportion","answerDisplay":"In class A girls are 59 of the class (about 55.6%); in class B girls are 47 (about 57.1%); 47 > 59 so class B has the greater proportion","accept":[],"parts":[],"solution":["In class A, girls make up 5 parts out of 5 + 4 = 9 parts, so 59 of the class.","Compare 59 and 47 using a common denominator of 63: 59 = 3563 and 47 = 3663.","3663 > 3563, so class B has the greater proportion of girls."],"diagram":null,"draw":false,"revision":"9f9896c3895d"},{"id":"g4-angles-in-parallel-lines-q01","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":4,"tier":"F","calc":false,"text":"

AB and CD are parallel straight lines.

A straight line crosses AB at E and crosses CD at F. On line AB, A is to the left of E and B is to the right of E. On line CD, C is to the left of F and D is to the right of F.

Angle AEF = 62°.

Work out the size of angle CFE.

Give a reason for each stage of your working.

","answerType":"exact","answer":"118°","answerDisplay":null,"accept":["118","118 degrees"],"parts":[],"solution":["Alternate angles lie between parallel lines on opposite sides of the crossing line. Co-interior angles lie between the parallel lines on the same side of the crossing line.","Angle EFD = 62° because angle AEF and angle EFD are alternate angles, and alternate angles are equal","Angle CFE = 180° − 62° = 118° because angles on a straight line add up to 180°","(Alternatively: angle AEF and angle CFE are co-interior angles, so angle CFE = 180° − 62° = 118°)"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[-4,3],"label":"A","style":"none","labelPos":"above"},{"at":[4,3],"label":"B","style":"none","labelPos":"above"},{"at":[-4,0],"label":"C","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"D","style":"none","labelPos":"below-right"},{"at":[0,3],"label":"E","style":"dot","labelPos":"above"},{"at":[-1.5951282949844363,0],"label":"F","style":"dot","labelPos":"below-left"}],"segments":[{"from":[-4,3],"to":[4,3],"parallel":1},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[1.5951282949844363,6],"to":[-3.1902565899688726,-3]}],"angles":[{"at":[0,3],"from":[-4,3],"to":[-1.5951282949844363,0],"label":"62°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a93191ded6d0"},{"id":"g4-angles-in-parallel-lines-q02","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":4,"tier":"F","calc":true,"text":"

PQ and RS are parallel straight lines.

A straight line crosses PQ at E and crosses RS at F. On line PQ, P is to the left of E and Q is to the right of E. On line RS, R is to the left of F and S is to the right of F.

Angle QEF = 74°.

  1. Work out the size of angle RFE. Give a reason for your answer.
  2. Work out the size of angle EFS. Give a reason for your answer.
","answerType":"multi","answer":"a) 74° b) 106°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"74°"},{"label":"b","answer":"106°"}],"solution":["a) Angle RFE = 74° because angle QEF and angle RFE are alternate angles, and alternate angles are equal","b) Angle EFS = 180° − 74° = 106° because angles on a straight line add up to 180°","(For part b you could also say angle QEF and angle EFS are co-interior angles, which add up to 180°)"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[-4,3],"label":"P","style":"none","labelPos":"above"},{"at":[4,3],"label":"Q","style":"none","labelPos":"above"},{"at":[-4,0],"label":"R","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"S","style":"none","labelPos":"below-right"},{"at":[0,3],"label":"E","style":"dot","labelPos":"above"},{"at":[0.8602361572764238,0],"label":"F","style":"dot","labelPos":"below-right"}],"segments":[{"from":[-4,3],"to":[4,3],"parallel":1},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[-0.8602361572764238,6],"to":[1.7204723145528475,-3]}],"angles":[{"at":[0,3],"from":[4,3],"to":[0.8602361572764238,0],"label":"74°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"73a3d40ab5ce"},{"id":"g4-angles-in-parallel-lines-q03","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":4,"tier":"F","calc":true,"text":"

AB and CD are parallel straight lines.

A straight line GEFH crosses AB at E and crosses CD at F, where G is above the line AB and H is below the line CD. On line AB, A is to the left of E and B is to the right of E. On line CD, C is to the left of F and D is to the right of F.

Angle GEB = 63°.

  1. Work out the size of angle EFD. Give a reason for your answer.
  2. Work out the size of angle CFH. Give a reason for your answer.
","answerType":"multi","answer":"a) 63° b) 63°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"63°"},{"label":"b","answer":"63°"}],"solution":["a) Angle EFD = 63° because angle GEB and angle EFD are corresponding angles, and corresponding angles are equal","b) Angle CFH = 63° because angle CFH and angle EFD are vertically opposite angles, and vertically opposite angles are equal"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[-4,3],"label":"A","style":"none","labelPos":"above"},{"at":[4,3],"label":"B","style":"none","labelPos":"above"},{"at":[-4,0],"label":"C","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"D","style":"none","labelPos":"below-right"},{"at":[0,3],"label":"E","style":"dot","labelPos":"above"},{"at":[-1.5285763484832866,0],"label":"F","style":"dot","labelPos":"below-left"},{"at":[1.5285763484832866,6],"label":"G","style":"none","labelPos":"above"},{"at":[-3.057152696966573,-3],"label":"H","style":"none","labelPos":"below-left"}],"segments":[{"from":[-4,3],"to":[4,3],"parallel":1},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[1.5285763484832866,6],"to":[-3.057152696966573,-3]}],"angles":[{"at":[0,3],"from":[1.5285763484832866,6],"to":[4,3],"label":"63°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"b6c0e9e363a6"},{"id":"g4-angles-in-parallel-lines-q04","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":4,"tier":"FH","calc":true,"text":"

DE and FG are parallel straight lines.

Triangle ABC is drawn so that A lies on DE, and B and C both lie on FG.

D is to the left of A and E is to the right of A.

Angle DAB = 48° and angle ACB = 63°.

Work out the size of angle BAC.

Give a reason for each stage of your working.

","answerType":"exact","answer":"69°","answerDisplay":null,"accept":["69","69 degrees"],"parts":[],"solution":["Angle ABC = 48° because angle DAB and angle ABC are alternate angles (DE is parallel to FG), and alternate angles are equal","Angle BAC = 180° − 48° − 63° = 69° because angles in a triangle add up to 180°"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,4],"label":"A","style":"none","labelPos":"above"},{"at":[-2,0],"label":"B","style":"none","labelPos":"below-left"},{"at":[2,0],"label":"C","style":"none","labelPos":"below-right"},{"at":[-4,4],"label":"D","style":"none","labelPos":"above"},{"at":[4,4],"label":"E","style":"none","labelPos":"above"},{"at":[-4,0],"label":"F","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"G","style":"none","labelPos":"below-right"}],"segments":[{"from":[-4,4],"to":[4,4],"parallel":1},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[0,4],"to":[-2,0],"ticks":0},{"from":[0,4],"to":[2,0],"ticks":0}],"angles":[{"at":[0,4],"from":[-4,4],"to":[-2,0],"label":"48°"},{"at":[2,0],"from":[0,4],"to":[-2,0],"label":"63°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ddf4069a6b3b"},{"id":"g4-angles-in-parallel-lines-q05","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

A straight line crosses line AB at the point P and crosses line CD at the point Q, where P is above Q. On line AB, B is to the right of P. On line CD, D is to the right of Q.

Angle QPB, between PQ and PB and inside the region between the two lines, is 104°.

Angle PQD, between QP and QD and inside the region between the two lines, is 76°.

Show that AB is parallel to CD.

Give reasons for your answer.

","answerType":"open","answer":"104° + 76° = 180°; angle QPB and angle PQD are co-interior angles; co-interior angles adding to 180° means the lines are parallel","answerDisplay":null,"accept":[],"parts":[],"solution":["Angle QPB and angle PQD are co-interior (allied) angles between the lines AB and CD","104° + 76° = 180°","Co-interior angles add up to 180° only when the two lines are parallel, so AB is parallel to CD"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[-4,3],"label":"A","style":"none","labelPos":"above"},{"at":[4,3],"label":"B","style":"none","labelPos":"above"},{"at":[-4,0],"label":"C","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"D","style":"none","labelPos":"below-right"},{"at":[0,3],"label":"P","style":"dot","labelPos":"above"},{"at":[-0.7479840085295426,0],"label":"Q","style":"dot","labelPos":"below-left"}],"segments":[{"from":[-4,3],"to":[4,3],"parallel":0},{"from":[-4,0],"to":[4,0],"parallel":0},{"from":[0.7479840085295426,6],"to":[-1.4959680170590852,-3]}],"angles":[{"at":[0,3],"from":[-0.7479840085295426,0],"to":[4,3],"label":"104°"},{"at":[-0.7479840085295426,0],"from":[0,3],"to":[4,0],"label":"76°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"386ce59828c2"},{"id":"g4-angles-in-parallel-lines-q06","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":5,"tier":"F","calc":true,"text":"

AB and CD are parallel straight lines.

E lies on AB. F and G both lie on CD, with F to the left of G.

A is to the left of E and B is to the right of E.

EF = EG and angle AEF = 64°.

Work out the size of angle FEG.

Give a reason for each stage of your working.

","answerType":"exact","answer":"52°","answerDisplay":null,"accept":["52","52 degrees"],"parts":[],"solution":["Angle EFG = 64° because angle AEF and angle EFG are alternate angles (AB is parallel to CD), and alternate angles are equal","Angle EGF = angle EFG = 64° because triangle EFG is isosceles (EF = EG), and base angles of an isosceles triangle are equal","Angle FEG = 180° − 64° − 64° = 52° because angles in a triangle add up to 180°"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,4],"label":"E","style":"none","labelPos":"above"},{"at":[-2,0],"label":"F","style":"none","labelPos":"below-left"},{"at":[2,0],"label":"G","style":"none","labelPos":"below-right"},{"at":[-4,4],"label":"A","style":"none","labelPos":"above"},{"at":[4,4],"label":"B","style":"none","labelPos":"above"},{"at":[-4,0],"label":"C","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"D","style":"none","labelPos":"below-right"}],"segments":[{"from":[-4,4],"to":[4,4],"parallel":1},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[0,4],"to":[-2,0],"ticks":1},{"from":[0,4],"to":[2,0],"ticks":1}],"angles":[{"at":[0,4],"from":[-4,4],"to":[-2,0],"label":"64°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"81c67e8d8c2f"},{"id":"g4-angles-in-parallel-lines-q07","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":5,"tier":"F","calc":false,"text":"

ABCD is a trapezium in which AB is parallel to DC.

The diagonal BD is drawn.

Angle DAB = 74° and angle ADB = 68°.

  1. Work out the size of angle ABD. Give a reason for your answer.
  2. Work out the size of angle BDC. Give a reason for your answer.
","answerType":"multi","answer":"a) 38° b) 38°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"38°"},{"label":"b","answer":"38°"}],"solution":["a) Angle ABD = 180° − 74° − 68° = 38° because angles in a triangle add up to 180°","b) Angle BDC = angle ABD = 38° because they are alternate angles (AB is parallel to DC, with BD as the transversal), and alternate angles are equal"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[7,4],"label":"C","style":"none","labelPos":"above"},{"at":[1,4],"label":"D","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"parallel":1},{"from":[6,0],"to":[7,4]},{"from":[7,4],"to":[1,4],"parallel":1},{"from":[1,4],"to":[0,0]},{"from":[6,0],"to":[1,4]}],"angles":[{"at":[0,0],"from":[1,4],"to":[6,0],"label":"74°"},{"at":[1,4],"from":[0,0],"to":[6,0],"label":"68°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"3b6f105568d6"},{"id":"g4-angles-in-parallel-lines-q08","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

AB and CD are parallel straight lines.

E lies on AB. F and G both lie on CD, with F to the left of G.

A is to the left of E and B is to the right of E.

Angle AEF = 67° and angle BEG = 67°.

Show that triangle EFG is an isosceles triangle.

Give a reason for each stage of your working.

","answerType":"open","answer":"Angle EFG = 67° (alternate with AEF) and angle EGF = 67° (alternate with BEG); two equal base angles mean triangle EFG is isosceles","answerDisplay":null,"accept":[],"parts":[],"solution":["Angle EFG = 67° because angle AEF and angle EFG are alternate angles (AB is parallel to CD), and alternate angles are equal","Angle EGF = 67° because angle BEG and angle EGF are alternate angles, and alternate angles are equal","Angle EFG = angle EGF, so triangle EFG has two equal base angles","A triangle with two equal angles is isosceles, so triangle EFG is isosceles (with EF = EG)"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,4],"label":"E","style":"none","labelPos":"above"},{"at":[-2,0],"label":"F","style":"none","labelPos":"below-left"},{"at":[2,0],"label":"G","style":"none","labelPos":"below-right"},{"at":[-4,4],"label":"A","style":"none","labelPos":"above"},{"at":[4,4],"label":"B","style":"none","labelPos":"above"},{"at":[-4,0],"label":"C","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"D","style":"none","labelPos":"below-right"}],"segments":[{"from":[-4,4],"to":[4,4],"parallel":1},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[0,4],"to":[-2,0],"ticks":0},{"from":[0,4],"to":[2,0],"ticks":0}],"angles":[{"at":[0,4],"from":[-4,4],"to":[-2,0],"label":"67°"},{"at":[0,4],"from":[4,4],"to":[2,0],"label":"67°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"0b402cab9c37"},{"id":"g4-angles-in-parallel-lines-q09","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"

AB and CD are parallel straight lines. A straight line crosses them both.

Two co-interior angles between the parallel lines are (3x + 20)° and (2x − 15)°.

Work out the size of the smaller of these two angles.

You must show all your working.

","answerType":"exact","answer":"55°","answerDisplay":null,"accept":["55","55 degrees"],"parts":[],"solution":["Co-interior angles between parallel lines add up to 180°, so (3x + 20) + (2x − 15) = 180","5x + 5 = 180, so 5x = 175 and x = 35","The two angles are 3(35) + 20 = 125° and 2(35) − 15 = 55°","Check: 125° + 55° = 180°. The smaller angle is 55°"],"diagram":null,"draw":false,"revision":"0cf5de2b1fea"},{"id":"g4-angles-in-parallel-lines-q10","topicId":"g4-angles-in-parallel-lines","topic":"Angles in Parallel Lines","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

AB and CD are parallel straight lines, with AB above CD.

E lies on AB and G lies on CD. F is a point between the two lines, to the right of both E and G, so that EFG forms a zigzag.

Angle BEF = 38° and angle DGF = 44°, where B is to the right of E and D is to the right of G.

Work out the size of angle EFG.

Give a reason for each stage of your working.

","answerType":"exact","answer":"82°","answerDisplay":null,"accept":["82","82 degrees"],"parts":[],"solution":["Draw a line through F parallel to AB and CD, splitting angle EFG into an upper part and a lower part","The upper part of angle EFG = 38° because it is alternate to angle BEF, and alternate angles are equal","The lower part of angle EFG = 44° because it is alternate to angle DGF, and alternate angles are equal","Angle EFG = 38° + 44° = 82°"],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[-1,5],"label":"A","style":"none","labelPos":"above"},{"at":[7,5],"label":"B","style":"none","labelPos":"above"},{"at":[-1,0],"label":"C","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"D","style":"none","labelPos":"below-right"},{"at":[0,5],"label":"E","style":"none","labelPos":"above"},{"at":[5,2.2],"label":"F","style":"none","labelPos":"below-right"},{"at":[0,0],"label":"G","style":"none","labelPos":"below-left"}],"segments":[{"from":[-1,5],"to":[7,5],"parallel":1},{"from":[-1,0],"to":[7,0],"parallel":1},{"from":[0,5],"to":[5,2.2]},{"from":[0,0],"to":[5,2.2]},{"from":[-1,2.2],"to":[7,2.2],"parallel":1,"dashed":true,"answer":true}],"angles":[{"at":[0,5],"from":[7,5],"to":[5,2.2],"label":"38°"},{"at":[0,0],"from":[7,0],"to":[5,2.2],"label":"44°"},{"at":[5,2.2],"from":[0,5],"to":[0,0],"label":"x","italic":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"71f4fec198e2"},{"id":"g4-angles-in-polygons-q01","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Each exterior angle of a regular polygon is 30°.

Work out the number of sides of the polygon.

","answerType":"exact","answer":"12","answerDisplay":null,"accept":["12 sides"],"parts":[],"solution":["The exterior angles of any polygon add up to 360°","Number of sides = 360 ÷ 30 = 12","Walking once around a convex polygon turns you through one complete turn, 360°. In a regular polygon each turn is equal, so divide 360° by the exterior angle."],"diagram":null,"draw":false,"revision":"78ea236346c0"},{"id":"g4-angles-in-polygons-q02","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

Work out the size of each interior angle of a regular decagon.

","answerType":"exact","answer":"144°","answerDisplay":null,"accept":["144","144 degrees"],"parts":[],"solution":["Draw diagonals from one vertex of a convex decagon: they divide it into 10 − 2 = 8 triangles. Each triangle contributes 180°, explaining the interior-angle sum formula.","A decagon has 10 sides, so the sum of its interior angles is (10 − 2) × 180° = 1440°","Each interior angle = 1440 ÷ 10 = 144°","(Alternatively: each exterior angle = 360 ÷ 10 = 36°, so each interior angle = 180 − 36 = 144°)"],"diagram":null,"draw":false,"revision":"29ea076d3c74"},{"id":"g4-angles-in-polygons-q03","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

Each interior angle of a regular polygon is 156°.

Work out the number of sides of the polygon.

","answerType":"exact","answer":"15","answerDisplay":null,"accept":["15 sides"],"parts":[],"solution":["Each exterior angle = 180° − 156° = 24° because an interior angle and its exterior angle lie on a straight line","The exterior angles of any polygon add up to 360°","Number of sides = 360 ÷ 24 = 15"],"diagram":null,"draw":false,"revision":"e9be76f12915"},{"id":"g4-angles-in-polygons-q04","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

In a regular polygon, each interior angle is 100° more than each exterior angle.

Work out the number of sides of the polygon.

","answerType":"exact","answer":"9","answerDisplay":null,"accept":["9 sides"],"parts":[],"solution":["Let the exterior angle be e. Then the interior angle is e + 100, and e + (e + 100) = 180 because they lie on a straight line","2e + 100 = 180, so e = 40°","Number of sides = 360 ÷ 40 = 9"],"diagram":null,"draw":false,"revision":"695112570959"},{"id":"g4-angles-in-polygons-q05","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

The interior angles of a pentagon are x°, (x + 30)°, (x + 40)°, 2x° and (2x + 50)°.

Work out the size of the largest interior angle of the pentagon.

","answerType":"exact","answer":"170°","answerDisplay":null,"accept":["170","170 degrees"],"parts":[],"solution":["The interior angles of a pentagon add up to (5 − 2) × 180° = 540°","x + (x + 30) + (x + 40) + 2x + (2x + 50) = 540, so 7x + 120 = 540","7x = 420, so x = 60","The angles are 60°, 90°, 100°, 120° and 170°. The largest angle is 2x + 50 = 170°"],"diagram":null,"draw":false,"revision":"33143b50c88c"},{"id":"g4-angles-in-polygons-q06","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":4,"tier":"F","calc":false,"text":"

In a pentagon, four of the interior angles are equal.

The fifth interior angle is 60° larger than each of the equal angles.

Work out the size of the largest interior angle of the pentagon.

","answerType":"exact","answer":"156°","answerDisplay":null,"accept":["156","156 degrees"],"parts":[],"solution":["The interior angles of a pentagon add up to 540°","Let each equal angle be a. Then 4a + (a + 60) = 540","5a + 60 = 540, so 5a = 480 and a = 96°","The largest angle = 96 + 60 = 156°. Check: 4 × 96 + 156 = 540°"],"diagram":null,"draw":false,"revision":"06a3ee816af2"},{"id":"g4-angles-in-polygons-q07","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":4,"tier":"FH","calc":true,"text":"

ABCDEF is a hexagon.

Angle A = angle D = 90°.

The other four interior angles are all equal.

Work out the size of angle B.

","answerType":"exact","answer":"135°","answerDisplay":null,"accept":["135","135 degrees"],"parts":[],"solution":["The interior angles of a hexagon add up to (6 − 2) × 180° = 720°","The four equal angles add up to 720 − 90 − 90 = 540°","Angle B = 540 ÷ 4 = 135°"],"diagram":null,"draw":false,"revision":"62877d686a1a"},{"id":"g4-angles-in-polygons-q08","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":4,"tier":"FH","calc":false,"text":"

A regular pentagon and a square have the same side length.

They are joined along one common edge PQ, on opposite sides of PQ, so they do not overlap.

At the vertex P, the pentagon has an interior angle and the square has an interior angle.

Work out the size of the angle at P that is outside both the pentagon and the square.

","answerType":"exact","answer":"162°","answerDisplay":null,"accept":["162","162 degrees"],"parts":[],"solution":["Each interior angle of a regular pentagon = 540 ÷ 5 = 108°","Each interior angle of a square = 90°","Angles around the point P add up to 360°","The remaining angle = 360 − 108 − 90 = 162°"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[4,0],[5.23606797749979,3.804226065180614],[2.0000000000000004,6.155367074350507],[-1.2360679774997894,3.804226065180615]],"labels":["P","Q","","",""]},{"pts":[[0,0],[4,0],[4,-4],[0,-4.000000000000001]],"labels":["","","",""]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"dd7a6d719cc4"},{"id":"g4-angles-in-polygons-q09","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

The sum of the interior angles of a polygon is 1800°.

  1. Work out the number of sides of the polygon.
  2. The polygon is regular. Work out the size of each interior angle.
","answerType":"multi","answer":"a) 12 b) 150°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"12"},{"label":"b","answer":"150°"}],"solution":["a) Sum of interior angles = (n − 2) × 180, so (n − 2) × 180 = 1800","n − 2 = 10, so n = 12","b) Each interior angle = 1800 ÷ 12 = 150°"],"diagram":null,"draw":false,"revision":"40a017ec4daa"},{"id":"g4-angles-in-polygons-q10","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":4,"tier":"F","calc":false,"text":"

ABCDE is a regular pentagon.

The diagonal AC is drawn.

Work out the size of angle ACD.

Give a reason for each stage of your working.

","answerType":"exact","answer":"72°","answerDisplay":null,"accept":["72","72 degrees"],"parts":[],"solution":["Each interior angle of a regular pentagon = 540 ÷ 5 = 108°","In triangle ABC, AB = BC because the pentagon is regular, so the triangle is isosceles","Angle BCA = (180 − 108) ÷ 2 = 36° because angles in a triangle add up to 180° and base angles of an isosceles triangle are equal","Angle ACD = angle BCD − angle BCA = 108 − 36 = 72°"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[4,0],[5.23606797749979,3.804226065180614],[2.0000000000000004,6.155367074350507],[-1.2360679774997894,3.804226065180615]],"labels":["A","B","C","D","E"]}],"segments":[{"from":[0,0],"to":[5.23606797749979,3.804226065180614]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ea7be1986b91"},{"id":"g4-angles-in-polygons-q11","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":11,"marks":5,"tier":"H","calc":false,"text":"

In a regular polygon, the ratio of each interior angle to each exterior angle is 13 : 2.

  1. Work out the number of sides of the polygon.
  2. Work out the sum of the interior angles of the polygon.
","answerType":"multi","answer":"a) 15 b) 2340°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"15"},{"label":"b","answer":"2340°"}],"solution":["a) An interior angle and its exterior angle add up to 180°, in the ratio 13 : 2, so the exterior angle = 180 × 215 = 24° and the interior angle = 156°","Number of sides = 360 ÷ 24 = 15","b) Sum of interior angles = (15 − 2) × 180 = 2340°. Check: 15 × 156 = 2340°"],"diagram":null,"draw":false,"revision":"5fa87ee307cd"},{"id":"g4-angles-in-polygons-q12","topicId":"g4-angles-in-polygons","topic":"Angles in Polygons","grade":"4","topicGrade":"4","strand":"G","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

Regular pentagon ABCDE and regular hexagon ABFGHI have the same side length.

They share the common edge AB and lie on opposite sides of AB, so they do not overlap.

  1. Work out the size of angle EAI.
  2. Work out the size of angle AEI.
","answerType":"multi","answer":"a) 132° b) 24°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"132°"},{"label":"b","answer":"24°"}],"solution":["a) Interior angle of a regular pentagon = 540 ÷ 5 = 108°; interior angle of a regular hexagon = 720 ÷ 6 = 120°","Angles around the point A add up to 360°, so angle EAI = 360 − 108 − 120 = 132°","b) AE and AI are both sides of the polygons, so AE = AI and triangle AEI is isosceles","Angle AEI = (180 − 132) ÷ 2 = 24°"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[4,0],[5.23606797749979,3.804226065180614],[2.0000000000000004,6.155367074350507],[-1.2360679774997894,3.804226065180615]],"labels":["A","B","C","D","E"]},{"pts":[[0,0],[4,0],[6,-3.4641016151377544],[4.000000000000001,-6.928203230275509],[8.881784197001252e-16,-6.92820323027551],[-2.000000000000001,-3.4641016151377557]],"labels":["","","F","G","H","I"]}],"segments":[{"from":[-1.2360679774997894,3.804226065180615],"to":[-2.000000000000001,-3.4641016151377557]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"66ad469f853a"},{"id":"g4-averages-from-frequency-tables-q01","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

The table shows the number of goals scored by a hockey team in each of its matches last season.

Number of goalsFrequency
07
111
28
34
  1. Write down the number of matches the team played.
  2. Ria says, "The median number of goals is 1.5, because 1.5 is halfway between 0 and 3." Explain why Ria is wrong.
","answerType":"multi","answer":"(a) 30 (b) The median is the middle of the 30 ordered results, the 15th and 16th values, which are both 1, so the median is 1","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"30"},{"label":"b","answer":"The median must come from the ordered data values, not halfway between the smallest and largest; the 15th and 16th values are both 1, so the median is 1"}],"solution":["Total matches = 7 + 11 + 8 + 4 = 30.","The cumulative frequencies are 7, 18, 26, 30.","The 15th and 16th values both lie in the group of matches with 1 goal, so the median is 1, not 1.5."],"diagram":null,"draw":false,"revision":"9fc4364c143e"},{"id":"g4-averages-from-frequency-tables-q02","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

The table shows the shoe sizes of the students in a year group.

Shoe sizeFrequency
45
59
612
78
86
  1. Write down the modal shoe size.
  2. Work out the number of students in the year group.
  3. Work out how many students have a shoe size of 7 or larger.
","answerType":"multi","answer":"(a) 6 (b) 40 (c) 14","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6"},{"label":"b","answer":"40"},{"label":"c","answer":"14"}],"solution":["The mode is the shoe size with the highest frequency: size 6 (frequency 12).","Total students = 5 + 9 + 12 + 8 + 6 = 40.","Size 7 or larger = 8 + 6 = 14."],"diagram":null,"draw":false,"revision":"869657ef1610"},{"id":"g4-averages-from-frequency-tables-q03","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

35 students were asked how many books they read last month. The table shows the results.

Number of booksFrequency
06
110
2x
37
44
  1. Work out the value of x.
  2. Write down the modal number of books.
","answerType":"multi","answer":"(a) x = 8 (b) 1","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8"},{"label":"b","answer":"1"}],"solution":["6 + 10 + 7 + 4 = 27.","x = 35 - 27 = 8.","The highest frequency is 10, for 1 book, so the mode is 1."],"diagram":null,"draw":false,"revision":"09ac395c5264"},{"id":"g4-averages-from-frequency-tables-q04","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"

The table shows the number of children in each of 20 families.

Number of childrenFrequency
03
18
26
33

Work out the mean number of children per family.

","answerType":"exact","answer":"1.45","answerDisplay":null,"accept":["29/20","1.45 children"],"parts":[],"solution":["Total children = (0 × 3) + (1 × 8) + (2 × 6) + (3 × 3) = 0 + 8 + 12 + 9 = 29.","Total families = 20.","Mean = 29 ÷ 20 = 1.45."],"diagram":null,"draw":false,"revision":"34fd631777b6"},{"id":"g4-averages-from-frequency-tables-q05","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

The table shows information about the time, t minutes, that 30 customers waited at a garage.

Time (t minutes)Frequency
0 < t ≤ 104
10 < t ≤ 209
20 < t ≤ 3012
30 < t ≤ 405

Work out an estimate for the mean waiting time.

","answerType":"exact","answer":"21 minutes","answerDisplay":null,"accept":["21","21 min"],"parts":[],"solution":["The individual measurements are unknown. Represent each class by its midpoint, halfway between its boundaries, then multiply by its frequency to estimate that class’s total. Divide the sum of these totals by the total frequency.","Use the midpoints 5, 15, 25 and 35.","(5 × 4) + (15 × 9) + (25 × 12) + (35 × 5) = 20 + 135 + 300 + 175 = 630.","Estimated mean = 630 ÷ 30 = 21 minutes."],"diagram":null,"draw":false,"revision":"62fc10eadab0"},{"id":"g4-averages-from-frequency-tables-q06","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":6,"marks":3,"tier":"FH","calc":true,"text":"

The table shows information about the masses, m kg, of 40 parcels handled by a courier.

Mass (m kg)Frequency
0 < m ≤ 27
2 < m ≤ 410
4 < m ≤ 614
6 < m ≤ 89

Calculate an estimate for the mean mass of the parcels.

","answerType":"exact","answer":"4.25 kg","answerDisplay":null,"accept":["4.25","17/4 kg"],"parts":[],"solution":["The individual measurements are unknown. Represent each class by its midpoint, halfway between its boundaries, then multiply by its frequency to estimate that class’s total. Divide the sum of these totals by the total frequency.","Use the midpoints 1, 3, 5 and 7.","(1 × 7) + (3 × 10) + (5 × 14) + (7 × 9) = 7 + 30 + 70 + 63 = 170.","Estimated mean = 170 ÷ 40 = 4.25 kg."],"diagram":null,"draw":false,"revision":"89226f418f29"},{"id":"g4-averages-from-frequency-tables-q07","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

The table shows the number of merit points earned by each of the students in a class group of 40.

Number of merit pointsFrequency
04
19
211
310
46
  1. Find the median number of merit points.
  2. Work out the total number of merit points earned by the class.
","answerType":"multi","answer":"(a) 2 (b) 85","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2"},{"label":"b","answer":"85"}],"solution":["There are 40 students, so the median is halfway between the 20th and 21st values.","Cumulative frequencies: 4, 13, 24, 34, 40. The 20th and 21st values are both 2, so the median is 2.","Total points = (0 × 4) + (1 × 9) + (2 × 11) + (3 × 10) + (4 × 6) = 0 + 9 + 22 + 30 + 24 = 85."],"diagram":null,"draw":false,"revision":"d28f207e3415"},{"id":"g4-averages-from-frequency-tables-q08","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

The table shows information about the lengths, l cm, of 40 courgettes grown on an allotment.

Length (l cm)Frequency
10 < l ≤ 206
20 < l ≤ 3011
30 < l ≤ 4013
40 < l ≤ 5010
  1. Write down the class interval that contains the median.
  2. Work out an estimate for the mean length.
","answerType":"multi","answer":"(a) 30 < l <= 40 (b) 31.75 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"30 < l <= 40"},{"label":"b","answer":"31.75 cm"}],"solution":["The individual measurements are unknown. Represent each class by its midpoint, halfway between its boundaries, then multiply by its frequency to estimate that class’s total. Divide the sum of these totals by the total frequency.","Cumulative frequencies: 6, 17, 30, 40. The median lies between the 20th and 21st values, which are in the class 30 < l ≤ 40.","Use midpoints 15, 25, 35, 45.","(15 × 6) + (25 × 11) + (35 × 13) + (45 × 10) = 90 + 275 + 455 + 450 = 1270.","Estimated mean = 1270 ÷ 40 = 31.75 cm."],"diagram":null,"draw":false,"revision":"52bc0a8bd968"},{"id":"g4-averages-from-frequency-tables-q09","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

The table shows the number of items bought by each of 40 customers at a corner shop.

Number of itemsFrequency
112
215
39
44
  1. Work out the mean number of items bought per customer.
  2. Kai says, "The mode is 15." Explain the mistake Kai has made.
","answerType":"multi","answer":"(a) 2.125 (b) 15 is the largest frequency; the mode is the data value with the largest frequency, which is 2 items","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2.125"},{"label":"b","answer":"Kai has given the largest frequency instead of the data value; the mode is 2"}],"solution":["Total items = (1 × 12) + (2 × 15) + (3 × 9) + (4 × 4) = 12 + 30 + 27 + 16 = 85.","Mean = 85 ÷ 40 = 2.125.","The mode is the number of items with the highest frequency, so the mode is 2, not 15."],"diagram":null,"draw":false,"revision":"3355b9905ebe"},{"id":"g4-averages-from-frequency-tables-q10","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"

The table shows information about the times, t seconds, taken by 40 swimmers to complete one length of a pool.

Time (t seconds)Frequency
20 < t ≤ 308
30 < t ≤ 4015
40 < t ≤ 5012
50 < t ≤ 605
  1. Work out an estimate for the mean time.
  2. Explain why your answer to part (a) is an estimate.
","answerType":"multi","answer":"(a) 38.5 seconds (b) The exact data values are not known, so the midpoint of each class is used","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"38.5 seconds"},{"label":"b","answer":"Because the data is grouped, the exact times are unknown and midpoints are used"}],"solution":["The individual measurements are unknown. Represent each class by its midpoint, halfway between its boundaries, then multiply by its frequency to estimate that class’s total. Divide the sum of these totals by the total frequency.","Use midpoints 25, 35, 45, 55.","(25 × 8) + (35 × 15) + (45 × 12) + (55 × 5) = 200 + 525 + 540 + 275 = 1540.","Estimated mean = 1540 ÷ 40 = 38.5 seconds.","The mean is an estimate because the individual times within each class are unknown, so midpoints are used to represent them."],"diagram":null,"draw":false,"revision":"98cd004bd662"},{"id":"g4-averages-from-frequency-tables-q11","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":11,"marks":5,"tier":"H","calc":true,"text":"

The table shows information about the distances, d km, that 40 employees travel to work.

Distance (d km)Frequency
0 < d ≤ 59
5 < d ≤ 1016
10 < d ≤ 1511
15 < d ≤ 204
  1. Work out an estimate for the mean distance travelled.
  2. Write down the modal class.
  3. Work out the fraction of the employees who travel more than 10 km. Give your fraction in its simplest form.
","answerType":"multi","answer":"(a) 8.75 km (b) 5 < d <= 10 (c) 3/8","answerDisplay":"(a) 8.75 km (b) 5 < d <= 10 (c) 38","accept":[],"parts":[{"label":"a","answer":"8.75 km"},{"label":"b","answer":"5 < d <= 10"},{"label":"c","answer":"3/8","answerDisplay":"38"}],"solution":["The individual measurements are unknown. Represent each class by its midpoint, halfway between its boundaries, then multiply by its frequency to estimate that class’s total. Divide the sum of these totals by the total frequency.","Use midpoints 2.5, 7.5, 12.5, 17.5.","(2.5 × 9) + (7.5 × 16) + (12.5 × 11) + (17.5 × 4) = 22.5 + 120 + 137.5 + 70 = 350.","Estimated mean = 350 ÷ 40 = 8.75 km.","The modal class is the class with the highest frequency: 5 < d ≤ 10.","More than 10 km: 11 + 4 = 15 employees, so the fraction is 1540 = 38."],"diagram":null,"draw":false,"revision":"35b2c282b419"},{"id":"g4-averages-from-frequency-tables-q12","topicId":"g4-averages-from-frequency-tables","topic":"Averages from Frequency Tables","grade":"4","topicGrade":"4","strand":"S","difficulty":12,"marks":6,"tier":"H","calc":true,"text":"

The table shows information about the heights, h cm, of 40 students.

Height (h cm)Frequency
140 < h ≤ 1506
150 < h ≤ 16014
160 < h ≤ 17016
170 < h ≤ 1804
  1. Work out an estimate for the mean height.
  2. Work out the percentage of the students who are more than 160 cm tall.
  3. Explain why your answer to part (a) is only an estimate.
","answerType":"multi","answer":"(a) 159.5 cm (b) 50% (c) The exact heights are unknown because the data is grouped; midpoints are used","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"159.5 cm"},{"label":"b","answer":"50%"},{"label":"c","answer":"The data is grouped so the exact heights are unknown; each class is represented by its midpoint"}],"solution":["The individual measurements are unknown. Represent each class by its midpoint, halfway between its boundaries, then multiply by its frequency to estimate that class’s total. Divide the sum of these totals by the total frequency.","Use midpoints 145, 155, 165, 175.","(145 × 6) + (155 × 14) + (165 × 16) + (175 × 4) = 870 + 2170 + 2640 + 700 = 6380.","Estimated mean = 6380 ÷ 40 = 159.5 cm.","Students more than 160 cm tall are in the last two classes: 16 + 4 = 20.","20 out of 40 = 50%.","The mean is an estimate because the individual heights within each class are not known."],"diagram":null,"draw":false,"revision":"663424636f9a"},{"id":"g4-bearings-q01","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Write down the three-figure bearing of the direction

  1. due east
  2. due south-west
","answerType":"multi","answer":"a) 090° b) 225°","answerDisplay":null,"accept":["090 and 225","90 and 225"],"parts":[{"label":"a","answer":"090°"},{"label":"b","answer":"225°"}],"solution":["Bearings are measured clockwise from north and written with three figures.","Due east is a quarter turn clockwise from north, so the bearing is 090°.","Due south-west is halfway between south (180°) and west (270°), so the bearing is 225°."],"diagram":null,"draw":false,"revision":"6cab4882bd2b"},{"id":"g4-bearings-q02","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

The bearing of a ship S from a lighthouse L is 038°.

Work out the bearing of L from S.

","answerType":"exact","answer":"218°","answerDisplay":null,"accept":["218","218 degrees"],"parts":[],"solution":["The back bearing differs from the original bearing by 180°.","038° is less than 180°, so add 180°.","38 + 180 = 218, so the bearing of L from S is 218°.","Looking back along the same route reverses your direction by half a turn, 180°. The north lines at the two places are parallel."],"diagram":null,"draw":false,"revision":"207779e89bb3"},{"id":"g4-bearings-q03","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":2,"tier":"FH","calc":false,"text":"

The bearing of a port Q from a port P is 295°.

Work out the bearing of P from Q.

","answerType":"exact","answer":"115°","answerDisplay":null,"accept":["115","115 degrees"],"parts":[],"solution":["The back bearing differs from the original bearing by 180°.","295° is more than 180°, so subtract 180°.","295 - 180 = 115, so the bearing of P from Q is 115°."],"diagram":null,"draw":false,"revision":"90e244ada64e"},{"id":"g4-bearings-q04","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

The scale of a map is 1 cm to 2 km.

Village B is 8 km from village A on a bearing of 110°.

Mark the position of B on the map with a cross.

","answerType":"open","answer":"B marked 4 cm from A on a bearing of 110 degrees (accept 108 to 112 degrees, 3.9 cm to 4.1 cm)","answerDisplay":null,"accept":[],"parts":[],"solution":["Convert the real distance to a map distance: 8 km ÷ 2 km per cm = 4 cm.","Place a protractor at A with 0° pointing north and measure 110° clockwise.","Draw a faint line from A along this direction and mark B exactly 4 cm from A."],"diagram":{"type":"shape","notAccurate":false,"grid":true,"bounds":[[-2,-4],[6,4]],"points":[{"at":[0,0],"label":"A","style":"dot","labelPos":"above"},{"at":[3.7587704831436337,-1.3680805733026749],"label":"B","style":"cross","answer":true,"labelPos":"below-right"}],"north":[{"at":[0,0],"len":2}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"51cc4a4cfd75"},{"id":"g4-bearings-q05","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

A plane flies in a straight line from airport A to airport B on a bearing of 068°.

It then flies in a straight line from B to airport C on a bearing of 155°.

Work out the size of angle ABC.

","answerType":"exact","answer":"93°","answerDisplay":null,"accept":["93","93 degrees"],"parts":[],"solution":["The bearing of A from B is the back bearing of 068°, which is 068 + 180 = 248°.","At B, the direction to C is 155° and the direction to A is 248°.","Angle ABC = 248 - 155 = 93°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"above"},{"at":[4.635919272833937,1.87303296707956],"label":"B","style":"none","answer":false,"labelPos":"above"},{"at":[6.749010581537434,-2.6585059681036896],"label":"C","style":"none","answer":false,"labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[4.635919272833937,1.87303296707956],"answer":false},{"from":[4.635919272833937,1.87303296707956],"to":[6.749010581537434,-2.6585059681036896],"answer":false}],"angles":[{"at":[0,0],"from":[0,2],"to":[4.635919272833937,1.87303296707956],"label":"68°","answer":false},{"at":[4.635919272833937,1.87303296707956],"from":[4.635919272833937,3.87303296707956],"to":[6.749010581537434,-2.6585059681036896],"label":"155°","answer":false}],"north":[{"at":[0,0],"len":2},{"at":[4.635919272833937,1.87303296707956],"len":2,"answer":false}],"bounds":[[-2,-6],[10,6]],"grid":false,"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a876004ba7fc"},{"id":"g4-bearings-q06","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":3,"tier":"H","calc":false,"text":"

B is on a bearing of 060° from A.

C is on a bearing of 140° from A.

AB = AC.

Work out the bearing of C from B.

","answerType":"exact","answer":"190°","answerDisplay":null,"accept":["190","190 degrees"],"parts":[],"solution":["Angle BAC = 140 - 60 = 80°.","Triangle ABC is isosceles with AB = AC, so angle ABC = angle ACB = (180 - 80) ÷ 2 = 50°.","The bearing of A from B is the back bearing of 060°, which is 240°.","C lies 50° anticlockwise from the direction BA, so the bearing of C from B is 240 - 50 = 190°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"above"},{"at":[4.330127018922194,2.4999999999999996],"label":"B","style":"none","labelPos":"above"},{"at":[3.2139380484326967,-3.83022221559489],"label":"C","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[4.330127018922194,2.4999999999999996],"ticks":1},{"from":[0,0],"to":[3.2139380484326967,-3.83022221559489],"ticks":1},{"from":[4.330127018922194,2.4999999999999996],"to":[3.2139380484326967,-3.83022221559489]}],"angles":[{"at":[0,0],"from":[0,3],"to":[4.330127018922194,2.4999999999999996],"label":"060°","r":1.3},{"at":[0,0],"from":[0,3],"to":[3.2139380484326967,-3.83022221559489],"label":"140°","r":2.2}],"north":[{"at":[0,0],"len":3},{"at":[4.330127018922194,2.4999999999999996],"len":2}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"5d34460d870e"},{"id":"g4-bearings-q07","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

A map has a scale of 1 : 50 000.

On the map, the distance from a church to a school is 6 cm.

The bearing of the school from the church is 132°.

  1. Work out the real distance from the church to the school. Give your answer in kilometres.
  2. Work out the bearing of the church from the school.
","answerType":"multi","answer":"a) 3 km b) 312°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3 km"},{"label":"b","answer":"312°"}],"solution":["a) Real distance = 6 cm × 50 000 = 300 000 cm.","300 000 cm = 3000 m = 3 km.","b) The back bearing differs by 180°: 132 + 180 = 312°."],"diagram":null,"draw":false,"revision":"2b902713cc74"},{"id":"g4-bearings-q08","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":5,"tier":"FH","calc":true,"text":"

A boat leaves a harbour H and sails 10 km on a bearing of 070° to a buoy B.

It then sails 8 km on a bearing of 150° to a point C.

  1. Using a scale of 1 cm to 2 km, make an accurate scale drawing of the journey.
  2. Use your drawing to find the distance and the bearing of C from H.
","answerType":"open","answer":"Accurate drawing (HB = 5 cm at 070 degrees, BC = 4 cm at 150 degrees); distance of C from H = 13.8 to 13.9 km; bearing of C from H = 104 to 106 degrees","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"HB drawn 5 cm at 070°, BC drawn 4 cm at 150°"},{"label":"b","answer":"about 13.9 km on a bearing of about 105°"}],"solution":["Convert to map lengths: 10 km = 5 cm and 8 km = 4 cm.","From H, draw a north line, measure 070° clockwise and draw HB = 5 cm.","From B, draw a north line, measure 150° clockwise and draw BC = 4 cm.","Measure HC: it is about 6.9 cm, which is 6.9 × 2 = 13.8 to 13.9 km.","Measure the angle at H from north to HC: about 105°, so the bearing of C from H is about 105°."],"diagram":{"type":"shape","notAccurate":false,"points":[{"at":[0,0],"label":"H","style":"none","labelPos":"above"},{"at":[4.698463103929543,1.7101007166283435],"label":"B","style":"none","answer":true,"labelPos":"above"},{"at":[6.698463103929543,-1.7540008985094109],"label":"C","style":"none","answer":true,"labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[4.698463103929543,1.7101007166283435],"label":"10 km","answer":true},{"from":[4.698463103929543,1.7101007166283435],"to":[6.698463103929543,-1.7540008985094109],"label":"8 km","answer":true}],"angles":[{"at":[0,0],"from":[0,2],"to":[4.698463103929543,1.7101007166283435],"label":"70°","answer":true},{"at":[4.698463103929543,1.7101007166283435],"from":[4.698463103929543,3.7101007166283435],"to":[6.698463103929543,-1.7540008985094109],"label":"150°","answer":true}],"north":[{"at":[0,0],"len":2},{"at":[4.698463103929543,1.7101007166283435],"len":2,"answer":true}],"bounds":[[-2,-6],[10,6]],"grid":true,"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"77b8d51e9f05"},{"id":"g4-bearings-q09","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

B is 15 km due east of A.

C is 9 km due south of B.

  1. Work out the distance AC. Give your answer correct to 1 decimal place.
  2. Work out the bearing of C from A. Give your answer correct to the nearest degree.
","answerType":"multi","answer":"a) 17.5 km b) 121°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"17.5 km"},{"label":"b","answer":"121°"}],"solution":["Triangle ABC has a right angle at B, with AB = 15 km and BC = 9 km.","a) AC² = 15² + 9² = 225 + 81 = 306, so AC = √306 = 17.492... = 17.5 km (1 dp).","b) The angle at A below due east satisfies tan x = 9 ÷ 15 = 0.6, so x = 30.96...°.","The bearing of C from A = 090 + 30.96... = 120.96...° = 121° to the nearest degree."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"above"},{"at":[7.5,0],"label":"B","style":"none","labelPos":"above"},{"at":[7.5,-4.5],"label":"C","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[7.5,0],"label":"15 km"},{"from":[7.5,0],"to":[7.5,-4.5],"label":"9 km"},{"from":[0,0],"to":[7.5,-4.5]}],"angles":[],"north":[{"at":[0,0],"len":2}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"859fba0dbe56"},{"id":"g4-bearings-q10","topicId":"g4-bearings","topic":"Bearings","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

A ship sails from a port P on a bearing of 052° for 9 km to a point Q.

It then sails on a bearing of 148° for 12 km to a point R.

  1. Work out the distance PR. Give your answer correct to 1 decimal place.
  2. Work out the bearing of R from P. Give your answer correct to the nearest degree.
","answerType":"multi","answer":"a) 14.2 km b) 109°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"14.2 km"},{"label":"b","answer":"109°"}],"solution":["At Q, the bearing of P is 052 + 180 = 232°, so angle PQR = 232 - 148 = 84°.","a) By the cosine rule, PR² = 9² + 12² - 2 × 9 × 12 × cos 84° = 225 - 216 × 0.1045... = 202.4...","PR = √202.4... = 14.227... = 14.2 km (1 dp).","b) By the sine rule, sin QPR ÷ 12 = sin 84° ÷ 14.227..., so sin QPR = 0.8389... and angle QPR = 57.0°.","The bearing of R from P = 052 + 57.0 = 109° to the nearest degree."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[3.546048391230249,2.770476638965462],"label":"Q","style":"none","answer":false,"labelPos":"above"},{"at":[6.725563976629479,-2.317811937973094],"label":"R","style":"none","answer":false,"labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[3.546048391230249,2.770476638965462],"label":"9 km","answer":false},{"from":[3.546048391230249,2.770476638965462],"to":[6.725563976629479,-2.317811937973094],"label":"12 km","answer":false}],"angles":[{"at":[0,0],"from":[0,2],"to":[3.546048391230249,2.770476638965462],"label":"52°","answer":false},{"at":[3.546048391230249,2.770476638965462],"from":[3.546048391230249,4.770476638965462],"to":[6.725563976629479,-2.317811937973094],"label":"148°","answer":false}],"north":[{"at":[0,0],"len":2},{"at":[3.546048391230249,2.770476638965462],"len":2,"answer":false}],"bounds":[[-2,-6],[10,6]],"grid":false,"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ddb68327683c"},{"id":"g4-compound-interest-and-depreciation-q01","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

Rosa puts £3000 into a savings account.

The account pays 2% compound interest per year.

Work out the value of her savings at the end of 2 years.

","answerType":"exact","answer":"£3121.20","answerDisplay":null,"accept":["3121.20","£3121.20","3121.2"],"parts":[],"solution":["Each year the balance becomes 102% of its previous value, so multiply by 1.02. In year 2 you earn interest on the new balance, including the first year’s interest.","End of year 1: 3000 × 1.02 = £3060","End of year 2: 3060 × 1.02 = £3121.20","Check with the multiplier method: 3000 × 1.022 = £3121.20"],"diagram":null,"draw":false,"revision":"07a1ba26b2f8"},{"id":"g4-compound-interest-and-depreciation-q02","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Ben buys a car for £16 000.

The value of the car depreciates by 15% each year.

Work out the value of the car at the end of 2 years.

","answerType":"exact","answer":"£11560","answerDisplay":null,"accept":["11560","£11,560","£11 560"],"parts":[],"solution":["End of year 1: 16 000 × 0.85 = £13 600","End of year 2: 13 600 × 0.85 = £11 560","Check: 16 000 × 0.852 = £11 560"],"diagram":null,"draw":false,"revision":"b76c95f9d5f3"},{"id":"g4-compound-interest-and-depreciation-q03","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

Zainab invests £2500 in an account paying 3% compound interest per year.

Work out the total interest she earns in 3 years. Give your answer to the nearest penny.

","answerType":"exact","answer":"£231.82","answerDisplay":null,"accept":["231.82","£231.82"],"parts":[],"solution":["End of year 1: 2500 × 1.03 = £2575","End of year 2: 2575 × 1.03 = £2652.25","End of year 3: 2652.25 × 1.03 = £2731.8175","Interest = 2731.8175 − 2500 = £231.82 to the nearest penny"],"diagram":null,"draw":false,"revision":"7b352a771e86"},{"id":"g4-compound-interest-and-depreciation-q04","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

Theo puts £4000 into a savings account for 2 years.

The account pays 4% compound interest in the first year and 2% compound interest in the second year.

Work out the value of the account at the end of 2 years.

","answerType":"exact","answer":"£4243.20","answerDisplay":null,"accept":["4243.20","£4243.20","4243.2"],"parts":[],"solution":["End of year 1: 4000 × 1.04 = £4160","End of year 2: 4160 × 1.02 = £4243.20"],"diagram":null,"draw":false,"revision":"dbcea794fab9"},{"id":"g4-compound-interest-and-depreciation-q05","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":5,"marks":3,"tier":"FH","calc":true,"text":"

A delivery van is bought for £24 500.

Its value depreciates by 18% each year.

Work out the value of the van at the end of 3 years. Give your answer to the nearest pound.

","answerType":"exact","answer":"£13509","answerDisplay":null,"accept":["13509","£13,509","£13 509"],"parts":[],"solution":["Multiplier for an 18% decrease is 0.82","Value = 24 500 × 0.823 = 24 500 × 0.551368 = £13 508.516","To the nearest pound this is £13 509"],"diagram":null,"draw":false,"revision":"42f7ba32ad74"},{"id":"g4-compound-interest-and-depreciation-q06","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":6,"marks":3,"tier":"FH","calc":true,"text":"

Freya buys a motorbike for £7200.

Its value depreciates by 12% each year.

Show that the value of the motorbike at the end of 3 years is less than £5000.

","answerType":"open","answer":"7200 x 0.88^3 = £4906.60 (to the nearest penny), which is less than £5000","answerDisplay":null,"accept":["£4906.60 < £5000","4906.5984"],"parts":[],"solution":["Multiplier for a 12% decrease is 0.88","Value = 7200 × 0.883 = 7200 × 0.681472 = £4906.5984","£4906.60 is less than £5000"],"diagram":null,"draw":false,"revision":"950ca868c300"},{"id":"g4-compound-interest-and-depreciation-q07","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

Idris puts £5000 into an account for 2 years.

The account pays 3% compound interest in the first year and r% compound interest in the second year.

At the end of 2 years the account is worth £5253.

Work out the value of r.

","answerType":"exact","answer":"2","answerDisplay":null,"accept":["r = 2","2%"],"parts":[],"solution":["End of year 1: 5000 × 1.03 = £5150","5253 ÷ 5150 = 1.02","So the second-year rate is 2%, giving r = 2","Check: 5150 × 1.02 = £5253"],"diagram":null,"draw":false,"revision":"b5ffb0650a06"},{"id":"g4-compound-interest-and-depreciation-q08","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

Meg has £2000 to invest for 3 years. She can choose one of two accounts.

Account A pays 2.5% compound interest every year.

Account B pays 3% compound interest in the first year, then 2.2% compound interest in each of the next 2 years.

Work out which account gives Meg more money at the end of 3 years, and by how much.

","answerType":"open","answer":"Account A by £2.14 (A gives £2153.78, B gives £2151.64)","answerDisplay":null,"accept":["Account A, £2.14","A by £2.14"],"parts":[],"solution":["Account A: 2000 × 1.0253 = 2000 × 1.076890625 = £2153.78","Account B: 2000 × 1.03 = £2060, then 2060 × 1.0222 = 2060 × 1.044484 = £2151.64","Account A gives more, by 2153.78 − 2151.64 = £2.14"],"diagram":null,"draw":false,"revision":"e1e031761654"},{"id":"g4-compound-interest-and-depreciation-q09","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

Ava invests £6000 at a rate of x% compound interest per year.

At the end of 2 years the investment is worth £6615.

Work out the value of x.

","answerType":"exact","answer":"5","answerDisplay":null,"accept":["x = 5","5%"],"parts":[],"solution":["6615 ÷ 6000 = 1.1025","The yearly multiplier is √1.1025 = 1.05","So the rate is 5%, giving x = 5","Check: 6000 × 1.05 = 6300, then 6300 × 1.05 = £6615"],"diagram":null,"draw":false,"revision":"5edc2dae932e"},{"id":"g4-compound-interest-and-depreciation-q10","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"

Tomasz puts £8000 into an account paying 4% compound interest per year.

At the end of the first year, after the interest has been added, he withdraws £1500.

He leaves the rest in the account for another year.

Work out how much is in the account at the end of the second year.

","answerType":"exact","answer":"£7092.80","answerDisplay":null,"accept":["7092.80","£7092.80","7092.8"],"parts":[],"solution":["End of year 1: 8000 × 1.04 = £8320","After the withdrawal: 8320 − 1500 = £6820","End of year 2: 6820 × 1.04 = £7092.80"],"diagram":null,"draw":false,"revision":"406dc489a877"},{"id":"g4-compound-interest-and-depreciation-q11","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":11,"marks":5,"tier":"H","calc":true,"text":"

Nina has £10 000 to invest for 3 years. She can choose one of two accounts.

Steady Saver pays 2.1% compound interest every year.

Flex Saver pays 3.5% compound interest in the first year, then 1.5% compound interest in each of the next 2 years.

Which account should Nina choose to get the most money after 3 years? You must show how much more it pays.

","answerType":"open","answer":"Flex Saver by £19.51 (Flex Saver £10662.83, Steady Saver £10643.32)","answerDisplay":null,"accept":["Flex Saver, £19.51"],"parts":[],"solution":["Steady Saver: 10 000 × 1.0213 = 10 000 × 1.064332261 = £10 643.32","Flex Saver: 10 000 × 1.035 = £10 350, then 10 350 × 1.0152 = 10 000 × 1.066282875 = £10 662.83","Flex Saver pays more, by 10 662.83 − 10 643.32 = £19.51","Nina should choose Flex Saver"],"diagram":null,"draw":false,"revision":"d055445f3b3c"},{"id":"g4-compound-interest-and-depreciation-q12","topicId":"g4-compound-interest-and-depreciation","topic":"Compound Interest and Depreciation","grade":"4","topicGrade":"4","strand":"R","difficulty":12,"marks":6,"tier":"H","calc":true,"text":"

Kira invests £8000 in an account paying 5% compound interest per year.

At the end of 2 years she uses all the money in the account to buy a car.

The value of the car depreciates by d% each year.

Exactly 2 years after she buys it, the car is worth £5644.80.

Work out the value of d.

","answerType":"exact","answer":"20","answerDisplay":null,"accept":["d = 20","20%"],"parts":[],"solution":["Value of the investment after 2 years: 8000 × 1.052 = 8000 × 1.1025 = £8820","So the car cost £8820","5644.80 ÷ 8820 = 0.64, so the 2-year depreciation multiplier is 0.64","Yearly multiplier = √0.64 = 0.8","A multiplier of 0.8 means a 20% decrease each year, so d = 20","Check: 8820 × 0.8 = 7056, then 7056 × 0.8 = £5644.80"],"diagram":null,"draw":false,"revision":"087fbafbba8a"},{"id":"g4-cylinders-q01","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":3,"tier":"F","calc":true,"text":"

A cylinder has a radius of 5 cm and a height of 12 cm.

Work out the volume of the cylinder.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"942 cm³","answerDisplay":null,"accept":["942","942cm3","942 cm^3"],"parts":[],"solution":["The circular cross-section has area πr². Multiplying by the perpendicular height gives the volume, just as base area × height does for a cuboid.","Volume of a cylinder = πr2h","Volume = π × 52 × 12 = 300π","Volume = 942.477... = 942 cm³ (3 significant figures)"],"diagram":null,"draw":false,"revision":"d9799b019a08"},{"id":"g4-cylinders-q02","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

A cylindrical tin has a diameter of 8 cm and a height of 10 cm.

Work out the volume of the tin.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"503 cm³","answerDisplay":null,"accept":["503","503cm3","503 cm^3"],"parts":[],"solution":["The radius = 8 ÷ 2 = 4 cm","Volume = πr2h = π × 42 × 10 = 160π","Volume = 502.654... = 503 cm³ (3 significant figures)"],"diagram":null,"draw":false,"revision":"23763bcd09a5"},{"id":"g4-cylinders-q03","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":4,"tier":"F","calc":true,"text":"

A cylindrical water butt has a radius of 25 cm and a height of 80 cm.

Work out the number of litres of water the water butt holds when it is full.

Give your answer to the nearest litre.

1 litre = 1000 cm3

","answerType":"exact","answer":"157 litres","answerDisplay":null,"accept":["157","157 l","157L"],"parts":[],"solution":["Volume = πr2h = π × 252 × 80 = 50000π cm³","Volume = 157079.6... cm³","157079.6 ÷ 1000 = 157.08 litres","The water butt holds 157 litres (nearest litre)"],"diagram":null,"draw":false,"revision":"c5e87c51ae6f"},{"id":"g4-cylinders-q04","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":4,"tier":"F","calc":false,"text":"

A solid cylinder has a radius of 6 cm and a height of 5 cm.

  1. Work out the volume of the cylinder. Give your answer in terms of π.
  2. Work out the total surface area of the cylinder. Give your answer in terms of π.
","answerType":"multi","answer":"a) 180π cm³ b) 132π cm²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"180π cm³"},{"label":"b","answer":"132π cm²"}],"solution":["a) Volume = πr2h = π × 36 × 5 = 180π cm³","b) The two circular ends have total area 2πr2 = 2 × π × 36 = 72π cm²","The curved surface has area 2πrh = 2 × π × 6 × 5 = 60π cm²","Total surface area = 72π + 60π = 132π cm²","Unrolling the curved surface produces a rectangle whose width is the circumference 2πr and whose height is h. Its area is therefore 2πrh."],"diagram":null,"draw":false,"revision":"c97004c7313b"},{"id":"g4-cylinders-q05","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

A cylindrical tank has a radius of 30 cm and a height of 100 cm.

The tank is empty.

Water flows into the tank at a rate of 20 litres per minute.

Work out how long it takes to fill the tank completely.

Give your answer in minutes, correct to 1 decimal place.

1 litre = 1000 cm3

","answerType":"exact","answer":"14.1 minutes","answerDisplay":null,"accept":["14.1","14.1 min","14.1 mins"],"parts":[],"solution":["Volume = πr2h = π × 302 × 100 = 90000π cm³","Volume = 282743.3... cm³ = 282.743... litres","Time = 282.743... ÷ 20 = 14.137... minutes","Time = 14.1 minutes (1 decimal place)"],"diagram":null,"draw":false,"revision":"3a51fa60d3bf"},{"id":"g4-cylinders-q06","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":4,"tier":"FH","calc":true,"text":"

A hall has 8 cylindrical pillars.

Each pillar has a radius of 0.25 m and a height of 3 m.

Ben paints the curved surface of every pillar.

One tin of paint covers 8 m2.

Work out the number of tins of paint Ben must buy.

","answerType":"exact","answer":"5","answerDisplay":null,"accept":["5 tins"],"parts":[],"solution":["Curved surface area of one pillar = 2πrh = 2 × π × 0.25 × 3 = 1.5π = 4.712... m²","Total area for 8 pillars = 8 × 1.5π = 12π = 37.699... m²","37.699... ÷ 8 = 4.71..., so 4 tins are not enough","Ben must buy 5 tins"],"diagram":null,"draw":false,"revision":"0989b53fb3c7"},{"id":"g4-cylinders-q07","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

A cylinder has a volume of 1500 cm3 and a radius of 6 cm.

Work out the height of the cylinder.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"13.3 cm","answerDisplay":null,"accept":["13.3","13.3cm"],"parts":[],"solution":["Volume = πr2h, so 1500 = π × 36 × h","h = 1500 ÷ (36π) = 1500 ÷ 113.097...","h = 13.262... = 13.3 cm (1 decimal place)"],"diagram":null,"draw":false,"revision":"1c7213959f0d"},{"id":"g4-cylinders-q08","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

A cylinder has a volume of 2000 cm3 and a height of 15 cm.

Work out the radius of the cylinder.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"6.5 cm","answerDisplay":null,"accept":["6.5","6.5cm"],"parts":[],"solution":["Volume = πr2h, so 2000 = π × r2 × 15","r2 = 2000 ÷ (15π) = 42.441...","r = √42.441... = 6.514... = 6.5 cm (1 decimal place)"],"diagram":null,"draw":false,"revision":"5251ebeb8172"},{"id":"g4-cylinders-q09","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

A cylindrical drinks dispenser has a radius of 20 cm and a height of 50 cm.

Priya wants to fill the dispenser completely with squash.

Squash is made by mixing cordial and water in the ratio 1 : 9 by volume.

Priya has 5.5 litres of cordial and an unlimited supply of water.

Does Priya have enough cordial to fill the dispenser? You must show your working.

1 litre = 1000 cm3

","answerType":"open","answer":"No - the dispenser holds 20000π ≈ 62.8 litres, which needs 62.8 ÷ 10 ≈ 6.28 litres of cordial, more than the 5.5 litres Priya has","answerDisplay":null,"accept":[],"parts":[],"solution":["Volume of the dispenser = πr2h = π × 202 × 50 = 20000π = 62831.8... cm³","62831.8... cm³ = 62.83... litres","In the ratio 1 : 9, cordial is 1 part out of 10, so cordial needed = 62.83... ÷ 10 = 6.28... litres","6.28 litres > 5.5 litres, so Priya does not have enough cordial"],"diagram":null,"draw":false,"revision":"f4508d5a81fe"},{"id":"g4-cylinders-q10","topicId":"g4-cylinders","topic":"Cylinders","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

A closed cylindrical oil tank stands on the ground.

The tank has a radius of 0.6 m and a height of 1.5 m.

Layla paints the curved surface and the top of the tank, but not the base.

1 litre of paint covers 2.5 m2.

Paint is sold in 1 litre tins. Each tin costs £8.75.

Work out the total cost of the tins of paint Layla must buy.

","answerType":"exact","answer":"£26.25","answerDisplay":null,"accept":["26.25"],"parts":[],"solution":["Curved surface area = 2πrh = 2 × π × 0.6 × 1.5 = 1.8π = 5.654... m²","Area of the top = πr2 = π × 0.36 = 1.131... m²","Total area = 5.654... + 1.131... = 6.785... m²","Paint needed = 6.785... ÷ 2.5 = 2.71... litres, so Layla must buy 3 tins","Total cost = 3 × £8.75 = £26.25"],"diagram":null,"draw":false,"revision":"70fa67c99a25"},{"id":"g4-expanding-and-factorising-q01","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":1,"marks":1,"tier":"H","calc":true,"text":"

Priya says that (5x − 3) − (2x − 7) = 3x − 10

Priya is wrong. Explain the mistake she has made.

","answerType":"open","answer":"She did not change the sign of -7 when subtracting; the answer should be 3x + 4","answerDisplay":null,"accept":["She subtracted 7 instead of adding 7","Subtracting -7 gives +7, so it should be 3x + 4"],"parts":[],"solution":["Subtracting the bracket changes the sign of every term inside it","(5x - 3) - (2x - 7) = 5x - 3 - 2x + 7 = 3x + 4","Priya kept -7 as -7 instead of changing it to +7"],"diagram":null,"draw":false,"revision":"433fafce4e67"},{"id":"g4-expanding-and-factorising-q02","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"
  1. Expand 4(2x − 5)
  2. Factorise 6y + 15
","answerType":"multi","answer":"a) 8x - 20; b) 3(2y + 5)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8x - 20"},{"label":"b","answer":"3(2y + 5)"}],"solution":["a) 4 times 2x = 8x and 4 times -5 = -20, so 8x - 20","b) The highest common factor of 6y and 15 is 3","6y + 15 = 3(2y + 5)","Factorising reverses expanding: divide every term by the common factor to find what belongs inside the bracket, then expand again to check."],"diagram":null,"draw":false,"revision":"40dcf595732d"},{"id":"g4-expanding-and-factorising-q03","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

Expand and simplify (x + 4)(x + 6)

","answerType":"exact","answer":"x^2 + 10x + 24","answerDisplay":null,"accept":["x² + 10x + 24","24 + 10x + x^2"],"parts":[],"solution":["(x + 4)(x + 6) = x^2 + 6x + 4x + 24","Collect like terms: 6x + 4x = 10x","The answer is x^2 + 10x + 24"],"diagram":null,"draw":false,"revision":"f83521eb357e"},{"id":"g4-expanding-and-factorising-q04","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"
  1. Factorise 5x + 20
  2. Factorise x2 − 9x
","answerType":"multi","answer":"a) 5(x + 4); b) x(x - 9)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5(x + 4)"},{"label":"b","answer":"x(x - 9)"}],"solution":["a) The highest common factor of 5x and 20 is 5, so 5x + 20 = 5(x + 4)","b) Both terms contain x, so x^2 - 9x = x(x - 9)"],"diagram":null,"draw":false,"revision":"cbe2642956e3"},{"id":"g4-expanding-and-factorising-q05","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":5,"marks":3,"tier":"FH","calc":true,"text":"
  1. Sam expands 3(2x + 4) and gets the answer 6x + 4
    Explain the mistake Sam has made.
  2. Factorise fully 12x2 + 18x
","answerType":"multi","answer":"a) he did not multiply the 4 by 3; b) 6x(2x + 3)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"He only multiplied the first term by 3; 3 times 4 = 12, so the answer should be 6x + 12"},{"label":"b","answer":"6x(2x + 3)"}],"solution":["a) Every term inside the bracket must be multiplied by 3","3(2x + 4) = 6x + 12, not 6x + 4","b) The highest common factor of 12x^2 and 18x is 6x","12x^2 + 18x = 6x(2x + 3)"],"diagram":null,"draw":false,"revision":"01f5648d936c"},{"id":"g4-expanding-and-factorising-q06","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":6,"marks":4,"tier":"F","calc":true,"text":"
  1. Expand and simplify 3(x + 2) + 2(x − 5)
  2. Solve 5(x − 3) = 20
","answerType":"multi","answer":"a) 5x - 4; b) x = 7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5x - 4"},{"label":"b","answer":"x = 7"}],"solution":["a) 3(x + 2) = 3x + 6 and 2(x - 5) = 2x - 10","3x + 6 + 2x - 10 = 5x - 4","b) Divide both sides by 5: x - 3 = 4","Add 3 to both sides: x = 7","Check: 5(7 - 3) = 20"],"diagram":null,"draw":false,"revision":"843f5cb85b1c"},{"id":"g4-expanding-and-factorising-q07","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"
  1. Expand and simplify (y − 3)(y − 8)
  2. Factorise fully 4a2b + 6ab2
","answerType":"multi","answer":"a) y^2 - 11y + 24; b) 2ab(2a + 3b)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"y^2 - 11y + 24"},{"label":"b","answer":"2ab(2a + 3b)"}],"solution":["a) (y - 3)(y - 8) = y^2 - 8y - 3y + 24","Collect like terms: y^2 - 11y + 24","b) The highest common factor of 4a^2b and 6ab^2 is 2ab","4a^2b + 6ab^2 = 2ab(2a + 3b)"],"diagram":null,"draw":false,"revision":"214cd884a398"},{"id":"g4-expanding-and-factorising-q08","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":8,"marks":4,"tier":"F","calc":false,"text":"
  1. Solve 2x + 35 = 3
  2. Factorise x2 + 8x + 15
","answerType":"multi","answer":"a) x = 6; b) (x + 3)(x + 5)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x = 6"},{"label":"b","answer":"(x + 3)(x + 5)"}],"solution":["a) Multiply both sides by 5: 2x + 3 = 15","Subtract 3: 2x = 12, so x = 6","Check: 12 + 35 = 3","b) Find two numbers with product 15 and sum 8: they are 3 and 5","x^2 + 8x + 15 = (x + 3)(x + 5)"],"diagram":null,"draw":false,"revision":"94f4836c425e"},{"id":"g4-expanding-and-factorising-q09","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":9,"marks":5,"tier":"F","calc":true,"text":"
  1. Simplify t4 × t5
  2. Simplify w8 ÷ w2
  3. Expand and simplify (x + 5)(x − 2)
  4. Factorise 9y − 12
","answerType":"multi","answer":"a) t^9; b) w^6; c) x^2 + 3x - 10; d) 3(3y - 4)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"t^9"},{"label":"b","answer":"w^6"},{"label":"c","answer":"x^2 + 3x - 10"},{"label":"d","answer":"3(3y - 4)"}],"solution":["a) Add the indices: t^(4+5) = t^9","b) Subtract the indices: w^(8-2) = w^6","c) (x + 5)(x - 2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10","d) The highest common factor of 9y and 12 is 3, so 9y - 12 = 3(3y - 4)"],"diagram":null,"draw":false,"revision":"1c249d267192"},{"id":"g4-expanding-and-factorising-q10","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":10,"marks":5,"tier":"FH","calc":true,"text":"
  1. Expand and simplify 5(2x + 1) − 3(x − 4)
  2. Factorise fully 18x2 − 27x
  3. Simplify (p2)5
","answerType":"multi","answer":"a) 7x + 17; b) 9x(2x - 3); c) p^10","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"7x + 17"},{"label":"b","answer":"9x(2x - 3)"},{"label":"c","answer":"p^10"}],"solution":["a) 5(2x + 1) = 10x + 5 and -3(x - 4) = -3x + 12","10x + 5 - 3x + 12 = 7x + 17","b) The highest common factor of 18x^2 and 27x is 9x","18x^2 - 27x = 9x(2x - 3)","c) Multiply the indices: p^(2 x 5) = p^10"],"diagram":null,"draw":false,"revision":"4b29055a11f2"},{"id":"g4-expanding-and-factorising-q11","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":11,"marks":6,"tier":"F","calc":true,"text":"
  1. Expand 3x(2x − 5)
  2. Factorise fully 20a2b − 8ab
  3. Make x the subject of y = 4x − 7
","answerType":"multi","answer":"a) 6x^2 - 15x; b) 4ab(5a - 2); c) x = (y + 7)/4","answerDisplay":"a) 6x2 - 15x; b) 4ab(5a - 2); c) x = y + 74","accept":[],"parts":[{"label":"a","answer":"6x^2 - 15x"},{"label":"b","answer":"4ab(5a - 2)"},{"label":"c","answer":"x = (y + 7)/4","answerDisplay":"x = y + 74"}],"solution":["a) 3x times 2x = 6x^2 and 3x times -5 = -15x, so 6x^2 - 15x","b) The highest common factor of 20a^2b and 8ab is 4ab","20a^2b - 8ab = 4ab(5a - 2)","c) Add 7 to both sides: y + 7 = 4x","Divide both sides by 4: x = y + 74"],"diagram":null,"draw":false,"revision":"0b416093f46a"},{"id":"g4-expanding-and-factorising-q12","topicId":"g4-expanding-and-factorising","topic":"Expanding and Factorising","grade":"4","topicGrade":"4","strand":"A","difficulty":12,"marks":6,"tier":"FH","calc":false,"text":"
  1. Expand and simplify (2x + 3)(x + 4)
  2. Factorise fully 10y2 − 15y
  3. Solve 4(2x − 1) = 3x + 16
","answerType":"multi","answer":"a) 2x^2 + 11x + 12; b) 5y(2y - 3); c) x = 4","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2x^2 + 11x + 12"},{"label":"b","answer":"5y(2y - 3)"},{"label":"c","answer":"x = 4"}],"solution":["a) (2x + 3)(x + 4) = 2x^2 + 8x + 3x + 12","Collect like terms: 2x^2 + 11x + 12","b) The highest common factor of 10y^2 and 15y is 5y","10y^2 - 15y = 5y(2y - 3)","c) Expand: 8x - 4 = 3x + 16","Subtract 3x: 5x - 4 = 16","Add 4: 5x = 20, so x = 4","Check: 4(8 - 1) = 28 and 3(4) + 16 = 28"],"diagram":null,"draw":false,"revision":"1874fa7da00c"},{"id":"g4-forming-and-solving-equations-q01","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":1,"marks":3,"tier":"F","calc":true,"text":"

The angles of a triangle are (x + 10)°, 2x° and (3x − 10)°.

Work out the size of the largest angle of the triangle.

","answerType":"exact","answer":"80°","answerDisplay":null,"accept":["80","80 degrees"],"parts":[],"solution":["Angles in a triangle add to 180: (x + 10) + 2x + (3x - 10) = 180","6x = 180, so x = 30","The angles are 40, 60 and 80 degrees","Check: 40 + 60 + 80 = 180","The largest angle is 80 degrees"],"diagram":null,"draw":false,"revision":"62940917479f"},{"id":"g4-forming-and-solving-equations-q02","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

Dan and Ella share 45 sweets.

Ella gets 7 more sweets than Dan.

Dan says, "I got 20 sweets."

Is Dan right? You must show how you get your answer.

","answerType":"open","answer":"No, Dan got 19 sweets","answerDisplay":null,"accept":["No, he got 19"],"parts":[],"solution":["Let d be the number of sweets Dan gets, so Ella gets d + 7","d + d + 7 = 45","2d = 38, so d = 19","Check: 19 + 26 = 45","Dan is wrong, he got 19 sweets, not 20"],"diagram":null,"draw":false,"revision":"c479a5a8bde6"},{"id":"g4-forming-and-solving-equations-q03","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

A trapezium has parallel sides of length (2x + 1) cm and (4x + 3) cm.

The perpendicular height of the trapezium is 6 cm.

Show that the area of the trapezium is 6(3x + 2) cm2.

","answerType":"open","answer":"Area = 18x + 12 = 6(3x + 2)","answerDisplay":null,"accept":[],"parts":[],"solution":["Area of a trapezium = half the sum of the parallel sides times the height","Area = (12)(2x + 1 + 4x + 3)(6)","Area = 3(6x + 4)","Area = 18x + 12 = 6(3x + 2), as required"],"diagram":null,"draw":false,"revision":"022d252b9ce4"},{"id":"g4-forming-and-solving-equations-q04","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":4,"marks":4,"tier":"F","calc":true,"text":"

A rectangle has length (3x + 2) cm and width (x − 1) cm.

The perimeter of the rectangle is 50 cm.

Work out the length of the rectangle.

","answerType":"exact","answer":"20 cm","answerDisplay":null,"accept":["20"],"parts":[],"solution":["Perimeter = 2(3x + 2) + 2(x - 1) = 6x + 4 + 2x - 2 = 8x + 2","8x + 2 = 50","8x = 48, so x = 6","Length = 3(6) + 2 = 20 cm","Check: width = 5 cm, perimeter = 2(20) + 2(5) = 50 cm"],"diagram":null,"draw":false,"revision":"3ae01762c74b"},{"id":"g4-forming-and-solving-equations-q05","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

The angles of a quadrilateral are x°, (x + 30)°, (2x − 10)° and 100°.

Work out the size of the largest angle of the quadrilateral.

","answerType":"exact","answer":"110°","answerDisplay":null,"accept":["110","110 degrees"],"parts":[],"solution":["Angles in a quadrilateral add to 360: x + (x + 30) + (2x - 10) + 100 = 360","4x + 120 = 360","4x = 240, so x = 60","The angles are 60, 90, 110 and 100 degrees","Check: 60 + 90 + 110 + 100 = 360","The largest angle is 110 degrees"],"diagram":null,"draw":false,"revision":"1337dfd33500"},{"id":"g4-forming-and-solving-equations-q06","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":6,"marks":4,"tier":"FH","calc":true,"text":"

Rectangle A has length 8 cm and width (x + 2) cm.

Rectangle B has length 12 cm and width x cm.

The two rectangles have equal areas.

Work out the area of rectangle A.

","answerType":"exact","answer":"48 cm^2","answerDisplay":null,"accept":["48"],"parts":[],"solution":["Area of A = 8(x + 2) = 8x + 16","Area of B = 12x","The areas are equal: 8x + 16 = 12x","16 = 4x, so x = 4","Area of A = 8(4 + 2) = 48 cm^2","Check: area of B = 12(4) = 48 cm^2"],"diagram":null,"draw":false,"revision":"6495016ed363"},{"id":"g4-forming-and-solving-equations-q07","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

A square has sides of length (x + 3) cm.

Another square has sides of length x cm.

The area of the larger square is 45 cm2 more than the area of the smaller square.

Find the value of x.

","answerType":"exact","answer":"x = 6","answerDisplay":null,"accept":["6"],"parts":[],"solution":["(x + 3)^2 = x^2 + 45","Expand: x^2 + 6x + 9 = x^2 + 45","6x + 9 = 45","6x = 36, so x = 6","Check: 9^2 - 6^2 = 81 - 36 = 45"],"diagram":null,"draw":false,"revision":"71c1af2c37b7"},{"id":"g4-forming-and-solving-equations-q08","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

A kite has two sides of length (2x + 1) cm and two sides of length (4x − 3) cm.

The perimeter of the kite is 44 cm.

Work out the length of one of the longer sides of the kite.

","answerType":"exact","answer":"13 cm","answerDisplay":null,"accept":["13"],"parts":[],"solution":["Perimeter = 2(2x + 1) + 2(4x - 3) = 4x + 2 + 8x - 6 = 12x - 4","12x - 4 = 44","12x = 48, so x = 4","The sides are 2(4) + 1 = 9 cm and 4(4) - 3 = 13 cm","Check: 2(9) + 2(13) = 44 cm","The longer sides are 13 cm"],"diagram":null,"draw":false,"revision":"900c0e35b18f"},{"id":"g4-forming-and-solving-equations-q09","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":9,"marks":5,"tier":"F","calc":true,"text":"

Three parcels have a total mass of 26 kg.

Parcel A has mass x kg.

Parcel B has mass (2x + 1) kg.

Parcel C has mass (3x − 5) kg.

Work out the mass of the heaviest parcel.

","answerType":"exact","answer":"11 kg","answerDisplay":null,"accept":["11"],"parts":[],"solution":["x + (2x + 1) + (3x - 5) = 26","6x - 4 = 26","6x = 30, so x = 5","Parcel A is 5 kg, parcel B is 11 kg, parcel C is 10 kg","Check: 5 + 11 + 10 = 26","The heaviest parcel is B at 11 kg"],"diagram":null,"draw":false,"revision":"272b08a3e493"},{"id":"g4-forming-and-solving-equations-q10","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":10,"marks":5,"tier":"F","calc":true,"text":"

A box contains only red counters, blue counters and green counters.

The number of blue counters is twice the number of red counters.

The number of green counters is 9 more than the number of red counters.

There are 49 counters in the box altogether.

Work out the ratio of red counters to green counters. Give your answer in its simplest form.

","answerType":"exact","answer":"10 : 19","answerDisplay":null,"accept":["10:19"],"parts":[],"solution":["Let r be the number of red counters","Blue = 2r and green = r + 9","r + 2r + r + 9 = 49","4r = 40, so r = 10","There are 10 red, 20 blue and 19 green counters","Check: 10 + 20 + 19 = 49","Ratio of red to green = 10 : 19, which is already in simplest form"],"diagram":null,"draw":false,"revision":"39fcf615a961"},{"id":"g4-forming-and-solving-equations-q11","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":11,"marks":5,"tier":"H","calc":false,"text":"

A rectangle has length (2x + 5) cm and width (x + 2) cm.

The perimeter of the rectangle is 38 cm.

Work out the area of the rectangle.

","answerType":"exact","answer":"78 cm^2","answerDisplay":null,"accept":["78"],"parts":[],"solution":["Perimeter = 2(2x + 5) + 2(x + 2) = 6x + 14","6x + 14 = 38","6x = 24, so x = 4","Length = 13 cm and width = 6 cm","Check: 2(13) + 2(6) = 38 cm","Area = 13 times 6 = 78 cm^2"],"diagram":null,"draw":false,"revision":"be22380e99ac"},{"id":"g4-forming-and-solving-equations-q12","topicId":"g4-forming-and-solving-equations","topic":"Forming and Solving Equations","grade":"4","topicGrade":"4","strand":"A","difficulty":12,"marks":5,"tier":"H","calc":false,"text":"

Cuboid A measures 2 cm by 5 cm by x cm.

Cuboid B measures 4 cm by 5 cm by (x − 3) cm.

The two cuboids have equal volumes.

Find the value of x.

","answerType":"exact","answer":"x = 6","answerDisplay":null,"accept":["6"],"parts":[],"solution":["Volume of A = 2 times 5 times x = 10x","Volume of B = 4 times 5 times (x - 3) = 20(x - 3) = 20x - 60","10x = 20x - 60","60 = 10x, so x = 6","Check: volume of A = 60 cm^3 and volume of B = 20(3) = 60 cm^3"],"diagram":null,"draw":false,"revision":"d90b0e888273"},{"id":"g4-indices-q01","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write down the value of 1003

","answerType":"exact","answer":"1000000","answerDisplay":null,"accept":["1000000","1,000,000","1 000 000"],"parts":[],"solution":["100^3 means 100 x 100 x 100.","100 x 100 = 10,000 and 10,000 x 100 = 1,000,000."],"diagram":null,"draw":false,"revision":"15c9749fb5cf"},{"id":"g4-indices-q02","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"
  1. Write 74 × 73 as a single power of 7
  2. Write 129 ÷ 124 as a single power of 12
","answerType":"multi","answer":"a) 7^7 b) 12^5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"7^7"},{"label":"b","answer":"12^5"}],"solution":["a) When multiplying powers of the same base, add the indices: 4 + 3 = 7, so the answer is 7^7.","b) When dividing powers of the same base, subtract the indices: 9 - 4 = 5, so the answer is 12^5.","The first product contains four factors of 7 followed by three more: seven altogether. For division, cancel four of the nine factors of 12, leaving five."],"diagram":null,"draw":false,"revision":"62a75eb47873"},{"id":"g4-indices-q03","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"
  1. Write (53)4 as a single power of 5
  2. Write down the value of 80
","answerType":"multi","answer":"a) 5^12 b) 1","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5^12"},{"label":"b","answer":"1"}],"solution":["a) For a power of a power, multiply the indices: 3 x 4 = 12, so the answer is 5^12.","b) Any non-zero number to the power 0 is 1.","For the zero power, 8³ ÷ 8³ = 1, and the division rule writes the same calculation as 8^(3−3) = 8⁰."],"diagram":null,"draw":false,"revision":"98296d2689dd"},{"id":"g4-indices-q04","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":4,"marks":2,"tier":"FH","calc":false,"text":"

Work out the value of (35 × 3−2) ÷ 3

","answerType":"exact","answer":"9","answerDisplay":null,"accept":["9","3^2"],"parts":[],"solution":["3^5 x 3^-2 = 3^(5-2) = 3^3.","3^3 divided by 3 = 3^(3-1) = 3^2.","3^2 = 9."],"diagram":null,"draw":false,"revision":"2064a0d4a1fb"},{"id":"g4-indices-q05","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":5,"marks":3,"tier":"FH","calc":true,"text":"
  1. Simplify  t8 ÷ t3
  2. Simplify  (k4)5
  3. Simplify  12x6y34x2y
","answerType":"multi","answer":"a) t^5 b) k^20 c) 3x^4y^2","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"t^5"},{"label":"b","answer":"k^20"},{"label":"c","answer":"3x^4y^2"}],"solution":["a) Subtract the indices: t^(8-3) = t^5.","b) Multiply the indices: k^(4x5) = k^20.","c) Divide the numbers: 12 / 4 = 3. Subtract indices: x^(6-2) = x^4 and y^(3-1) = y^2, giving 3x^4y^2.","Keep the original restriction: t, x and y must be non-zero wherever they occur in a denominator or with a negative power."],"diagram":null,"draw":false,"revision":"dfa448810635"},{"id":"g4-indices-q06","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":6,"marks":3,"tier":"FH","calc":true,"text":"
  1. Simplify  5a3 × 4a6
  2. Simplify  (2c4)3
","answerType":"multi","answer":"a) 20a^9 b) 8c^12","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"20a^9"},{"label":"b","answer":"8c^12"}],"solution":["a) Multiply the numbers: 5 x 4 = 20. Add the indices: a^(3+6) = a^9, giving 20a^9.","b) Cube the 2 as well as the power: 2^3 = 8 and (c^4)^3 = c^12, giving 8c^12."],"diagram":null,"draw":false,"revision":"bbb436985d34"},{"id":"g4-indices-q07","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":7,"marks":3,"tier":"FH","calc":false,"text":"
  1. Simplify  n−5 × n8
  2. Write down the value of 2−3
","answerType":"multi","answer":"a) n^3 b) 1/8","answerDisplay":"a) n^3 b) 18","accept":[],"parts":[{"label":"a","answer":"n^3"},{"label":"b","answer":"1/8","answerDisplay":"18"}],"solution":["a) Add the indices: n^(-5+8) = n^3.","b) 2^-3 = 1 / 2^3 = 18.","A negative power means a reciprocal, not a negative answer: 2³ × 2⁻³ must equal 2⁰ = 1, so 2⁻³ = 1/8. Here n must be non-zero because n⁻⁵ occurs in the original expression."],"diagram":null,"draw":false,"revision":"9dc7c2a56625"},{"id":"g4-indices-q08","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":8,"marks":3,"tier":"H","calc":false,"text":"

26 × 2k23  =  211

Find the value of k.

","answerType":"exact","answer":"8","answerDisplay":null,"accept":["8","k = 8"],"parts":[],"solution":["Using the index laws, the left-hand side is 2^(6 + k - 3) = 2^(k + 3).","So k + 3 = 11.","k = 8."],"diagram":null,"draw":false,"revision":"3aaf90c5e486"},{"id":"g4-indices-q09","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":9,"marks":4,"tier":"FH","calc":true,"text":"
  1. Simplify  (h5)2
  2. Simplify  15d7 ÷ 3d2
  3. Expand  3t(2t + 5)
","answerType":"multi","answer":"a) h^10 b) 5d^5 c) 6t^2 + 15t","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"h^10"},{"label":"b","answer":"5d^5"},{"label":"c","answer":"6t^2 + 15t"}],"solution":["a) Multiply the indices: h^(5x2) = h^10.","b) 15 / 3 = 5 and d^(7-2) = d^5, giving 5d^5.","c) 3t x 2t = 6t^2 and 3t x 5 = 15t, giving 6t^2 + 15t.","Keep the original restriction: d must be non-zero wherever they occur in a denominator or with a negative power."],"diagram":null,"draw":false,"revision":"79bc1f2a4b3c"},{"id":"g4-indices-q10","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":10,"marks":4,"tier":"FH","calc":false,"text":"
  1. Write down the value of 170
  2. Write down the value of 4−2
  3. Simplify  (m−3)4 × m14
","answerType":"multi","answer":"a) 1 b) 1/16 c) m^2","answerDisplay":"a) 1 b) 116 c) m^2","accept":[],"parts":[{"label":"a","answer":"1"},{"label":"b","answer":"1/16","answerDisplay":"116"},{"label":"c","answer":"m^2"}],"solution":["a) Any non-zero number to the power 0 is 1.","b) 4^-2 = 1 / 4^2 = 116.","c) (m^-3)^4 = m^-12, then m^-12 x m^14 = m^(-12+14) = m^2.","Keep the original restriction: m must be non-zero wherever they occur in a denominator or with a negative power."],"diagram":null,"draw":false,"revision":"07e17851e210"},{"id":"g4-indices-q11","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":11,"marks":4,"tier":"H","calc":true,"text":"
  1. Write (72)5 as a single power of 7
  2. Zara writes

    34 × 35 = 99

    Explain what Zara has done wrong, and write down the correct answer as a single power.

","answerType":"multi","answer":"a) 7^10 b) Zara multiplied the bases as well as adding the indices - the base must stay as 3. The correct answer is 3^9.","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"7^10"},{"label":"b","answer":"3^9 (Zara wrongly changed the base from 3 to 9)"}],"solution":["a) Multiply the indices: 7^(2x5) = 7^10.","b) When multiplying powers of the same base you add the indices but keep the base the same.","b) Zara multiplied the bases (3 x 3 = 9) as well - the correct answer is 3^(4+5) = 3^9."],"diagram":null,"draw":false,"revision":"9ca9c1d4c1b0"},{"id":"g4-indices-q12","topicId":"g4-indices","topic":"Indices","grade":"4","topicGrade":"4","strand":"N","difficulty":12,"marks":5,"tier":"H","calc":false,"text":"
  1. Simplify  (5p4q)2
  2. Simplify  24m8 ÷ 6m2
  3. Solve  4x − 7 > 2x + 3
","answerType":"multi","answer":"a) 25p^8q^2 b) 4m^6 c) x > 5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"25p^8q^2"},{"label":"b","answer":"4m^6"},{"label":"c","answer":"x > 5"}],"solution":["a) Square each factor: 5^2 = 25, (p^4)^2 = p^8, q^2 = q^2, giving 25p^8q^2.","b) 24 / 6 = 4 and m^(8-2) = m^6, giving 4m^6.","c) Subtract 2x from both sides: 2x - 7 > 3. Add 7: 2x > 10. Divide by 2: x > 5.","Keep the original restriction: m must be non-zero wherever they occur in a denominator or with a negative power."],"diagram":null,"draw":false,"revision":"1e9c63f6b601"},{"id":"g4-inequalities-q01","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

n is an integer.

−2 < n ≤ 3

Write down all the possible values of n.

","answerType":"exact","answer":"-1, 0, 1, 2, 3","answerDisplay":null,"accept":["-1,0,1,2,3"],"parts":[],"solution":["n must be greater than -2, so -2 itself is not included","n can be less than or equal to 3, so 3 is included","The integers are -1, 0, 1, 2, 3"],"diagram":null,"draw":false,"revision":"5510e2449e8e"},{"id":"g4-inequalities-q02","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":2,"marks":2,"tier":"FH","calc":true,"text":"

Solve 4x + 3 < 19

","answerType":"exact","answer":"x < 4","answerDisplay":null,"accept":["x<4"],"parts":[],"solution":["Subtract 3 from both sides: 4x < 16","Divide both sides by 4: x < 4"],"diagram":null,"draw":false,"revision":"3518825c0c04"},{"id":"g4-inequalities-q03","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":3,"marks":2,"tier":"H","calc":true,"text":"

n is an integer such that n > −3 and 2n + 1 ≤ 5

Write down all the possible values of n.

","answerType":"exact","answer":"-2, -1, 0, 1, 2","answerDisplay":null,"accept":["-2,-1,0,1,2"],"parts":[],"solution":["Solve 2n + 1 <= 5: 2n <= 4, so n <= 2","Combine with n > -3: -3 < n <= 2","The integers are -2, -1, 0, 1, 2"],"diagram":null,"draw":false,"revision":"65aa3fd8b232"},{"id":"g4-inequalities-q04","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":4,"marks":2,"tier":"F","calc":false,"text":"

The diagram shows an inequality on a number line.

Write down the inequality shown, using x.

","answerType":"exact","answer":"-1 <= x < 4","answerDisplay":null,"accept":["-1 ≤ x < 4","x >= -1 and x < 4"],"parts":[],"solution":["The solid circle at -1 means -1 is included, so x >= -1","The open circle at 4 means 4 is not included, so x < 4","Together: -1 <= x < 4"],"diagram":{"type":"numberline","min":-3,"max":6,"step":1,"ranges":[{"from":-1,"to":4,"fromStyle":"solid","toStyle":"open"}]},"draw":false,"revision":"47bc27bbff39"},{"id":"g4-inequalities-q05","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

Solve 5x − 2 > 3x + 10

","answerType":"exact","answer":"x > 6","answerDisplay":null,"accept":["x>6"],"parts":[],"solution":["Subtract 3x from both sides: 2x - 2 > 10","Add 2 to both sides: 2x > 12","Divide both sides by 2: x > 6"],"diagram":null,"draw":false,"revision":"ea5197af533c"},{"id":"g4-inequalities-q06","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

Solve 2x5 − 1 < 3

","answerType":"exact","answer":"x < 10","answerDisplay":null,"accept":["x<10"],"parts":[],"solution":["Add 1 to both sides: 2x5 < 4","Multiply both sides by 5: 2x < 20","Divide both sides by 2: x < 10"],"diagram":null,"draw":false,"revision":"64e1ba6ed085"},{"id":"g4-inequalities-q07","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"
  1. Solve 3(x − 1) ≥ 6
  2. Show your solution on the number line.
","answerType":"multi","answer":"a) x >= 3; b) solid circle at 3 with arrow to the right","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x >= 3"},{"label":"b","answer":"Solid circle at 3 with a line and arrow pointing right"}],"solution":["a) Divide both sides by 3: x - 1 >= 2","Add 1 to both sides: x >= 3","b) Draw a solid circle at 3, because 3 is included, with an arrow pointing right"],"diagram":{"type":"numberline","min":-2,"max":8,"step":1,"ranges":[{"from":3,"fromStyle":"solid","answer":true}]},"draw":true,"revision":"8dfe3b8b23c2"},{"id":"g4-inequalities-q08","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"

Chloe wants to show x > 2 on a number line.

She draws a solid circle at 2 with an arrow pointing to the right.

  1. Explain the mistake Chloe has made.
  2. Write down the greatest integer that satisfies 4x − 3 < 18
","answerType":"multi","answer":"a) she should use an open circle because 2 is not included; b) 5","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"The circle at 2 should be open, not solid, because x > 2 does not include 2"},{"label":"b","answer":"5"}],"solution":["a) A solid circle means the value is included. x > 2 does not include 2, so she needs an open circle","b) Solve 4x - 3 < 18: 4x < 21, so x < 5.25","The greatest integer less than 5.25 is 5"],"diagram":null,"draw":false,"revision":"9aa0f2220c36"},{"id":"g4-inequalities-q09","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":9,"marks":4,"tier":"H","calc":false,"text":"
  1. Solve −3 < 2x + 1 ≤ 9
  2. Write down all the integer values of x that satisfy your answer to part a.
","answerType":"multi","answer":"a) -2 < x <= 4; b) -1, 0, 1, 2, 3, 4","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-2 < x <= 4"},{"label":"b","answer":"-1, 0, 1, 2, 3, 4"}],"solution":["a) Subtract 1 from all three parts: -4 < 2x <= 8","Divide all three parts by 2: -2 < x <= 4","b) The integers strictly greater than -2 and at most 4 are -1, 0, 1, 2, 3, 4"],"diagram":null,"draw":false,"revision":"b0e091e879ad"},{"id":"g4-inequalities-q10","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":10,"marks":5,"tier":"F","calc":true,"text":"
  1. n is an integer with −4 < n ≤ −1
    Write down all the possible values of n.
  2. Solve 7y − 3 > 2y + 22 and show your solution on the number line.
","answerType":"multi","answer":"a) -3, -2, -1; b) y > 5, open circle at 5 with arrow right","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-3, -2, -1"},{"label":"b","answer":"y > 5"}],"solution":["a) n is greater than -4 but at most -1, so n = -3, -2 or -1","b) Subtract 2y from both sides: 5y - 3 > 22","Add 3 to both sides: 5y > 25","Divide by 5: y > 5","Show with an open circle at 5 and an arrow pointing right"],"diagram":{"type":"numberline","min":0,"max":10,"step":1,"ranges":[{"from":5,"fromStyle":"open","answer":true}]},"draw":true,"revision":"8984dcdd5d43"},{"id":"g4-inequalities-q11","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":11,"marks":5,"tier":"F","calc":false,"text":"
  1. Solve 5 − 2x ≥ 1
  2. Factorise x2 + 5x + 6
","answerType":"multi","answer":"a) x <= 2; b) (x + 2)(x + 3)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x <= 2"},{"label":"b","answer":"(x + 2)(x + 3)"}],"solution":["a) Subtract 5 from both sides: -2x >= -4","Divide both sides by -2 and reverse the inequality sign: x <= 2","Dividing by a negative number reverses order: for example 6 > 2, but −3 < −1 after dividing both by −2. This is why the inequality sign reverses here.","b) Find two numbers with product 6 and sum 5: they are 2 and 3","x^2 + 5x + 6 = (x + 2)(x + 3)"],"diagram":null,"draw":false,"revision":"5cd8978bb839"},{"id":"g4-inequalities-q12","topicId":"g4-inequalities","topic":"Inequalities","grade":"4","topicGrade":"4","strand":"A","difficulty":12,"marks":6,"tier":"FH","calc":true,"text":"
  1. n is an integer with −5 ≤ n < −1
    Write down all the possible values of n.
  2. Solve 4(x + 3) > 2x + 17
  3. Write down the smallest integer that satisfies your answer to part b.
","answerType":"multi","answer":"a) -5, -4, -3, -2; b) x > 2.5; c) 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-5, -4, -3, -2"},{"label":"b","answer":"x > 2.5"},{"label":"c","answer":"3"}],"solution":["a) n is at least -5 and strictly less than -1, so n = -5, -4, -3, -2","b) Expand: 4x + 12 > 2x + 17","Subtract 2x from both sides: 2x + 12 > 17","Subtract 12: 2x > 5, so x > 2.5","c) The smallest integer greater than 2.5 is 3"],"diagram":null,"draw":false,"revision":"4f9add245e8a"},{"id":"g4-loci-and-construction-q01","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Sam is asked to shade the region of all points that are less than 4 cm from a point P.

Sam shades a square with sides of length 8 cm, with P at the centre of the square.

Explain why Sam's region is wrong.

","answerType":"open","answer":"The set of points less than 4 cm from P is the inside of a circle, centre P, radius 4 cm - not a square. The corners of Sam's square are further than 4 cm from P.","answerDisplay":null,"accept":[],"parts":[],"solution":["All points less than 4 cm from P lie inside a circle with centre P and radius 4 cm","Sam's square is wrong because its corners are about 5.7 cm from P, which is more than 4 cm, so the square includes points that should not be shaded"],"diagram":null,"draw":false,"revision":"b52f4d2d95f6"},{"id":"g4-loci-and-construction-q02","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

Describe fully the locus of all points that are exactly 5 cm from a fixed point A.

","answerType":"open","answer":"A circle with centre A and radius 5 cm","answerDisplay":null,"accept":["circle centre A radius 5 cm"],"parts":[],"solution":["Every point at a fixed distance from a single point lies on a circle","The locus is a circle, centre A, radius 5 cm"],"diagram":null,"draw":false,"revision":"56ff5e26e94e"},{"id":"g4-loci-and-construction-q03","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

Draw a line segment AB of length 8 cm.

Using a ruler and compasses only, construct the perpendicular bisector of AB.

You must show all your construction lines.

","answerType":"open","answer":"Arcs of equal radius (more than 4 cm) drawn from A and from B, crossing above and below AB; the straight line through the two crossing points is the perpendicular bisector, cutting AB at its midpoint at 90 degrees","answerDisplay":null,"accept":[],"parts":[],"solution":["Open the compasses to a radius more than half of AB (more than 4 cm)","With the point on A, draw arcs above and below the line. Keep the same radius, put the point on B and draw arcs crossing the first pair","Draw the straight line through the two points where the arcs cross - this is the perpendicular bisector of AB","Each arc intersection is equally far from A and B because the compass radius stayed the same. Joining these equidistant points gives the line through the midpoint at 90°, called the perpendicular bisector."],"diagram":{"type":"shape","notAccurate":false,"bounds":[[-2,-6],[12,8]],"segments":[{"from":[0,0],"to":[8,0],"answer":false},{"from":[4,-5],"to":[4,5],"answer":true}],"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[8,0],"label":"B","style":"none","labelPos":"below-right"}],"circles":[{"c":[0,0],"r":5,"answer":true},{"c":[8,0],"r":5,"answer":true}],"arcs":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"85e59bb6cb64"},{"id":"g4-loci-and-construction-q04","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":2,"tier":"F","calc":false,"text":"

Draw an angle ABC of 70° using a protractor, where B is the vertex of the angle.

Using a ruler and compasses only, construct the bisector of angle ABC.

You must show all your construction lines.

","answerType":"open","answer":"An arc from B crossing BA and BC, then equal arcs from those two crossing points meeting inside the angle; the line from B through that meeting point bisects the angle into two 35 degree angles","answerDisplay":null,"accept":[],"parts":[],"solution":["With the compass point on B, draw an arc crossing BA at P and BC at Q","With the same radius, draw arcs from P and from Q that cross at a point R inside the angle","Draw the straight line from B through R - this bisects angle ABC into two angles of 35°"],"diagram":{"type":"shape","notAccurate":false,"bounds":[[-2,-6],[12,8]],"segments":[{"from":[0,0],"to":[7,0],"answer":false},{"from":[0,0],"to":[2.3941410032796817,6.5778483455013586],"answer":false},{"from":[0,0],"to":[6.553216354311934,4.588611490808368],"answer":true}],"points":[{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[0,0],"label":"B","style":"none","labelPos":"below-left"},{"at":[2.3941410032796817,6.5778483455013586],"label":"C","style":"none","labelPos":"above"}],"circles":[],"arcs":[{"c":[0,0],"r":3,"from":0,"to":70,"answer":true},{"c":[3,0],"r":3,"from":0,"to":150,"answer":true},{"c":[1.0260604299770064,2.819077862357725],"r":3,"from":-70,"to":80,"answer":true}],"angles":[{"at":[0,0],"from":[7,0],"to":[2.3941410032796817,6.5778483455013586],"label":"70°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"e0b731deb067"},{"id":"g4-loci-and-construction-q05","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":2,"tier":"F","calc":true,"text":"

Draw a straight line 10 cm long. Mark a point P about 4 cm above the line.

Using a ruler and compasses only, construct the perpendicular from P to the line.

You must show all your construction lines.

","answerType":"open","answer":"An arc from P crossing the line at two points, then arcs of equal radius from those two points crossing below the line; the line from P through that crossing point is perpendicular to the original line","answerDisplay":null,"accept":[],"parts":[],"solution":["With the compass point on P, draw an arc that crosses the line at two points, X and Y","With the same radius, draw arcs from X and from Y that cross on the far side of the line from P","Draw the straight line from P through this crossing point - it meets the original line at 90°"],"diagram":{"type":"shape","notAccurate":false,"bounds":[[-2,-6],[12,8]],"segments":[{"from":[0,0],"to":[10,0],"answer":false},{"from":[5,5],"to":[5,-5],"answer":true}],"points":[{"at":[5,4],"label":"P","style":"dot","labelPos":"below-right"}],"circles":[{"c":[5,4],"r":5,"answer":true},{"c":[2,0],"r":5,"answer":true},{"c":[8,0],"r":5,"answer":true}],"arcs":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"fd697f85b299"},{"id":"g4-loci-and-construction-q06","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":2,"tier":"F","calc":true,"text":"

Draw a straight line 10 cm long. Mark a point M on the line, about halfway along.

Using a ruler and compasses only, construct the line perpendicular to the line at the point M.

You must show all your construction lines.

","answerType":"open","answer":"Equal arcs from M crossing the line either side at X and Y, then larger equal arcs from X and Y crossing above the line; the line through M and that crossing point is perpendicular to the original line at M","answerDisplay":null,"accept":[],"parts":[],"solution":["With the compass point on M, mark two points X and Y on the line at equal distances either side of M","Open the compasses wider. With the point on X, then on Y, draw two arcs of the same radius crossing above the line","Draw the straight line from M through the point where the arcs cross - it is perpendicular to the line at M"],"diagram":{"type":"shape","notAccurate":false,"bounds":[[-2,-6],[12,8]],"segments":[{"from":[0,0],"to":[10,0],"answer":false},{"from":[5,-1],"to":[5,5],"answer":true}],"points":[{"at":[5,0],"label":"M","style":"dot","labelPos":"below-right"}],"circles":[{"c":[5,0],"r":2,"answer":true},{"c":[3,0],"r":3,"answer":true},{"c":[7,0],"r":3,"answer":true}],"arcs":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"6bc222d3ebe1"},{"id":"g4-loci-and-construction-q07","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":2,"tier":"FH","calc":true,"text":"

Using a ruler and compasses only, construct an isosceles triangle ABC in which AB = 7 cm and AC = BC = 5 cm.

You must show all your construction arcs.

","answerType":"open","answer":"Draw AB = 7 cm; arcs of radius 5 cm from A and from B cross at C; joining AC and BC gives the required isosceles triangle","answerDisplay":null,"accept":[],"parts":[],"solution":["Draw the base AB of length 7 cm with a ruler","Open the compasses to 5 cm. With the point on A, draw an arc above AB; with the point on B and the same radius, draw a second arc crossing the first at C","Join AC and BC to complete the triangle. AC = BC = 5 cm, so the triangle is isosceles"],"diagram":{"type":"shape","notAccurate":false,"bounds":[[-2,-6],[12,8]],"segments":[{"from":[0,0],"to":[7,0],"answer":false},{"from":[0,0],"to":[3.5,3.570714214271425],"answer":true},{"from":[7,0],"to":[3.5,3.570714214271425],"answer":true}],"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[3.5,3.570714214271425],"label":"C","style":"none","answer":true,"labelPos":"above"}],"circles":[],"arcs":[{"c":[0,0],"r":5,"from":20,"to":100,"answer":true},{"c":[7,0],"r":5,"from":80,"to":160,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"7a8834ddd842"},{"id":"g4-loci-and-construction-q08","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":2,"tier":"H","calc":true,"text":"

Two towns, Ashford and Bexley, are 60 km apart.

A new phone mast must be built so that it is exactly the same distance from Ashford as it is from Bexley.

Describe fully the locus of all possible positions of the mast.

","answerType":"open","answer":"The perpendicular bisector of the line segment joining Ashford and Bexley - the straight line at right angles to it, passing through its midpoint, 30 km from each town","answerDisplay":null,"accept":[],"parts":[],"solution":["Points equidistant from two fixed points lie on the perpendicular bisector of the segment joining them","The locus is the straight line perpendicular to the line joining Ashford and Bexley, passing through its midpoint (30 km from each town)"],"diagram":null,"draw":false,"revision":"d7d3d3652fb1"},{"id":"g4-loci-and-construction-q09","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":3,"tier":"H","calc":true,"text":"

A goat is tethered to a post in the middle of a large flat field by a rope of length 7 m.

The rope stays taut or slack, so the goat can reach any point up to 7 m from the post.

  1. Describe fully the boundary of the region the goat can reach.
  2. Work out the area of grass the goat can graze. Give your answer correct to 1 decimal place.
","answerType":"multi","answer":"a) A circle, centre the post, radius 7 m b) 153.9 m²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"a circle, centre the post, radius 7 m"},{"label":"b","answer":"153.9 m²"}],"solution":["a) The goat can reach every point up to 7 m from the post, so the boundary is a circle with centre the post and radius 7 m","b) Area = πr2 = π × 72 = 49π","Area = 153.938... = 153.9 m² (1 decimal place)"],"diagram":null,"draw":false,"revision":"fb4ae58bf6c0"},{"id":"g4-loci-and-construction-q10","topicId":"g4-loci-and-construction","topic":"Loci and Construction","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"

The diagram shows a scale drawing of a rectangular garden ABCD, with AB = 10 cm and BC = 6 cm on the drawing.

The scale is 1 cm to 2 m.

A pond will be dug in the garden so that it is

more than 6 m from corner A

and closer to side AB than to side DC.

Shade the region of the garden where the pond can be dug. You must show all your construction lines.

","answerType":"open","answer":"Shade the part of the garden between AB and the mid-line parallel to AB (the line 3 cm from AB on the drawing) and outside the arc of radius 3 cm centred at A (6 m at scale 1 cm : 2 m)","answerDisplay":null,"accept":[],"parts":[],"solution":["6 m in real life is 6 ÷ 2 = 3 cm on the drawing. Draw an arc of radius 3 cm, centre A - the pond must be outside this arc","Points closer to AB than to DC lie on the AB side of the line midway between AB and DC. Draw this mid-line parallel to AB, 3 cm from it","Shade the region between AB and the mid-line that lies outside the arc centred at A","Do not include the circular arc or the mid-line: “more than” and “closer than” are strict conditions, so draw these boundaries dashed."],"diagram":{"type":"shape","notAccurate":false,"bounds":[[-1,-1],[11,7]],"segments":[{"from":[0,3],"to":[10,3],"answer":true,"dashed":true}],"points":[],"circles":[],"arcs":[{"c":[0,0],"r":3,"from":0,"to":90,"answer":true,"dashed":true}],"polygons":[{"pts":[[0,0],[10,0],[10,6],[0,6]],"labels":["A","B","C","D"],"sides":["10 cm","6 cm"]},{"pts":[[3,0],[10,0],[10,3],[0,3],[1.8369701987210297e-16,3],[0.1570078687288319,2.9958886042637216],[0.31358538980296036,2.9835656861048196],[0.46930339512069275,2.9630650217854133],[0.6237350724532784,2.9344428022014166],[0.7764571353075622,2.897777478867205],[0.9270509831248424,2.8531695488854605],[1.075103848635901,2.8007412794916053],[1.2202099292274007,2.7406363729278027],[1.3619714992186407,2.6730195725651034],[1.5000000000000004,2.598076211353316],[1.6339171050450816,2.5160117038362717],[1.7633557568774194,2.4270509831248424],[1.8879611731495125,2.3314378843709123],[2.0073918190765747,2.229434476432183],[2.121320343559643,2.1213203435596424],[2.229434476432183,2.0073918190765747],[2.3314378843709127,1.8879611731495123],[2.4270509831248424,1.7633557568774194],[2.516011703836272,1.6339171050450814],[2.598076211353316,1.4999999999999998],[2.673019572565104,1.3619714992186402],[2.7406363729278027,1.2202099292274005],[2.8007412794916053,1.0751038486359008],[2.8531695488854605,0.9270509831248421],[2.897777478867205,0.7764571353075622],[2.934442802201417,0.623735072453278],[2.9630650217854133,0.46930339512069263],[2.9835656861048196,0.31358538980296036],[2.9958886042637216,0.1570078687288315],[3,0]],"fill":true,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"bdda052c83c5"},{"id":"g4-plans-and-elevations-q01","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

A solid cuboid is 5 cm long, 3 cm wide and 2 cm high.

On centimetre-squared paper, draw

  1. the plan of the cuboid
  2. the front elevation of the cuboid, seen with the 5 cm edge facing you.
","answerType":"open","answer":"a) a 5 cm by 3 cm rectangle b) a 5 cm by 2 cm rectangle","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"rectangle 5 cm by 3 cm"},{"label":"b","answer":"rectangle 5 cm by 2 cm"}],"solution":["The plan is the view from directly above, so it shows the length and the width: a 5 cm by 3 cm rectangle.","The front elevation is the view from the front, so it shows the length and the height: a 5 cm by 2 cm rectangle."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Given solid","diagram":{"type":"solid","kind":"cuboid","dims":{"l":5,"w":3,"h":2},"labels":{"l":"5 cm","w":"3 cm","h":"2 cm"},"caption":"Front: along the labelled length"}},{"label":"Plan","diagram":{"type":"axes","x":{"min":-1,"max":8,"step":1},"y":{"min":-1,"max":6,"step":1},"square":true,"polygons":[{"pts":[[0,0],[5,0],[5,3],[0,3]],"answer":true}]}},{"label":"Front elevation","diagram":{"type":"axes","x":{"min":-1,"max":8,"step":1},"y":{"min":-1,"max":6,"step":1},"square":true,"polygons":[{"pts":[[0,0],[5,0],[5,2],[0,2]],"answer":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"efd6facc3b5c"},{"id":"g4-plans-and-elevations-q02","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

A solid cylinder has radius 3 cm and height 7 cm. It stands on one of its circular faces.

Draw

  1. the plan of the cylinder
  2. the front elevation of the cylinder.
","answerType":"open","answer":"a) a circle of radius 3 cm b) a rectangle 6 cm wide and 7 cm tall","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"circle, radius 3 cm"},{"label":"b","answer":"rectangle 6 cm by 7 cm"}],"solution":["From above, a cylinder standing on its circular face looks like a circle of radius 3 cm.","From the front, it looks like a rectangle: the width is the diameter, 2 × 3 = 6 cm, and the height is 7 cm."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Given solid","diagram":{"type":"solid","kind":"cylinder","dims":{"r":3,"h":7},"labels":{"r":"3 cm","h":"7 cm"}}},{"label":"Plan","diagram":{"type":"shape","notAccurate":false,"bounds":[[-4,-4],[4,4]],"grid":true,"circles":[{"c":[0,0],"r":3,"answer":true}]}},{"label":"Front elevation","diagram":{"type":"axes","x":{"min":-1,"max":8,"step":1},"y":{"min":-1,"max":8,"step":1},"square":true,"polygons":[{"pts":[[0,0],[6,0],[6,7],[0,7]],"answer":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"52d13ee19619"},{"id":"g4-plans-and-elevations-q03","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

A solid is made from four centimetre cubes.

Three of the cubes are in a row on a table. The fourth cube sits on top of the left-hand cube.

The long side of the row faces you.

On centimetre-squared paper, draw

  1. the front elevation of the solid
  2. the plan of the solid.
","answerType":"open","answer":"a) an L shape: a row of 3 squares with 1 extra square on top of the left square b) a row of 3 squares","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"L shape - 3 squares in a row, 1 square on top left"},{"label":"b","answer":"3 squares in a row"}],"solution":["From the front you see all three cubes of the row, plus the extra cube on top at the left: an L shape made of 4 squares.","From above you see one square for each column of cubes: the stacked cube is directly over the left cube, so the plan is just a row of 3 squares."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Given solid","diagram":{"type":"solid","kind":"cubes","cells":[[0,0,0],[1,0,0],[2,0,0],[0,1,0]],"frontLabel":"Front"}},{"label":"Plan","diagram":{"type":"axes","x":{"min":-1,"max":4,"step":1},"y":{"min":-1,"max":4,"step":1},"square":true,"polygons":[{"pts":[[0,0],[3,0],[3,1],[0,1]],"answer":true}]}},{"label":"Front elevation","diagram":{"type":"axes","x":{"min":-1,"max":4,"step":1},"y":{"min":-1,"max":4,"step":1},"square":true,"polygons":[{"pts":[[0,0],[3,0],[3,1],[1,1],[1,2],[0,2]],"answer":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"fdc7234125d8"},{"id":"g4-plans-and-elevations-q04","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

The diagram shows the plan and the front elevation of a prism, drawn on centimetre-squared paper.

Draw the side elevation of the prism, seen from the right-hand side.

","answerType":"open","answer":"a rectangle 2 cm wide and 3 cm tall, with a horizontal line across it at height 1 cm","answerDisplay":null,"accept":[],"parts":[],"solution":["The plan shows the prism is 4 cm long and 2 cm deep.","The front elevation shows the left half is 3 cm tall and the right half is 1 cm tall, and this step runs the full depth of the prism.","From the right you see the depth (2 cm) as the width. The nearest part is 1 cm tall but the taller part behind rises to 3 cm, so the outline is a 2 cm by 3 cm rectangle with a line at height 1 cm showing the step."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Given plan","diagram":{"type":"axes","x":{"min":-1,"max":5,"step":1},"y":{"min":-1,"max":4,"step":1},"square":true,"polygons":[{"pts":[[0,0],[4,0],[4,2],[0,2]]}],"lines":[{"x":2,"from":0,"to":2}]}},{"label":"Given front elevation","diagram":{"type":"axes","x":{"min":-1,"max":5,"step":1},"y":{"min":-1,"max":4,"step":1},"square":true,"polygons":[{"pts":[[0,0],[4,0],[4,1],[2,1],[2,3],[0,3]]}]}},{"label":"Right elevation","diagram":{"type":"axes","x":{"min":-1,"max":4,"step":1},"y":{"min":-1,"max":4,"step":1},"square":true,"polygons":[{"pts":[[0,0],[2,0],[2,3],[0,3]],"answer":true}],"lines":[{"y":1,"from":0,"to":2,"answer":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"a17218475189"},{"id":"g4-plans-and-elevations-q05","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":2,"tier":"FH","calc":false,"text":"

The plan of a solid is a circle. The front elevation and the side elevation of the solid are both isosceles triangles.

  1. Write down a possible name of the solid.

The plan, the front elevation and the side elevation of a different solid are all squares of the same size.

  1. Write down a possible name of this solid.
","answerType":"multi","answer":"a) cone b) cube","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"cone"},{"label":"b","answer":"cube"}],"solution":["a) One solid with these views is a right circular cone: its plan is a circle and its front and side outlines are isosceles triangles.","b) A cube has three square views of the same size, so it is a valid example. These outlines alone do not prove that every hidden part of an unknown solid has the shape of a cube."],"diagram":null,"draw":false,"revision":"9da5661b79f8"},{"id":"g4-plans-and-elevations-q06","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

A solid cuboid is 6 cm long, 4 cm wide and 3 cm high.

Priya draws the front elevation of the cuboid, seen with the 6 cm edge facing her, as a rectangle 6 cm by 4 cm.

  1. Explain why Priya's front elevation is wrong.
  2. On centimetre-squared paper, draw the plan of the cuboid.
","answerType":"open","answer":"a) the front elevation should show the height, so it should be 6 cm by 3 cm - the 4 cm width is only seen from above b) a 6 cm by 4 cm rectangle","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"the elevation should be 6 cm by 3 cm; Priya has used the width instead of the height"},{"label":"b","answer":"rectangle 6 cm by 4 cm"}],"solution":["a) A front elevation shows the length and the height of the cuboid. The height is 3 cm, not 4 cm, so the rectangle should be 6 cm by 3 cm. Priya has drawn the plan measurement (width) instead of the height.","b) The plan is the view from above, showing length and width: a 6 cm by 4 cm rectangle."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Given solid","diagram":{"type":"solid","kind":"cuboid","dims":{"l":6,"w":4,"h":3},"labels":{"l":"6 cm","w":"4 cm","h":"3 cm"},"caption":"Front: along the labelled length"}},{"label":"Plan","diagram":{"type":"axes","x":{"min":-1,"max":8,"step":1},"y":{"min":-1,"max":6,"step":1},"square":true,"polygons":[{"pts":[[0,0],[6,0],[6,4],[0,4]],"answer":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"24cb22c2f906"},{"id":"g4-plans-and-elevations-q07","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

A solid cuboid has volume 48 cm3 and height 3 cm.

The plan of the cuboid is a square.

On centimetre-squared paper, draw the plan of the cuboid, full size.

","answerType":"open","answer":"a 4 cm by 4 cm square","answerDisplay":null,"accept":[],"parts":[],"solution":["Volume = area of plan × height, so the area of the plan = 48 ÷ 3 = 16 cm².","The plan is a square of area 16 cm², so each side is √16 = 4 cm.","Draw a 4 cm by 4 cm square."],"diagram":{"type":"axes","x":{"min":-1,"max":8,"step":1},"y":{"min":-1,"max":8,"step":1},"square":true,"polygons":[{"pts":[[0,0],[4,0],[4,4],[0,4]],"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"ad3b0028f9ea"},{"id":"g4-plans-and-elevations-q08","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":3,"tier":"H","calc":false,"text":"

A solid is made from six centimetre cubes.

Four cubes form a 2 by 2 square layer on a table. Two more cubes are stacked on top of the back-left cube, making a column 3 cubes tall.

On centimetre-squared paper, draw

  1. the plan of the solid
  2. the front elevation of the solid
  3. the side elevation of the solid, seen from the left.
","answerType":"open","answer":"a) a 2 by 2 square b) an L shape: 2 squares wide at the base, left column 3 squares tall, right column 1 square tall c) the same L shape: 2 squares wide at the base, back column 3 tall, front column 1 tall","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2 by 2 square"},{"label":"b","answer":"L shape, base 2 wide, left column 3 tall"},{"label":"c","answer":"L shape, base 2 wide, back column 3 tall"}],"solution":["a) From above there is a cube (or column) in each of the four positions, so the plan is a 2 by 2 square.","b) From the front, the left side has a column 3 cubes tall behind a single cube, so the outline on the left is 3 squares tall; the right side is 1 square tall. This gives an L shape.","c) From the left, the back position is 3 cubes tall and the front position is 1 cube tall, giving the same L shape with the tall column at the back."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Given solid","diagram":{"type":"solid","kind":"cubes","cells":[[0,0,0],[1,0,0],[0,0,1],[1,0,1],[0,1,1],[0,2,1]],"frontLabel":"Front"}},{"label":"Plan","diagram":{"type":"axes","x":{"min":-1,"max":4,"step":1},"y":{"min":-1,"max":4,"step":1},"square":true,"polygons":[{"pts":[[0,0],[2,0],[2,2],[0,2]],"answer":true}]}},{"label":"Front elevation","diagram":{"type":"axes","x":{"min":-1,"max":4,"step":1},"y":{"min":-1,"max":4,"step":1},"square":true,"polygons":[{"pts":[[0,0],[2,0],[2,1],[1,1],[1,3],[0,3]],"answer":true}]}},{"label":"Left elevation (front at left)","diagram":{"type":"axes","x":{"min":-1,"max":4,"step":1},"y":{"min":-1,"max":4,"step":1},"square":true,"polygons":[{"pts":[[0,0],[2,0],[2,3],[1,3],[1,1],[0,1]],"answer":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"5befa6071af6"},{"id":"g4-plans-and-elevations-q09","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":6,"tier":"FH","calc":true,"text":"

A solid square-based pyramid has base edges of 10 cm and vertical height 12 cm. The apex is directly above the centre of the base.

  1. Draw an accurate front elevation of the pyramid.
  2. Work out the total surface area of the pyramid.
","answerType":"multi","answer":"a) an isosceles triangle with base 10 cm and height 12 cm b) 360 cm²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"isosceles triangle, base 10 cm, height 12 cm"},{"label":"b","answer":"360 cm²"}],"solution":["a) From the front the pyramid looks like an isosceles triangle with base 10 cm and vertical height 12 cm.","b) The slant height of each triangular face runs from the apex to the midpoint of a base edge. The horizontal distance from the centre of the base to the midpoint of an edge is 5 cm.","Slant height = √(12² + 5²) = √169 = 13 cm.","Each triangular face has area 12 × 10 × 13 = 65 cm², so the four faces have area 4 × 65 = 260 cm².","The base has area 10 × 10 = 100 cm², so the total surface area = 260 + 100 = 360 cm²."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Given solid","diagram":{"type":"solid","kind":"pyramid","dims":{"b":10,"h":12},"labels":{"b":"10 cm","h":"12 cm"}}},{"label":"Front elevation","diagram":{"type":"axes","x":{"min":-1,"max":12,"step":1},"y":{"min":-1,"max":14,"step":1},"square":true,"polygons":[{"pts":[[0,0],[10,0],[5,12]],"answer":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"9afd32e5edf3"},{"id":"g4-plans-and-elevations-q10","topicId":"g4-plans-and-elevations","topic":"Plans and Elevations","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":6,"tier":"H","calc":true,"text":"

A solid is made from a cuboid with a square-based pyramid on top.

The cuboid has a square base of edge 8 cm and height 3 cm. The pyramid has a base of edge 8 cm and vertical height 3 cm, and its base fits exactly on top of the cuboid. The apex is directly above the centre of the base.

  1. Draw an accurate front elevation of the solid.
  2. Work out the total surface area of the solid.
","answerType":"multi","answer":"a) an 8 cm by 3 cm rectangle with an isosceles triangle of base 8 cm and height 3 cm on top b) 240 cm²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"rectangle 8 cm by 3 cm with isosceles triangle (base 8 cm, height 3 cm) on top"},{"label":"b","answer":"240 cm²"}],"solution":["a) From the front you see the cuboid as an 8 cm by 3 cm rectangle, with the pyramid as an isosceles triangle of base 8 cm and height 3 cm sitting on top of it.","b) The slant height of each pyramid face = √(3² + 4²) = √25 = 5 cm, since the apex is 3 cm above the centre and the centre is 4 cm from each edge midpoint.","Pyramid faces: 4 × 12 × 8 × 5 = 80 cm².","Cuboid sides: 4 × (8 × 3) = 96 cm².","Base of cuboid: 8 × 8 = 64 cm². The top of the cuboid is covered by the pyramid, so it is not counted.","Total surface area = 80 + 96 + 64 = 240 cm²."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Given solid","diagram":{"type":"shape","notAccurate":true,"points":[],"segments":[{"from":[0,0],"to":[8,0],"label":"8 cm"},{"from":[8,0],"to":[11.6,2.4],"label":"8 cm"},{"from":[11.6,2.4],"to":[3.6,2.4]},{"from":[3.6,2.4],"to":[0,0]},{"from":[0,0],"to":[0,3],"label":"3 cm"},{"from":[8,0],"to":[8,3]},{"from":[11.6,2.4],"to":[11.6,5.4]},{"from":[3.6,2.4],"to":[3.6,5.4]},{"from":[0,3],"to":[8,3]},{"from":[8,3],"to":[11.6,5.4]},{"from":[11.6,5.4],"to":[3.6,5.4]},{"from":[3.6,5.4],"to":[0,3]},{"from":[0,3],"to":[5.8,7.2]},{"from":[8,3],"to":[5.8,7.2]},{"from":[11.6,5.4],"to":[5.8,7.2]},{"from":[3.6,5.4],"to":[5.8,7.2]},{"from":[5.8,4.2],"to":[5.8,7.2],"label":"3 cm","dashed":true}],"angles":[]}},{"label":"Front elevation","diagram":{"type":"axes","x":{"min":-1,"max":10,"step":1},"y":{"min":-1,"max":8,"step":1},"square":true,"polygons":[{"pts":[[0,0],[8,0],[8,3],[4,6],[0,3]],"answer":true}],"lines":[{"y":3,"from":0,"to":8,"answer":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"e54ea667741d"},{"id":"g4-prime-factors-hcf-and-lcm-q01","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":1,"marks":1,"tier":"FH","calc":true,"text":"

Write down the lowest common multiple (LCM) of

3    6    9    18    54

","answerType":"exact","answer":"54","answerDisplay":null,"accept":["54"],"parts":[],"solution":["54 is a multiple of every number in the list: 54 = 3 x 18 = 6 x 9 = 9 x 6 = 18 x 3 = 54 x 1.","Since 54 is itself in the list, no smaller number can work, so the LCM is 54."],"diagram":null,"draw":false,"revision":"dec6e0d9d39d"},{"id":"g4-prime-factors-hcf-and-lcm-q02","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

Write 84 as a product of its prime factors.

","answerType":"exact","answer":"2^2 x 3 x 7","answerDisplay":null,"accept":["2^2 x 3 x 7","2 x 2 x 3 x 7","2²×3×7"],"parts":[],"solution":["84 = 2 x 42.","42 = 2 x 21 and 21 = 3 x 7.","So 84 = 2 x 2 x 3 x 7 = 2^2 x 3 x 7."],"diagram":null,"draw":false,"revision":"babd84ab2d8a"},{"id":"g4-prime-factors-hcf-and-lcm-q03","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":3,"marks":2,"tier":"FH","calc":true,"text":"

Write 264 as a product of its prime factors.

","answerType":"exact","answer":"2^3 x 3 x 11","answerDisplay":null,"accept":["2^3 x 3 x 11","2 x 2 x 2 x 3 x 11","2³×3×11"],"parts":[],"solution":["264 = 2 x 132, 132 = 2 x 66, 66 = 2 x 33.","33 = 3 x 11, and 11 is prime.","So 264 = 2 x 2 x 2 x 3 x 11 = 2^3 x 3 x 11."],"diagram":null,"draw":false,"revision":"a09b4513f006"},{"id":"g4-prime-factors-hcf-and-lcm-q04","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

Find the highest common factor (HCF) of 54 and 90

","answerType":"exact","answer":"18","answerDisplay":null,"accept":["18"],"parts":[],"solution":["54 = 2 x 3^3 and 90 = 2 x 3^2 x 5.","Take the lowest power of each common prime: 2 and 3^2.","HCF = 2 x 9 = 18.","A common factor must divide both numbers, so it cannot use more copies of a prime than either number contains. That is why we choose the lower of the two powers."],"diagram":null,"draw":false,"revision":"d4070dbb7971"},{"id":"g4-prime-factors-hcf-and-lcm-q05","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":5,"marks":2,"tier":"FH","calc":false,"text":"

Find the highest common factor (HCF) of 48 and 84

","answerType":"exact","answer":"12","answerDisplay":null,"accept":["12"],"parts":[],"solution":["48 = 2^4 x 3 and 84 = 2^2 x 3 x 7.","Take the lowest power of each common prime: 2^2 and 3.","HCF = 4 x 3 = 12."],"diagram":null,"draw":false,"revision":"5d5cca6ebf87"},{"id":"g4-prime-factors-hcf-and-lcm-q06","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":6,"marks":2,"tier":"H","calc":true,"text":"

Write down two numbers, both greater than 20, whose highest common factor (HCF) is 7

","answerType":"open","answer":"Any valid pair, e.g. 21 and 28 (21 = 3 x 7 and 28 = 4 x 7, and 3 and 4 share no common factor greater than 1, so HCF = 7).","answerDisplay":null,"accept":["21 and 28","28 and 35","21 and 35"],"parts":[],"solution":["Both numbers must be multiples of 7 that are greater than 20.","The multiples of 7 used must not share any other common factor.","For example 21 = 3 x 7 and 28 = 4 x 7: since 3 and 4 have no common factor greater than 1, HCF(21, 28) = 7."],"diagram":null,"draw":false,"revision":"295f53221590"},{"id":"g4-prime-factors-hcf-and-lcm-q07","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":7,"marks":3,"tier":"FH","calc":false,"text":"

Find the lowest common multiple (LCM) of 15 and 18

","answerType":"exact","answer":"90","answerDisplay":null,"accept":["90"],"parts":[],"solution":["15 = 3 x 5 and 18 = 2 x 3^2.","Take the highest power of every prime that appears: 2, 3^2 and 5.","LCM = 2 x 9 x 5 = 90.","A common multiple must contain enough prime factors to be divisible by both numbers. Taking the larger power of each prime provides exactly enough, giving the lowest such multiple."],"diagram":null,"draw":false,"revision":"09ab7e29ad2b"},{"id":"g4-prime-factors-hcf-and-lcm-q08","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":8,"marks":3,"tier":"FH","calc":true,"text":"

A = 23 × 3 × 52

B = 22 × 32 × 5

  1. Find the highest common factor (HCF) of A and B.
  2. Find the lowest common multiple (LCM) of A and B.

    Give your answers as products of powers of primes.

","answerType":"multi","answer":"a) 2^2 x 3 x 5 (= 60) b) 2^3 x 3^2 x 5^2 (= 1800)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2^2 x 3 x 5 = 60"},{"label":"b","answer":"2^3 x 3^2 x 5^2 = 1800"}],"solution":["a) For the HCF take the lowest power of each common prime: 2^2, 3 and 5, so HCF = 2^2 x 3 x 5 = 60.","b) For the LCM take the highest power of each prime: 2^3, 3^2 and 5^2, so LCM = 2^3 x 3^2 x 5^2 = 1800."],"diagram":null,"draw":false,"revision":"7a2de5d1d155"},{"id":"g4-prime-factors-hcf-and-lcm-q09","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":9,"marks":3,"tier":"FH","calc":true,"text":"

Write 360 as a product of powers of its prime factors.

","answerType":"exact","answer":"2^3 x 3^2 x 5","answerDisplay":null,"accept":["2^3 x 3^2 x 5","2³×3²×5"],"parts":[],"solution":["360 = 2 x 180, 180 = 2 x 90, 90 = 2 x 45.","45 = 3 x 15 and 15 = 3 x 5.","So 360 = 2 x 2 x 2 x 3 x 3 x 5 = 2^3 x 3^2 x 5."],"diagram":null,"draw":false,"revision":"8bce5a02530d"},{"id":"g4-prime-factors-hcf-and-lcm-q10","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":10,"marks":3,"tier":"FH","calc":true,"text":"

Nadia is making table decorations.

Each decoration needs one candle and one holder.

Candles are sold in packs of 9

Holders are sold in packs of 12

Nadia buys exactly the same number of candles as holders, with none left over.

Work out the smallest number of packs of candles and the smallest number of packs of holders she can buy.

","answerType":"multi","answer":"4 packs of candles and 3 packs of holders (36 of each)","answerDisplay":null,"accept":[],"parts":[{"label":"candles","answer":"4 packs"},{"label":"holders","answer":"3 packs"}],"solution":["The number of candles must equal the number of holders, so it must be a common multiple of 9 and 12.","9 = 3^2 and 12 = 2^2 x 3, so LCM = 2^2 x 3^2 = 36.","36 candles need 36 / 9 = 4 packs, and 36 holders need 36 / 12 = 3 packs."],"diagram":null,"draw":false,"revision":"d217ac04bcdf"},{"id":"g4-prime-factors-hcf-and-lcm-q11","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":11,"marks":3,"tier":"H","calc":true,"text":"

Two warning buoys are anchored in a harbour.

One buoy flashes every 24 seconds.

The other buoy flashes every 40 seconds.

At 8 pm the two buoys flash at the same time.

Work out how many seconds later the two buoys next flash at the same time.

","answerType":"exact","answer":"120 seconds","answerDisplay":null,"accept":["120","120 seconds","2 minutes"],"parts":[],"solution":["The buoys flash together at common multiples of 24 and 40.","24 = 2^3 x 3 and 40 = 2^3 x 5, so LCM = 2^3 x 3 x 5 = 120.","They next flash together 120 seconds (2 minutes) later."],"diagram":null,"draw":false,"revision":"031235d013e2"},{"id":"g4-prime-factors-hcf-and-lcm-q12","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":12,"marks":4,"tier":"FH","calc":true,"text":"
  1. Write 126 as a product of its prime factors.
  2. Find the lowest common multiple (LCM) of 126 and 60
","answerType":"multi","answer":"a) 2 x 3^2 x 7 b) 1260","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2 x 3^2 x 7"},{"label":"b","answer":"1260"}],"solution":["a) 126 = 2 x 63, 63 = 9 x 7 = 3^2 x 7, so 126 = 2 x 3^2 x 7.","b) 60 = 2^2 x 3 x 5.","b) Take the highest power of each prime: 2^2, 3^2, 5 and 7, so LCM = 4 x 9 x 5 x 7 = 1260."],"diagram":null,"draw":false,"revision":"d9e9b01e1f66"},{"id":"g4-prime-factors-hcf-and-lcm-q13","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":13,"marks":4,"tier":"FH","calc":true,"text":"
  1. Write 448 as a product of powers of its prime factors.
  2. Find the highest common factor (HCF) of 448 and 168
","answerType":"multi","answer":"a) 2^6 x 7 b) 56","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2^6 x 7"},{"label":"b","answer":"56"}],"solution":["a) 448 = 2 x 224 = 2 x 2 x 112 = 2 x 2 x 2 x 56 = 2 x 2 x 2 x 2 x 28 = 2 x 2 x 2 x 2 x 2 x 14 = 2^6 x 7.","b) 168 = 2^3 x 3 x 7.","b) Take the lowest power of each common prime: 2^3 and 7, so HCF = 8 x 7 = 56."],"diagram":null,"draw":false,"revision":"7b0e2bc6876e"},{"id":"g4-prime-factors-hcf-and-lcm-q14","topicId":"g4-prime-factors-hcf-and-lcm","topic":"Prime Factors, HCF and LCM","grade":"4","topicGrade":"4","strand":"N","difficulty":14,"marks":4,"tier":"FH","calc":false,"text":"
  1. Find the highest common factor (HCF) of 72 and 120
  2. Find the lowest common multiple (LCM) of 72 and 120
","answerType":"multi","answer":"a) 24 b) 360","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"24"},{"label":"b","answer":"360"}],"solution":["72 = 2^3 x 3^2 and 120 = 2^3 x 3 x 5.","a) HCF: lowest powers of the common primes are 2^3 and 3, so HCF = 8 x 3 = 24.","b) LCM: highest powers of all primes are 2^3, 3^2 and 5, so LCM = 8 x 9 x 5 = 360."],"diagram":null,"draw":false,"revision":"8e634114b428"},{"id":"g4-probability-q01","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

A spinner can land on red, blue, green or yellow.

The table shows the probabilities for three of the colours.

Colourredbluegreenyellow
Probability0.20.350.1

Work out the probability that the spinner lands on yellow.

","answerType":"exact","answer":"0.35","answerDisplay":null,"accept":["35/100","7/20"],"parts":[],"solution":["The four probabilities must add up to 1.","0.2 + 0.35 + 0.1 = 0.65.","P(yellow) = 1 - 0.65 = 0.35"],"diagram":null,"draw":false,"revision":"5b3ba580a238"},{"id":"g4-probability-q02","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Amy and Ben each spin the same spinner and record how many times it lands on red.

Number of spinsNumber of reds
Amy5018
Ben15042

Using all of the results, work out the best estimate for the probability that the spinner lands on red.

","answerType":"exact","answer":"3/10","answerDisplay":"310","accept":["60/200","0.3","30%"],"parts":[],"solution":["Combining both sets of results gives the largest number of trials, so it gives the best estimate.","Total reds = 18 + 42 = 60, total spins = 50 + 150 = 200.","Best estimate for P(red) = 60200 = 310 = 0.3"],"diagram":null,"draw":false,"revision":"fe36db930402"},{"id":"g4-probability-q03","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":3,"marks":2,"tier":"FH","calc":true,"text":"

The probability that a bus is late on any day is 0.15.

Nadia catches this bus on 40 days.

Work out an estimate for the number of days the bus is late.

","answerType":"exact","answer":"6","answerDisplay":null,"accept":["6 days"],"parts":[],"solution":["Expected number of late days = probability × number of days.","0.15 × 40 = 6"],"diagram":null,"draw":false,"revision":"d8b8114e85ce"},{"id":"g4-probability-q04","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":4,"marks":3,"tier":"F","calc":false,"text":"

A bag contains only red, blue and green counters.

16 of the counters are green.

The rest of the counters are red and blue in the ratio red : blue = 3 : 2.

A counter is taken from the bag at random.

Work out the probability that the counter is blue.

","answerType":"exact","answer":"1/3","answerDisplay":"13","accept":["10/30","2/6"],"parts":[],"solution":["The fraction of counters that are red or blue = 1 - 16 = 56.","Blue counters are 2 parts out of 5, so the blue fraction of the rest = 25.","P(blue) = 25 × 56 = 1030 = 13"],"diagram":null,"draw":false,"revision":"2d184739537e"},{"id":"g4-probability-q05","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

A five-sided spinner can land on 1, 2, 3, 4 or 5.

The table shows some of the probabilities.

Number12345
Probability0.10.20.250.15
  1. Work out the probability that the spinner lands on 5.
  2. The spinner is spun 300 times. Work out an estimate for the number of times it lands on 5.
","answerType":"multi","answer":"a) 0.3; b) 90","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"0.3"},{"label":"b","answer":"90"}],"solution":["0.1 + 0.2 + 0.25 + 0.15 = 0.7, and the probabilities must sum to 1.","P(5) = 1 - 0.7 = 0.3.","Estimate for the number of 5s = 0.3 × 300 = 90"],"diagram":null,"draw":false,"revision":"ae28cda562a0"},{"id":"g4-probability-q06","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":6,"marks":4,"tier":"FH","calc":true,"text":"

A box contains white, black, silver and gold beads.

The probability of picking each colour at random is shown in the table.

Colourwhiteblacksilvergold
Probability0.160.240.40.2

There are 12 white beads in the box.

  1. Work out the total number of beads in the box.
  2. Work out the number of black beads in the box.
","answerType":"multi","answer":"a) 75; b) 18","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"75"},{"label":"b","answer":"18"}],"solution":["P(white) = 0.16 and there are 12 white beads, so the total = 12 ÷ 0.16 = 75.","Number of black beads = 0.24 × 75 = 18.","Check: the probabilities 0.16 + 0.24 + 0.4 + 0.2 = 1."],"diagram":null,"draw":false,"revision":"78c8e29eccc0"},{"id":"g4-probability-q07","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":7,"marks":4,"tier":"FH","calc":true,"text":"

A jar contains only red, blue and green marbles.

The probability of picking a red marble at random is 0.3.

There are 24 red marbles in the jar.

The rest of the marbles are blue and green in the ratio blue : green = 5 : 3.

Work out the number of green marbles in the jar.

","answerType":"exact","answer":"21","answerDisplay":null,"accept":["21 green marbles"],"parts":[],"solution":["Total marbles = 24 ÷ 0.3 = 80.","Marbles that are blue or green = 80 - 24 = 56.","Each ratio part = 56 ÷ (5 + 3) = 7.","Green marbles = 3 × 7 = 21"],"diagram":null,"draw":false,"revision":"d3ba1ecc7ea5"},{"id":"g4-probability-q08","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":8,"marks":4,"tier":"FH","calc":false,"text":"

A four-sided spinner can land on A, B, C or D.

P(A) = 0.1 and P(D) = 0.3.

The probabilities of landing on B and on C are in the ratio 2 : 1.

Complete the probability table.

LetterABCD
Probability0.10.3
","answerType":"multi","answer":"P(B) = 0.4, P(C) = 0.2","answerDisplay":null,"accept":[],"parts":[{"label":"B","answer":"0.4"},{"label":"C","answer":"0.2"}],"solution":["P(B) + P(C) = 1 - 0.1 - 0.3 = 0.6.","B and C are in the ratio 2 : 1, so split 0.6 into 3 parts of 0.2.","P(B) = 2 × 0.2 = 0.4 and P(C) = 0.2"],"diagram":null,"draw":false,"revision":"4862df57cf18"},{"id":"g4-probability-q09","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

Aisha and Blake each take one shot at a target.

The probability that Aisha hits the target is 0.8.

The probability that Blake hits the target is 0.65.

The two events are independent.

  1. Work out the probability that they both hit the target.
  2. Work out the probability that neither of them hits the target.
","answerType":"multi","answer":"a) 0.52; b) 0.07","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"0.52"},{"label":"b","answer":"0.07"}],"solution":["P(both hit) = 0.8 × 0.65 = 0.52.","P(Aisha misses) = 0.2 and P(Blake misses) = 0.35.","P(neither hits) = 0.2 × 0.35 = 0.07"],"diagram":null,"draw":false,"revision":"05e84beeba2a"},{"id":"g4-probability-q10","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

A four-sided spinner can land on 1, 2, 3 or 4.

The table shows the probabilities.

Number1234
Probabilityx2xx + 0.10.3
  1. Work out the value of x.
  2. The spinner is spun 200 times. Work out an estimate for the number of times it lands on 2.
","answerType":"multi","answer":"a) x = 0.15; b) 60","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"0.15"},{"label":"b","answer":"60"}],"solution":["The probabilities sum to 1: x + 2x + x + 0.1 + 0.3 = 1.","4x + 0.4 = 1, so 4x = 0.6 and x = 0.15.","P(2) = 2x = 0.3.","Estimate for the number of 2s = 0.3 × 200 = 60"],"diagram":null,"draw":false,"revision":"a4305f986b54"},{"id":"g4-probability-q11","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":11,"marks":5,"tier":"FH","calc":false,"text":"

A three-colour spinner can land on red, white or blue.

The spinner is spun 100 times. The table shows the results.

Colourredwhiteblue
Frequency253540
  1. Work out the best estimate for the probability that the spinner lands on blue.
  2. Two of these identical spinners are spun together. Use the results to estimate the probability that both spinners land on red.

Assume the two spinners operate independently.

","answerType":"multi","answer":"a) 2/5; b) 1/16","answerDisplay":"a) 25; b) 116","accept":[],"parts":[{"label":"a","answer":"2/5 (or 0.4)","answerDisplay":"25 (or 0.4)"},{"label":"b","answer":"1/16 (or 0.0625)","answerDisplay":"116 (or 0.0625)"}],"solution":["Estimate for P(blue) = 40100 = 25.","Estimate for P(red) = 25100 = 14.","The spinners are independent, so P(both red) = 14 × 14 = 116"],"diagram":null,"draw":false,"revision":"cede9ebe5b37"},{"id":"g4-probability-q12","topicId":"g4-probability","topic":"Probability","grade":"4","topicGrade":"4","strand":"P","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

A bag contains only red, green and yellow counters.

The number of green counters is 4 times the number of red counters.

The number of yellow counters is 6 more than the number of red counters.

A counter is taken from the bag at random.

The probability that it is red is 18.

Work out the number of yellow counters in the bag.

","answerType":"exact","answer":"9","answerDisplay":null,"accept":["9 yellow counters"],"parts":[],"solution":["Let the number of red counters be x. Then green = 4x and yellow = x + 6.","Total = x + 4x + x + 6 = 6x + 6.","P(red) = x6x + 6 = 18, so 8x = 6x + 6.","2x = 6, so x = 3.","Yellow counters = 3 + 6 = 9.","Check: red 3, green 12, yellow 9, total 24, and P(red) = 324 = 18."],"diagram":null,"draw":false,"revision":"aba140c69f9c"},{"id":"g4-pythagoras-q01","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Triangle ABC is a right-angled triangle with the right angle at B.

AB = 6 cm and BC = 8 cm.

Work out the length of AC.

","answerType":"exact","answer":"10 cm","answerDisplay":null,"accept":["10","10cm","10 centimetres"],"parts":[],"solution":["The hypotenuse is the side opposite the right angle and is the longest side. Pythagoras relates the areas of squares on the three sides: the two smaller areas add to the largest. It applies to right-angled triangles.","AC is the hypotenuse, so use Pythagoras' theorem: AC2 = AB2 + BC2","AC2 = 62 + 82 = 36 + 64 = 100","AC = √100 = 10 cm"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,6],[0,0],[8,0]],"labels":["A","B","C"],"sides":["6 cm","8 cm",null],"fill":false}],"angles":[{"at":[0,0],"from":[8,0],"to":[0,6],"right":true}]},"draw":false,"revision":"f6bfed6f132b"},{"id":"g4-pythagoras-q02","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Triangle PQR is a right-angled triangle with the right angle at Q.

PQ = 7 cm and QR = 9 cm.

Work out the length of PR.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"11.4 cm","answerDisplay":null,"accept":["11.4","11.4cm"],"parts":[],"solution":["PR is the hypotenuse, so PR2 = PQ2 + QR2","PR2 = 72 + 92 = 49 + 81 = 130","PR = √130 = 11.401... = 11.4 cm (1 decimal place)"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,7],[0,0],[9,0]],"labels":["P","Q","R"],"sides":["7 cm","9 cm",null],"fill":false}],"angles":[{"at":[0,0],"from":[9,0],"to":[0,7],"right":true}]},"draw":false,"revision":"ce2daaa243c9"},{"id":"g4-pythagoras-q03","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":2,"tier":"FH","calc":true,"text":"

Triangle DEF is a right-angled triangle with the right angle at E.

The hypotenuse DF = 20 cm and DE = 8 cm.

Work out the length of EF.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"18.3 cm","answerDisplay":null,"accept":["18.3","18.3cm"],"parts":[],"solution":["EF is a shorter side, so rearrange Pythagoras' theorem: EF2 = DF2 − DE2","EF2 = 202 − 82 = 400 − 64 = 336","EF = √336 = 18.330... = 18.3 cm (1 decimal place)"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,8],[0,0],[18.33,0]],"labels":["D","E","F"],"sides":["8 cm",null,"20 cm"],"fill":false}],"angles":[{"at":[0,0],"from":[18.33,0],"to":[0,8],"right":true}]},"draw":false,"revision":"4960f8786fca"},{"id":"g4-pythagoras-q04","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":2,"tier":"H","calc":true,"text":"

A is the point with coordinates (2, 3).

B is the point with coordinates (10, 9).

Work out the length of AB.

","answerType":"exact","answer":"10","answerDisplay":null,"accept":["10 units","AB = 10"],"parts":[],"solution":["The horizontal distance from A to B is 10 − 2 = 8","The vertical distance from A to B is 9 − 3 = 6","AB2 = 82 + 62 = 64 + 36 = 100, so AB = √100 = 10"],"diagram":null,"draw":false,"revision":"29c24ad50999"},{"id":"g4-pythagoras-q05","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":2,"tier":"FH","calc":true,"text":"

A rectangle is 9 cm long and 6 cm wide.

Work out the length of a diagonal of the rectangle.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"10.8 cm","answerDisplay":null,"accept":["10.8","10.8cm"],"parts":[],"solution":["The diagonal is the hypotenuse of a right-angled triangle with sides 9 cm and 6 cm","diagonal2 = 92 + 62 = 81 + 36 = 117","diagonal = √117 = 10.816... = 10.8 cm (1 decimal place)"],"diagram":null,"draw":false,"revision":"523c7356e988"},{"id":"g4-pythagoras-q06","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

A ladder is 6.5 m long.

The ladder leans against a vertical wall, with the foot of the ladder on horizontal ground 2.5 m from the base of the wall.

Work out how far up the wall the top of the ladder reaches.

","answerType":"exact","answer":"6 m","answerDisplay":null,"accept":["6","6m","6 metres"],"parts":[],"solution":["The ladder, the wall and the ground form a right-angled triangle with the ladder as the hypotenuse","height2 = 6.52 − 2.52 = 42.25 − 6.25 = 36","height = √36 = 6 m"],"diagram":{"type":"shape","notAccurate":true,"segments":[{"from":[0,0],"to":[0,7]},{"from":[-1,0],"to":[4,0]},{"from":[2.5,0],"to":[0,6],"label":"6.5 m","labelPos":"above-right"}],"angles":[{"at":[0,0],"from":[2.5,0],"to":[0,6],"right":true}],"texts":[{"at":[1.25,-0.5],"text":"2.5 m"},{"at":[-0.7,3.5],"text":"wall"},{"at":[3.3,-0.5],"text":"ground"}]},"draw":false,"revision":"bef5cfbf7d8e"},{"id":"g4-pythagoras-q07","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

A triangle has sides of length 10 cm, 24 cm and 26 cm.

Show that the triangle is a right-angled triangle.

","answerType":"open","answer":"10^2 + 24^2 = 100 + 576 = 676 = 26^2, so by the converse of Pythagoras' theorem the triangle is right-angled","answerDisplay":null,"accept":[],"parts":[],"solution":["102 + 242 = 100 + 576 = 676","262 = 676","Since 102 + 242 = 262, the sides satisfy Pythagoras' theorem, so the triangle is right-angled (the right angle is opposite the 26 cm side)"],"diagram":null,"draw":false,"revision":"5a41b3560dff"},{"id":"g4-pythagoras-q08","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":4,"tier":"FH","calc":true,"text":"

ABCD is a trapezium in which AB is parallel to DC.

Angle ADC = angle DAB = 90°.

AB = 9 cm, AD = 12 cm and DC = 14 cm.

Work out the perimeter of the trapezium.

","answerType":"exact","answer":"48 cm","answerDisplay":null,"accept":["48","48cm"],"parts":[],"solution":["Draw the line from B perpendicular to DC, meeting DC at E. Then DE = AB = 9 cm and BE = AD = 12 cm","EC = 14 − 9 = 5 cm","BC2 = 122 + 52 = 144 + 25 = 169, so BC = 13 cm","Perimeter = 9 + 13 + 14 + 12 = 48 cm"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,12],[9,12],[14,0],[0,0]],"labels":["A","B","C","D"],"sides":["9 cm",null,"14 cm","12 cm"],"fill":false}],"segments":[{"from":[0,12],"to":[9,12],"parallel":1},{"from":[0,0],"to":[14,0],"parallel":1},{"from":[9,12],"to":[9,0],"dashed":true,"answer":true}],"angles":[{"at":[0,12],"from":[9,12],"to":[0,0],"right":true},{"at":[0,0],"from":[14,0],"to":[0,12],"right":true},{"at":[9,0],"from":[9,12],"to":[14,0],"right":true,"answer":true}],"points":[{"at":[9,0],"label":"E","style":"none","labelPos":"below","answer":true}]},"draw":false,"revision":"c6383f7d3d4b"},{"id":"g4-pythagoras-q09","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

A rectangle measures 8 cm by 15 cm.

A circle is drawn that passes through all four vertices of the rectangle, so a diagonal of the rectangle is a diameter of the circle.

Work out the circumference of the circle.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"53.4 cm","answerDisplay":null,"accept":["53.4","53.4cm"],"parts":[],"solution":["diagonal2 = 82 + 152 = 64 + 225 = 289","diagonal = √289 = 17 cm, so the diameter of the circle is 17 cm","Circumference = π × d = π × 17","Circumference = 53.407... = 53.4 cm (1 decimal place)"],"diagram":null,"draw":false,"revision":"9292a9b04b1c"},{"id":"g4-pythagoras-q10","topicId":"g4-pythagoras","topic":"Pythagoras","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

A field ABCD is a trapezium in which AB is parallel to DC.

Angle DAB = angle ADC = 90°.

AB = 25 m, AD = 15 m and DC = 33 m.

Joe wants to put a fence around the whole perimeter of the field.

Fencing is sold in 12 m rolls. Each roll costs £19.50.

Work out the total cost of the rolls of fencing Joe must buy.

","answerType":"exact","answer":"£156","answerDisplay":null,"accept":["156","£156.00","156.00"],"parts":[],"solution":["Draw the line from B perpendicular to DC, meeting DC at E. Then EC = 33 − 25 = 8 m and BE = 15 m","BC2 = 82 + 152 = 64 + 225 = 289, so BC = 17 m","Perimeter = 25 + 17 + 33 + 15 = 90 m","90 ÷ 12 = 7.5, so Joe must buy 8 rolls","Total cost = 8 × £19.50 = £156"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,15],[25,15],[33,0],[0,0]],"labels":["A","B","C","D"],"sides":["25 m",null,"33 m","15 m"],"fill":false}],"segments":[{"from":[0,15],"to":[25,15],"parallel":1},{"from":[0,0],"to":[33,0],"parallel":1},{"from":[25,15],"to":[25,0],"dashed":true,"answer":true}],"angles":[{"at":[0,15],"from":[25,15],"to":[0,0],"right":true},{"at":[0,0],"from":[33,0],"to":[0,15],"right":true},{"at":[25,0],"from":[25,15],"to":[33,0],"right":true,"answer":true}],"points":[{"at":[25,0],"label":"E","style":"none","labelPos":"below","answer":true}]},"draw":false,"revision":"ef890e3ac6a3"},{"id":"g4-real-life-and-distance-time-graphs-q01","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

A graph shows the total cost of hiring a ladder plotted against the number of days of hire.

The graph is a straight line with gradient 6.

Interpret what the gradient tells you about the cost of hiring the ladder.

Cost is measured in pounds and hire time in days.

","answerType":"open","answer":"The cost increases by £6 for each extra day of hire","answerDisplay":null,"accept":["It costs £6 per day","£6 for each day of hire"],"parts":[],"solution":["The gradient of a cost graph is the rate of change of cost","A gradient of 6 means the cost goes up by £6 for each extra day"],"diagram":null,"draw":false,"revision":"0b22724e2d5a"},{"id":"g4-real-life-and-distance-time-graphs-q02","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":2,"marks":1,"tier":"H","calc":false,"text":"

Water is poured into a tank at a steady rate.

The graph of depth of water, in cm, against time is a straight line that crosses the vertical axis at 20.

Interpret the value 20.

","answerType":"open","answer":"There was already 20 cm of water in the tank at the start","answerDisplay":null,"accept":["The depth of water at time 0 was 20 cm","The tank started with a depth of 20 cm"],"parts":[],"solution":["The vertical intercept is the value when time = 0","So the depth of water at the start was 20 cm"],"diagram":null,"draw":false,"revision":"67db47ff2020"},{"id":"g4-real-life-and-distance-time-graphs-q03","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

Amy cycles from her house to the library.

She leaves home at 09:00 and arrives at the library at 09:45.

The library is 12 km from her house and she cycles at a steady speed.

Work out Amy's average speed in km/h.

","answerType":"exact","answer":"16 km/h","answerDisplay":null,"accept":["16"],"parts":[],"solution":["Time taken = 45 minutes = 0.75 hours","Speed = distance divided by time = 120.75","Speed = 16 km/h"],"diagram":null,"draw":false,"revision":"418cf76cfc8e"},{"id":"g4-real-life-and-distance-time-graphs-q04","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

The distance-time graph shows Marek's car journey from home to a garden centre and back.

  1. For how many minutes does Marek stay at the garden centre?
  2. Work out his speed, in km/h, on the way to the garden centre.
  3. Work out his speed, in km/h, on the way home.
","answerType":"multi","answer":"a) 30 minutes; b) 40 km/h; c) 30 km/h","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"30 minutes"},{"label":"b","answer":"40 km/h"},{"label":"c","answer":"30 km/h"}],"solution":["a) The horizontal section runs from 10:00 to 10:30, so 30 minutes","b) He travels 40 km between 09:00 and 10:00, so speed = 401 = 40 km/h","c) He travels 40 km between 10:30 and 11:50, which is 80 minutes = 43 hours","Speed = 40 divided by 43 = 30 km/h"],"diagram":{"type":"axes","height":280,"x":{"min":540,"max":720,"step":30,"minor":3,"format":"time","label":"Time"},"y":{"min":0,"max":50,"step":10,"minor":2,"label":"Distance from home (km)"},"curves":[{"pts":[[540,0],[600,40],[630,40],[710,0]]}]},"draw":false,"revision":"b304ba7a3931"},{"id":"g4-real-life-and-distance-time-graphs-q05","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

A graph shows Kelly's total pay plotted against the number of hours she works.

The graph is a straight line through (0, 0) and through the point (8, 96), where pay is in pounds.

  1. Work out Kelly's hourly rate of pay.
  2. Work out Kelly's total pay for working 11 hours.
","answerType":"multi","answer":"a) £12 per hour; b) £132","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£12 per hour"},{"label":"b","answer":"£132"}],"solution":["a) 8 hours of work pays £96, so hourly rate = 968 = £12","b) Pay for 11 hours = 11 times 12 = £132"],"diagram":{"type":"axes","height":280,"x":{"min":0,"max":12,"step":1,"label":"Hours worked"},"y":{"min":0,"max":150,"step":10,"label":"Total pay (£)"},"lines":[{"through":[[0,0],[8,96]],"from":0,"to":12}],"points":[{"at":[8,96],"style":"dot"}]},"draw":false,"revision":"471afc98a034"},{"id":"g4-real-life-and-distance-time-graphs-q06","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":false,"text":"

A taxi company shows its fares on a graph of cost against distance.

The graph is a straight line through (0, 3) and (10, 18), where cost is in pounds and distance is in miles.

  1. Write down the fixed charge for any journey.
  2. Work out the cost per mile.
  3. Work out the fare for a journey of 6 miles.
","answerType":"multi","answer":"a) £3; b) £1.50 per mile; c) £12","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£3"},{"label":"b","answer":"£1.50"},{"label":"c","answer":"£12"}],"solution":["a) The vertical intercept is 3, so the fixed charge is £3","b) Gradient = 18 - 310 - 0 = 1510 = £1.50 per mile","c) Fare = 3 + 6 times 1.50 = 3 + 9 = £12"],"diagram":{"type":"axes","height":280,"x":{"min":0,"max":10,"step":1,"label":"Distance (miles)"},"y":{"min":0,"max":20,"step":2,"minor":2,"label":"Cost (£)"},"lines":[{"through":[[0,3],[10,18]],"from":0,"to":10}]},"draw":false,"revision":"fb91eae4edf5"},{"id":"g4-real-life-and-distance-time-graphs-q07","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

An electricity company's monthly bill is shown on a graph of total bill against units used.

The graph is a straight line through (0, 15) and (400, 95), where the bill is in pounds.

  1. Work out the gradient of the line.
  2. Interpret what the gradient tells you.
","answerType":"multi","answer":"a) 0.2; b) each unit of electricity costs 20p","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"0.2"},{"label":"b","answer":"Each unit of electricity used adds 20p to the bill"}],"solution":["a) Gradient = 95 - 15400 - 0 = 80400 = 0.2","b) The bill rises by £0.20, that is 20p, for each extra unit of electricity used"],"diagram":{"type":"axes","height":280,"x":{"min":0,"max":400,"step":50,"label":"Units used"},"y":{"min":0,"max":100,"step":10,"label":"Total bill (£)"},"lines":[{"through":[[0,15],[400,95]],"from":0,"to":400}]},"draw":false,"revision":"9d4cdd741608"},{"id":"g4-real-life-and-distance-time-graphs-q08","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

The graph shows the temperature of a cup of coffee as it cools.

  1. Write down the temperature of the coffee at the start.
  2. Find the temperature of the coffee after 20 minutes.
  3. Find how long the coffee takes to cool to 30°C.
","answerType":"multi","answer":"a) 90°C; b) 40°C; c) 35 minutes","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"90°C"},{"label":"b","answer":"40°C"},{"label":"c","answer":"35 minutes"}],"solution":["a) Read the curve at time 0: the temperature is 90°C","b) Read up from 20 minutes to the curve and across: 40°C","c) Read across from 30°C to the curve and down: 35 minutes"],"diagram":{"type":"axes","height":300,"x":{"min":0,"max":40,"step":5,"minor":5,"label":"Time (minutes)"},"y":{"min":0,"max":100,"step":10,"minor":2,"label":"Temperature (°C)"},"curves":[{"pts":[[0,90],[10,60],[20,40],[30,32],[35,30],[40,28]],"smooth":true}]},"draw":false,"revision":"e7a0cae4c5d1"},{"id":"g4-real-life-and-distance-time-graphs-q09","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":9,"marks":4,"tier":"FH","calc":true,"text":"

Ben drives from Leeds to visit his aunt.

He leaves Leeds at 09:30 and drives 60 km in 1 hour.

He then stops at a service station for 30 minutes.

He then drives a further 40 km in 1 hour to arrive at his aunt's house.

  1. Work out the time Ben arrives at his aunt's house.
  2. Work out Ben's average speed, in km/h, for the whole journey from Leeds to his aunt's house.
","answerType":"multi","answer":"a) 12:00; b) 40 km/h","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"12:00"},{"label":"b","answer":"40 km/h"}],"solution":["a) 09:30 + 1 hour = 10:30, + 30 minutes = 11:00, + 1 hour = 12:00","b) Total distance = 60 + 40 = 100 km","Total time = 2.5 hours, including the stop","Average speed = 1002.5 = 40 km/h"],"diagram":null,"draw":false,"revision":"2a98ac1e379d"},{"id":"g4-real-life-and-distance-time-graphs-q10","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"

The graph shows the depth of water in a harbour, in metres, during the 6 hours after midday.

By drawing a suitable tangent to the curve, estimate the rate at which the depth is increasing 3 hours after midday.

Give your answer in metres per hour.

","answerType":"open","answer":"About 2.5 metres per hour (accept 2 to 3)","answerDisplay":null,"accept":["2.5 m/h","any value from 2 to 3 m per hour with a tangent drawn"],"parts":[],"solution":["A tangent is a straight line that follows the curve’s direction at the chosen point. Its gradient estimates how quickly the depth is changing at that instant, rather than an average over the whole time interval.","Draw a tangent to the curve at t = 3","Work out the gradient of the tangent using a large triangle, for example rise 5 m over run 2 hours","Gradient of tangent is approximately 52 = 2.5 metres per hour"],"diagram":{"type":"axes","height":300,"x":{"min":0,"max":6,"step":1,"minor":2,"label":"Time after midday (hours)"},"y":{"min":0,"max":14,"step":2,"minor":2,"label":"Depth of water (m)"},"curves":[{"pts":[[0,2],[1,3],[2,5],[3,7.5],[4,10],[5,12],[6,13]],"smooth":true}],"lines":[{"through":[[3,7.5],[5,12.5]],"from":1,"to":5.5,"answer":true,"label":"tangent","labelAt":[4.6,11.5],"labelPos":"below-right"}]},"draw":true,"revision":"c078adde7714"},{"id":"g4-real-life-and-distance-time-graphs-q11","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":11,"marks":5,"tier":"F","calc":true,"text":"

Sara walks from home to a park 3 km away.

She leaves home at 14:00 and walks at a steady speed, arriving at the park at 14:40.

She stays at the park for 30 minutes.

She then walks straight home at a steady speed of 4 km/h.

  1. Work out Sara's speed, in km/h, on the way to the park.
  2. Work out the time Sara arrives back home.
  3. Work out the total time, in hours and minutes, that Sara is away from home.
","answerType":"multi","answer":"a) 4.5 km/h; b) 15:55; c) 1 hour 55 minutes","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"4.5 km/h"},{"label":"b","answer":"15:55"},{"label":"c","answer":"1 hour 55 minutes"}],"solution":["a) 40 minutes = 23 hour, so speed = 3 divided by 23 = 4.5 km/h","b) She leaves the park at 14:40 + 30 minutes = 15:10","Walking home takes 34 hour = 45 minutes at 4 km/h","15:10 + 45 minutes = 15:55","c) From 14:00 to 15:55 is 1 hour 55 minutes"],"diagram":null,"draw":false,"revision":"b144b23296ab"},{"id":"g4-real-life-and-distance-time-graphs-q12","topicId":"g4-real-life-and-distance-time-graphs","topic":"Real Life and Distance Time Graphs","grade":"4","topicGrade":"4","strand":"A","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

Jamal leaves home at 10:00 and drives at a steady speed of 60 km/h for 1 hour.

He rests for 30 minutes.

He then drives straight home at a steady speed of 40 km/h.

  1. Draw a distance-time graph for Jamal's journey.
  2. Work out the time Jamal arrives back home.
","answerType":"multi","answer":"a) line to (11:00, 60), horizontal to (11:30, 60), line to (13:00, 0); b) 13:00","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Straight line from (10:00, 0) to (11:00, 60), horizontal to (11:30, 60), straight line down to (13:00, 0)"},{"label":"b","answer":"13:00"}],"solution":["a) In the first hour he travels 60 km, so draw a line from (10:00, 0) to (11:00, 60)","The rest is a horizontal line from (11:00, 60) to (11:30, 60)","b) The journey home is 60 km at 40 km/h, which takes 6040 = 1.5 hours","11:30 + 1 hour 30 minutes = 13:00, so the final line goes down to (13:00, 0)"],"diagram":{"type":"axes","height":300,"x":{"min":600,"max":810,"step":30,"minor":3,"format":"time","label":"Time"},"y":{"min":0,"max":70,"step":10,"minor":2,"label":"Distance from home (km)"},"curves":[{"pts":[[600,0],[660,60],[690,60],[780,0]],"answer":true}]},"draw":true,"revision":"1550768be087"},{"id":"g4-sampling-q01","topicId":"g4-sampling","topic":"Sampling","grade":"4","topicGrade":"4","strand":"S","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

There are 800 students at a school.

Mia asks a sample of 40 students how they travel to school. 12 of the students in the sample walk to school.

Work out an estimate for the number of students at the school who walk to school.

","answerType":"exact","answer":"240","answerDisplay":null,"accept":["240 students"],"parts":[],"solution":["Fraction of the sample who walk = 1240.","Estimate = 800 × 1240 = 800 × 0.3 = 240.","This assumes the sample’s proportion is representative of the whole school. It is an estimate, so the actual number may differ."],"diagram":null,"draw":false,"revision":"599e01274c3e"},{"id":"g4-sampling-q02","topicId":"g4-sampling","topic":"Sampling","grade":"4","topicGrade":"4","strand":"S","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

1200 people visited a supermarket one Saturday.

A manager took a sample of 50 of these people. 18 of the people in the sample used a discount voucher.

  1. Work out an estimate for the total number of people who used a discount voucher that Saturday.
  2. Write down one assumption you made in working out your estimate.
","answerType":"multi","answer":"(a) 432 (b) The sample is representative of all 1200 people","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"432"},{"label":"b","answer":"The sample is representative of all the people who visited"}],"solution":["Fraction of the sample using a voucher = 1850.","Estimate = 1200 × 1850 = 1200 × 0.36 = 432.","This assumes the sample is representative of all the Saturday shoppers."],"diagram":null,"draw":false,"revision":"5b2022155f96"},{"id":"g4-sampling-q03","topicId":"g4-sampling","topic":"Sampling","grade":"4","topicGrade":"4","strand":"S","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

A town has 5600 residents.

A council officer asks a sample of 80 residents whether they support a new cycle lane. 28 of the residents in the sample support the cycle lane.

  1. Work out an estimate for the number of residents of the town who support the cycle lane.
  2. State one assumption you made. Explain how your estimate would be affected if the assumption is not true.
","answerType":"multi","answer":"(a) 1960 (b) Assumed the sample is representative of the whole town; if it is not, the estimate could be too high or too low","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1960"},{"label":"b","answer":"The sample is representative; if not, the estimate would be unreliable"}],"solution":["Fraction of the sample in support = 2880.","Estimate = 5600 × 2880 = 5600 × 0.35 = 1960.","This assumes the 80 residents are representative of all 5600; if the sample is biased, the estimate could be too high or too low."],"diagram":null,"draw":false,"revision":"34556ca3c79b"},{"id":"g4-sampling-q04","topicId":"g4-sampling","topic":"Sampling","grade":"4","topicGrade":"4","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

A farm packs apples in batches of 1500.

A quality checker takes a sample of apples from a batch. She finds that 3 out of every 20 apples in the sample are bruised.

  1. Work out an estimate for the number of bruised apples in the batch.
  2. Give one reason why the estimate might not be reliable.
","answerType":"multi","answer":"(a) 225 (b) The sample may be too small or not representative of the whole batch","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"225"},{"label":"b","answer":"The sample may not be representative of the batch (or it may be too small)"}],"solution":["Fraction of bruised apples in the sample = 320.","Estimate = 1500 × 320 = 225.","If the sample was small or all taken from one part of the batch, it may not represent the whole batch."],"diagram":null,"draw":false,"revision":"ebe41658ff15"},{"id":"g4-sampling-q05","topicId":"g4-sampling","topic":"Sampling","grade":"4","topicGrade":"4","strand":"S","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

Ben wants to find out how many of the 3400 adults in a village travel to work by car.

  1. Ben plans to survey adults standing on the platform at the village railway station at 8 am. Explain why this sample is likely to be biased.
  2. Instead, Ben surveys a fair sample of 120 adults from the village. 42 of them travel to work by car. Work out an estimate for the number of adults in the village who travel to work by car.
","answerType":"multi","answer":"(a) People at the station are catching trains, so they are less likely to travel by car - the sample would under-represent car users (b) 1190","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"People at a railway station at 8 am are mostly commuting by train, so car users would be under-represented"},{"label":"b","answer":"1190"}],"solution":["People at the station at 8 am are mostly travelling by train, so the sample would not represent all adults in the village.","Fraction of the fair sample using a car = 42120.","Estimate = 3400 × 42120 = 3400 × 0.35 = 1190."],"diagram":null,"draw":false,"revision":"15c23f4f0548"},{"id":"g4-sampling-q06","topicId":"g4-sampling","topic":"Sampling","grade":"4","topicGrade":"4","strand":"S","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

A leisure centre has 2400 members.

The manager takes a random sample of 60 members.

24 of the members in the sample are female.

15 of the members in the sample are female members who attend a fitness class each week.

  1. Work out an estimate for the number of female members of the leisure centre.
  2. Work out an estimate for the number of members who are female and attend a fitness class each week.
","answerType":"multi","answer":"(a) 960 (b) 600","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"960"},{"label":"b","answer":"600"}],"solution":["Estimate of female members = 2400 × 2460 = 960.","Estimate of female members attending a class = 2400 × 1560 = 600."],"diagram":null,"draw":false,"revision":"a25570d874e6"},{"id":"g4-sampling-q07","topicId":"g4-sampling","topic":"Sampling","grade":"4","topicGrade":"4","strand":"S","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

A factory fills 5000 jars of jam each day.

One day, a supervisor checks a sample of 150 jars. 6 of the jars in the sample are underfilled.

  1. Work out an estimate for the number of underfilled jars filled that day.
  2. The sample of 150 jars was taken during the first hour of production only. Explain why this could make the estimate unreliable.
","answerType":"multi","answer":"(a) 200 (b) Jars from the first hour may not represent the whole day - for example the machines may perform differently later on","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"200"},{"label":"b","answer":"The sample only covers the first hour, so it may not be representative of the whole day's production"}],"solution":["Fraction underfilled in the sample = 6150.","Estimate = 5000 × 6150 = 200.","Machines can drift or behave differently through the day, so jars filled later may have a different underfill rate to the first hour."],"diagram":null,"draw":false,"revision":"8bc26ecd475f"},{"id":"g4-sampling-q08","topicId":"g4-sampling","topic":"Sampling","grade":"4","topicGrade":"4","strand":"S","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

Jo wants to estimate how many of the 1800 students at her school would buy food from a new salad bar.

She asks a sample of 90 students. 54 of them say they would buy food from the salad bar.

  1. Work out an estimate for the number of students at the school who would buy food from the salad bar.
  2. Jo only asked students in Year 7. Explain how this could affect your estimate.
  3. Describe one way Jo could improve her sample.
","answerType":"multi","answer":"(a) 1080 (b) Year 7 students may not be typical of the whole school, so the estimate could be too high or too low (c) Take a random sample of students from all year groups","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1080"},{"label":"b","answer":"Year 7 students may have different tastes from other year groups, so the sample is biased and the estimate may be too high or too low"},{"label":"c","answer":"Choose students at random from every year group (a random sample of the whole school)"}],"solution":["Fraction of the sample saying yes = 5490.","Estimate = 1800 × 5490 = 1800 × 0.6 = 1080.","A sample of only Year 7 students is biased because their preferences may differ from older students.","Select students randomly from a list covering all year groups. This reduces the Year 7 selection bias and makes a representative sample more likely, though no random sample is guaranteed to match the population exactly."],"diagram":null,"draw":false,"revision":"b04eed14f253"},{"id":"g4-scatter-graphs-q01","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

Write down the type of correlation you would expect to see in a scatter graph of each of the following.

  1. The daily temperature and the number of ice creams sold at a kiosk.
  2. The age of a car and its value.
","answerType":"multi","answer":"(a) positive correlation (b) negative correlation","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"positive correlation"},{"label":"b","answer":"negative correlation"}],"solution":["As temperature rises, more ice creams are sold, so positive correlation.","As a car gets older, its value falls, so negative correlation."],"diagram":null,"draw":false,"revision":"1db1a732e66e"},{"id":"g4-scatter-graphs-q02","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

The table shows the number of hours that each of 8 students spent revising, and their scores in a test.

Hours revising12345678
Test score2228353947525865

The data is plotted on a scatter graph.

  1. Describe the type of correlation the scatter graph would show.
  2. Interpret this correlation for these students.
","answerType":"multi","answer":"(a) positive correlation (b) The more hours a student spent revising, the higher their test score","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"positive correlation"},{"label":"b","answer":"Students who spent longer revising scored higher marks"}],"solution":["As the hours increase from 1 to 8, the scores rise steadily from 22 to 65, so the correlation is positive.","In context, more revision time is associated with a higher test score."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":9,"step":1,"label":"Hours revising"},"y":{"min":0,"max":70,"step":10,"minor":2,"label":"Test score"},"points":[{"at":[1,22]},{"at":[2,28]},{"at":[3,35]},{"at":[4,39]},{"at":[5,47]},{"at":[6,52]},{"at":[7,58]},{"at":[8,65]}]},"draw":false,"revision":"f791a2c04804"},{"id":"g4-scatter-graphs-q03","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":3,"marks":2,"tier":"FH","calc":true,"text":"

A garage records the engine size, in litres, and the top speed of a number of cars. The engine sizes range from 1.0 litres to 2.5 litres. A scatter graph of the data shows strong positive correlation and a line of best fit has been drawn.

  1. Priya uses the line of best fit to estimate the top speed of a car with a 4.5 litre engine. Explain why her estimate may not be reliable.
  2. Dev says, "The graph proves that a bigger engine causes a higher top speed." Give one reason why Dev may be wrong.
","answerType":"open","answer":"(a) 4.5 litres is outside the range of the data (1.0 to 2.5 litres), so this is extrapolation and the trend may not continue. (b) Correlation does not prove causation; another factor (e.g. car weight or design) could affect top speed.","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"4.5 litres is outside the range of the data, so the line of best fit may not apply (extrapolation)"},{"label":"b","answer":"Correlation does not show causation; other factors could explain the link"}],"solution":["The data only covers engine sizes from 1.0 to 2.5 litres, so using the line at 4.5 litres assumes the trend continues outside the data range, which may not be true.","A correlation between two variables does not prove that one causes the other; a third factor may be involved."],"diagram":null,"draw":false,"revision":"9c396599cc72"},{"id":"g4-scatter-graphs-q04","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

The scatter graph shows the number of hours of sunshine and the number of visitors to a country park on each of 10 days.

  1. Describe the correlation shown by the scatter graph.
  2. On another day there were 7.5 hours of sunshine. By drawing a line of best fit, estimate the number of visitors to the park on this day.
","answerType":"multi","answer":"(a) positive correlation (b) about 160 visitors (accept 150 to 175)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"positive correlation"},{"label":"b","answer":"approximately 160 (accept 150-175)"}],"solution":["The points rise from bottom left to top right, so the correlation is positive.","Draw a line of best fit through the middle of the points.","Reading up from 7.5 hours to the line and across gives roughly 160 visitors (any value from 150 to 175 with a sensible line)."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":12,"step":1,"label":"Hours of sunshine"},"y":{"min":0,"max":240,"step":20,"minor":2,"label":"Number of visitors"},"points":[{"at":[2,55]},{"at":[3,70]},{"at":[4,95]},{"at":[5,110]},{"at":[6,130]},{"at":[7,150]},{"at":[8,170]},{"at":[9,185]},{"at":[10,210]},{"at":[11,225]}],"lines":[{"through":[[2,55],[11,225]],"from":0,"to":12,"answer":true},{"x":7.5,"from":0,"to":159,"dashed":true,"answer":true},{"y":159,"from":0,"to":7.5,"dashed":true,"answer":true}]},"draw":true,"revision":"1172c57f6738"},{"id":"g4-scatter-graphs-q05","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

The table shows the maths score and the science score of 8 students.

Maths score1020253035404550
Science score1826323840445054

The first six points have been plotted on the scatter graph.

  1. Complete the scatter graph by plotting the last two points.
  2. Describe the correlation between the maths scores and the science scores.
  3. Interpret this correlation.
","answerType":"multi","answer":"(a) points (45,50) and (50,54) plotted (b) positive correlation (c) Students with higher maths scores tend to have higher science scores","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(45,50) and (50,54) plotted correctly"},{"label":"b","answer":"positive correlation"},{"label":"c","answer":"The higher a student's maths score, the higher their science score tends to be"}],"solution":["Plot (45, 50) and (50, 54) on the grid.","The points rise from bottom left to top right, so the correlation is positive.","In context, students who do well in maths also tend to do well in science."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":60,"step":5,"label":"Maths score"},"y":{"min":0,"max":60,"step":5,"label":"Science score"},"points":[{"at":[10,18]},{"at":[20,26]},{"at":[25,32]},{"at":[30,38]},{"at":[35,40]},{"at":[40,44]},{"at":[45,50],"answer":true},{"at":[50,54],"answer":true}]},"draw":true,"revision":"8e5e534b8f64"},{"id":"g4-scatter-graphs-q06","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":6,"marks":3,"tier":"FH","calc":true,"text":"

A factory records the age, in years, and the weekly maintenance cost, in pounds, of its machines. A scatter graph of the data is drawn and a line of best fit passes through the points (1, 20) and (9, 60).

  1. What type of correlation does the graph show?
  2. Use the line of best fit to find an estimate for the weekly maintenance cost of a machine that is 5 years old.
","answerType":"multi","answer":"(a) positive correlation (b) £40","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"positive correlation"},{"label":"b","answer":"£40"}],"solution":["Cost rises as age rises, so the correlation is positive.","The line rises by 60 - 20 = 40 pounds over 9 - 1 = 8 years, which is £5 per year.","At 5 years: 20 + (5 - 1) × 5 = 20 + 20 = £40."],"diagram":null,"draw":false,"revision":"962839041e2a"},{"id":"g4-scatter-graphs-q07","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

The scatter graph shows the outside temperature, in °C, and the number of portions of soup sold by a cafe on each of 8 days.

  1. One of the points is an outlier. Write down the coordinates of this point.
  2. Ignoring the outlier, use the graph to estimate the number of portions of soup sold on a day when the outside temperature is 9 °C.
","answerType":"multi","answer":"(a) (11, 10) (b) about 47 portions (accept 44 to 50)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(11, 10)"},{"label":"b","answer":"approximately 47 (accept 44-50)"}],"solution":["The general trend is negative, with points falling steadily as temperature rises; the point (11, 10) lies far below this trend, so it is the outlier.","Draw a line of best fit through the remaining points.","Reading from 9 °C to the line gives roughly 47 portions (accept 44 to 50)."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":20,"step":2,"label":"Temperature (°C)"},"y":{"min":0,"max":70,"step":5,"label":"Portions of soup sold"},"points":[{"at":[4,62]},{"at":[6,55]},{"at":[8,50]},{"at":[10,44]},{"at":[12,38]},{"at":[14,30]},{"at":[16,24]},{"at":[11,10]},{"at":[11,10],"style":"ring","answer":true}],"lines":[{"through":[[4,62],[16,24]],"from":0,"to":20,"answer":true},{"x":9,"from":0,"to":46,"dashed":true,"answer":true},{"y":46,"from":0,"to":9,"dashed":true,"answer":true}]},"draw":false,"revision":"93089a6728f7"},{"id":"g4-scatter-graphs-q08","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":8,"marks":4,"tier":"FH","calc":true,"text":"

The table shows the age, in years, and the resale value, in pounds, of 8 laptops of the same model.

Age (years)122.5344.556
Value (£)52046043038030090210150

The data is plotted on a scatter graph.

  1. Describe the correlation between the age of a laptop and its value.
  2. One laptop does not fit the general trend. Which laptop is this?
  3. Interpret the correlation for these laptops.
  4. Give a reason why the scatter graph should not be used to estimate the value of a 10 year old laptop.
","answerType":"multi","answer":"(a) negative correlation (b) The 4.5 year old laptop valued at £90 (c) The older a laptop is, the less it is worth (d) 10 years is outside the range of the data, so this would be unreliable extrapolation","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"negative correlation"},{"label":"b","answer":"the 4.5 year old laptop at £90"},{"label":"c","answer":"As laptops get older their value decreases"},{"label":"d","answer":"10 years is outside the range of the data (1 to 6 years), so the trend may not continue"}],"solution":["As age increases the value falls, so the correlation is negative.","The 4.5 year old laptop at £90 lies well below the trend of the other points (its neighbours at 4 and 5 years are worth £300 and £210), so it is the outlier.","In context, older laptops are worth less.","The data only covers ages 1 to 6 years, so estimating at 10 years is extrapolation and may not be reliable."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":7,"step":1,"minor":2,"label":"Age (years)"},"y":{"min":0,"max":600,"step":100,"minor":2,"label":"Value (£)"},"points":[{"at":[1,520]},{"at":[2,460]},{"at":[2.5,430]},{"at":[3,380]},{"at":[4,300]},{"at":[4.5,90]},{"at":[5,210]},{"at":[6,150]}]},"draw":false,"revision":"d9df9cbcac0a"},{"id":"g4-scatter-graphs-q09","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"

The scatter graph shows the rainfall, in mm, and the number of umbrellas sold at a market stall on each of 8 days.

  1. Describe the correlation shown by the graph.
  2. One point is an outlier. Circle this point and write down its coordinates.
  3. Ignoring the outlier, estimate the number of umbrellas sold on a day with 7 mm of rainfall.
","answerType":"multi","answer":"(a) positive correlation (b) (3, 45) (c) about 35 umbrellas (accept 32 to 38)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"positive correlation"},{"label":"b","answer":"(3, 45)"},{"label":"c","answer":"approximately 35 (accept 32-38)"}],"solution":["Sales rise as rainfall rises, so the correlation is positive.","The point (3, 45) lies far above the trend of the other points, so it is the outlier.","A line of best fit through the other points gives about 35 umbrellas at 7 mm (accept 32 to 38)."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":12,"step":1,"label":"Rainfall (mm)"},"y":{"min":0,"max":60,"step":5,"label":"Umbrellas sold"},"points":[{"at":[2,14]},{"at":[4,22]},{"at":[5,26]},{"at":[6,31]},{"at":[8,40]},{"at":[9,44]},{"at":[10,50]},{"at":[3,45]},{"at":[3,45],"style":"ring","answer":true}],"lines":[{"through":[[2,14],[9,44]],"from":0,"to":12,"answer":true},{"x":7,"from":0,"to":35.4,"dashed":true,"answer":true},{"y":35.4,"from":0,"to":7,"dashed":true,"answer":true}]},"draw":true,"revision":"bd41bb6b06c0"},{"id":"g4-scatter-graphs-q10","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"

The scatter graph shows the number of hours each of 9 students spent revising and their scores in an exam marked out of 80.

  1. Write down the type of correlation shown.
  2. Draw a line of best fit on the scatter graph.
  3. Use your line of best fit to estimate the exam score of a student who revised for 5.5 hours.
  4. Amir revised for 20 hours. Explain why the line of best fit should not be used to estimate his score.
","answerType":"multi","answer":"(a) positive correlation (b) suitable straight line through the points (c) about 53 (accept 49 to 57) (d) 20 hours is far outside the range of the data, so this is unreliable extrapolation","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"positive correlation"},{"label":"b","answer":"a straight line following the trend of the points"},{"label":"c","answer":"approximately 53 (accept 49-57)"},{"label":"d","answer":"20 hours is outside the range of the data (0 to 9 hours), so the trend may not continue"}],"solution":["The points rise together, so the correlation is positive.","Draw a straight line through the middle of the points, following the trend.","Reading from 5.5 hours to the line gives roughly 53 marks (accept 49 to 57).","The data only goes up to 9 hours of revision, so using the line at 20 hours is extrapolation and may give a score that is impossible (over 80) or simply wrong."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":10,"step":1,"label":"Hours revising"},"y":{"min":0,"max":80,"step":10,"minor":2,"label":"Exam score"},"points":[{"at":[1,25]},{"at":[2,33]},{"at":[3,36]},{"at":[4,45]},{"at":[5,50]},{"at":[6,55]},{"at":[7,63]},{"at":[8,66]},{"at":[9,75]}],"lines":[{"through":[[1,25],[9,75]],"from":0,"to":10,"answer":true},{"x":5.5,"from":0,"to":53.1,"dashed":true,"answer":true},{"y":53.1,"from":0,"to":5.5,"dashed":true,"answer":true}]},"draw":true,"revision":"85142871d880"},{"id":"g4-scatter-graphs-q11","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":11,"marks":5,"tier":"H","calc":false,"text":"

The scatter graph shows the age, in years, and the value, in £1000s, of 9 cars of the same model.

  1. One point is an outlier. Write down the coordinates of this point.
  2. Ignoring the outlier, draw a line of best fit on the graph.
  3. Use your line of best fit to find an estimate for the value of a car that is 512 years old.
  4. Explain why it would not be sensible to use your line of best fit to estimate the value of a 30 year old car.
  5. Describe the relationship between the age of a car and its value.
","answerType":"multi","answer":"(a) (3, 4) (b) suitable straight line (c) about £5800 (accept £5400 to £6200) (d) 30 years is far outside the data range, so this is extrapolation; the line would give a negative value, which is impossible (e) negative correlation: the older the car, the lower its value","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(3, 4)"},{"label":"b","answer":"a straight line following the trend, ignoring the outlier"},{"label":"c","answer":"approximately £5800 (accept £5400-£6200)"},{"label":"d","answer":"30 years is outside the range of the data, so the trend cannot be assumed to continue (the line would predict a negative value)"},{"label":"e","answer":"negative correlation - as age increases, value decreases"}],"solution":["The trend falls steadily from about £9200 at 1 year to £4000 at 8 years; the point (3, 4) lies far below this trend, so it is the outlier.","Draw a straight line through the remaining points.","Reading from 5.5 years to the line gives about 5.8 thousand pounds, so roughly £5800 (accept £5400 to £6200).","The data covers ages 1 to 8 years; at 30 years the line would predict a negative value, so extrapolating is not sensible.","The graph shows negative correlation: older cars are worth less."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":10,"step":1,"label":"Age (years)"},"y":{"min":0,"max":10,"step":1,"minor":2,"label":"Value (£1000s)"},"points":[{"at":[1,9.2]},{"at":[2,8.4]},{"at":[3,7.8]},{"at":[4,7]},{"at":[5,6.1]},{"at":[6,5.6]},{"at":[7,4.8]},{"at":[8,4]},{"at":[3,4]},{"at":[3,4],"style":"ring","answer":true}],"lines":[{"through":[[1,9.2],[8,4]],"from":0,"to":10,"answer":true},{"x":5.5,"from":0,"to":5.86,"dashed":true,"answer":true},{"y":5.86,"from":0,"to":5.5,"dashed":true,"answer":true}]},"draw":true,"revision":"08b948b3c24c"},{"id":"g4-scatter-graphs-q12","topicId":"g4-scatter-graphs","topic":"Scatter Graphs","grade":"4","topicGrade":"4","strand":"S","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

A beach kiosk records the maximum temperature, in °C, and the number of cold drinks sold each day for two weeks. The temperatures ranged from 15 °C to 27 °C. On a scatter graph of the data, the line of best fit passes through the points (15, 90) and (27, 210).

  1. Use the line of best fit to estimate the number of cold drinks sold on a day when the maximum temperature is 22 °C.
  2. The kiosk owner uses the line to make two predictions:
    Prediction 1: sales on a day with a maximum temperature of 24 °C.
    Prediction 2: sales on a day with a maximum temperature of 35 °C.
    Which prediction is more reliable? Give a reason for your answer.
  3. The owner says, "The graph proves that hot weather makes people buy cold drinks." Comment on this statement.
","answerType":"multi","answer":"(a) 160 drinks (b) Prediction 1, because 24 degrees C is within the range of the data (interpolation) while 35 degrees C is outside it (extrapolation) (c) The graph shows correlation, but correlation does not prove causation","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"160"},{"label":"b","answer":"Prediction 1; 24 degrees C is inside the data range so interpolation is reliable, 35 degrees C is outside the range so extrapolation is not"},{"label":"c","answer":"Correlation does not prove causation; the graph shows an association only"}],"solution":["The line rises by 210 - 90 = 120 drinks over 27 - 15 = 12 degrees, which is 10 drinks per degree.","At 22 °C: 90 + (22 - 15) × 10 = 90 + 70 = 160 drinks.","24 °C lies within the data range (15 to 27), so prediction 1 uses interpolation and is more reliable; 35 °C is outside the range, so prediction 2 relies on the trend continuing, which cannot be assumed.","The correlation shows an association between temperature and sales but does not by itself prove that one causes the other."],"diagram":null,"draw":false,"revision":"50f66af7ff83"},{"id":"g4-sequences-nth-term-q01","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Here are the first four terms of an arithmetic sequence.

5   9   13   17

Find an expression, in terms of n, for the nth term of the sequence.

","answerType":"exact","answer":"4n + 1","answerDisplay":null,"accept":["1 + 4n","4n+1"],"parts":[],"solution":["n is the position in the sequence, starting with n = 1. The nth-term rule lets you calculate a term directly from its position.","The common difference is 4, so the nth term contains 4n","The sequence 4n gives 4, 8, 12, 16, which is 1 less than each term","The nth term is 4n + 1","Check: when n = 3, 4(3) + 1 = 13"],"diagram":null,"draw":false,"revision":"a6cb024dd033"},{"id":"g4-sequences-nth-term-q02","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Here are the first four terms of an arithmetic sequence.

20   17   14   11

Find an expression, in terms of n, for the nth term of the sequence.

","answerType":"exact","answer":"23 - 3n","answerDisplay":null,"accept":["-3n + 23","23-3n"],"parts":[],"solution":["The common difference is -3, so the nth term contains -3n","The sequence -3n gives -3, -6, -9, -12, which is 23 less than each term","The nth term is 23 - 3n","Check: when n = 2, 23 - 6 = 17"],"diagram":null,"draw":false,"revision":"75f973569a35"},{"id":"g4-sequences-nth-term-q03","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Here are the first four terms of an arithmetic sequence.

7   12   17   22

Explain why 81 is not a term of this sequence.

","answerType":"open","answer":"Every term of the sequence ends in 2 or 7, and 81 ends in 1","answerDisplay":null,"accept":["The nth term is 5n + 2 and 5n + 2 = 81 gives n = 15.8, which is not a whole number"],"parts":[],"solution":["The nth term is 5n + 2","If 5n + 2 = 81 then 5n = 79, so n = 15.8, which is not a whole number","Alternatively, every term ends in a 2 or a 7, but 81 ends in a 1","So 81 is not a term of the sequence"],"diagram":null,"draw":false,"revision":"02a72e6e532a"},{"id":"g4-sequences-nth-term-q04","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":4,"marks":2,"tier":"F","calc":true,"text":"

The nth term of a sequence is 6n − 1

Write down the first three terms of the sequence.

","answerType":"exact","answer":"5, 11, 17","answerDisplay":null,"accept":["5,11,17"],"parts":[],"solution":["When n = 1: 6(1) - 1 = 5","When n = 2: 6(2) - 1 = 11","When n = 3: 6(3) - 1 = 17"],"diagram":null,"draw":false,"revision":"7ca63408636c"},{"id":"g4-sequences-nth-term-q05","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":5,"marks":2,"tier":"FH","calc":true,"text":"

Here are the first four terms of an arithmetic sequence.

3   10   17   24

Find an expression, in terms of n, for the nth term of the sequence.

","answerType":"exact","answer":"7n - 4","answerDisplay":null,"accept":["-4 + 7n","7n-4"],"parts":[],"solution":["The common difference is 7, so the nth term contains 7n","The sequence 7n gives 7, 14, 21, 28, which is 4 more than each term","The nth term is 7n - 4","Check: when n = 4, 7(4) - 4 = 24"],"diagram":null,"draw":false,"revision":"4b3de43c2a62"},{"id":"g4-sequences-nth-term-q06","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":6,"marks":2,"tier":"H","calc":false,"text":"

Here are the first five terms of a quadratic sequence.

2   5   10   17   26

Find an expression, in terms of n, for the nth term of the sequence.

","answerType":"exact","answer":"n^2 + 1","answerDisplay":null,"accept":["n² + 1","1 + n^2"],"parts":[],"solution":["The differences are 3, 5, 7, 9, so the second differences are all 2, meaning the sequence is quadratic with n^2","The square numbers are 1, 4, 9, 16, 25, which is 1 less than each term","The nth term is n^2 + 1","Check: when n = 4, 16 + 1 = 17"],"diagram":null,"draw":false,"revision":"465d2becf5a8"},{"id":"g4-sequences-nth-term-q07","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":7,"marks":3,"tier":"F","calc":true,"text":"

Here are the first four terms of an arithmetic sequence.

4   7   10   13

  1. Write down the next term of the sequence.
  2. Find an expression, in terms of n, for the nth term of the sequence.
","answerType":"multi","answer":"a) 16; b) 3n + 1","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"16"},{"label":"b","answer":"3n + 1"}],"solution":["a) The common difference is 3, so the next term is 13 + 3 = 16","b) The nth term contains 3n, and 3n gives 3, 6, 9, 12, which is 1 less than each term","The nth term is 3n + 1"],"diagram":null,"draw":false,"revision":"5e3f831c316a"},{"id":"g4-sequences-nth-term-q08","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":8,"marks":3,"tier":"F","calc":true,"text":"

The nth term of a sequence is 5n − 3

  1. Work out the 12th term of the sequence.
  2. Is 122 a term of the sequence? You must show how you get your answer.
","answerType":"multi","answer":"a) 57; b) Yes, it is the 25th term","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"57"},{"label":"b","answer":"Yes, 5n - 3 = 122 gives n = 25, a whole number"}],"solution":["a) 5(12) - 3 = 60 - 3 = 57","b) Solve 5n - 3 = 122: 5n = 125, so n = 25","n = 25 is a whole number, so 122 is the 25th term of the sequence"],"diagram":null,"draw":false,"revision":"0662e58077da"},{"id":"g4-sequences-nth-term-q09","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":9,"marks":3,"tier":"H","calc":false,"text":"

Sequence A has nth term n2 + 4

Sequence B has nth term 4n + 1

Show that 13 is a term of both sequences.

","answerType":"open","answer":"n^2 + 4 = 13 gives n = 3; 4n + 1 = 13 gives n = 3; both are whole numbers","answerDisplay":null,"accept":[],"parts":[],"solution":["Sequence A: n^2 + 4 = 13 gives n^2 = 9, so n = 3, a whole number","So 13 is the 3rd term of sequence A","Sequence B: 4n + 1 = 13 gives 4n = 12, so n = 3, a whole number","So 13 is the 3rd term of sequence B","Therefore 13 is a term of both sequences"],"diagram":null,"draw":false,"revision":"9a51ff55b937"},{"id":"g4-sequences-nth-term-q10","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":10,"marks":4,"tier":"F","calc":true,"text":"

Here are the first four terms of an arithmetic sequence.

8   15   22   29

  1. Find an expression, in terms of n, for the nth term of the sequence.
  2. Explain why 150 is not a term of the sequence.
","answerType":"multi","answer":"a) 7n + 1; b) 7n + 1 = 150 gives n = 149/7, which is not a whole number","answerDisplay":"a) 7n + 1; b) 7n + 1 = 150 gives n = 1497, which is not a whole number","accept":[],"parts":[{"label":"a","answer":"7n + 1"},{"label":"b","answer":"7n + 1 = 150 gives 7n = 149, and 149 is not divisible by 7"}],"solution":["a) The common difference is 7, and 7n gives 7, 14, 21, 28, which is 1 less than each term","The nth term is 7n + 1","b) If 7n + 1 = 150 then 7n = 149","149 divided by 7 is not a whole number, so 150 is not a term"],"diagram":null,"draw":false,"revision":"0aa0b8832024"},{"id":"g4-sequences-nth-term-q11","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":11,"marks":4,"tier":"F","calc":true,"text":"

The first term of an arithmetic sequence is 3.

The term-to-term rule is add 6.

  1. Work out the 5th term of the sequence.
  2. Find an expression, in terms of n, for the nth term of the sequence.
  3. Explain why every term of the sequence is an odd number.
","answerType":"multi","answer":"a) 27; b) 6n - 3; c) 6n is always even and an even number minus 3 is odd","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"27"},{"label":"b","answer":"6n - 3"},{"label":"c","answer":"6n is even, so 6n - 3 is always odd"}],"solution":["a) The terms are 3, 9, 15, 21, 27, so the 5th term is 27","b) The common difference is 6, and 6n gives 6, 12, 18, which is 3 more than each term, so the nth term is 6n - 3","c) 6n is a multiple of 6, so it is always even","An even number minus 3 is always odd, so every term is odd"],"diagram":null,"draw":false,"revision":"8e78ebe20b98"},{"id":"g4-sequences-nth-term-q12","topicId":"g4-sequences-nth-term","topic":"Sequences (Nth Term)","grade":"4","topicGrade":"4","strand":"A","difficulty":12,"marks":4,"tier":"FH","calc":false,"text":"

Here are the first four terms of an arithmetic sequence.

100   93   86   79

  1. Write down the next two terms of the sequence.
  2. Find an expression, in terms of n, for the nth term of the sequence.
  3. Explain why 0 is not a term of the sequence.
","answerType":"multi","answer":"a) 72 and 65; b) 107 - 7n; c) 107 - 7n = 0 gives n = 107/7, which is not a whole number","answerDisplay":"a) 72 and 65; b) 107 - 7n; c) 107 - 7n = 0 gives n = 1077, which is not a whole number","accept":[],"parts":[{"label":"a","answer":"72 and 65"},{"label":"b","answer":"107 - 7n"},{"label":"c","answer":"107 - 7n = 0 gives 7n = 107, and 107 is not divisible by 7"}],"solution":["a) The common difference is -7, so the next terms are 79 - 7 = 72 and 72 - 7 = 65","b) The nth term contains -7n, and -7n gives -7, -14, -21, which is 107 less than each term, so the nth term is 107 - 7n","Check: when n = 1, 107 - 7 = 100","c) If 107 - 7n = 0 then 7n = 107","107 divided by 7 is not a whole number, so 0 is not a term"],"diagram":null,"draw":false,"revision":"18b69d18305d"},{"id":"g4-surface-area-q01","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":3,"tier":"F","calc":false,"text":"

A cuboid measures 5 cm by 3 cm by 2 cm.

Work out the total surface area of the cuboid.

State the units of your answer.

","answerType":"exact","answer":"62 cm²","answerDisplay":null,"accept":["62","62cm2","62 cm^2","62 square centimetres"],"parts":[],"solution":["The three different faces have areas 5 × 3 = 15 cm², 5 × 2 = 10 cm² and 3 × 2 = 6 cm²","Each face appears twice, so total surface area = 2 × (15 + 10 + 6)","Total surface area = 2 × 31 = 62 cm²"],"diagram":null,"draw":false,"revision":"f8d748b2dffb"},{"id":"g4-surface-area-q02","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

A cube has a volume of 64 cm3.

Work out the total surface area of the cube.

","answerType":"exact","answer":"96 cm²","answerDisplay":null,"accept":["96","96cm2","96 cm^2"],"parts":[],"solution":["The side length of the cube is the cube root of 64, which is 4 cm (since 4 × 4 × 4 = 64)","Each face has area 4 × 4 = 16 cm²","A cube has 6 faces, so total surface area = 6 × 16 = 96 cm²"],"diagram":null,"draw":false,"revision":"09e9feea5bc0"},{"id":"g4-surface-area-q03","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

The cross-section of a triangular prism is a right-angled triangle with base 3 cm, height 4 cm and hypotenuse 5 cm.

The length of the prism is 10 cm.

Work out the total surface area of the prism.

","answerType":"exact","answer":"132 cm²","answerDisplay":null,"accept":["132","132cm2","132 cm^2"],"parts":[],"solution":["The two triangular ends each have area 12 × 3 × 4 = 6 cm², giving 12 cm² in total","The three rectangular faces have areas 3 × 10 = 30 cm², 4 × 10 = 40 cm² and 5 × 10 = 50 cm²","Total surface area = 12 + 30 + 40 + 50 = 132 cm²"],"diagram":{"type":"solid","kind":"prism","dims":{"b":3,"h":4,"len":10,"right":true},"labels":{"b":"3 cm","h":"4 cm","len":"10 cm","slant":"5 cm"},"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"5cc0c02b26dd"},{"id":"g4-surface-area-q04","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":4,"tier":"F","calc":false,"text":"

A cube has a total surface area of 150 cm2.

Show that the volume of the cube is 125 cm3.

","answerType":"open","answer":"Face area = 150 ÷ 6 = 25 cm², so side = 5 cm and volume = 5³ = 125 cm³","answerDisplay":null,"accept":[],"parts":[],"solution":["A cube has 6 identical square faces, so each face has area 150 ÷ 6 = 25 cm²","The side length is √25 = 5 cm","Volume = 5 × 5 × 5 = 125 cm³"],"diagram":null,"draw":false,"revision":"400180bead71"},{"id":"g4-surface-area-q05","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":4,"tier":"FH","calc":false,"text":"

A solid is made by placing a cube of side 4 cm centrally on top of a cuboid.

The cuboid measures 8 cm by 8 cm by 2 cm, and the cube sits on the 8 cm by 8 cm face.

Work out the total surface area of the solid.

","answerType":"exact","answer":"256 cm²","answerDisplay":null,"accept":["256","256cm2","256 cm^2"],"parts":[],"solution":["Base of the cuboid: 8 × 8 = 64 cm²","Four vertical faces of the cuboid: 4 × (8 × 2) = 64 cm²","Exposed top of the cuboid: 64 − (4 × 4) = 48 cm² because the cube covers a 4 cm by 4 cm square","Four vertical faces of the cube: 4 × (4 × 4) = 64 cm², and the top of the cube: 4 × 4 = 16 cm²","Total surface area = 64 + 64 + 48 + 64 + 16 = 256 cm²"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[8,0],[8,2],[0,2]],"fill":true},{"pts":[[0,2],[8,2],[11.6,4.4],[3.6,4.4]],"fill":true},{"pts":[[8,0],[11.6,2.4],[11.6,4.4],[8,2]],"fill":true},{"pts":[[2.9,2.6],[6.9,2.6],[6.9,6.6],[2.9,6.6]],"fill":true},{"pts":[[2.9,6.6],[6.9,6.6],[8.7,7.8],[4.7,7.8]],"fill":true},{"pts":[[6.9,2.6],[8.7,3.8],[8.7,7.8],[6.9,6.6]],"fill":true}],"texts":[{"at":[4,-1],"text":"8 cm"},{"at":[12,1],"text":"8 cm"},{"at":[-1,1],"text":"2 cm"},{"at":[2,5],"text":"4 cm"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"534d17910857"},{"id":"g4-surface-area-q06","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A cube has sides of length 6 cm.

A sphere has the same total surface area as the cube.

The surface area of a sphere with radius r is 4πr2.

Work out the radius of the sphere.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"4.15 cm","answerDisplay":null,"accept":["4.15","4.15cm"],"parts":[],"solution":["Surface area of the cube = 6 × 62 = 216 cm²","4πr2 = 216, so r2 = 216 ÷ (4π) = 17.188...","r = √17.188... = 4.1459...","r = 4.15 cm (3 significant figures)"],"diagram":null,"draw":false,"revision":"1bce6c66df0b"},{"id":"g4-surface-area-q07","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

A closed wooden box is a cuboid measuring 40 cm by 25 cm by 20 cm.

Amira varnishes all 6 faces of each box.

One tin of varnish covers 5000 cm2.

Amira has 3 tins of varnish.

Work out the greatest number of boxes Amira can varnish completely.

","answerType":"exact","answer":"3","answerDisplay":null,"accept":["3 boxes"],"parts":[],"solution":["Surface area of one box = 2 × (40 × 25 + 40 × 20 + 25 × 20) = 2 × (1000 + 800 + 500) = 4600 cm²","3 tins cover 3 × 5000 = 15000 cm²","15000 ÷ 4600 = 3.26..., so Amira can varnish 3 complete boxes"],"diagram":null,"draw":false,"revision":"94d06d281ebd"},{"id":"g4-surface-area-q08","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":false,"text":"

A cube has a total surface area of 384 cm2.

Work out the volume of the cube.

","answerType":"exact","answer":"512 cm³","answerDisplay":null,"accept":["512","512cm3","512 cm^3"],"parts":[],"solution":["Each of the 6 faces has area 384 ÷ 6 = 64 cm²","The side length is √64 = 8 cm","Volume = 8 × 8 × 8 = 512 cm³"],"diagram":null,"draw":false,"revision":"5e5cb5dfdd87"},{"id":"g4-surface-area-q09","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

Cuboid A measures 8 cm by 2 cm by 3.2 cm.

Cube B has the same total surface area as cuboid A.

Work out the difference between the volume of cube B and the volume of cuboid A.

","answerType":"exact","answer":"12.8 cm³","answerDisplay":null,"accept":["12.8","12.8cm3","12.8 cm^3"],"parts":[],"solution":["Surface area of cuboid A = 2 × (8 × 2 + 8 × 3.2 + 2 × 3.2) = 2 × (16 + 25.6 + 6.4) = 96 cm²","Cube B has surface area 96 cm², so each face has area 96 ÷ 6 = 16 cm² and the side length is 4 cm","Volume of cube B = 4 × 4 × 4 = 64 cm³","Volume of cuboid A = 8 × 2 × 3.2 = 51.2 cm³","Difference = 64 − 51.2 = 12.8 cm³"],"diagram":null,"draw":false,"revision":"e10888478655"},{"id":"g4-surface-area-q10","topicId":"g4-surface-area","topic":"Surface Area","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

A cuboid has a square base of side 5 cm and height h cm.

The total surface area of the cuboid is 190 cm2.

Work out the volume of the cuboid.

","answerType":"exact","answer":"175 cm³","answerDisplay":null,"accept":["175","175cm3","175 cm^3"],"parts":[],"solution":["The top and bottom faces have total area 2 × 5 × 5 = 50 cm²","The four vertical faces have total area 4 × 5h = 20h cm²","50 + 20h = 190, so 20h = 140 and h = 7 cm","Volume = 5 × 5 × 7 = 175 cm³"],"diagram":null,"draw":false,"revision":"e8ec66c97b16"},{"id":"g4-volume-of-a-prism-q01","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

A cuboid measures 6 cm by 4 cm by 3 cm.

Work out the volume of the cuboid.

State the units of your answer.

","answerType":"exact","answer":"72 cm³","answerDisplay":null,"accept":["72","72cm3","72 cm^3","72 cubic centimetres"],"parts":[],"solution":["Volume of a cuboid = length × width × height","Volume = 6 × 4 × 3 = 72 cm³"],"diagram":null,"draw":false,"revision":"b07cdf674e07"},{"id":"g4-volume-of-a-prism-q02","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

The cross-section of a prism is a right-angled triangle with base 8 cm and height 5 cm.

The length of the prism is 12 cm.

Work out the volume of the prism.

","answerType":"exact","answer":"240 cm³","answerDisplay":null,"accept":["240","240cm3","240 cm^3"],"parts":[],"solution":["Area of the triangular cross-section = 12 × 8 × 5 = 20 cm²","Volume of a prism = area of cross-section × length","Volume = 20 × 12 = 240 cm³","A prism has the same cross-section all along its length. Imagine stacking identical thin slices: cross-sectional area × length gives the volume."],"diagram":{"type":"solid","kind":"prism","dims":{"b":8,"h":5,"len":12,"right":true},"labels":{"b":"8 cm","h":"5 cm","len":"12 cm","slant":null},"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"1c051bba1a96"},{"id":"g4-volume-of-a-prism-q03","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

A cuboid has a volume of 180 cm3.

The base of the cuboid measures 6 cm by 5 cm.

Work out the height of the cuboid.

","answerType":"exact","answer":"6 cm","answerDisplay":null,"accept":["6","6cm"],"parts":[],"solution":["Area of the base = 6 × 5 = 30 cm²","Volume = base area × height, so 30 × height = 180","Height = 180 ÷ 30 = 6 cm"],"diagram":null,"draw":false,"revision":"57215b3ee01a"},{"id":"g4-volume-of-a-prism-q04","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

The cross-section of a prism is a trapezium.

The parallel sides of the trapezium are 6 cm and 10 cm, and the perpendicular distance between them is 4 cm.

The length of the prism is 15 cm.

Work out the volume of the prism.

","answerType":"exact","answer":"480 cm³","answerDisplay":null,"accept":["480","480cm3","480 cm^3"],"parts":[],"solution":["Area of the trapezium = 12 × (6 + 10) × 4 = 12 × 16 × 4 = 32 cm²","Volume of a prism = area of cross-section × length","Volume = 32 × 15 = 480 cm³"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[3.2,2],[13.2,2],[11.2,6],[5.2,6]]},{"pts":[[0,0],[10,0],[8,4],[2,4]],"fill":true,"sides":["10 cm","","6 cm",""]}],"segments":[{"from":[0,0],"to":[3.2,2]},{"from":[10,0],"to":[13.2,2],"label":"15 cm"},{"from":[8,4],"to":[11.2,6]},{"from":[2,4],"to":[5.2,6]},{"from":[2,0],"to":[2,4],"label":"4 cm","dashed":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"bd58ab500da1"},{"id":"g4-volume-of-a-prism-q05","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

A box is a cuboid measuring 60 cm by 40 cm by 30 cm.

Packets of tea are cuboids measuring 10 cm by 8 cm by 6 cm.

The packets are packed into the box so that they fill it completely with no gaps.

Work out the number of packets that fill the box.

","answerType":"exact","answer":"150","answerDisplay":null,"accept":["150 packets"],"parts":[],"solution":["Along the 60 cm edge: 60 ÷ 10 = 6 packets; along the 40 cm edge: 40 ÷ 8 = 5 packets; along the 30 cm edge: 30 ÷ 6 = 5 packets","Number of packets = 6 × 5 × 5 = 150","Check by volume: 60 × 40 × 30 = 72000 cm³ and 10 × 8 × 6 = 480 cm³; 72000 ÷ 480 = 150"],"diagram":null,"draw":false,"revision":"33cde8ed945b"},{"id":"g4-volume-of-a-prism-q06","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":6,"marks":4,"tier":"F","calc":true,"text":"

A fish tank is a cuboid with a base measuring 40 cm by 25 cm and a height of 20 cm.

The tank is filled with water to a depth of 15 cm.

All of the water is poured into cups. Each cup holds 250 ml.

Work out the number of cups that can be completely filled.

1 ml = 1 cm3

","answerType":"exact","answer":"60","answerDisplay":null,"accept":["60 cups"],"parts":[],"solution":["Volume of water = 40 × 25 × 15 = 15000 cm³","15000 cm³ = 15000 ml","Number of cups = 15000 ÷ 250 = 60"],"diagram":null,"draw":false,"revision":"8099a4c1bd18"},{"id":"g4-volume-of-a-prism-q07","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

Three faces of a cuboid that meet at a vertex have areas 12 cm2, 8 cm2 and 6 cm2.

Work out the volume of the cuboid.

","answerType":"exact","answer":"24 cm³","answerDisplay":null,"accept":["24","24cm3","24 cm^3"],"parts":[],"solution":["Let the edges be a, b and c, so ab = 12, ac = 8 and bc = 6","Multiplying: (ab)(ac)(bc) = (abc)2 = 12 × 8 × 6 = 576","Volume = abc = √576 = 24 cm³","Check: a = 4, b = 3, c = 2 gives face areas 12, 8 and 6 and volume 24 cm³"],"diagram":null,"draw":false,"revision":"a40f227aff92"},{"id":"g4-volume-of-a-prism-q08","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":8,"marks":5,"tier":"F","calc":true,"text":"

A water tank is a cuboid measuring 1.5 m by 0.8 m by 0.6 m.

The tank is empty. Dev fills the tank completely with water.

Water costs £1.25 for every 100 litres.

Work out the cost of the water needed to fill the tank.

1 m3 = 1000 litres

","answerType":"exact","answer":"£9.00","answerDisplay":null,"accept":["9","£9","9.00"],"parts":[],"solution":["Volume of the tank = 1.5 × 0.8 × 0.6 = 0.72 m³","0.72 m³ = 0.72 × 1000 = 720 litres","720 ÷ 100 = 7.2 lots of 100 litres","Cost = 7.2 × £1.25 = £9.00"],"diagram":null,"draw":false,"revision":"961928af3fad"},{"id":"g4-volume-of-a-prism-q09","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":9,"marks":5,"tier":"FH","calc":true,"text":"

The cross-section of a prism is a right-angled triangle.

The hypotenuse of the triangle is 13 cm and one of the shorter sides is 5 cm.

The length of the prism is 20 cm.

Work out the volume of the prism.

","answerType":"exact","answer":"600 cm³","answerDisplay":null,"accept":["600","600cm3","600 cm^3"],"parts":[],"solution":["Use Pythagoras' theorem to find the other short side: side2 = 132 − 52 = 169 − 25 = 144, so the side is 12 cm","Area of the cross-section = 12 × 5 × 12 = 30 cm²","Volume = 30 × 20 = 600 cm³"],"diagram":{"type":"solid","kind":"prism","dims":{"b":5,"h":12,"len":20,"right":true},"labels":{"b":"5 cm","h":null,"len":"20 cm","slant":"13 cm"},"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"3e37a74d16a5"},{"id":"g4-volume-of-a-prism-q10","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

A cuboid has a base measuring 8 cm by 5 cm.

The volume of the cuboid is 320 cm3.

Work out the total surface area of the cuboid.

","answerType":"exact","answer":"288 cm²","answerDisplay":null,"accept":["288","288cm2","288 cm^2"],"parts":[],"solution":["Base area = 8 × 5 = 40 cm², so the height = 320 ÷ 40 = 8 cm","The three different faces have areas 8 × 5 = 40 cm², 8 × 8 = 64 cm² and 5 × 8 = 40 cm²","Total surface area = 2 × (40 + 64 + 40) = 2 × 144 = 288 cm²"],"diagram":null,"draw":false,"revision":"8098ca556e7d"},{"id":"g4-volume-of-a-prism-q11","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":11,"marks":5,"tier":"H","calc":false,"text":"

A cube has sides of length 6 cm.

A pyramid has a square base of side 9 cm and height h cm.

Volume of a pyramid = ⅓ × base area × height

The volume of the pyramid is equal to the volume of the cube.

Work out the value of h.

","answerType":"exact","answer":"8","answerDisplay":null,"accept":["8 cm","h = 8"],"parts":[],"solution":["Volume of the cube = 6 × 6 × 6 = 216 cm³","Volume of the pyramid = ⅓ × 9 × 9 × h = 27h cm³","27h = 216, so h = 216 ÷ 27 = 8"],"diagram":null,"draw":false,"revision":"6a2f9be572af"},{"id":"g4-volume-of-a-prism-q12","topicId":"g4-volume-of-a-prism","topic":"Volume of a Prism","grade":"4","topicGrade":"4","strand":"G","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

The cross-section of a prism is an L-shape.

The L-shape is made by starting with a rectangle 12 cm wide and 9 cm tall, then removing a 7 cm by 5 cm rectangle from the top right corner.

The length of the prism is 24 cm.

Show that the volume of the prism is 1752 cm3.

","answerType":"open","answer":"Cross-section area = 12 × 9 - 7 × 5 = 108 - 35 = 73 cm²; volume = 73 × 24 = 1752 cm³","answerDisplay":null,"accept":[],"parts":[],"solution":["Area of the full rectangle = 12 × 9 = 108 cm²","Area removed = 7 × 5 = 35 cm²","Area of the L-shaped cross-section = 108 − 35 = 73 cm²","Volume = area of cross-section × length = 73 × 24 = 1752 cm³"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[3.2,2],[15.2,2],[15.2,6],[8.2,6],[8.2,11],[3.2,11]]},{"pts":[[0,0],[12,0],[12,4],[5,4],[5,9],[0,9]],"fill":true,"sides":["12 cm","","7 cm","5 cm","","9 cm"]}],"segments":[{"from":[0,0],"to":[3.2,2]},{"from":[12,0],"to":[15.2,2],"label":"24 cm"},{"from":[12,4],"to":[15.2,6]},{"from":[5,4],"to":[8.2,6]},{"from":[5,9],"to":[8.2,11]},{"from":[0,9],"to":[3.2,11]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"36465971768c"},{"id":"g5-changing-the-subject-of-a-formula-q01","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

Make x the subject of the formula

y = 2x + 9

","answerType":"exact","answer":"x = (y - 9)/2","answerDisplay":"x = y - 92","accept":["x = (y-9)/2","x = (y - 9) / 2","x = y/2 - 9/2","x = y/2 - 4.5"],"parts":[],"solution":["Making x the subject means leaving x alone on one side. Apply the same inverse operation to both sides to keep the equality true.","Subtract 9 from both sides: y - 9 = 2x","Divide both sides by 2: x = y - 92"],"diagram":null,"draw":false,"revision":"32ef23f7777a"},{"id":"g5-changing-the-subject-of-a-formula-q02","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":1,"tier":"FH","calc":false,"text":"

Chloe is making x the subject of the formula y = 3x + 5.

Here is her working.

Step 1:  y - 5 = 3x

Step 2:  x = y3 - 5

Explain the mistake Chloe has made.

","answerType":"open","answer":"In Step 2 she has only divided y by 3. She should divide the whole of y - 5 by 3, giving x = (y - 5)/3.","answerDisplay":"In Step 2 she has only divided y by 3. She should divide the whole of y - 5 by 3, giving x = y - 53.","accept":["She divided only y by 3 instead of the whole expression y - 5","The correct answer is x = (y - 5)/3"],"parts":[],"solution":["Step 1 is correct: y - 5 = 3x","To find x, the whole of the left-hand side must be divided by 3","So x = y - 53, not y3 - 5"],"diagram":null,"draw":false,"revision":"9b71e7a181bc"},{"id":"g5-changing-the-subject-of-a-formula-q03","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Make a the subject of the formula

P = 3a + 2b

","answerType":"exact","answer":"a = (P - 2b)/3","answerDisplay":"a = P - 2b3","accept":["a = (P-2b)/3","a = (P - 2b) / 3","a = P/3 - 2b/3"],"parts":[],"solution":["Subtract 2b from both sides: P - 2b = 3a","Divide both sides by 3: a = P - 2b3"],"diagram":null,"draw":false,"revision":"79f1d4c7db77"},{"id":"g5-changing-the-subject-of-a-formula-q04","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

Make k the subject of the formula

m = k5 - 2

","answerType":"exact","answer":"k = 5(m + 2)","answerDisplay":null,"accept":["k = 5m + 10","k = 5(m+2)"],"parts":[],"solution":["Add 2 to both sides: m + 2 = k5","Multiply both sides by 5: k = 5(m + 2) = 5m + 10"],"diagram":null,"draw":false,"revision":"94a0c177b28f"},{"id":"g5-changing-the-subject-of-a-formula-q05","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":2,"tier":"FH","calc":true,"text":"

Make c the subject of the formula

a = 4(c + 2d)

","answerType":"exact","answer":"c = a/4 - 2d","answerDisplay":"c = a4 - 2d","accept":["c = (a - 8d)/4","c = (a-8d)/4","c = a/4 - 2d"],"parts":[],"solution":["Divide both sides by 4: a4 = c + 2d","Subtract 2d from both sides: c = a4 - 2d","Equivalently, expand first: a = 4c + 8d, so c = a - 8d4"],"diagram":null,"draw":false,"revision":"6b6558422035"},{"id":"g5-changing-the-subject-of-a-formula-q06","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":2,"tier":"H","calc":false,"text":"

Make x the subject of the formula

y = 3x - 52

","answerType":"exact","answer":"x = (2y + 5)/3","answerDisplay":"x = 2y + 53","accept":["x = (2y+5)/3","x = (2y + 5) / 3","x = 2y/3 + 5/3"],"parts":[],"solution":["Multiply both sides by 2: 2y = 3x - 5","Add 5 to both sides: 2y + 5 = 3x","Divide both sides by 3: x = 2y + 53"],"diagram":null,"draw":false,"revision":"5c1aacb35979"},{"id":"g5-changing-the-subject-of-a-formula-q07","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"

P = 5q - 3r

  1. Work out the value of P when q = 4 and r = -2
  2. Make q the subject of the formula
","answerType":"multi","answer":"a) P = 26; b) q = (P + 3r)/5","answerDisplay":"a) P = 26; b) q = P + 3r5","accept":["a) 26 b) q = (P+3r)/5","a) 26 b) q = P/5 + 3r/5"],"parts":[{"label":"a","answer":"26"},{"label":"b","answer":"q = (P + 3r)/5","answerDisplay":"q = P + 3r5"}],"solution":["a) P = 5 × 4 - 3 × (-2) = 20 + 6 = 26","b) Add 3r to both sides: P + 3r = 5q","b) Divide both sides by 5: q = P + 3r5"],"diagram":null,"draw":false,"revision":"84d96f3f5554"},{"id":"g5-changing-the-subject-of-a-formula-q08","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

Make g the subject of the formula

h = 5 + 3g

","answerType":"exact","answer":"g = 3/(h - 5)","answerDisplay":"g = 3h - 5","accept":["g = 3/(h-5)","g = 3 / (h - 5)"],"parts":[],"solution":["Subtract 5 from both sides: h - 5 = 3g","Multiply both sides by g: g(h - 5) = 3","Divide both sides by (h - 5): g = 3h - 5","The original formula requires g ≠ 0. Its rearranged form requires h ≠ 5; division by zero is undefined."],"diagram":null,"draw":false,"revision":"6017ebede9f3"},{"id":"g5-changing-the-subject-of-a-formula-q09","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":9,"marks":3,"tier":"FH","calc":true,"text":"

Make a the subject of the formula

b = √(a - 6) + 2

","answerType":"exact","answer":"a = (b - 2)^2 + 6","answerDisplay":null,"accept":["a = (b-2)^2 + 6","a = (b - 2)² + 6","a = b^2 - 4b + 10"],"parts":[],"solution":["Subtract 2 from both sides: b - 2 = √(a - 6)","Square both sides: (b - 2)2 = a - 6","Add 6 to both sides: a = (b - 2)2 + 6","The square-root sign means the non-negative root, so b − 2 ≥ 0, or b ≥ 2. Keep this restriction when using the rearranged formula."],"diagram":null,"draw":false,"revision":"f517fb8c8024"},{"id":"g5-changing-the-subject-of-a-formula-q10","topicId":"g5-changing-the-subject-of-a-formula","topic":"Changing the Subject of a Formula","grade":"5","topicGrade":"5","strand":"A","difficulty":10,"marks":3,"tier":"H","calc":false,"text":"

Make x the subject of the formula

5(x + 2y) = w + 3x

","answerType":"exact","answer":"x = (w - 10y)/2","answerDisplay":"x = w - 10y2","accept":["x = (w-10y)/2","x = (w - 10y) / 2","x = w/2 - 5y"],"parts":[],"solution":["Expand the bracket: 5x + 10y = w + 3x","Subtract 3x from both sides: 2x + 10y = w","Subtract 10y from both sides: 2x = w - 10y","Divide both sides by 2: x = w - 10y2"],"diagram":null,"draw":false,"revision":"ba82d51aaa16"},{"id":"g5-direct-and-inverse-proportion-q01","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

4 identical pumps can drain a flooded basement in 18 hours.

Work out how long it would take 6 of these pumps to drain the basement.

","answerType":"exact","answer":"12 hours","answerDisplay":null,"accept":["12","12 h"],"parts":[],"solution":["Total work = 4 × 18 = 72 pump-hours","Time for 6 pumps = 72 ÷ 6 = 12 hours","Check: 6 × 12 = 72, the same total work"],"diagram":null,"draw":false,"revision":"b591fc4fae39"},{"id":"g5-direct-and-inverse-proportion-q02","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

The weight of a stack of identical paving slabs is directly proportional to the number of slabs.

5 slabs weigh 11 kg.

Work out the weight of 8 slabs.

","answerType":"exact","answer":"17.6 kg","answerDisplay":null,"accept":["17.6","17.6kg"],"parts":[],"solution":["Weight of 1 slab = 11 ÷ 5 = 2.2 kg","Weight of 8 slabs = 8 × 2.2 = 17.6 kg","Check: k = 2.2 both ways, since 5 × 2.2 = 11"],"diagram":null,"draw":false,"revision":"e205f25dd206"},{"id":"g5-direct-and-inverse-proportion-q03","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

3 identical machines take 10 hours to print an order of leaflets.

Work out how long 5 of these machines would take to print the same order.

","answerType":"exact","answer":"6 hours","answerDisplay":null,"accept":["6","6 h"],"parts":[],"solution":["Total work = 3 × 10 = 30 machine-hours","Time for 5 machines = 30 ÷ 5 = 6 hours"],"diagram":null,"draw":false,"revision":"d32012d52649"},{"id":"g5-direct-and-inverse-proportion-q04","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

5 identical pipes can fill a storage tank in 4.8 hours.

Only 3 of the pipes are working.

Work out how long the 3 pipes will take to fill the tank.

","answerType":"exact","answer":"8 hours","answerDisplay":null,"accept":["8","8 h"],"parts":[],"solution":["Total work = 5 × 4.8 = 24 pipe-hours","Time for 3 pipes = 24 ÷ 3 = 8 hours","Check: 3 × 8 = 24, the same total work"],"diagram":null,"draw":false,"revision":"83c320d12656"},{"id":"g5-direct-and-inverse-proportion-q05","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":5,"marks":3,"tier":"FH","calc":true,"text":"

It takes 8 bricklayers 15 days to build a wall.

  1. Work out how many days it would take 12 bricklayers to build the same wall.
  2. State one assumption you have made.
","answerType":"multi","answer":"a) 10 days, b) e.g. all bricklayers work at the same rate","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"10 days"},{"label":"b","answer":"All the bricklayers work at the same rate"}],"solution":["a) Total work = 8 × 15 = 120 worker-days","Time for 12 bricklayers = 120 ÷ 12 = 10 days","b) Assumption: every bricklayer works at the same steady rate"],"diagram":null,"draw":false,"revision":"522ef44decbe"},{"id":"g5-direct-and-inverse-proportion-q06","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":6,"marks":2,"tier":"H","calc":false,"text":"

x is a positive quantity. For each relationship, write down what happens to y when x is doubled.

  1. y is directly proportional to x
  2. y is inversely proportional to x
  3. y is directly proportional to x2
","answerType":"multi","answer":"a) y is doubled, b) y is halved, c) y is multiplied by 4","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"y is doubled"},{"label":"b","answer":"y is halved"},{"label":"c","answer":"y is multiplied by 4"}],"solution":["a) y = kx, so doubling x doubles y","b) y = kx, so doubling x halves y","c) y = kx2, so doubling x multiplies y by 22 = 4"],"diagram":null,"draw":false,"revision":"417776aca466"},{"id":"g5-direct-and-inverse-proportion-q07","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

y is inversely proportional to x.

When x = 4, y = 9.

  1. Find the value of y when x = 6.
  2. Find the value of y when x = 12.
","answerType":"multi","answer":"a) y = 6, b) y = 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6"},{"label":"b","answer":"3"}],"solution":["y = kx with k = 4 × 9 = 36","a) y = 36 ÷ 6 = 6","b) y = 36 ÷ 12 = 3","Check: 6 × 6 = 36 and 12 × 3 = 36, so k is constant both ways"],"diagram":null,"draw":false,"revision":"011b72cefbb9"},{"id":"g5-direct-and-inverse-proportion-q08","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":8,"marks":2,"tier":"H","calc":false,"text":"

y is inversely proportional to x.

When x = 3, y = 20.

Work out the value of y when x = 5.

","answerType":"exact","answer":"12","answerDisplay":null,"accept":["y = 12"],"parts":[],"solution":["k = 3 × 20 = 60, so y = 60x","When x = 5, y = 60 ÷ 5 = 12"],"diagram":null,"draw":false,"revision":"ff51c999d800"},{"id":"g5-direct-and-inverse-proportion-q09","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":9,"marks":3,"tier":"H","calc":true,"text":"

6 identical machines can complete an order in 14 days.

The 6 machines work for 4 days. Then 2 more identical machines are added.

Work out how many more days are needed to complete the order.

","answerType":"exact","answer":"7.5 days","answerDisplay":null,"accept":["7.5","7 and a half days"],"parts":[],"solution":["Total work = 6 × 14 = 84 machine-days","Work done in the first 4 days = 6 × 4 = 24 machine-days","Work remaining = 84 − 24 = 60 machine-days","With 8 machines: 60 ÷ 8 = 7.5 more days"],"diagram":null,"draw":false,"revision":"5107e605f725"},{"id":"g5-direct-and-inverse-proportion-q10","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":10,"marks":3,"tier":"H","calc":true,"text":"

The area a tin of paint covers is directly proportional to the amount of paint used.

3 litres of paint covers 40 m2.

Work out how many litres of paint are needed to cover 140 m2.

","answerType":"exact","answer":"10.5 litres","answerDisplay":null,"accept":["10.5","10.5 l"],"parts":[],"solution":["1 m2 needs 3 ÷ 40 = 0.075 litres","140 m2 needs 140 × 0.075 = 10.5 litres","Check the other way: 10.5 ÷ 3 = 3.5 and 140 ÷ 40 = 3.5, the same scale factor"],"diagram":null,"draw":false,"revision":"1b201882dcbe"},{"id":"g5-direct-and-inverse-proportion-q11","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":11,"marks":2,"tier":"H","calc":true,"text":"

The distance d metres a dropped object falls is directly proportional to the square of the time t seconds.

When t = 3, d = 45.

Work out the value of d when t = 6.

","answerType":"exact","answer":"180","answerDisplay":null,"accept":["d = 180","180 m"],"parts":[],"solution":["d = kt2 with 45 = k × 9, so k = 5","When t = 6, d = 5 × 36 = 180","Check: doubling t multiplies d by 4, and 45 × 4 = 180"],"diagram":null,"draw":false,"revision":"e91cb5a79b4b"},{"id":"g5-direct-and-inverse-proportion-q12","topicId":"g5-direct-and-inverse-proportion","topic":"Direct and Inverse Proportion","grade":"5","topicGrade":"5","strand":"R","difficulty":12,"marks":3,"tier":"H","calc":false,"text":"

y is directly proportional to the square of x.

x is increased by 20%.

Show that y increases by 44%.

Assume x and y are positive.

","answerType":"open","answer":"New y = k(1.2x)^2 = 1.44kx^2, which is a 44% increase","answerDisplay":null,"accept":[],"parts":[],"solution":["y = kx2","The new value of x is 1.2x","New y = k(1.2x)2 = 1.44kx2","1.44 times the original y is a 44% increase"],"diagram":null,"draw":false,"revision":"8cce1d9421dc"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q01","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Complete the table of values for y = x3 - 2

x-2-1012
y-10-26
","answerType":"multi","answer":"y = -3 when x = -1; y = -1 when x = 1","answerDisplay":null,"accept":["-3 and -1"],"parts":[{"label":"x = -1","answer":"-3"},{"label":"x = 1","answer":"-1"}],"solution":["When x = -1: y = (-1)3 - 2 = -1 - 2 = -3","When x = 1: y = 13 - 2 = 1 - 2 = -1"],"diagram":null,"draw":false,"revision":"8c9fdcb61331"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q02","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Here are four equations.

A  y = 3x + 2     B  y = x2     C  y = 4x     D  y = x3

  1. Write down the letter of the equation whose graph is a reciprocal curve.
  2. Write down the letter of the equation whose graph is a straight line.
","answerType":"multi","answer":"a) C; b) A","answerDisplay":null,"accept":["a) C b) A"],"parts":[{"label":"a","answer":"C"},{"label":"b","answer":"A"}],"solution":["a) A reciprocal graph has the form y = kx, so the answer is C","b) A straight line has the form y = mx + c, so the answer is A"],"diagram":null,"draw":false,"revision":"46e7d7835cef"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q03","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":2,"tier":"FH","calc":false,"text":"

The diagram shows four sketch graphs, labelled A, B, C and D.

  1. Write down the letter of the graph of y = x3
  2. Write down the letter of the graph of y = 2x
","answerType":"multi","answer":"a) C; b) D","answerDisplay":null,"accept":["a) C b) D"],"parts":[{"label":"a","answer":"C"},{"label":"b","answer":"D"}],"solution":["a) The graph of y = x³ is an S-shaped curve through the origin: graph C. For example, x = −1, 0 and 1 give y = −1, 0 and 1.","b) For y = 2/x, x cannot be zero and y never equals zero. Positive x gives positive y, and negative x gives negative y, so the two branches approach the axes without touching them: graph D."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"A","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"x"}]}},{"label":"B","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"x*x"}]}},{"label":"C","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"x*x*x"}]}},{"label":"D","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"2/x"}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"8a899e6784ba"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q04","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

Complete the table of values for y = 12x

x1234612
y1242
","answerType":"multi","answer":"y = 6 (x = 2), y = 3 (x = 4), y = 1 (x = 12)","answerDisplay":null,"accept":["6, 3, 1"],"parts":[{"label":"x = 2","answer":"6"},{"label":"x = 4","answer":"3"},{"label":"x = 12","answer":"1"}],"solution":["When x = 2: y = 122 = 6","When x = 4: y = 124 = 3","When x = 12: y = 1212 = 1"],"diagram":null,"draw":false,"revision":"d1133545137e"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q05","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

The diagram shows five sketch graphs, labelled A, B, C, D and E.

Match each equation below to the letter of its graph.

  1. y = -x2
  2. y = x3
  3. y = 5x
","answerType":"multi","answer":"a) C; b) D; c) E","answerDisplay":null,"accept":["a) C b) D c) E"],"parts":[{"label":"a","answer":"C"},{"label":"b","answer":"D"},{"label":"c","answer":"E"}],"solution":["a) y = -x2 is an n-shaped (upside down) parabola: graph C","b) y = x3 is an S-shaped cubic curve through the origin: graph D","c) y = 5x is a reciprocal curve with two branches: graph E"],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"A","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"x"}]}},{"label":"B","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"x*x"}]}},{"label":"C","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"-x*x"}]}},{"label":"D","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"x*x*x"}]}},{"label":"E","diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":4,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"5/x"}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"51388262a9cb"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q06","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":false,"text":"

A curve has equation y = 8x

  1. Explain why there is no point on the curve where x = 0
  2. Work out the value of y when x = -4
  3. Does the point (2, 4) lie on the curve? You must show how you get your answer.
","answerType":"multi","answer":"a) You cannot divide by 0, so y is not defined at x = 0; b) y = -2; c) Yes, because 8/2 = 4","answerDisplay":"a) You cannot divide by 0, so y is not defined at x = 0; b) y = -2; c) Yes, because 82 = 4","accept":["a) division by zero is not possible b) -2 c) yes"],"parts":[{"label":"a","answer":"You cannot divide 8 by 0, so there is no value of y when x = 0"},{"label":"b","answer":"-2"},{"label":"c","answer":"Yes, because 8/2 = 4","answerDisplay":"Yes, because 82 = 4"}],"solution":["a) y = 8x requires dividing by x, and division by 0 is not possible, so the curve never meets the line x = 0","b) When x = -4: y = 8-4 = -2","c) When x = 2: y = 82 = 4, so (2, 4) does lie on the curve"],"diagram":null,"draw":false,"revision":"ba9cea07d57c"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q07","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"
  1. Complete the table of values for y = x3 - 3x
    x-2-1012
    y-2-2
  2. Explain how the table shows that the graph of y = x3 - 3x passes through the origin.
","answerType":"multi","answer":"a) y = 2 (x = -1), y = 0 (x = 0), y = 2 (x = 2); b) When x = 0, y = 0, so the graph passes through (0, 0)","answerDisplay":null,"accept":["a) 2, 0, 2 b) y = 0 when x = 0"],"parts":[{"label":"a (x = -1)","answer":"2"},{"label":"a (x = 0)","answer":"0"},{"label":"a (x = 2)","answer":"2"},{"label":"b","answer":"When x = 0, y = 0, so the curve passes through (0, 0)"}],"solution":["a) When x = -1: y = -1 + 3 = 2; when x = 0: y = 0; when x = 2: y = 8 - 6 = 2","b) The table gives y = 0 when x = 0, so the point (0, 0) is on the curve"],"diagram":null,"draw":false,"revision":"286f9c78edd1"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q08","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":4,"tier":"F","calc":true,"text":"
  1. Complete the table of values for y = x3 + 2
    x-2-1012
    y-6210
  2. On the grid, draw the graph of y = x3 + 2 for values of x from -2 to 2
","answerType":"multi","answer":"a) y = 1 (x = -1), y = 3 (x = 1); b) smooth S-shaped curve through all five points","answerDisplay":null,"accept":["a) 1 and 3"],"parts":[{"label":"a (x = -1)","answer":"1"},{"label":"a (x = 1)","answer":"3"},{"label":"b","answer":"correct smooth cubic curve through (-2, -6), (-1, 1), (0, 2), (1, 3), (2, 10)"}],"solution":["a) When x = -1: y = -1 + 2 = 1; when x = 1: y = 1 + 2 = 3","b) Plot the five points and join them with a smooth curve"],"diagram":{"type":"axes","x":{"min":-2,"max":2,"step":1},"y":{"min":-8,"max":12,"step":2},"curves":[{"fn":"x*x*x+2","from":-2,"to":2,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"a10bc5ded535"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q09","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":9,"marks":4,"tier":"FH","calc":true,"text":"
  1. Complete the table of values for y = x3 - 2x2
    x-10123
    y09
  2. Use the table to write down two values of x for which y = 0
","answerType":"multi","answer":"a) y = -3 (x = -1), y = -1 (x = 1), y = 0 (x = 2); b) x = 0 and x = 2","answerDisplay":null,"accept":["a) -3, -1, 0 b) 0 and 2"],"parts":[{"label":"a (x = -1)","answer":"-3"},{"label":"a (x = 1)","answer":"-1"},{"label":"a (x = 2)","answer":"0"},{"label":"b","answer":"x = 0 and x = 2"}],"solution":["a) When x = -1: y = -1 - 2 = -3; when x = 1: y = 1 - 2 = -1; when x = 2: y = 8 - 8 = 0","b) The table shows y = 0 at x = 0 and at x = 2"],"diagram":null,"draw":false,"revision":"70a6ca1481a7"},{"id":"g5-drawing-other-graphs-cubic-reciprocal-q10","topicId":"g5-drawing-other-graphs-cubic-reciprocal","topic":"Drawing Other Graphs: Cubic/Reciprocal","grade":"5","topicGrade":"5","strand":"A","difficulty":10,"marks":4,"tier":"H","calc":false,"text":"
  1. On the axes, sketch the graph of y = 6x for negative and positive values of x
  2. Write down the coordinates of any points where the graph crosses the coordinate axes.
","answerType":"multi","answer":"a) Two-branch reciprocal curve in the first and third quadrants; b) The graph never crosses either axis","answerDisplay":null,"accept":["b) none / it does not cross the axes"],"parts":[{"label":"a","answer":"reciprocal curve with one branch in the first quadrant and one branch in the third quadrant, approaching but never touching the axes"},{"label":"b","answer":"There are no such points - the graph never crosses the axes"}],"solution":["a) For positive x, y is positive (branch in the first quadrant); for negative x, y is negative (branch in the third quadrant)","a) As x gets large the curve gets closer to the x-axis; as x gets close to 0 the curve gets closer to the y-axis","b) y = 6x can never equal 0, and x can never equal 0, so the graph never crosses either axis"],"diagram":{"type":"axes","x":{"min":-6,"max":6,"step":1},"y":{"min":-6,"max":6,"step":1},"curves":[{"fn":"6/x","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"47390af1126e"},{"id":"g5-drawing-quadratic-graphs-q01","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

Complete the table of values for y = x2 - 2

x-2-1012
y2-22
","answerType":"multi","answer":"y = -1 when x = -1; y = -1 when x = 1","answerDisplay":null,"accept":["-1 and -1"],"parts":[{"label":"x = -1","answer":"-1"},{"label":"x = 1","answer":"-1"}],"solution":["When x = -1: y = (-1)2 - 2 = 1 - 2 = -1","When x = 1: y = 12 - 2 = -1"],"diagram":null,"draw":false,"revision":"06654bd71194"},{"id":"g5-drawing-quadratic-graphs-q02","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":2,"tier":"FH","calc":true,"text":"

The curve y = x2 - 6x + 8 crosses the x-axis at x = 2 and at x = 4.

  1. Write down the solutions of x2 - 6x + 8 = 0
  2. Write down the x-coordinate of the turning point of the curve.
","answerType":"multi","answer":"a) x = 2 and x = 4; b) x = 3","answerDisplay":null,"accept":["a) 2 and 4 b) 3"],"parts":[{"label":"a","answer":"x = 2 and x = 4"},{"label":"b","answer":"x = 3"}],"solution":["a) The solutions of x2 - 6x + 8 = 0 are the x-values where the curve crosses the x-axis: x = 2 and x = 4","b) By symmetry, the turning point is halfway between the roots: x = 2 + 42 = 3"],"diagram":{"type":"axes","x":{"min":0,"max":6,"step":1},"y":{"min":-2,"max":10,"step":1},"curves":[{"fn":"x*x-6*x+8","from":0,"to":6,"answer":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"c6382800e61d"},{"id":"g5-drawing-quadratic-graphs-q03","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

The diagram shows the graph of y = x2 - 2x - 3

  1. Write down the coordinates of the point where the graph crosses the y-axis.
  2. Write down the solutions of x2 - 2x - 3 = 0
  3. Write down the coordinates of the turning point of the graph.
","answerType":"multi","answer":"a) (0, -3); b) x = -1 and x = 3; c) (1, -4)","answerDisplay":null,"accept":["a) (0,-3) b) -1 and 3 c) (1,-4)"],"parts":[{"label":"a","answer":"(0, -3)"},{"label":"b","answer":"x = -1 and x = 3"},{"label":"c","answer":"(1, -4)"}],"solution":["a) The graph crosses the y-axis where x = 0: y = -3, so (0, -3)","b) The graph crosses the x-axis at x = -1 and x = 3","c) The lowest point of the curve is at (1, -4)"],"diagram":{"type":"axes","x":{"min":-3,"max":5,"step":1},"y":{"min":-5,"max":6,"step":1},"curves":[{"fn":"x*x-2*x-3","from":-3,"to":5,"answer":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ceca61000fbd"},{"id":"g5-drawing-quadratic-graphs-q04","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":4,"tier":"F","calc":true,"text":"
  1. Complete the table of values for y = x2 - 3x + 1
    x-101234
    y51-1
  2. On the grid, draw the graph of y = x2 - 3x + 1 for values of x from -1 to 4
","answerType":"multi","answer":"a) y = -1 (x = 1), y = 1 (x = 3), y = 5 (x = 4); b) smooth curve through all six points","answerDisplay":null,"accept":["a) -1, 1, 5"],"parts":[{"label":"a (x = 1)","answer":"-1"},{"label":"a (x = 3)","answer":"1"},{"label":"a (x = 4)","answer":"5"},{"label":"b","answer":"correct smooth parabola through (-1, 5), (0, 1), (1, -1), (2, -1), (3, 1), (4, 5)"}],"solution":["a) When x = 1: y = 1 - 3 + 1 = -1","a) When x = 3: y = 9 - 9 + 1 = 1","a) When x = 4: y = 16 - 12 + 1 = 5","b) Plot all six points and join them with a smooth curve"],"diagram":{"type":"axes","x":{"min":-1,"max":4,"step":1},"y":{"min":-2,"max":6,"step":1},"curves":[{"fn":"x*x-3*x+1","from":-1,"to":4,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"2f51ac86ab77"},{"id":"g5-drawing-quadratic-graphs-q05","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":4,"tier":"FH","calc":false,"text":"
  1. Complete the table of values for y = 6 - x2
    x-3-2-10123
    y-355-3
  2. Write down the coordinates of the maximum point of the graph of y = 6 - x2
  3. Use the table to write down the two consecutive whole numbers that the positive solution of 6 - x2 = 0 lies between.
","answerType":"multi","answer":"a) y = 2 (x = -2), y = 6 (x = 0), y = 2 (x = 2); b) (0, 6); c) between 2 and 3","answerDisplay":null,"accept":["a) 2, 6, 2 b) (0,6) c) 2 and 3"],"parts":[{"label":"a (x = -2)","answer":"2"},{"label":"a (x = 0)","answer":"6"},{"label":"a (x = 2)","answer":"2"},{"label":"b","answer":"(0, 6)"},{"label":"c","answer":"between 2 and 3"}],"solution":["a) When x = -2: y = 6 - 4 = 2; when x = 0: y = 6; when x = 2: y = 6 - 4 = 2","b) The largest value of y is 6, at x = 0, so the maximum point is (0, 6)","c) Between x = 2 and x = 3 the y-values change from 2 to -3, so y = 0 somewhere between them"],"diagram":null,"draw":false,"revision":"214976e0a8bb"},{"id":"g5-drawing-quadratic-graphs-q06","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":4,"tier":"FH","calc":true,"text":"
  1. Complete the table of values for y = x2 + 2x - 8
    x-4-3-2-1012
    y0-5-9-5
  2. Use the table to write down the solutions of x2 + 2x - 8 = 0
  3. Write down the equation of the line of symmetry of the graph of y = x2 + 2x - 8
","answerType":"multi","answer":"a) y = -8 (x = -2), y = -8 (x = 0), y = 0 (x = 2); b) x = -4 and x = 2; c) x = -1","answerDisplay":null,"accept":["a) -8, -8, 0 b) -4 and 2 c) x = -1"],"parts":[{"label":"a (x = -2)","answer":"-8"},{"label":"a (x = 0)","answer":"-8"},{"label":"a (x = 2)","answer":"0"},{"label":"b","answer":"x = -4 and x = 2"},{"label":"c","answer":"x = -1"}],"solution":["a) When x = -2: y = 4 - 4 - 8 = -8; when x = 0: y = -8; when x = 2: y = 4 + 4 - 8 = 0","b) y = 0 in the table at x = -4 and x = 2, so these are the solutions","c) The line of symmetry is halfway between the roots: x = -4 + 22 = -1"],"diagram":null,"draw":false,"revision":"d95b15ebe403"},{"id":"g5-drawing-quadratic-graphs-q07","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":5,"tier":"F","calc":true,"text":"
  1. Complete the table of values for y = x2 - 4x
    x-1012345
    y50-35
  2. On the grid, draw the graph of y = x2 - 4x for values of x from -1 to 5
  3. Write down the coordinates of the turning point of the graph.
","answerType":"multi","answer":"a) y = -3 (x = 1), y = -4 (x = 2), y = 0 (x = 4); b) smooth curve; c) (2, -4)","answerDisplay":null,"accept":["a) -3, -4, 0 c) (2,-4)"],"parts":[{"label":"a (x = 1)","answer":"-3"},{"label":"a (x = 2)","answer":"-4"},{"label":"a (x = 4)","answer":"0"},{"label":"b","answer":"correct smooth parabola through (-1, 5), (0, 0), (1, -3), (2, -4), (3, -3), (4, 0), (5, 5)"},{"label":"c","answer":"(2, -4)"}],"solution":["a) When x = 1: y = 1 - 4 = -3; when x = 2: y = 4 - 8 = -4; when x = 4: y = 16 - 16 = 0","b) Plot the seven points and join with a smooth curve","c) The minimum point of the curve is (2, -4)"],"diagram":{"type":"axes","x":{"min":-1,"max":5,"step":1},"y":{"min":-5,"max":6,"step":1},"curves":[{"fn":"x*x-4*x","from":-1,"to":5,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"64ef536df01a"},{"id":"g5-drawing-quadratic-graphs-q08","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":5,"tier":"FH","calc":true,"text":"
  1. Complete the table of values for y = x2 - 8x + 11
    x012345678
    y114-4-4411
  2. Write down the coordinates of the turning point of the graph of y = x2 - 8x + 11
  3. Write down the equation of the line of symmetry of the graph.
","answerType":"multi","answer":"a) y = -1 (x = 2), y = -5 (x = 4), y = -1 (x = 6); b) (4, -5); c) x = 4","answerDisplay":null,"accept":["a) -1, -5, -1 b) (4,-5) c) x = 4"],"parts":[{"label":"a (x = 2)","answer":"-1"},{"label":"a (x = 4)","answer":"-5"},{"label":"a (x = 6)","answer":"-1"},{"label":"b","answer":"(4, -5)"},{"label":"c","answer":"x = 4"}],"solution":["a) When x = 2: y = 4 - 16 + 11 = -1; when x = 4: y = 16 - 32 + 11 = -5; when x = 6: y = 36 - 48 + 11 = -1","b) The smallest y-value is -5, at x = 4, so the turning point is (4, -5)","c) The line of symmetry passes through the turning point: x = 4"],"diagram":null,"draw":false,"revision":"208e30570cb7"},{"id":"g5-drawing-quadratic-graphs-q09","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"
  1. Complete the table of values for y = 2x2 - 3
    x-2-1012
    y5-1
  2. Write down the coordinates of the turning point of the graph of y = 2x2 - 3
  3. Use the table to write down the two consecutive whole numbers that the positive solution of 2x2 - 3 = 0 lies between.
","answerType":"multi","answer":"a) y = -1 (x = -1), y = -3 (x = 0), y = 5 (x = 2); b) (0, -3); c) between 1 and 2","answerDisplay":null,"accept":["a) -1, -3, 5 b) (0,-3) c) 1 and 2"],"parts":[{"label":"a (x = -1)","answer":"-1"},{"label":"a (x = 0)","answer":"-3"},{"label":"a (x = 2)","answer":"5"},{"label":"b","answer":"(0, -3)"},{"label":"c","answer":"between 1 and 2"}],"solution":["a) When x = -1: y = 2 - 3 = -1; when x = 0: y = -3; when x = 2: y = 8 - 3 = 5","b) The minimum value of y is -3, at x = 0, so the turning point is (0, -3)","c) Between x = 1 and x = 2 the y-values change from -1 to 5, so y = 0 somewhere between them"],"diagram":null,"draw":false,"revision":"5e9c7d763539"},{"id":"g5-drawing-quadratic-graphs-q10","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":10,"marks":6,"tier":"F","calc":true,"text":"
  1. Complete the table of values for y = x2 - 2x - 4
    x-2-101234
    y4-4-44
  2. On the grid, draw the graph of y = x2 - 2x - 4 for values of x from -2 to 4
  3. Use your graph to estimate the solutions of x2 - 2x - 4 = 0
","answerType":"multi","answer":"a) y = -1 (x = -1), y = -5 (x = 1), y = -1 (x = 3); b) smooth curve; c) x is approximately -1.2 and 3.2","answerDisplay":null,"accept":["c) accept x between -1.4 and -1.1 and x between 3.1 and 3.4"],"parts":[{"label":"a (x = -1)","answer":"-1"},{"label":"a (x = 1)","answer":"-5"},{"label":"a (x = 3)","answer":"-1"},{"label":"b","answer":"correct smooth parabola through (-2, 4), (-1, -1), (0, -4), (1, -5), (2, -4), (3, -1), (4, 4)"},{"label":"c","answer":"x is approximately -1.2 and x is approximately 3.2"}],"solution":["a) When x = -1: y = 1 + 2 - 4 = -1; when x = 1: y = 1 - 2 - 4 = -5; when x = 3: y = 9 - 6 - 4 = -1","b) Plot the seven points and join with a smooth curve","c) Read where the curve crosses the x-axis: approximately x = -1.2 and x = 3.2 (exact values are 1 ± √5)"],"diagram":{"type":"axes","x":{"min":-2,"max":4,"step":1},"y":{"min":-6,"max":5,"step":1},"curves":[{"fn":"x*x-2*x-4","from":-2,"to":4,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"34bccb07f0f3"},{"id":"g5-drawing-quadratic-graphs-q11","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":11,"marks":6,"tier":"H","calc":false,"text":"
  1. Complete the table of values for y = x2 + x - 6
    x-4-3-2-10123
    y60-6-46
  2. On the grid, draw the graph of y = x2 + x - 6 for values of x from -4 to 3
  3. Write down the solutions of x2 + x - 6 = 0
  4. Use your graph to estimate the solutions of x2 + x - 6 = -5
","answerType":"multi","answer":"a) y = -4 (x = -2), y = -6 (x = 0), y = 0 (x = 2); b) smooth curve; c) x = -3 and x = 2; d) x is approximately -1.6 and 0.6","answerDisplay":null,"accept":["d) accept x between -1.8 and -1.4 and x between 0.4 and 0.8"],"parts":[{"label":"a (x = -2)","answer":"-4"},{"label":"a (x = 0)","answer":"-6"},{"label":"a (x = 2)","answer":"0"},{"label":"b","answer":"correct smooth parabola through (-4, 6), (-3, 0), (-2, -4), (-1, -6), (0, -6), (1, -4), (2, 0), (3, 6)"},{"label":"c","answer":"x = -3 and x = 2"},{"label":"d","answer":"x is approximately -1.6 and x is approximately 0.6"}],"solution":["a) When x = -2: y = 4 - 2 - 6 = -4; when x = 0: y = -6; when x = 2: y = 4 + 2 - 6 = 0","b) Plot the eight points and join with a smooth curve","c) The curve crosses the x-axis at x = -3 and x = 2","d) Draw the line y = -5 and read where it meets the curve: approximately x = -1.6 and x = 0.6 (exact values are -1 ± √52)","For the final equation, draw the horizontal line y = −5 and read the x-coordinates where it meets the curve. Keep the negative sign: the line is five units below the x-axis."],"diagram":{"type":"axes","x":{"min":-4,"max":3,"step":1},"y":{"min":-7,"max":7,"step":1},"curves":[{"fn":"x*x+x-6","from":-4,"to":3,"answer":true}],"alt":"Question diagram. Use the labels and information in the question.","lines":[{"m":0,"c":-5,"answer":true,"label":"y = −5"}]},"draw":true,"revision":"2a7d3a7dddab"},{"id":"g5-drawing-quadratic-graphs-q12","topicId":"g5-drawing-quadratic-graphs","topic":"Drawing Quadratic Graphs","grade":"5","topicGrade":"5","strand":"A","difficulty":12,"marks":6,"tier":"H","calc":true,"text":"
  1. Complete the table of values for y = 2x2 - 4x - 1
    x-10123
    y5-35
  2. On the grid, draw the graph of y = 2x2 - 4x - 1 for values of x from -1 to 3
  3. Write down the coordinates of the turning point of the graph.
  4. Use your graph to estimate the solutions of 2x2 - 4x - 1 = 0
","answerType":"multi","answer":"a) y = -1 (x = 0), y = -1 (x = 2); b) smooth curve; c) (1, -3); d) x is approximately -0.2 and 2.2","answerDisplay":null,"accept":["d) accept x between -0.4 and -0.1 and x between 2.1 and 2.4"],"parts":[{"label":"a (x = 0)","answer":"-1"},{"label":"a (x = 2)","answer":"-1"},{"label":"b","answer":"correct smooth parabola through (-1, 5), (0, -1), (1, -3), (2, -1), (3, 5)"},{"label":"c","answer":"(1, -3)"},{"label":"d","answer":"x is approximately -0.2 and x is approximately 2.2"}],"solution":["a) When x = 0: y = -1; when x = 2: y = 8 - 8 - 1 = -1","b) Plot the five points and join with a smooth curve","c) The minimum point of the curve is (1, -3)","d) Read where the curve crosses the x-axis: approximately x = -0.2 and x = 2.2"],"diagram":{"type":"axes","x":{"min":-1,"max":3,"step":1},"y":{"min":-4,"max":6,"step":1},"curves":[{"fn":"2*x*x-4*x-1","from":-1,"to":3,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"7c327c0426a0"},{"id":"g5-equation-of-a-line-q01","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

A straight line has gradient 4 and passes through the point (0, 5).

Find the equation of the line.

","answerType":"exact","answer":"y = 4x + 5","answerDisplay":null,"accept":["y = 4x + 5","y = 5 + 4x"],"parts":[],"solution":["A straight line has equation y = mx + c, where m is the gradient and c is the y-intercept.","The gradient is 4 and the line crosses the y-axis at (0, 5), so c = 5.","The equation is y = 4x + 5."],"diagram":null,"draw":false,"revision":"07c167e2d113"},{"id":"g5-equation-of-a-line-q02","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

A straight line passes through the points (0, -3) and (2, 5).

Find the equation of the line.

Give your answer in the form y = mx + c

","answerType":"exact","answer":"y = 4x - 3","answerDisplay":null,"accept":["y = 4x - 3","y = -3 + 4x"],"parts":[],"solution":["Gradient = change in y divided by change in x = 5 - (-3)2 - 0 = 8 ÷ 2 = 4.","The line crosses the y-axis at (0, -3), so c = -3.","The equation is y = 4x - 3. Check with (2, 5): 4 × 2 - 3 = 5, correct."],"diagram":null,"draw":false,"revision":"19b925e7f042"},{"id":"g5-equation-of-a-line-q03","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":3,"tier":"F","calc":true,"text":"

A straight line passes through the points (1, 7) and (3, 13).

Find the equation of the line.

Give your answer in the form y = mx + c

","answerType":"exact","answer":"y = 3x + 4","answerDisplay":null,"accept":["y = 3x + 4","y = 4 + 3x"],"parts":[],"solution":["Gradient = 13 - 73 - 1 = 6 ÷ 2 = 3.","Substitute (1, 7) into y = 3x + c: 7 = 3 + c, so c = 4.","The equation is y = 3x + 4. Check with (3, 13): 3 × 3 + 4 = 13, correct."],"diagram":null,"draw":false,"revision":"e1a64e471ca7"},{"id":"g5-equation-of-a-line-q04","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":3,"tier":"FH","calc":true,"text":"

A straight line has gradient -2 and passes through the point (3, 1).

Find the equation of the line.

","answerType":"exact","answer":"y = -2x + 7","answerDisplay":null,"accept":["y = -2x + 7","y = 7 - 2x"],"parts":[],"solution":["The equation has the form y = -2x + c.","Substitute (3, 1): 1 = -2 × 3 + c, so 1 = -6 + c and c = 7.","The equation is y = -2x + 7."],"diagram":null,"draw":false,"revision":"0330340116c4"},{"id":"g5-equation-of-a-line-q05","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":3,"tier":"FH","calc":false,"text":"

A straight line passes through the points (-2, 1) and (2, 9).

Find the equation of the line.

Give your answer in the form y = mx + c

","answerType":"exact","answer":"y = 2x + 5","answerDisplay":null,"accept":["y = 2x + 5","y = 5 + 2x"],"parts":[],"solution":["Gradient = 9 - 12 - (-2) = 8 ÷ 4 = 2.","Substitute (2, 9) into y = 2x + c: 9 = 4 + c, so c = 5.","The equation is y = 2x + 5. Check with (-2, 1): 2 × (-2) + 5 = 1, correct."],"diagram":null,"draw":false,"revision":"bd82e9d34058"},{"id":"g5-equation-of-a-line-q06","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

A straight line passes through the points (4, -1) and (6, -7).

Find the equation of the line.

","answerType":"exact","answer":"y = -3x + 11","answerDisplay":null,"accept":["y = -3x + 11","y = 11 - 3x"],"parts":[],"solution":["Gradient = -7 - (-1)6 - 4 = -6 ÷ 2 = -3.","Substitute (4, -1) into y = -3x + c: -1 = -12 + c, so c = 11.","The equation is y = -3x + 11. Check with (6, -7): -3 × 6 + 11 = -7, correct."],"diagram":null,"draw":false,"revision":"2a13398ba15f"},{"id":"g5-equation-of-a-line-q07","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":4,"tier":"FH","calc":true,"text":"

The straight line L has equation y = 5x - 3

  1. Write down the gradient of L.
  2. Find the equation of the straight line that is parallel to L and passes through the point (2, 11).
","answerType":"multi","answer":"a) 5 b) y = 5x + 1","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5"},{"label":"b","answer":"y = 5x + 1"}],"solution":["The gradient of L is the coefficient of x, which is 5.","Parallel lines have equal gradients, so the new line has the form y = 5x + c.","Substitute (2, 11): 11 = 10 + c, so c = 1.","The equation is y = 5x + 1."],"diagram":null,"draw":false,"revision":"579d20f5a884"},{"id":"g5-equation-of-a-line-q08","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

The straight line A has equation y = 3x - 4

The straight line B has equation 2x + y = 11

Find the coordinates of the point where line A and line B intersect.

","answerType":"exact","answer":"(3, 5)","answerDisplay":null,"accept":["(3, 5)","x = 3, y = 5","x = 3 and y = 5"],"parts":[],"solution":["At the intersection point, both equations are true, so substitute y = 3x - 4 into 2x + y = 11.","2x + 3x - 4 = 11, so 5x = 15 and x = 3.","y = 3 × 3 - 4 = 5.","The lines intersect at (3, 5). Check in line B: 2 × 3 + 5 = 11, correct."],"diagram":null,"draw":false,"revision":"e7c92d7d16b5"},{"id":"g5-equation-of-a-line-q09","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

A is the point (2, 3) and B is the point (6, 11).

M is the midpoint of AB.

Find the equation of the straight line with gradient 12 that passes through M.

","answerType":"exact","answer":"y = x/2 + 5","answerDisplay":"y = x2 + 5","accept":["y = x/2 + 5","y = 0.5x + 5","y = (1/2)x + 5","2y = x + 10"],"parts":[],"solution":["The midpoint of AB is (2 + 62, 3 + 112) = (4, 7).","The line has the form y = x2 + c.","Substitute (4, 7): 7 = 2 + c, so c = 5.","The equation is y = x2 + 5."],"diagram":null,"draw":false,"revision":"1ba1e7bd02ac"},{"id":"g5-equation-of-a-line-q10","topicId":"g5-equation-of-a-line","topic":"Equation of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

A is the point (1, 2) and B is the point (9, 18).

The point P lies on AB so that AP : PB = 3 : 1

Show that P lies on the straight line with equation y = 2x

","answerType":"open","answer":"P = (7, 14) and 2 x 7 = 14, so P lies on y = 2x","answerDisplay":null,"accept":[],"parts":[],"solution":["AP : PB = 3 : 1 means P is 34 of the way from A to B.","Change in x from A to B is 9 - 1 = 8, so the x-coordinate of P is 1 + 34 × 8 = 7.","Change in y from A to B is 18 - 2 = 16, so the y-coordinate of P is 2 + 34 × 16 = 14.","P is (7, 14). Substituting x = 7 into y = 2x gives y = 14, so P lies on the line y = 2x."],"diagram":null,"draw":false,"revision":"4cd8c90b1ed0"},{"id":"g5-exact-trig-values-q01","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Write down the exact value of sin 30°.

","answerType":"exact","answer":"1/2","answerDisplay":"12","accept":["0.5","½"],"parts":[],"solution":["sin 30° = 12 is one of the exact trig values to learn.","Bisect an equilateral triangle of side 2. Each right triangle has a 30° angle opposite a side of length 1 and a hypotenuse of length 2, so sin 30° = opposite ÷ hypotenuse = 1/2."],"diagram":null,"draw":false,"revision":"7d41a4554ff6"},{"id":"g5-exact-trig-values-q02","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":2,"marks":1,"tier":"F","calc":false,"text":"

Write down the exact value of tan 45°.

","answerType":"exact","answer":"1","answerDisplay":null,"accept":[],"parts":[],"solution":["tan 45° = 1. In a right-angled isosceles triangle the two shorter sides are equal, so opposite ÷ adjacent = 1."],"diagram":null,"draw":false,"revision":"d252d69fc871"},{"id":"g5-exact-trig-values-q03","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":3,"marks":2,"tier":"FH","calc":false,"text":"

Work out the exact value of sin 60° × cos 30°.

Give your answer as a fraction.

","answerType":"exact","answer":"3/4","answerDisplay":"34","accept":[],"parts":[],"solution":["sin 60° = √32 and cos 30° = √32.","√32 × √32 = 34."],"diagram":null,"draw":false,"revision":"a24d43820038"},{"id":"g5-exact-trig-values-q04","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":4,"marks":2,"tier":"H","calc":false,"text":"

Work out the exact value of (sin 30° + cos 60°) × tan 60°.

","answerType":"exact","answer":"√3","answerDisplay":null,"accept":["root 3","sqrt(3)"],"parts":[],"solution":["sin 30° = 12 and cos 60° = 12, so sin 30° + cos 60° = 1.","tan 60° = √3.","1 × √3 = √3."],"diagram":null,"draw":false,"revision":"a1090f625939"},{"id":"g5-exact-trig-values-q05","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":5,"marks":3,"tier":"F","calc":false,"text":"

In triangle ABC, angle ABC = 90° and angle BAC = 30°.

AC = 14 cm.

Without using a calculator, work out the length of BC.

You must show your working.

","answerType":"exact","answer":"7 cm","answerDisplay":null,"accept":["7"],"parts":[],"solution":["BC is opposite the 30° angle and AC = 14 cm is the hypotenuse, so sin 30° = BC ÷ 14.","sin 30° = 12.","BC = 14 × 12 = 7 cm."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6,3.4641016151377544],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0]},{"from":[6,0],"to":[6,3.4641016151377544]},{"from":[6,3.4641016151377544],"to":[0,0],"label":"14 cm"}],"angles":[{"at":[6,0],"from":[6,3.4641016151377544],"to":[0,0],"label":"","right":true},{"at":[0,0],"from":[6,0],"to":[6,3.4641016151377544],"label":"30°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"708458a995dc"},{"id":"g5-exact-trig-values-q06","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":6,"marks":3,"tier":"FH","calc":false,"text":"

In triangle PQR, angle PQR = 90° and angle QPR = 45°.

PQ = 8 cm.

Without using a calculator, work out the exact length of PR.

Give your answer in the form k√2.

","answerType":"exact","answer":"8√2 cm","answerDisplay":null,"accept":["8root2","8sqrt(2)"],"parts":[],"solution":["PQ = 8 cm is adjacent to the 45° angle and PR is the hypotenuse, so cos 45° = 8 ÷ PR.","cos 45° = √22, so PR = 8 ÷ (√22) = 16 ÷ √2.","Multiply numerator and denominator by √2. This multiplies the fraction by 1, so its value is unchanged, and √2 × √2 = 2.","Rationalise: 16√2 = 16√22 = 8√2 cm."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"Q","style":"none","labelPos":"below-right"},{"at":[6,5.999999999999999],"label":"R","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"8 cm"},{"from":[6,0],"to":[6,5.999999999999999]},{"from":[6,5.999999999999999],"to":[0,0]}],"angles":[{"at":[6,0],"from":[6,5.999999999999999],"to":[0,0],"label":"","right":true},{"at":[0,0],"from":[6,0],"to":[6,5.999999999999999],"label":"45°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"c738ca1cdba4"},{"id":"g5-exact-trig-values-q07","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":7,"marks":3,"tier":"H","calc":false,"text":"

In triangle DEF, angle DEF = 90° and angle EDF = 60°.

DE = 5 cm.

Without using a calculator, work out the exact length of EF.

","answerType":"exact","answer":"5√3 cm","answerDisplay":null,"accept":["5root3","5sqrt(3)"],"parts":[],"solution":["EF is opposite the 60° angle and DE = 5 cm is adjacent to it, so tan 60° = EF ÷ 5.","tan 60° = √3.","EF = 5 × √3 = 5√3 cm."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"D","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"E","style":"none","labelPos":"below-right"},{"at":[6,10.39230484541326],"label":"F","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"5 cm"},{"from":[6,0],"to":[6,10.39230484541326]},{"from":[6,10.39230484541326],"to":[0,0]}],"angles":[{"at":[6,0],"from":[6,10.39230484541326],"to":[0,0],"label":"","right":true},{"at":[0,0],"from":[6,0],"to":[6,10.39230484541326],"label":"60°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"eac32861dcc9"},{"id":"g5-exact-trig-values-q08","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":8,"marks":3,"tier":"H","calc":false,"text":"

In triangle JKL, angle JKL = 90°.

JK = 6 cm and JL = 12 cm.

Without using a calculator, find the size of angle JLK.

You must show your working.

","answerType":"exact","answer":"30°","answerDisplay":null,"accept":["30","30 degrees"],"parts":[],"solution":["JK = 6 cm is opposite angle JLK and JL = 12 cm is the hypotenuse.","sin JLK = 6 ÷ 12 = 12.","The exact value sin 30° = 12, so angle JLK = 30°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"J","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"K","style":"none","labelPos":"below-right"},{"at":[6,10.39230484541326],"label":"L","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"6 cm"},{"from":[6,0],"to":[6,10.39230484541326]},{"from":[6,10.39230484541326],"to":[0,0],"label":"12 cm"}],"angles":[{"at":[6,0],"from":[6,10.39230484541326],"to":[0,0],"label":"","right":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"7e43e1ab0e74"},{"id":"g5-exact-trig-values-q09","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":9,"marks":3,"tier":"H","calc":false,"text":"

Work out the exact value of

tan 30° × tan 60° + cos 45° × sin 45°

Give your answer as a fraction.

","answerType":"exact","answer":"3/2","answerDisplay":"32","accept":["1 1/2"],"parts":[],"solution":["tan 30° = √33 = 1√3 and tan 60° = √3, so tan 30° × tan 60° = 1.","cos 45° = √22 and sin 45° = √22, so cos 45° × sin 45° = 24 = 12.","1 + 12 = 32."],"diagram":null,"draw":false,"revision":"6249eca3a1d8"},{"id":"g5-exact-trig-values-q10","topicId":"g5-exact-trig-values","topic":"Exact Trig Values","grade":"5","topicGrade":"5","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

ABCD is a trapezium in which AB is parallel to DC.

Angle ADC = 90°, angle BCD = 60°, AB = 6 cm and DC = 10 cm.

Without using a calculator, work out the exact area of the trapezium.

Give your answer in the form k√3 cm2.

","answerType":"exact","answer":"32√3 cm²","answerDisplay":null,"accept":["32root3","32sqrt(3)"],"parts":[],"solution":["Let h be the height of the trapezium (the length AD).","Drop a perpendicular from B to DC, meeting it at E. Then DE = AB = 6 cm, so EC = 10 − 6 = 4 cm.","In right-angled triangle BEC, tan 60° = BE ÷ EC, so h = BE = 4 × tan 60° = 4√3 cm.","Area of the trapezium = 12(AB + DC) × h = 12(6 + 10) × 4√3.","Area = 8 × 4√3 = 32√3 cm²."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,4],"label":"A","style":"none","labelPos":"above"},{"at":[6,4],"label":"B","style":"none","labelPos":"above"},{"at":[10,0],"label":"C","style":"none","labelPos":"below-right"},{"at":[0,0],"label":"D","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"E","style":"none","labelPos":"below","answer":true}],"segments":[{"from":[0,4],"to":[6,4],"label":"6 cm","parallel":1},{"from":[6,4],"to":[10,0]},{"from":[10,0],"to":[0,0],"label":"10 cm","parallel":1},{"from":[0,0],"to":[0,4]},{"from":[6,4],"to":[6,0],"dashed":true,"answer":true}],"angles":[{"at":[0,0],"from":[0,4],"to":[10,0],"label":"","right":true},{"at":[10,0],"from":[6,4],"to":[0,0],"label":"60°"},{"at":[6,0],"from":[6,4],"to":[10,0],"right":true,"label":"","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"75feb41cef49"},{"id":"g5-expanding-and-factorising-quadratics-q01","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Factorise  x2 - 49

","answerType":"exact","answer":"(x + 7)(x - 7)","answerDisplay":null,"accept":["(x+7)(x-7)","(x - 7)(x + 7)","(x-7)(x+7)"],"parts":[],"solution":["x2 - 49 is a difference of two squares, since 49 = 72","x2 - 49 = (x + 7)(x - 7)","Check by expanding: (x + 7)(x − 7) = x² − 7x + 7x − 49. The two middle terms cancel, leaving x² − 49."],"diagram":null,"draw":false,"revision":"ca3816cfc572"},{"id":"g5-expanding-and-factorising-quadratics-q02","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

Expand and simplify  (x + 4)(x + 6)

","answerType":"exact","answer":"x^2 + 10x + 24","answerDisplay":null,"accept":["x² + 10x + 24","x^2+10x+24"],"parts":[],"solution":["(x + 4)(x + 6) = x2 + 6x + 4x + 24","Collect like terms: x2 + 10x + 24"],"diagram":null,"draw":false,"revision":"0623eb5165d1"},{"id":"g5-expanding-and-factorising-quadratics-q03","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

Expand and simplify  (x - 3)(x + 8)

","answerType":"exact","answer":"x^2 + 5x - 24","answerDisplay":null,"accept":["x² + 5x - 24","x^2+5x-24"],"parts":[],"solution":["(x - 3)(x + 8) = x2 + 8x - 3x - 24","Collect like terms: x2 + 5x - 24"],"diagram":null,"draw":false,"revision":"93f965a66df4"},{"id":"g5-expanding-and-factorising-quadratics-q04","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

Factorise  x2 + 9x + 14

","answerType":"exact","answer":"(x + 2)(x + 7)","answerDisplay":null,"accept":["(x+2)(x+7)","(x + 7)(x + 2)","(x+7)(x+2)"],"parts":[],"solution":["Expanding (x + a)(x + b) gives x² + (a + b)x + ab. Therefore the two numbers must add to the x coefficient and multiply to the constant.","Find two numbers that multiply to 14 and add to 9: they are 2 and 7","x2 + 9x + 14 = (x + 2)(x + 7)"],"diagram":null,"draw":false,"revision":"7408a5623bc1"},{"id":"g5-expanding-and-factorising-quadratics-q05","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":false,"text":"

Without working out the squares, show that

532 - 472 = 600

","answerType":"open","answer":"53^2 - 47^2 = (53 + 47)(53 - 47) = 100 x 6 = 600","answerDisplay":null,"accept":["(53+47)(53-47) = 100 × 6 = 600"],"parts":[],"solution":["Use the difference of two squares: a2 - b2 = (a + b)(a - b)","532 - 472 = (53 + 47)(53 - 47)","= 100 × 6 = 600"],"diagram":null,"draw":false,"revision":"726550c372ee"},{"id":"g5-expanding-and-factorising-quadratics-q06","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":false,"text":"

A rectangular card measures (x + 5) cm by (x + 3) cm.

A square of side x cm is cut from the card.

Show that the area of card remaining is (8x + 15) cm2.

","answerType":"open","answer":"(x + 5)(x + 3) - x^2 = x^2 + 8x + 15 - x^2 = 8x + 15","answerDisplay":null,"accept":["(x+5)(x+3) - x^2 = 8x + 15"],"parts":[],"solution":["Area of rectangle = (x + 5)(x + 3) = x2 + 3x + 5x + 15 = x2 + 8x + 15","Area of square removed = x2","Remaining area = x2 + 8x + 15 - x2 = 8x + 15"],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[8,0],[8,6],[0,6]],"sides":["(x + 5) cm","(x + 3) cm"]},{"pts":[[1,1],[4,1],[4,4],[1,4]],"sides":["x cm","x cm"],"dashed":true}],"texts":[{"at":[2.5,2.5],"text":"Cut out"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"4464b26dcf0e"},{"id":"g5-expanding-and-factorising-quadratics-q07","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":4,"tier":"F","calc":true,"text":"
  1. Expand and simplify  (x + 9)(x - 2)
  2. Factorise  x2 - 3x - 28
","answerType":"multi","answer":"a) x^2 + 7x - 18; b) (x - 7)(x + 4)","answerDisplay":null,"accept":["a) x² + 7x - 18 b) (x + 4)(x - 7)"],"parts":[{"label":"a","answer":"x^2 + 7x - 18"},{"label":"b","answer":"(x - 7)(x + 4)"}],"solution":["a) (x + 9)(x - 2) = x2 - 2x + 9x - 18 = x2 + 7x - 18","b) Find two numbers that multiply to -28 and add to -3: they are -7 and 4","b) x2 - 3x - 28 = (x - 7)(x + 4)"],"diagram":null,"draw":false,"revision":"a5e2c2b7a17a"},{"id":"g5-expanding-and-factorising-quadratics-q08","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":4,"tier":"FH","calc":true,"text":"
  1. Expand and simplify  (2x + 3)(x - 4)
  2. Factorise fully  6y2 + 9y
","answerType":"multi","answer":"a) 2x^2 - 5x - 12; b) 3y(2y + 3)","answerDisplay":null,"accept":["a) 2x² - 5x - 12 b) 3y(2y + 3)"],"parts":[{"label":"a","answer":"2x^2 - 5x - 12"},{"label":"b","answer":"3y(2y + 3)"}],"solution":["a) (2x + 3)(x - 4) = 2x2 - 8x + 3x - 12 = 2x2 - 5x - 12","b) The highest common factor of 6y2 and 9y is 3y","b) 6y2 + 9y = 3y(2y + 3)"],"diagram":null,"draw":false,"revision":"4d1776dcd237"},{"id":"g5-expanding-and-factorising-quadratics-q09","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":9,"marks":4,"tier":"F","calc":true,"text":"
  1. Expand and simplify  (y + 7)(y - 7)
  2. Factorise  x2 + x - 30
","answerType":"multi","answer":"a) y^2 - 49; b) (x + 6)(x - 5)","answerDisplay":null,"accept":["a) y² - 49 b) (x - 5)(x + 6)"],"parts":[{"label":"a","answer":"y^2 - 49"},{"label":"b","answer":"(x + 6)(x - 5)"}],"solution":["a) (y + 7)(y - 7) = y2 - 7y + 7y - 49 = y2 - 49","b) Find two numbers that multiply to -30 and add to 1: they are 6 and -5","b) x2 + x - 30 = (x + 6)(x - 5)"],"diagram":null,"draw":false,"revision":"2884770d274c"},{"id":"g5-expanding-and-factorising-quadratics-q10","topicId":"g5-expanding-and-factorising-quadratics","topic":"Expanding and Factorising Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

Amy is asked to factorise x2 - 4x - 12.

She writes  x2 - 4x - 12 = (x - 6)(x - 2)

  1. Explain why Amy's answer is wrong, and write down the correct factorisation.
  2. Factorise fully  2x2 + 6x - 20
","answerType":"multi","answer":"a) (x - 6)(x - 2) expands to x^2 - 8x + 12, not x^2 - 4x - 12; correct is (x - 6)(x + 2). b) 2(x + 5)(x - 2)","answerDisplay":null,"accept":["a) correct factorisation (x + 2)(x - 6) b) 2(x - 2)(x + 5)"],"parts":[{"label":"a","answer":"(x - 6)(x + 2)"},{"label":"b","answer":"2(x + 5)(x - 2)"}],"solution":["a) Expanding Amy's answer: (x - 6)(x - 2) = x2 - 8x + 12, which is not x2 - 4x - 12","a) The two numbers must multiply to -12 and add to -4: they are -6 and 2, so x2 - 4x - 12 = (x - 6)(x + 2)","b) Take out the common factor 2 first: 2x2 + 6x - 20 = 2(x2 + 3x - 10)","b) Factorise the quadratic: x2 + 3x - 10 = (x + 5)(x - 2)","b) So 2x2 + 6x - 20 = 2(x + 5)(x - 2)"],"diagram":null,"draw":false,"revision":"0fbfca2015ca"},{"id":"g5-gradient-of-a-line-q01","topicId":"g5-gradient-of-a-line","topic":"Gradient of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

A straight line has equation y = 5x + 2

Write down the gradient of the line.

","answerType":"exact","answer":"5","answerDisplay":null,"accept":["5","m = 5","gradient = 5"],"parts":[],"solution":["The equation is in the form y = mx + c, where m is the gradient.","Comparing y = 5x + 2 with y = mx + c gives m = 5.","The gradient is 5.","Gradient measures change in y for each 1-unit increase in x. Here y increases by 5 whenever x increases by 1."],"diagram":null,"draw":false,"revision":"06b86aeb451a"},{"id":"g5-gradient-of-a-line-q02","topicId":"g5-gradient-of-a-line","topic":"Gradient of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":1,"tier":"F","calc":true,"text":"

A straight line has equation y = 7 - 2x

Write down the gradient of the line.

","answerType":"exact","answer":"-2","answerDisplay":null,"accept":["-2","m = -2","gradient = -2"],"parts":[],"solution":["Rewrite the equation as y = -2x + 7 so it matches the form y = mx + c.","The coefficient of x is -2, so the gradient is -2.","A common error is to answer 7, which is the y-intercept, not the gradient."],"diagram":null,"draw":false,"revision":"3681eb33db32"},{"id":"g5-gradient-of-a-line-q03","topicId":"g5-gradient-of-a-line","topic":"Gradient of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":2,"tier":"FH","calc":true,"text":"

A straight line has equation 2y = 6x + 10

  1. Write down the gradient of the line.
  2. Write down the coordinates of the point where the line crosses the y-axis.
","answerType":"multi","answer":"a) 3 b) (0, 5)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3"},{"label":"b","answer":"(0, 5)"}],"solution":["Divide every term by 2 to get the equation in the form y = mx + c: y = 3x + 5.","The gradient is the coefficient of x, so the gradient is 3.","The line crosses the y-axis where x = 0, giving y = 5, so the point is (0, 5)."],"diagram":null,"draw":false,"revision":"61d47ad31fed"},{"id":"g5-gradient-of-a-line-q04","topicId":"g5-gradient-of-a-line","topic":"Gradient of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

A straight line passes through the points A (2, 5) and B (6, 17).

Work out the gradient of the line.

","answerType":"exact","answer":"3","answerDisplay":null,"accept":["3","m = 3","gradient = 3"],"parts":[],"solution":["Gradient = change in y divided by change in x.","Change in y = 17 - 5 = 12. Change in x = 6 - 2 = 4.","Gradient = 12 ÷ 4 = 3."],"diagram":null,"draw":false,"revision":"1ab01c2c0f1a"},{"id":"g5-gradient-of-a-line-q05","topicId":"g5-gradient-of-a-line","topic":"Gradient of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":2,"tier":"FH","calc":false,"text":"

A straight line passes through the points P (-3, 4) and Q (5, -8).

Work out the gradient of the line.

","answerType":"exact","answer":"-3/2","answerDisplay":"-32","accept":["-3/2","-1.5","m = -3/2","m = -1.5","-12/8"],"parts":[],"solution":["Gradient = change in y divided by change in x.","Change in y = -8 - 4 = -12. Change in x = 5 - (-3) = 8.","Gradient = -12 ÷ 8 = -32 = -1.5."],"diagram":null,"draw":false,"revision":"5a792cdcc4d3"},{"id":"g5-gradient-of-a-line-q06","topicId":"g5-gradient-of-a-line","topic":"Gradient of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":3,"tier":"FH","calc":true,"text":"

A straight line passes through the points A (1, 3) and B (k, 11).

The gradient of the line is 2.

Find the value of k.

","answerType":"exact","answer":"k = 5","answerDisplay":null,"accept":["5","k = 5"],"parts":[],"solution":["Gradient = change in y divided by change in x, so 11 - 3k - 1 = 2.","8 = 2(k - 1), so k - 1 = 4.","k = 5. Check: from (1, 3) to (5, 11) the line rises 8 across 4, and 8 ÷ 4 = 2."],"diagram":null,"draw":false,"revision":"7380e56feda3"},{"id":"g5-gradient-of-a-line-q07","topicId":"g5-gradient-of-a-line","topic":"Gradient of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

Water is draining from a tank at a constant rate.

The depth of the water is 120 cm when the tank starts to drain.

After 40 minutes the depth of the water is 40 cm.

The graph of depth against time is a straight line.

  1. Work out the gradient of the line.
  2. Interpret what your gradient means in this situation.
","answerType":"multi","answer":"a) -2 b) The depth of water decreases by 2 cm every minute","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"-2"},{"label":"b","answer":"The depth of the water decreases by 2 cm each minute"}],"solution":["The line passes through (0, 120) and (40, 40).","Gradient = 40 - 12040 - 0 = -80 ÷ 40 = -2.","The gradient tells you the rate of change: the depth of water falls by 2 cm every minute."],"diagram":{"type":"axes","x":{"min":0,"max":60,"step":10,"label":"Time (minutes)"},"y":{"min":0,"max":140,"step":20,"label":"Depth (cm)"},"curves":[{"pts":[[0,120],[40,40],[60,0]]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"1de9672b83d0"},{"id":"g5-gradient-of-a-line-q08","topicId":"g5-gradient-of-a-line","topic":"Gradient of a Line","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

A straight line passes through the points C (-2, 7) and D (4, a).

The gradient of the line is -32

Find the value of a.

","answerType":"exact","answer":"a = -2","answerDisplay":null,"accept":["-2","a = -2"],"parts":[],"solution":["Gradient = change in y divided by change in x, so a - 74 - (-2) = -32.","Change in x = 6, so a - 7 = -32 × 6 = -9.","a = 7 - 9 = -2. Check: from (-2, 7) to (4, -2) the line falls 9 across 6, and -9 ÷ 6 = -32."],"diagram":null,"draw":false,"revision":"0b10ff68ea4a"},{"id":"g5-probability-trees-q01","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

The probability that it rains in a town on Saturday is 0.3.

The probability that it rains in the town on Sunday is 0.4.

The two events are independent.

Work out the probability that it rains on both days.

","answerType":"exact","answer":"0.12","answerDisplay":null,"accept":["12/100","3/25"],"parts":[],"solution":["For independent events, multiply the probabilities.","P(rain both days) = 0.3 × 0.4 = 0.12"],"diagram":null,"draw":false,"revision":"3571e852f94c"},{"id":"g5-probability-trees-q02","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":2,"marks":2,"tier":"FH","calc":false,"text":"

Zak plays two games of darts. He draws a probability tree diagram.

On the first pair of branches he writes P(win) = 0.6 and P(lose) = 0.5.

He then works out P(win both games) = 0.6 + 0.6 = 1.2.

Write down two mistakes Zak has made.

The games are independent, with the same probability of winning each game. There are no draws.

","answerType":"open","answer":"The first pair of branch probabilities do not add up to 1 (0.6 + 0.5 = 1.1); he should have multiplied along the branches, not added (and a probability cannot be more than 1).","answerDisplay":null,"accept":[],"parts":[],"solution":["The probabilities on any pair of branches must sum to 1, but 0.6 + 0.5 = 1.1.","To combine two events along a path you multiply the probabilities, so P(win both) should be 0.6 × 0.6 = 0.36, and 1.2 is impossible because a probability cannot be greater than 1."],"diagram":null,"draw":false,"revision":"633d0a0d1167"},{"id":"g5-probability-trees-q03","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

Femi plays two games of table tennis.

The probability that he wins each game is 23.

The results of the games are independent.

Work out the probability that Femi loses at least one of the games.

","answerType":"exact","answer":"5/9","answerDisplay":"59","accept":[],"parts":[],"solution":["P(wins both games) = 23 × 23 = 49.","Losing at least one game is the opposite of winning both.","P(loses at least one) = 1 - 49 = 59"],"diagram":null,"draw":false,"revision":"d9c4cf63f538"},{"id":"g5-probability-trees-q04","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

A bag contains 3 red counters and 7 blue counters.

Kira takes a counter at random, notes its colour and puts it back.

She then takes a second counter at random.

  1. Complete the probability tree diagram for the two picks.
  2. Work out the probability that both counters are blue.
","answerType":"multi","answer":"a) each stage: P(red) = 0.3, P(blue) = 0.7; b) 0.49","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"P(red) = 0.3 and P(blue) = 0.7 on every pair of branches"},{"label":"b","answer":"0.49"}],"solution":["P(red) = 310 = 0.3 and P(blue) = 710 = 0.7.","The counter is replaced, so the second pick has the same probabilities.","P(both blue) = 0.7 × 0.7 = 0.49"],"diagram":{"type":"tree","stage1":{"label":"First counter","outcomes":["Red","Blue"],"p":[null,null]},"stage2":{"label":"Second counter","outcomes":["Red","Blue"],"p":[[null,null],[null,null]]},"answer":{"stage1":["0.3","0.7"],"stage2":[["0.3","0.7"],["0.3","0.7"]]}},"draw":false,"revision":"538e78162511"},{"id":"g5-probability-trees-q05","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

The probability that Jon's bus is on time on any day is 0.9.

Whether the bus is on time on one day is independent of any other day.

Jon catches the bus on Monday and on Tuesday.

  1. Complete the probability tree diagram.
  2. Work out the probability that the bus is on time on both days.
","answerType":"multi","answer":"a) missing branches all show P(on time) = 0.9, P(late) = 0.1; b) 0.81","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"P(late) = 0.1 on each pair of branches, P(on time) = 0.9 on the second day branches"},{"label":"b","answer":"0.81"}],"solution":["Each pair of branches sums to 1, so P(late) = 1 - 0.9 = 0.1.","P(on time both days) = 0.9 × 0.9 = 0.81"],"diagram":{"type":"tree","stage1":{"label":"Monday","outcomes":["On time","Late"],"p":["0.9",null]},"stage2":{"label":"Tuesday","outcomes":["On time","Late"],"p":[[null,null],[null,null]]},"answer":{"stage1":["0.9","0.1"],"stage2":[["0.9","0.1"],["0.9","0.1"]]}},"draw":false,"revision":"5554d3284089"},{"id":"g5-probability-trees-q06","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":6,"marks":4,"tier":"FH","calc":true,"text":"

Amy and Beth each take one penalty kick.

The probability that Amy scores is 0.7.

The probability that Beth scores is 0.55.

The two events are independent.

Work out the probability that neither of them scores.

","answerType":"exact","answer":"0.135","answerDisplay":null,"accept":["27/200"],"parts":[],"solution":["P(Amy misses) = 1 - 0.7 = 0.3.","P(Beth misses) = 1 - 0.55 = 0.45.","P(neither scores) = 0.3 × 0.45 = 0.135"],"diagram":null,"draw":false,"revision":"e1c0aa552385"},{"id":"g5-probability-trees-q07","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

A bag contains 2 green sweets and 3 yellow sweets.

Cal takes a sweet at random, notes its colour and puts it back in the bag.

He then takes a second sweet at random.

Work out the probability that the two sweets are the same colour.

","answerType":"exact","answer":"13/25","answerDisplay":"1325","accept":["0.52"],"parts":[],"solution":["P(green) = 25 and P(yellow) = 35 on each pick, because the sweet is replaced.","P(both green) = 25 × 25 = 425.","P(both yellow) = 35 × 35 = 925.","P(same colour) = 425 + 925 = 1325","Add the two paths because both green and both yellow cannot happen together in the same two picks. These are the only ways to obtain the same colour."],"diagram":null,"draw":false,"revision":"cb5e20c298bf"},{"id":"g5-probability-trees-q08","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

A warehouse has two independent fire alarms.

The probability that each alarm works correctly in a test is 0.95.

Both alarms are tested.

Work out the probability that at least one of the alarms works correctly.

","answerType":"exact","answer":"0.9975","answerDisplay":null,"accept":["399/400"],"parts":[],"solution":["P(an alarm fails) = 1 - 0.95 = 0.05.","P(both alarms fail) = 0.05 × 0.05 = 0.0025.","P(at least one works) = 1 - 0.0025 = 0.9975"],"diagram":null,"draw":false,"revision":"496c3a050baa"},{"id":"g5-probability-trees-q09","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

Cara and Dan each take a driving test.

The probability that Cara passes is 0.6.

The probability that Dan passes is 0.75.

The two events are independent.

Work out the probability that exactly one of them passes.

","answerType":"exact","answer":"0.45","answerDisplay":null,"accept":["9/20","45%"],"parts":[],"solution":["P(Cara passes and Dan fails) = 0.6 × 0.25 = 0.15.","P(Cara fails and Dan passes) = 0.4 × 0.75 = 0.3.","P(exactly one passes) = 0.15 + 0.3 = 0.45"],"diagram":null,"draw":false,"revision":"aef9f1c40205"},{"id":"g5-probability-trees-q10","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"

A biased coin is flipped twice. The flips are independent.

The probability of getting tails on both flips is 0.36.

  1. Work out the probability that the coin lands on heads on one flip.
  2. Work out the probability of getting exactly one tail in the two flips.
","answerType":"multi","answer":"a) 0.4; b) 0.48","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"0.4"},{"label":"b","answer":"0.48"}],"solution":["Let P(tails) = t. Then t × t = 0.36, so t = 0.6.","P(heads) = 1 - 0.6 = 0.4.","P(exactly one tail) = P(TH) + P(HT) = 0.6 × 0.4 + 0.4 × 0.6 = 0.24 + 0.24 = 0.48"],"diagram":null,"draw":false,"revision":"7374264da406"},{"id":"g5-probability-trees-q11","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":11,"marks":5,"tier":"H","calc":true,"text":"

Ella plays a tennis match on Saturday and a badminton match on Sunday.

The probability that she wins the tennis match is 0.65.

The probability that she wins the badminton match is 0.4.

The results are independent.

  1. Complete the probability tree diagram.
  2. Work out the probability that Ella wins exactly one of the two matches.
","answerType":"multi","answer":"a) tennis: 0.65 win, 0.35 lose; badminton: 0.4 win, 0.6 lose on each pair; b) 0.53","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"P(lose tennis) = 0.35; badminton branches 0.4 win and 0.6 lose from each tennis outcome"},{"label":"b","answer":"0.53"}],"solution":["P(lose tennis) = 1 - 0.65 = 0.35 and P(lose badminton) = 1 - 0.4 = 0.6.","P(win tennis, lose badminton) = 0.65 × 0.6 = 0.39.","P(lose tennis, win badminton) = 0.35 × 0.4 = 0.14.","P(exactly one win) = 0.39 + 0.14 = 0.53"],"diagram":{"type":"tree","stage1":{"label":"Tennis","outcomes":["Win","Lose"],"p":["0.65",null]},"stage2":{"label":"Badminton","outcomes":["Win","Lose"],"p":[[null,null],[null,null]]},"answer":{"stage1":["0.65","0.35"],"stage2":[["0.4","0.6"],["0.4","0.6"]]}},"draw":false,"revision":"61f68ebad31d"},{"id":"g5-probability-trees-q12","topicId":"g5-probability-trees","topic":"Probability Trees","grade":"5","topicGrade":"5","strand":"P","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

A box contains 5 red pens and 4 blue pens.

Milo takes a pen at random and does not replace it.

He then takes a second pen at random.

  1. Complete the probability tree diagram.
  2. Work out the probability that Milo takes at least one red pen.
","answerType":"multi","answer":"a) first pick: 5/9 red, 4/9 blue; after red: 4/8 red, 4/8 blue; after blue: 5/8 red, 3/8 blue; b) 5/6","answerDisplay":"a) first pick: 59 red, 49 blue; after red: 48 red, 48 blue; after blue: 58 red, 38 blue; b) 56","accept":[],"parts":[{"label":"a","answer":"first pick 5/9 and 4/9; second pick 4/8, 4/8 after red and 5/8, 3/8 after blue","answerDisplay":"first pick 59 and 49; second pick 48, 48 after red and 58, 38 after blue"},{"label":"b","answer":"5/6","answerDisplay":"56"}],"solution":["First pick: P(red) = 59, P(blue) = 49.","The pen is not replaced, so 8 pens remain: after a red, P(red) = 48 and P(blue) = 48; after a blue, P(red) = 58 and P(blue) = 38.","P(no red pens) = P(blue, blue) = 49 × 38 = 1272 = 16.","P(at least one red) = 1 - 16 = 56"],"diagram":{"type":"tree","stage1":{"label":"First pen","outcomes":["Red","Blue"],"p":[null,null]},"stage2":{"label":"Second pen","outcomes":["Red","Blue"],"p":[[null,null],[null,null]]},"answer":{"stage1":["5/9","4/9"],"stage2":[["4/8","4/8"],["5/8","3/8"]]}},"draw":false,"revision":"465c9e4eaa87"},{"id":"g5-reverse-percentages-q01","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

In a sale, all prices are reduced by 15%.

The sale price of a jumper is £68.

Work out the normal price of the jumper.

","answerType":"exact","answer":"£80","answerDisplay":null,"accept":["80","£80.00"],"parts":[],"solution":["The sale price is 100% − 15% = 85% of the normal price","The original price was multiplied by 0.85 to obtain the sale price. Undo that multiplication by dividing by 0.85; adding 15% of the sale price would use the wrong starting amount.","Normal price = 68 ÷ 0.85 = £80","Check: 80 × 0.85 = £68"],"diagram":null,"draw":false,"revision":"a4c98a99585e"},{"id":"g5-reverse-percentages-q02","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":2,"marks":2,"tier":"F","calc":false,"text":"

The price of a rail ticket increased by 20%.

The ticket now costs £54.

Work out the price of the ticket before the increase.

","answerType":"exact","answer":"£45","answerDisplay":null,"accept":["45","£45.00"],"parts":[],"solution":["The new price is 120% of the old price","Old price = 54 ÷ 1.2 = £45","Check: 45 × 1.2 = £54"],"diagram":null,"draw":false,"revision":"045e51798349"},{"id":"g5-reverse-percentages-q03","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":3,"marks":2,"tier":"F","calc":false,"text":"

The price of a jacket is reduced by 10%.

The reduced price is £63.

Work out the price of the jacket before the reduction.

","answerType":"exact","answer":"£70","answerDisplay":null,"accept":["70","£70.00"],"parts":[],"solution":["The reduced price is 90% of the original price","Original price = 63 ÷ 0.9 = £70","Check: 70 × 0.9 = £63"],"diagram":null,"draw":false,"revision":"420b2ff1b792"},{"id":"g5-reverse-percentages-q04","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

The population of a town increased by 12% between 2015 and 2025.

The population in 2025 was 26 880.

Work out the population in 2015.

","answerType":"exact","answer":"24000","answerDisplay":null,"accept":["24 000","24,000"],"parts":[],"solution":["The 2025 population is 112% of the 2015 population","2015 population = 26 880 ÷ 1.12 = 24 000","Check: 24 000 × 1.12 = 26 880"],"diagram":null,"draw":false,"revision":"7e4214a30980"},{"id":"g5-reverse-percentages-q05","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

In a sale there is 30% off all prices.

Kofi pays £59.36 for a coat in the sale.

Work out the price of the coat before the sale.

","answerType":"exact","answer":"£84.80","answerDisplay":null,"accept":["84.80","84.8"],"parts":[],"solution":["The sale price is 70% of the original price","Original price = 59.36 ÷ 0.7 = £84.80","Check: 84.80 × 0.7 = £59.36"],"diagram":null,"draw":false,"revision":"7e8ff97ed6d0"},{"id":"g5-reverse-percentages-q06","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":6,"marks":2,"tier":"H","calc":false,"text":"

Dana gets a 5% pay rise.

Her new rate of pay is £13.65 per hour.

Work out her rate of pay before the rise.

","answerType":"exact","answer":"£13","answerDisplay":null,"accept":["13","£13.00"],"parts":[],"solution":["The new rate is 105% of the old rate","Old rate = 13.65 ÷ 1.05 = £13 per hour","Check: 13 × 1.05 = £13.65"],"diagram":null,"draw":false,"revision":"1be801951474"},{"id":"g5-reverse-percentages-q07","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

The value of a car depreciates by 20% each year.

At the end of 2 years the car is worth £10 368.

Work out the value of the car 2 years ago.

","answerType":"exact","answer":"£16200","answerDisplay":null,"accept":["16200","£16,200","£16 200"],"parts":[],"solution":["The 2-year multiplier is 0.8 × 0.8 = 0.64","Original value = 10 368 ÷ 0.64 = £16 200","Check: 16 200 × 0.8 = 12 960, then 12 960 × 0.8 = £10 368"],"diagram":null,"draw":false,"revision":"866be78423f1"},{"id":"g5-reverse-percentages-q08","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

The value of a house increased by 8% in 2024 and then by 5% in 2025.

At the end of 2025 the house was worth £226 800.

Work out the value of the house at the start of 2024.

","answerType":"exact","answer":"£200000","answerDisplay":null,"accept":["200000","£200,000","£200 000"],"parts":[],"solution":["The overall multiplier is 1.08 × 1.05 = 1.134","Value at the start of 2024 = 226 800 ÷ 1.134 = £200 000","Check: 200 000 × 1.08 = 216 000, then 216 000 × 1.05 = £226 800"],"diagram":null,"draw":false,"revision":"07a1999878b6"},{"id":"g5-reverse-percentages-q09","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

The price of a television was reduced by 20% in a sale.

In a final clearance, the sale price was reduced by a further 10%.

The clearance price is £216.

  1. Work out the original price of the television.
  2. Show that the two reductions together are equivalent to a single reduction of 28%.
","answerType":"multi","answer":"a) £300, b) 0.8 x 0.9 = 0.72 = 72% of the original, a 28% reduction","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£300"},{"label":"b","answer":"0.8 x 0.9 = 0.72, so the final price is 72% of the original, a 28% reduction"}],"solution":["a) Overall multiplier = 0.8 × 0.9 = 0.72","Original price = 216 ÷ 0.72 = £300","Check: 300 × 0.8 = 240, then 240 × 0.9 = £216","b) A multiplier of 0.72 means the final price is 72% of the original, which is a reduction of 100% − 72% = 28%"],"diagram":null,"draw":false,"revision":"7e2df0aeba13"},{"id":"g5-reverse-percentages-q10","topicId":"g5-reverse-percentages","topic":"Reverse Percentages","grade":"5","topicGrade":"5","strand":"R","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

Ines put a sum of money into a savings account for 3 years.

The account paid 4% compound interest in the first year, then 2.5% compound interest in each of the next 2 years.

At the end of the 3 years the account held £8741.20.

  1. Work out the sum of money Ines put into the account.
  2. Work out the total interest she earned.
","answerType":"multi","answer":"a) £8000, b) £741.20","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£8000"},{"label":"b","answer":"£741.20"}],"solution":["a) Overall multiplier = 1.04 × 1.025 × 1.025 = 1.09265.","Original sum = 8741.20 ÷ (1.04 × 1.0252) = £8000","Check forwards: 8000 × 1.04 = 8320, then 8320 × 1.025 = 8528, then 8528 × 1.025 = £8741.20","b) Total interest = 8741.20 − 8000 = £741.20"],"diagram":null,"draw":false,"revision":"3ebe0ed695d0"},{"id":"g5-sector-areas-and-arc-lengths-q01","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":1,"marks":4,"tier":"H","calc":true,"text":"

A sector of a circle has radius 7.5 cm and angle 140°.

  1. Work out the arc length of the sector.
  2. Work out the area of the sector.

Give each answer correct to 3 significant figures.

","answerType":"multi","answer":"a) 18.3 cm b) 68.7 cm²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"18.3 cm"},{"label":"b","answer":"68.7 cm²"}],"solution":["A full turn is 360°. This sector is 140/360 of the full circle, so both its arc and its area are that fraction of the full circumference and area respectively.","a) Arc length = (140360) × 2π × 7.5 = (718) × 15π = 18.32... = 18.3 cm (3 sf).","b) Area = (140360) × π × 7.5² = (718) × 56.25π = 68.72... = 68.7 cm² (3 sf)."],"diagram":null,"draw":false,"revision":"8d0993a8309f"},{"id":"g5-sector-areas-and-arc-lengths-q02","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":2,"marks":2,"tier":"H","calc":false,"text":"

A sector of a circle has radius 6 cm and area 15π cm².

Work out the angle of the sector.

","answerType":"exact","answer":"150°","answerDisplay":null,"accept":["150","150 degrees"],"parts":[],"solution":["Area of a sector = (angle360) × πr².","The full circle has area π × 36 = 36π cm².","angle360 = 15π ÷ 36π = 512.","Angle = 512 × 360 = 150°."],"diagram":null,"draw":false,"revision":"28aadd24af3b"},{"id":"g5-sector-areas-and-arc-lengths-q03","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

A sector of a circle has radius 5 cm.

The arc length of the sector is 8.4 cm.

Find the angle of the sector.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"96.3°","answerDisplay":null,"accept":["96.3","96.3 degrees"],"parts":[],"solution":["Arc length = (angle360) × 2πr, and the full circumference = 2π × 5 = 10π = 31.415... cm.","angle360 = 8.4 ÷ 31.415... = 0.2673...","Angle = 0.2673... × 360 = 96.25... = 96.3° (1 dp)."],"diagram":null,"draw":false,"revision":"73a9a78df6e9"},{"id":"g5-sector-areas-and-arc-lengths-q04","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

A sector of a circle has radius 6 cm.

The perimeter of the sector is 42 cm.

The angle of the sector is reflex.

Work out the angle of the sector.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"286.5°","answerDisplay":null,"accept":["286.5","286.5 degrees"],"parts":[],"solution":["The perimeter is made of two radii and the arc, so the arc length = 42 − 6 − 6 = 30 cm.","The full circumference = 2π × 6 = 37.699... cm.","angle360 = 30 ÷ 37.699... = 0.7957...","Angle = 0.7957... × 360 = 286.47... = 286.5° (1 dp), which is reflex as required."],"diagram":null,"draw":false,"revision":"9d143edfb5f4"},{"id":"g5-sector-areas-and-arc-lengths-q05","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":5,"marks":4,"tier":"H","calc":false,"text":"

A sector of a circle has angle 45°.

The arc length of the sector is 3π cm.

Work out the area of the sector.

Give your answer in terms of π.

","answerType":"exact","answer":"18π cm²","answerDisplay":null,"accept":["18pi","18π"],"parts":[],"solution":["Arc length = (45360) × 2πr = πr4.","πr4 = 3π, so r = 12 cm.","Area = (45360) × π × 12² = 18 × 144π = 18π cm²."],"diagram":null,"draw":false,"revision":"41e7dffcd639"},{"id":"g5-sector-areas-and-arc-lengths-q06","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A sector of a circle has angle 120° and area 48π cm².

Work out the perimeter of the sector.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"49.1 cm","answerDisplay":null,"accept":["49.1"],"parts":[],"solution":["Area = (120360) × πr² = πr²3.","πr²3 = 48π, so r² = 144 and r = 12 cm.","Arc length = (120360) × 2π × 12 = 8π cm.","Perimeter = 12 + 12 + 8π = 24 + 8π = 49.13... = 49.1 cm (3 sf)."],"diagram":null,"draw":false,"revision":"27ae7f6941d6"},{"id":"g5-sector-areas-and-arc-lengths-q07","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":7,"marks":4,"tier":"FH","calc":true,"text":"

OAB is a sector of a circle, centre O, with radius 10 cm and angle AOB = 60°.

The chord AB divides the sector into a triangle and a segment.

Calculate the area of the segment.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"9.06 cm²","answerDisplay":null,"accept":["9.06"],"parts":[],"solution":["Area of the sector = (60360) × π × 10² = 50π3 = 52.359... cm².","Area of triangle OAB = 12 × 10 × 10 × sin 60° = 50 × 0.8660... = 43.301... cm².","Area of the segment = 52.359... − 43.301... = 9.058... = 9.06 cm² (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[5,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[2.5000000000000004,4.330127018922193],"label":"B","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[5,0],"label":"10 cm"},{"from":[0,0],"to":[2.5000000000000004,4.330127018922193]},{"from":[5,0],"to":[2.5000000000000004,4.330127018922193]}],"angles":[{"at":[0,0],"from":[5,0],"to":[2.5000000000000004,4.330127018922193],"label":"60°"}],"arcs":[{"c":[0,0],"r":5,"from":0,"to":60}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"495d40e942a3"},{"id":"g5-sector-areas-and-arc-lengths-q08","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

OAB is a sector of a circle, centre O, with angle AOB = 40°.

The chord AB has length 9 cm.

Calculate the area of the sector.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"60.4 cm²","answerDisplay":null,"accept":["60.4"],"parts":[],"solution":["Draw OM perpendicular to chord AB, with M on AB. The two right triangles have equal hypotenuses OA = OB and common side OM, so AM = MB = 4.5 cm and angle AOM = 20°.","In right-angled triangle AOM, sin 20° = 4.5/r, so r = 4.5 ÷ sin 20° = 13.157... cm. Use degree mode.","The sector is 40/360 = 1/9 of a full circle. Its area is (1/9) × π × (4.5 ÷ sin 20°)² = 60.426... cm².","Keep the full calculator value until the last step: area = 60.4 cm² to 3 significant figures."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[5,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[3.83022221559489,3.2139380484326963],"label":"B","style":"none","labelPos":"above"},{"at":[4.415111107797445,1.6069690242163481],"label":"M","style":"none","labelPos":"right","answer":true}],"segments":[{"from":[0,0],"to":[5,0]},{"from":[0,0],"to":[3.83022221559489,3.2139380484326963]},{"from":[5,0],"to":[3.83022221559489,3.2139380484326963],"label":"9 cm"},{"from":[0,0],"to":[4.415111107797445,1.6069690242163481],"dashed":true,"answer":true}],"angles":[{"at":[0,0],"from":[5,0],"to":[3.83022221559489,3.2139380484326963],"label":"40°"},{"at":[4.415111107797445,1.6069690242163481],"from":[0,0],"to":[5,0],"right":true,"label":"","answer":true}],"arcs":[{"c":[0,0],"r":5,"from":0,"to":40}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"9c8ceecc12f9"},{"id":"g5-sector-areas-and-arc-lengths-q09","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"

A sector of a circle has radius x cm and angle 144°.

A circle has radius 6 cm.

The area of the sector is equal to the area of the circle.

Work out the value of x.

Give your answer in the form k√10.

","answerType":"exact","answer":"x = 3√10","answerDisplay":null,"accept":["3root10","3√10","3sqrt(10)"],"parts":[],"solution":["Area of the sector = (144360) × πx² = (25)πx².","Area of the circle = π × 6² = 36π.","(25)πx² = 36π, so x² = 36 × 52 = 90.","x = √90 = √9 × √10 = 3√10."],"diagram":null,"draw":false,"revision":"77e2dfdf6d5e"},{"id":"g5-sector-areas-and-arc-lengths-q10","topicId":"g5-sector-areas-and-arc-lengths","topic":"Sector Areas and Arc Lengths","grade":"5","topicGrade":"5","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

The diagram shows two circles, each of radius 6 cm, with centres A and B.

Each circle passes through the centre of the other circle, so AB = 6 cm.

The two circles overlap in a shaded region.

Calculate the area of the shaded region.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"44.2 cm²","answerDisplay":null,"accept":["44.2"],"parts":[],"solution":["Let the circles cross at P and Q. In triangle APB, AP = BP = AB = 6 cm, so the triangles are equilateral and the chord PQ subtends an angle of 120° at each centre.","The shaded region is made of two equal segments, one from each circle, each cut off by the chord PQ.","Area of one 120° sector = (120360) × π × 36 = 12π = 37.699... cm².","Area of the triangle in that sector = 12 × 6 × 6 × sin 120° = 15.588... cm².","Area of one segment = 37.699... − 15.588... = 22.110... cm².","Shaded area = 2 × 22.110... = 44.22... = 44.2 cm² (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"circles":[{"c":[0,0],"r":3},{"c":[3,0],"r":3}],"polygons":[{"pts":[[1.5000000000000004,-2.598076211353316],[1.5897577926996147,-2.544144288469278],[1.6775787104122404,-2.4871127176651253],[1.7633557568774194,-2.4270509831248424],[1.846984425976975,-2.3640322608201663],[1.9283628290596182,-2.298133329356934],[2.0073918190765747,-2.229434476432183],[2.083975111376992,-2.1580194010159532],[2.1580194010159537,-2.0839751113769918],[2.229434476432183,-2.0073918190765747],[2.298133329356934,-1.9283628290596178],[2.3640322608201663,-1.8469844259769745],[2.4270509831248424,-1.7633557568774194],[2.487112717665125,-1.6775787104122406],[2.544144288469278,-1.5897577926996147],[2.598076211353316,-1.4999999999999998],[2.648842778576781,-1.4084146883576725],[2.6963821388975013,-1.3151134403672322],[2.7406363729278027,-1.2202099292274005],[2.7815515637003623,-1.123819780247736],[2.8190778623577253,-1.0260604299770062],[2.8531695488854605,-0.9270509831248421],[2.8837850878149567,-0.8269120674509975],[2.9108871788279895,-0.7257656867990032],[2.934442802201417,-0.623735072453278],[2.954423259036624,-0.520944533000791],[2.9708042062247113,-0.41751930288019634],[2.9835656861048196,-0.31358538980296036],[2.9926921507794724,-0.2092694212323759],[2.998172481057287,-0.1046984901075029],[3,0],[2.998172481057287,0.1046984901075029],[2.9926921507794724,0.2092694212323759],[2.9835656861048196,0.31358538980296036],[2.9708042062247113,0.41751930288019634],[2.954423259036624,0.520944533000791],[2.934442802201417,0.623735072453278],[2.9108871788279895,0.7257656867990032],[2.8837850878149567,0.8269120674509975],[2.8531695488854605,0.9270509831248421],[2.8190778623577253,1.0260604299770062],[2.7815515637003623,1.123819780247736],[2.7406363729278027,1.2202099292274005],[2.6963821388975013,1.3151134403672322],[2.648842778576781,1.4084146883576725],[2.598076211353316,1.4999999999999998],[2.544144288469278,1.5897577926996147],[2.487112717665125,1.6775787104122406],[2.4270509831248424,1.7633557568774194],[2.3640322608201663,1.8469844259769745],[2.298133329356934,1.9283628290596178],[2.229434476432183,2.0073918190765747],[2.1580194010159537,2.0839751113769918],[2.083975111376992,2.1580194010159532],[2.0073918190765747,2.229434476432183],[1.9283628290596182,2.298133329356934],[1.846984425976975,2.3640322608201663],[1.7633557568774194,2.4270509831248424],[1.6775787104122404,2.4871127176651253],[1.5897577926996147,2.544144288469278],[1.5000000000000004,2.598076211353316],[1.5000000000000007,2.598076211353316],[1.4102422073003855,2.544144288469278],[1.3224212895877598,2.4871127176651253],[1.2366442431225808,2.4270509831248424],[1.153015574023025,2.3640322608201663],[1.0716371709403818,2.298133329356934],[0.9926081809234253,2.229434476432183],[0.9160248886230091,2.158019401015954],[0.8419805989840459,2.0839751113769913],[0.7705655235678175,2.007391819076575],[0.7018666706430663,1.9283628290596184],[0.6359677391798342,1.8469844259769752],[0.5729490168751581,1.7633557568774196],[0.5128872823348751,1.6775787104122406],[0.455855711530722,1.5897577926996147],[0.401923788646684,1.4999999999999998],[0.3511572214232199,1.4084146883576731],[0.30361786110249867,1.3151134403672318],[0.25936362707219773,1.2202099292274013],[0.21844843629963817,1.1238197802477368],[0.18092213764227516,1.0260604299770066],[0.14683045111453952,0.9270509831248426],[0.1162149121850442,0.826912067450999],[0.08911282117201047,0.7257656867990032],[0.06555719779858293,0.623735072453278],[0.04557674096337605,0.5209445330007908],[0.029195793775289136,0.41751930288019723],[0.016434313895180352,0.3135853898029612],[0.007307849220527629,0.20926942123237657],[0.0018275189427128247,0.1046984901075021],[0,3.6739403974420594e-16],[0.0018275189427128247,-0.1046984901075027],[0.007307849220527185,-0.2092694212323745],[0.01643431389517991,-0.31358538980295914],[0.029195793775289136,-0.41751930288019656],[0.04557674096337605,-0.5209445330007914],[0.06555719779858293,-0.6237350724532772],[0.08911282117201047,-0.7257656867990026],[0.11621491218504332,-0.826912067450997],[0.14683045111453952,-0.9270509831248432],[0.18092213764227472,-1.026060429977006],[0.21844843629963773,-1.123819780247736],[0.25936362707219685,-1.2202099292273996],[0.30361786110249867,-1.3151134403672313],[0.3511572214232195,-1.4084146883576727],[0.401923788646684,-1.5000000000000004],[0.455855711530722,-1.5897577926996145],[0.5128872823348747,-1.6775787104122402],[0.5729490168751576,-1.7633557568774192],[0.6359677391798333,-1.8469844259769737],[0.7018666706430654,-1.9283628290596178],[0.770565523567817,-2.0073918190765747],[0.8419805989840468,-2.083975111376992],[0.9160248886230073,-2.1580194010159524],[0.9926081809234244,-2.229434476432182],[1.0716371709403816,-2.2981333293569337],[1.153015574023026,-2.3640322608201663],[1.2366442431225804,-2.427050983124842],[1.3224212895877583,-2.4871127176651244],[1.410242207300385,-2.544144288469278],[1.4999999999999987,-2.5980762113533156]],"fill":true}],"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[3,0],"label":"B","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[3,0],"label":"6 cm"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ce9d17011042"},{"id":"g5-similar-shapes-lengths-q01","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Rectangle A is 4 cm by 6 cm. Rectangle B is 8 cm by 10 cm.

Sam says, \"Each side of rectangle B is 4 cm longer than the matching side of rectangle A, so the rectangles are similar.\"

Explain why Sam is wrong.

","answerType":"open","answer":"Similar shapes need the same scale factor (ratio) for every pair of sides, not the same added amount: 8 divided by 4 = 2 but 10 divided by 6 is not 2, so the rectangles are not similar","answerDisplay":null,"accept":[],"parts":[],"solution":["For two shapes to be similar, corresponding sides must be in the same ratio (multiplied by the same scale factor), not increased by the same amount.","8 ÷ 4 = 2 but 10 ÷ 6 = 53, so the ratios are different and the rectangles are not similar."],"diagram":null,"draw":false,"revision":"8722690293f9"},{"id":"g5-similar-shapes-lengths-q02","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

Triangle ABC is similar to triangle PQR.

AB corresponds to PQ, and BC corresponds to QR.

AB = 6 cm, PQ = 9 cm and BC = 8 cm.

Work out the length of QR.

","answerType":"exact","answer":"12 cm","answerDisplay":null,"accept":["12"],"parts":[],"solution":["Scale factor from ABC to PQR = PQ ÷ AB = 9 ÷ 6 = 1.5.","QR = BC × 1.5 = 8 × 1.5 = 12 cm."],"diagram":null,"draw":false,"revision":"badfea32e19f"},{"id":"g5-similar-shapes-lengths-q03","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":3,"marks":2,"tier":"H","calc":false,"text":"

In triangle ABC, angle A = 35° and angle B = 80°.

In triangle DEF, angle D = 80° and angle E = 65°.

Show that triangle ABC is similar to triangle DEF.

","answerType":"open","answer":"Angle C = 180 - 35 - 80 = 65 degrees, so both triangles have angles 35, 80 and 65 degrees; triangles with the same three angles are similar","answerDisplay":null,"accept":[],"parts":[],"solution":["Angles in a triangle sum to 180°, so angle C = 180 − 35 − 80 = 65°.","Also angle F = 180 − 80 − 65 = 35°.","Both triangles have angles of 35°, 80° and 65°. Two triangles with the same three angles are similar."],"diagram":null,"draw":false,"revision":"a0adede50e98"},{"id":"g5-similar-shapes-lengths-q04","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

A photograph is 15 cm long and 10 cm wide.

The photograph is enlarged to make a print that is mathematically similar to it.

The print is 24 cm long.

  1. Work out the scale factor of the enlargement.
  2. Work out the width of the print.
","answerType":"multi","answer":"a) 1.6 b) 16 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1.6"},{"label":"b","answer":"16 cm"}],"solution":["a) Scale factor = 24 ÷ 15 = 1.6.","b) Width of the print = 10 × 1.6 = 16 cm."],"diagram":null,"draw":false,"revision":"155c51acc1a4"},{"id":"g5-similar-shapes-lengths-q05","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":5,"marks":4,"tier":"FH","calc":true,"text":"

Triangle ABC is similar to triangle DEF.

AB corresponds to DE, BC corresponds to EF, and AC corresponds to DF.

AB = 5 cm, BC = 7 cm, DE = 7.5 cm and DF = 12 cm.

  1. Work out the length of EF.
  2. Work out the length of AC.
","answerType":"multi","answer":"a) 10.5 cm b) 8 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"10.5 cm"},{"label":"b","answer":"8 cm"}],"solution":["Scale factor from ABC to DEF = DE ÷ AB = 7.5 ÷ 5 = 1.5.","a) EF = BC × 1.5 = 7 × 1.5 = 10.5 cm.","b) Going the other way, divide by the scale factor: AC = DF ÷ 1.5 = 12 ÷ 1.5 = 8 cm."],"diagram":null,"draw":false,"revision":"3742656988b6"},{"id":"g5-similar-shapes-lengths-q06","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

In triangle ABC, D is a point on AB and E is a point on AC.

DE is parallel to BC.

AD = 4 cm, DB = 6 cm and DE = 5 cm.

Work out the length of BC.

","answerType":"exact","answer":"12.5 cm","answerDisplay":null,"accept":["12.5"],"parts":[],"solution":["The triangles share angle A, and the parallel lines give equal corresponding angles at D and B. Matching angles establish similarity before using a scale factor.","Because DE is parallel to BC, triangle ADE is similar to triangle ABC.","AB = AD + DB = 4 + 6 = 10 cm, so the scale factor from ADE to ABC = AB ÷ AD = 10 ÷ 4 = 2.5.","BC = DE × 2.5 = 5 × 2.5 = 12.5 cm."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,5],"label":"A","style":"none","labelPos":"above"},{"at":[-4,0],"label":"B","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"C","style":"none","labelPos":"below-right"},{"at":[-1.6,3],"label":"D","style":"none","labelPos":"above"},{"at":[1.6,3],"label":"E","style":"none","labelPos":"above"}],"segments":[{"from":[0,5],"to":[-1.6,3],"label":"4 cm"},{"from":[-1.6,3],"to":[-4,0],"label":"6 cm"},{"from":[0,5],"to":[4,0]},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[-1.6,3],"to":[1.6,3],"label":"5 cm","parallel":1}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"13067cd2a985"},{"id":"g5-similar-shapes-lengths-q07","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

AB is parallel to DE.

The straight lines AE and BD cross at the point C.

AC = 9 cm, CE = 6 cm, AB = 12 cm and CD = 5 cm.

  1. Work out the length of DE.
  2. Work out the length of BC.
","answerType":"multi","answer":"a) 8 cm b) 7.5 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8 cm"},{"label":"b","answer":"7.5 cm"}],"solution":["Because AB is parallel to DE, alternate angles are equal and the vertically opposite angles at C are equal, so triangle ABC is similar to triangle EDC.","The scale factor from triangle ABC to triangle EDC = CE ÷ AC = 6 ÷ 9 = 23.","a) DE corresponds to AB, so DE = 12 × 23 = 8 cm.","b) BC corresponds to DC, so BC = CD ÷ (23) = 5 × 32 = 7.5 cm."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[-4,3],"label":"A","style":"none","labelPos":"above"},{"at":[4,3],"label":"B","style":"none","labelPos":"above"},{"at":[0,0],"label":"C","style":"none","labelPos":"below-right"},{"at":[-2.6666666666666665,-2],"label":"D","style":"none","labelPos":"below-left"},{"at":[2.6666666666666665,-2],"label":"E","style":"none","labelPos":"below-right"}],"segments":[{"from":[-4,3],"to":[4,3],"label":"12 cm","parallel":1},{"from":[-2.6666666666666665,-2],"to":[2.6666666666666665,-2],"parallel":1},{"from":[-4,3],"to":[0,0],"label":"9 cm"},{"from":[0,0],"to":[2.6666666666666665,-2],"label":"6 cm"},{"from":[4,3],"to":[0,0]},{"from":[0,0],"to":[-2.6666666666666665,-2],"label":"5 cm"}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"5d95efd7cc60"},{"id":"g5-similar-shapes-lengths-q08","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":false,"text":"

In triangle ADE, B is a point on AD and C is a point on AE.

BC is parallel to DE.

AB = 6 cm, BD = 3 cm and BC = 8 cm.

The area of triangle ABC is 24 cm2.

  1. Work out the length of DE.
  2. Work out the area of the trapezium BDEC.
","answerType":"multi","answer":"a) 12 cm b) 30 cm²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"12 cm"},{"label":"b","answer":"30 cm²"}],"solution":["Because BC is parallel to DE, triangle ABC is similar to triangle ADE.","AD = 6 + 3 = 9 cm, so the scale factor = 9 ÷ 6 = 1.5.","a) DE = 8 × 1.5 = 12 cm.","b) Areas of similar shapes scale by the square of the scale factor: area of ADE = 24 × 1.5² = 24 × 2.25 = 54 cm².","Area of trapezium BDEC = 54 − 24 = 30 cm²."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,5],"label":"A","style":"none","labelPos":"above"},{"at":[-4,0],"label":"D","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"E","style":"none","labelPos":"below-right"},{"at":[-2.6666666666666665,1.666666666666667],"label":"B","style":"none","labelPos":"above"},{"at":[2.6666666666666665,1.666666666666667],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,5],"to":[-2.6666666666666665,1.666666666666667],"label":"6 cm"},{"from":[-2.6666666666666665,1.666666666666667],"to":[-4,0],"label":"3 cm"},{"from":[0,5],"to":[4,0]},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[-2.6666666666666665,1.666666666666667],"to":[2.6666666666666665,1.666666666666667],"label":"8 cm","parallel":1}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"fc80664e676c"},{"id":"g5-similar-shapes-lengths-q09","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

In triangle PQR, angle PQR = 90°, PQ = 6 cm, QR = 8 cm and PR = 10 cm.

S is the point on PR such that QS is perpendicular to PR.

Work out the length of QS.

","answerType":"exact","answer":"4.8 cm","answerDisplay":null,"accept":["4.8"],"parts":[],"solution":["Triangles PSQ and PQR share angle P, and each has a right angle (at S and at Q), so they are similar.","In similar triangles, QS ÷ QR = PQ ÷ PR.","QS = QR × PQ ÷ PR = 8 × 6 ÷ 10 = 4.8 cm.","Check using area: 12 × 6 × 8 = 24 cm² and 12 × 10 × 4.8 = 24 cm², which agree."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"Q","style":"none","labelPos":"below-right"},{"at":[6,8],"label":"R","style":"none","labelPos":"above"},{"at":[2.16,2.88],"label":"S","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"6 cm"},{"from":[6,0],"to":[6,8],"label":"8 cm"},{"from":[0,0],"to":[6,8],"label":"10 cm"},{"from":[6,0],"to":[2.16,2.88]}],"angles":[{"at":[6,0],"from":[0,0],"to":[6,8],"label":"","right":true},{"at":[2.16,2.88],"from":[6,0],"to":[6,8],"label":"","right":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"16ea239ba46a"},{"id":"g5-similar-shapes-lengths-q10","topicId":"g5-similar-shapes-lengths","topic":"Similar Shapes (Lengths)","grade":"5","topicGrade":"5","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

In triangle ABC, D is a point on AB.

Angle ACD = angle ABC.

AC = 8 cm and AD = 4 cm.

  1. Prove that triangle ACD is similar to triangle ABC.
  2. Work out the length of BD.
","answerType":"multi","answer":"a) proof (see solution) b) 12 cm","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"shared angle A and angle ACD = angle ABC, so the triangles have two equal angles and are similar"},{"label":"b","answer":"12 cm"}],"solution":["a) In triangles ACD and ABC: angle A is common to both triangles, and angle ACD = angle ABC (given).","Two pairs of equal angles means the third pair must also be equal, so triangle ACD is similar to triangle ABC.","b) Corresponding sides: AC in ACD corresponds to AB in ABC, and AD in ACD corresponds to AC in ABC.","So AC ÷ AB = AD ÷ AC, giving 8 ÷ AB = 4 ÷ 8.","AB = 8 × 8 ÷ 4 = 16 cm.","BD = AB − AD = 16 − 4 = 12 cm."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[8,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[2,3.4641016151377544],"label":"C","style":"none","labelPos":"above"},{"at":[2,0],"label":"D","style":"none","labelPos":"below-left"}],"segments":[{"from":[0,0],"to":[2,0],"label":"4 cm"},{"from":[2,0],"to":[8,0]},{"from":[0,0],"to":[2,3.4641016151377544],"label":"8 cm"},{"from":[8,0],"to":[2,3.4641016151377544]},{"from":[2,3.4641016151377544],"to":[2,0]}],"angles":[{"at":[2,3.4641016151377544],"from":[0,0],"to":[2,0],"label":"","double":true},{"at":[8,0],"from":[0,0],"to":[2,3.4641016151377544],"label":"","double":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"8f8d36ee8cea"},{"id":"g5-simultaneous-equations-q01","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":3,"tier":"F","calc":true,"text":"

Solve the simultaneous equations

x + y = 11

x - y = 3

","answerType":"exact","answer":"x = 7, y = 4","answerDisplay":null,"accept":["x = 7 and y = 4","y = 4, x = 7","(7, 4)"],"parts":[],"solution":["We need one pair (x, y) that makes both equations true. Adding the equations makes +y and −y cancel, leaving an equation containing only x.","Add the two equations: 2x = 14, so x = 7","Substitute into x + y = 11: 7 + y = 11, so y = 4","Check in the second equation: 7 - 4 = 3 ✓"],"diagram":null,"draw":false,"revision":"006dcf3485b3"},{"id":"g5-simultaneous-equations-q02","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":3,"tier":"F","calc":true,"text":"

Solve the simultaneous equations

3x + y = 17

x + y = 7

","answerType":"exact","answer":"x = 5, y = 2","answerDisplay":null,"accept":["x = 5 and y = 2","y = 2, x = 5","(5, 2)"],"parts":[],"solution":["Subtract the second equation from the first: 2x = 10, so x = 5","Substitute into x + y = 7: 5 + y = 7, so y = 2","Check in the first equation: 15 + 2 = 17 ✓"],"diagram":null,"draw":false,"revision":"2dce8edced90"},{"id":"g5-simultaneous-equations-q03","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

Solve the simultaneous equations

2x + 3y = 12

2x + y = 8

","answerType":"exact","answer":"x = 3, y = 2","answerDisplay":null,"accept":["x = 3 and y = 2","y = 2, x = 3","(3, 2)"],"parts":[],"solution":["Subtract the second equation from the first: 2y = 4, so y = 2","Substitute into 2x + y = 8: 2x + 2 = 8, so 2x = 6 and x = 3","Check in the first equation: 6 + 6 = 12 ✓"],"diagram":null,"draw":false,"revision":"9bbe60f19025"},{"id":"g5-simultaneous-equations-q04","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":3,"tier":"FH","calc":true,"text":"

Solve the simultaneous equations

4x - y = 14

2x + y = 10

","answerType":"exact","answer":"x = 4, y = 2","answerDisplay":null,"accept":["x = 4 and y = 2","y = 2, x = 4","(4, 2)"],"parts":[],"solution":["Add the two equations to eliminate y: 6x = 24, so x = 4","Substitute into 2x + y = 10: 8 + y = 10, so y = 2","Check in the first equation: 16 - 2 = 14 ✓"],"diagram":null,"draw":false,"revision":"edc142d204a3"},{"id":"g5-simultaneous-equations-q05","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":3,"tier":"FH","calc":false,"text":"

Solve the simultaneous equations

3x + 2y = 19

x + 4y = 13

","answerType":"exact","answer":"x = 5, y = 2","answerDisplay":null,"accept":["x = 5 and y = 2","y = 2, x = 5","(5, 2)"],"parts":[],"solution":["Multiply the first equation by 2: 6x + 4y = 38","Subtract the second equation: 5x = 25, so x = 5","Substitute into x + 4y = 13: 5 + 4y = 13, so y = 2","Check in the first equation: 15 + 4 = 19 ✓"],"diagram":null,"draw":false,"revision":"7cdfc5f2f4bf"},{"id":"g5-simultaneous-equations-q06","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":3,"tier":"FH","calc":true,"text":"

Solve the simultaneous equations

x + 2y = 4

3x + 5y = 9

","answerType":"exact","answer":"x = -2, y = 3","answerDisplay":null,"accept":["x = -2 and y = 3","y = 3, x = -2","(-2, 3)"],"parts":[],"solution":["Multiply the first equation by 3: 3x + 6y = 12","Subtract the second equation: y = 3","Substitute into x + 2y = 4: x + 6 = 4, so x = -2","Check in the second equation: -6 + 15 = 9 ✓"],"diagram":null,"draw":false,"revision":"24fe7657bd4a"},{"id":"g5-simultaneous-equations-q07","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":4,"tier":"FH","calc":false,"text":"

Solve the simultaneous equations

3x + 2y = 24

2x + 5y = 27

","answerType":"exact","answer":"x = 6, y = 3","answerDisplay":null,"accept":["x = 6 and y = 3","y = 3, x = 6","(6, 3)"],"parts":[],"solution":["Multiply the first equation by 5 and the second by 2: 15x + 10y = 120 and 4x + 10y = 54","Subtract: 11x = 66, so x = 6","Substitute into 3x + 2y = 24: 18 + 2y = 24, so y = 3","Check in the second equation: 12 + 15 = 27 ✓"],"diagram":null,"draw":false,"revision":"6177ade46c27"},{"id":"g5-simultaneous-equations-q08","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":false,"text":"

Solve the simultaneous equations

4x + 3y = 5

6x - 2y = 14

","answerType":"exact","answer":"x = 2, y = -1","answerDisplay":null,"accept":["x = 2 and y = -1","y = -1, x = 2","(2, -1)"],"parts":[],"solution":["Multiply the first equation by 2 and the second by 3: 8x + 6y = 10 and 18x - 6y = 42","Add: 26x = 52, so x = 2","Substitute into 4x + 3y = 5: 8 + 3y = 5, so 3y = -3 and y = -1","Check in the second equation: 12 + 2 = 14 ✓"],"diagram":null,"draw":false,"revision":"f80afd87a420"},{"id":"g5-simultaneous-equations-q09","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":9,"marks":4,"tier":"H","calc":false,"text":"

Solve the simultaneous equations

6x + 5y = 10

4x - 3y = 13

","answerType":"exact","answer":"x = 2.5, y = -1","answerDisplay":null,"accept":["x = 2.5 and y = -1","x = 5/2, y = -1","y = -1, x = 2.5","(2.5, -1)"],"parts":[],"solution":["Multiply the first equation by 3 and the second by 5: 18x + 15y = 30 and 20x - 15y = 65","Add: 38x = 95, so x = 9538 = 2.5","Substitute into 6x + 5y = 10: 15 + 5y = 10, so 5y = -5 and y = -1","Check in the second equation: 10 + 3 = 13 ✓"],"diagram":null,"draw":false,"revision":"779976a3783e"},{"id":"g5-simultaneous-equations-q10","topicId":"g5-simultaneous-equations","topic":"Simultaneous Equations","grade":"5","topicGrade":"5","strand":"A","difficulty":10,"marks":5,"tier":"F","calc":true,"text":"

A cafe sells teas and coffees.

3 teas and 2 coffees cost £7.80

2 teas and 3 coffees cost £8.20

Work out the cost of one tea and the cost of one coffee.

","answerType":"multi","answer":"Tea = £1.40, coffee = £1.80","answerDisplay":null,"accept":["tea 1.40, coffee 1.80","t = 1.40, c = 1.80"],"parts":[{"label":"tea","answer":"£1.40"},{"label":"coffee","answer":"£1.80"}],"solution":["Let t be the cost of a tea and c the cost of a coffee, in pounds: 3t + 2c = 7.80 and 2t + 3c = 8.20","Multiply the first equation by 3 and the second by 2: 9t + 6c = 23.40 and 4t + 6c = 16.40","Subtract: 5t = 7.00, so t = 1.40","Substitute into 3t + 2c = 7.80: 4.20 + 2c = 7.80, so 2c = 3.60 and c = 1.80","Check in the second equation: 2 × 1.40 + 3 × 1.80 = 2.80 + 5.40 = £8.20 ✓"],"diagram":null,"draw":false,"revision":"29cad46de7b8"},{"id":"g5-sohcahtoa-trigonometry-q01","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

In triangle ABC, angle ABC = 90° and angle BAC = 34°.

AB = 12 cm.

Work out the length of BC.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"8.09 cm","answerDisplay":null,"accept":["8.09"],"parts":[],"solution":["Use degree mode. The hypotenuse is opposite the right angle; relative to the 34° angle, the opposite side is across from it and the adjacent side touches it but is not the hypotenuse.","BC is opposite the 34° angle and AB = 12 cm is adjacent to it, so use tan.","tan 34° = BC ÷ 12.","BC = 12 × tan 34° = 12 × 0.6745... = 8.094... = 8.09 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6,4.04705110105456],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"12 cm"},{"from":[6,0],"to":[6,4.04705110105456]},{"from":[6,4.04705110105456],"to":[0,0]}],"angles":[{"at":[6,0],"from":[6,4.04705110105456],"to":[0,0],"label":"","right":true},{"at":[0,0],"from":[6,0],"to":[6,4.04705110105456],"label":"34°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"efbf9b64975c"},{"id":"g5-sohcahtoa-trigonometry-q02","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":2,"marks":2,"tier":"FH","calc":true,"text":"

In triangle PQR, angle PQR = 90° and angle PRQ = 52°.

PQ = 9 cm.

Work out the length of PR.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"11.4 cm","answerDisplay":null,"accept":["11.4"],"parts":[],"solution":["PQ = 9 cm is opposite the 52° angle and PR is the hypotenuse, so use sin.","sin 52° = 9 ÷ PR.","PR = 9 ÷ sin 52° = 9 ÷ 0.7880... = 11.42... = 11.4 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"Q","style":"none","labelPos":"below-right"},{"at":[6,4.687713759040304],"label":"R","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"9 cm"},{"from":[6,0],"to":[6,4.687713759040304]},{"from":[6,4.687713759040304],"to":[0,0]}],"angles":[{"at":[6,0],"from":[6,4.687713759040304],"to":[0,0],"label":"","right":true},{"at":[6,4.687713759040304],"from":[0,0],"to":[6,0],"label":"52°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"5c9f1273387e"},{"id":"g5-sohcahtoa-trigonometry-q03","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

In triangle DEF, angle DEF = 90°.

DE = 10 cm and EF = 7 cm.

Work out the size of angle FDE.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"35.0°","answerDisplay":null,"accept":["35.0","35.0 degrees"],"parts":[],"solution":["EF = 7 cm is opposite angle FDE and DE = 10 cm is adjacent to it, so use tan.","tan FDE = 7 ÷ 10 = 0.7.","Angle FDE = tan-1(0.7) = 34.99... = 35.0° (1 dp).","tan⁻¹ is the inverse tangent button: it finds the angle from its tangent ratio. It does not mean 1/tan. Use degree mode."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"D","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"E","style":"none","labelPos":"below-right"},{"at":[6,4.2],"label":"F","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"10 cm"},{"from":[6,0],"to":[6,4.2],"label":"7 cm"},{"from":[6,4.2],"to":[0,0]}],"angles":[{"at":[6,0],"from":[6,4.2],"to":[0,0],"label":"","right":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ab58f8721546"},{"id":"g5-sohcahtoa-trigonometry-q04","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":4,"marks":2,"tier":"FH","calc":true,"text":"

In triangle JKL, angle JKL = 90°.

JK = 5 cm and JL = 13 cm.

Work out the size of angle KJL.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"67.4°","answerDisplay":null,"accept":["67.4","67.4 degrees"],"parts":[],"solution":["JK = 5 cm is adjacent to angle KJL and JL = 13 cm is the hypotenuse, so use cos.","cos KJL = 5 ÷ 13 = 0.3846...","Angle KJL = cos-1(513) = 67.38... = 67.4° (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"J","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"K","style":"none","labelPos":"below-right"},{"at":[6,14.399999999999997],"label":"L","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"5 cm"},{"from":[6,0],"to":[6,14.399999999999997]},{"from":[6,14.399999999999997],"to":[0,0],"label":"13 cm"}],"angles":[{"at":[6,0],"from":[6,14.399999999999997],"to":[0,0],"label":"","right":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"5ccdcc2549fe"},{"id":"g5-sohcahtoa-trigonometry-q05","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

In triangle RST, angle RST = 90° and angle SRT = 41°.

RT = 20 cm.

Calculate the length of RS.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"15.1 cm","answerDisplay":null,"accept":["15.1"],"parts":[],"solution":["RS is adjacent to the 41° angle and RT = 20 cm is the hypotenuse, so use cos.","cos 41° = RS ÷ 20.","RS = 20 × cos 41° = 20 × 0.7547... = 15.09... = 15.1 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"R","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"S","style":"none","labelPos":"below-right"},{"at":[6,5.2157204268973585],"label":"T","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0]},{"from":[6,0],"to":[6,5.2157204268973585]},{"from":[6,5.2157204268973585],"to":[0,0],"label":"20 cm"}],"angles":[{"at":[6,0],"from":[6,5.2157204268973585],"to":[0,0],"label":"","right":true},{"at":[0,0],"from":[6,0],"to":[6,5.2157204268973585],"label":"41°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"fa54cc14f3d4"},{"id":"g5-sohcahtoa-trigonometry-q06","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":6,"marks":2,"tier":"F","calc":true,"text":"

A ladder of length 6 m leans against a vertical wall.

The ladder makes an angle of 68° with the horizontal ground.

Work out how far up the wall the ladder reaches.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"5.56 m","answerDisplay":null,"accept":["5.56"],"parts":[],"solution":["The height up the wall is opposite the 68° angle and the ladder is the hypotenuse, so use sin.","sin 68° = height ÷ 6.","Height = 6 × sin 68° = 6 × 0.9271... = 5.563... = 5.56 m (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[4,7],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[4,0]},{"from":[4,0],"to":[4,7]},{"from":[0,0],"to":[4,7],"label":"6 m"}],"angles":[{"at":[0,0],"from":[4,0],"to":[4,7],"label":"68°"},{"at":[4,0],"from":[0,0],"to":[4,7],"label":"","right":true}],"texts":[{"at":[5,4],"text":"Wall"},{"at":[2,-1],"text":"Ground"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a99ad970be52"},{"id":"g5-sohcahtoa-trigonometry-q07","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":7,"marks":3,"tier":"FH","calc":true,"text":"

In triangle ABC, angle ABC = 90°.

AB = 15 cm and BC = 8 cm.

  1. Work out the size of angle BAC. Give your answer correct to 1 decimal place.
  2. Write down the size of angle BCA, correct to 1 decimal place.
","answerType":"multi","answer":"a) 28.1° b) 61.9°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"28.1°"},{"label":"b","answer":"61.9°"}],"solution":["a) BC = 8 cm is opposite angle BAC and AB = 15 cm is adjacent, so tan BAC = 815.","Angle BAC = tan-1(815) = 28.07... = 28.1° (1 dp).","b) The acute angles total 90°. Use the unrounded angle: BCA = 90° − tan⁻¹(8/15) = 61.927...°, giving 61.9° to 1 decimal place."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6,3.1999999999999993],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"15 cm"},{"from":[6,0],"to":[6,3.1999999999999993],"label":"8 cm"},{"from":[6,3.1999999999999993],"to":[0,0]}],"angles":[{"at":[6,0],"from":[6,3.1999999999999993],"to":[0,0],"label":"","right":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ab62d3418ddb"},{"id":"g5-sohcahtoa-trigonometry-q08","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

tan 40° = (x + 3) ÷ 8

Work out the value of x.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"x = 3.71","answerDisplay":null,"accept":["3.71"],"parts":[],"solution":["Multiply both sides by 8: x + 3 = 8 × tan 40°.","8 × tan 40° = 8 × 0.8390... = 6.712...","x = 6.712... − 3 = 3.712... = 3.71 (3 sf)."],"diagram":null,"draw":false,"revision":"62749e7fbdb1"},{"id":"g5-sohcahtoa-trigonometry-q09","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":9,"marks":4,"tier":"FH","calc":true,"text":"

In triangle ABC, angle ABC = 90° and angle BAC = 27°. AB = 11 cm.

  1. Work out the length of BC. Give your answer correct to 3 significant figures.

In triangle PQR, angle PQR = 90°. PQ = 9 cm and PR = 14 cm.

  1. Work out the size of angle PRQ. Give your answer correct to 1 decimal place.
","answerType":"multi","answer":"a) 5.60 cm b) 40.0°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5.60 cm"},{"label":"b","answer":"40.0°"}],"solution":["a) BC is opposite the 27° angle and AB is adjacent, so BC = 11 × tan 27° = 11 × 0.5095... = 5.604... = 5.60 cm (3 sf).","b) PQ = 9 cm is opposite angle PRQ and PR = 14 cm is the hypotenuse, so sin PRQ = 914 = 0.6428...","Angle PRQ = sin-1(914) = 40.00... = 40.0° (1 dp)."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"(a)","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6,3.0571526969665728],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"11 cm"},{"from":[6,0],"to":[6,3.0571526969665728]},{"from":[6,3.0571526969665728],"to":[0,0]}],"angles":[{"at":[6,0],"from":[6,3.0571526969665728],"to":[0,0],"label":"","right":true},{"at":[0,0],"from":[6,0],"to":[6,3.0571526969665728],"label":"27°","right":false}]}},{"label":"(b)","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"Q","style":"none","labelPos":"below-right"},{"at":[6,7.149203529842399],"label":"R","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"9 cm"},{"from":[6,0],"to":[6,7.149203529842399]},{"from":[6,7.149203529842399],"to":[0,0],"label":"14 cm"}],"angles":[{"at":[6,0],"from":[6,7.149203529842399],"to":[0,0],"label":"","right":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"201a382e3a6b"},{"id":"g5-sohcahtoa-trigonometry-q10","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":10,"marks":3,"tier":"H","calc":true,"text":"

In triangle ABC, D is the point on AC such that BD is perpendicular to AC.

Angle BAD = 40°, AD = 7 cm and angle BCD = 55°.

Work out the length of BC.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"7.17 cm","answerDisplay":null,"accept":["7.17"],"parts":[],"solution":["In right-angled triangle ABD, BD is opposite the 40° angle and AD = 7 cm is adjacent, so BD = 7 × tan 40° = 5.873... cm.","In right-angled triangle BCD, BD is opposite the 55° angle and BC is the hypotenuse, so sin 55° = BD ÷ BC.","BC = 5.873... ÷ sin 55° = 5.873... ÷ 0.8191... = 7.170... = 7.17 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[5,0],"label":"D","style":"none","labelPos":"below-right"},{"at":[5,4.2],"label":"B","style":"none","labelPos":"above"},{"at":[8,0],"label":"C","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[5,0],"label":"7 cm"},{"from":[5,0],"to":[8,0]},{"from":[0,0],"to":[5,4.2]},{"from":[5,4.2],"to":[8,0]},{"from":[5,4.2],"to":[5,0]}],"angles":[{"at":[0,0],"from":[5,4.2],"to":[5,0],"label":"40°"},{"at":[8,0],"from":[5,4.2],"to":[5,0],"label":"55°"},{"at":[5,0],"from":[5,4.2],"to":[0,0],"label":"","right":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"4822bba54700"},{"id":"g5-sohcahtoa-trigonometry-q11","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":11,"marks":4,"tier":"H","calc":true,"text":"

In triangle ABC, angle ABC = 90°.

The two acute angles of the triangle are in the ratio 2 : 3.

AC = 15 cm.

Work out the length of the side opposite the larger acute angle.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"12.1 cm","answerDisplay":null,"accept":["12.1"],"parts":[],"solution":["The two acute angles sum to 90°. Dividing 90° in the ratio 2 : 3 gives 90 ÷ 5 = 18, so the angles are 36° and 54°.","The side opposite the 54° angle satisfies sin 54° = side ÷ 15, since AC = 15 cm is the hypotenuse.","Side = 15 × sin 54° = 15 × 0.8090... = 12.13... = 12.1 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6,10.39230484541326],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0]},{"from":[6,0],"to":[6,10.39230484541326]},{"from":[6,10.39230484541326],"to":[0,0],"label":"15 cm"}],"angles":[{"at":[6,0],"from":[6,10.39230484541326],"to":[0,0],"label":"","right":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"97df4df16f98"},{"id":"g5-sohcahtoa-trigonometry-q12","topicId":"g5-sohcahtoa-trigonometry","topic":"SOHCAHTOA (Trigonometry)","grade":"5","topicGrade":"5","strand":"G","difficulty":12,"marks":5,"tier":"H","calc":true,"text":"

The acute angle θ appears in two right-angled triangles.

In the first triangle, the side opposite θ is x cm and the side adjacent to θ is 2 cm.

In the second triangle, the side opposite θ is (x + 5) cm and the side adjacent to θ is (x − 1) cm.

Work out the value of x.

","answerType":"exact","answer":"x = 5","answerDisplay":null,"accept":["5"],"parts":[],"solution":["Both triangles give tan θ, so x ÷ 2 = (x + 5) ÷ (x − 1).","Cross multiply: x(x − 1) = 2(x + 5).","x² − x = 2x + 10, so x² − 3x − 10 = 0.","Factorise: (x − 5)(x + 2) = 0, so x = 5 or x = −2.","Lengths must be positive (and x − 1 > 0), so x = 5.","Check: 52 = 2.5 and 104 = 2.5, so both triangles give the same tan θ."],"diagram":null,"draw":false,"revision":"57af3016abde"},{"id":"g5-solving-quadratics-q01","topicId":"g5-solving-quadratics","topic":"Solving Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

Solve  (x + 3)(x - 9) = 0

","answerType":"exact","answer":"x = -3 or x = 9","answerDisplay":null,"accept":["x = 9 or x = -3","x = -3, x = 9","-3 and 9"],"parts":[],"solution":["If a product of two brackets is 0, one of the brackets must equal 0","x + 3 = 0 gives x = -3","x - 9 = 0 gives x = 9"],"diagram":null,"draw":false,"revision":"617c839fadc3"},{"id":"g5-solving-quadratics-q02","topicId":"g5-solving-quadratics","topic":"Solving Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

Solve  x2 + 8x + 12 = 0

","answerType":"exact","answer":"x = -2 or x = -6","answerDisplay":null,"accept":["x = -6 or x = -2","x = -2, x = -6","-2 and -6"],"parts":[],"solution":["Find two numbers that multiply to 12 and add to 8: they are 2 and 6","Factorise: (x + 2)(x + 6) = 0","So x = -2 or x = -6"],"diagram":null,"draw":false,"revision":"56cf97e3de38"},{"id":"g5-solving-quadratics-q03","topicId":"g5-solving-quadratics","topic":"Solving Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":3,"tier":"FH","calc":false,"text":"

Solve  x2 - 5x - 24 = 0

","answerType":"exact","answer":"x = 8 or x = -3","answerDisplay":null,"accept":["x = -3 or x = 8","x = 8, x = -3","8 and -3"],"parts":[],"solution":["Find two numbers that multiply to -24 and add to -5: they are -8 and 3","Factorise: (x - 8)(x + 3) = 0","So x = 8 or x = -3"],"diagram":null,"draw":false,"revision":"6e428b9cee0a"},{"id":"g5-solving-quadratics-q04","topicId":"g5-solving-quadratics","topic":"Solving Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":3,"tier":"FH","calc":true,"text":"

Solve  x2 - 11x + 28 = 0

","answerType":"exact","answer":"x = 4 or x = 7","answerDisplay":null,"accept":["x = 7 or x = 4","x = 4, x = 7","4 and 7"],"parts":[],"solution":["Find two numbers that multiply to 28 and add to -11: they are -4 and -7","Factorise: (x - 4)(x - 7) = 0","So x = 4 or x = 7"],"diagram":null,"draw":false,"revision":"3bee397191be"},{"id":"g5-solving-quadratics-q05","topicId":"g5-solving-quadratics","topic":"Solving Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":false,"text":"

Solve  x2 = 7x

","answerType":"exact","answer":"x = 0 or x = 7","answerDisplay":null,"accept":["x = 7 or x = 0","x = 0, x = 7","0 and 7"],"parts":[],"solution":["Rearrange to make one side zero: x2 - 7x = 0","Factorise: x(x - 7) = 0","So x = 0 or x = 7 (do not divide both sides by x, or the solution x = 0 is lost)"],"diagram":null,"draw":false,"revision":"6bb46b6e99e5"},{"id":"g5-solving-quadratics-q06","topicId":"g5-solving-quadratics","topic":"Solving Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":false,"text":"

Solve  x2 - 2x = 35

","answerType":"exact","answer":"x = 7 or x = -5","answerDisplay":null,"accept":["x = -5 or x = 7","x = 7, x = -5","7 and -5"],"parts":[],"solution":["Rearrange to make one side zero: x2 - 2x - 35 = 0","Find two numbers that multiply to -35 and add to -2: they are -7 and 5","Factorise: (x - 7)(x + 5) = 0, so x = 7 or x = -5"],"diagram":null,"draw":false,"revision":"b5b1f53563eb"},{"id":"g5-solving-quadratics-q07","topicId":"g5-solving-quadratics","topic":"Solving Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

The width of a rectangle is w cm.

The length of the rectangle is 3 cm more than its width.

The area of the rectangle is 40 cm2.

Find the value of w.

","answerType":"exact","answer":"w = 5","answerDisplay":null,"accept":["5","w = 5 cm","5 cm"],"parts":[],"solution":["The length is (w + 3) cm, so the area gives w(w + 3) = 40","Expand and rearrange: w2 + 3w - 40 = 0","Factorise: (w + 8)(w - 5) = 0, so w = -8 or w = 5","A width cannot be negative, so w = 5","Check: 5 × 8 = 40 ✓"],"diagram":null,"draw":false,"revision":"dd7c6722401c"},{"id":"g5-solving-quadratics-q08","topicId":"g5-solving-quadratics","topic":"Solving Quadratics","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":4,"tier":"FH","calc":true,"text":"

Two positive whole numbers differ by 5.

The product of the two numbers is 84.

By forming and solving a quadratic equation, find the two numbers.

","answerType":"exact","answer":"7 and 12","answerDisplay":null,"accept":["12 and 7","n = 7, so the numbers are 7 and 12"],"parts":[],"solution":["Let the smaller number be n, so the larger number is n + 5","Then n(n + 5) = 84, so n2 + 5n - 84 = 0","Find two numbers that multiply to -84 and add to 5: they are 12 and -7","Factorise: (n + 12)(n - 7) = 0, so n = -12 or n = 7","The numbers are positive, so n = 7 and the numbers are 7 and 12","Check: 12 - 7 = 5 and 7 × 12 = 84 ✓"],"diagram":null,"draw":false,"revision":"098fcaf4c792"},{"id":"g5-solving-simultaneous-equations-graphically-q01","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":1,"marks":1,"tier":"F","calc":true,"text":"

The graphs of x + y = 6 and y = 2x cross at the point (2, 4).

Write down the solution of the simultaneous equations

x + y = 6

y = 2x

","answerType":"exact","answer":"x = 2, y = 4","answerDisplay":null,"accept":["x = 2 and y = 4","(2, 4)","y = 4, x = 2"],"parts":[],"solution":["The solution of a pair of simultaneous equations is the point where their graphs cross","The graphs cross at (2, 4), so x = 2 and y = 4"],"diagram":null,"draw":false,"revision":"f8c67defba2b"},{"id":"g5-solving-simultaneous-equations-graphically-q02","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":2,"marks":1,"tier":"FH","calc":false,"text":"

The graphs of x + y = 10 and y = x + 2 cross at the point (4, 6).

Write down the solution of the simultaneous equations

x + y = 10

y = x + 2

","answerType":"exact","answer":"x = 4, y = 6","answerDisplay":null,"accept":["x = 4 and y = 6","(4, 6)","y = 6, x = 4"],"parts":[],"solution":["The crossing point of the two graphs gives the solution","So x = 4 and y = 6"],"diagram":null,"draw":false,"revision":"91b6c087ec0f"},{"id":"g5-solving-simultaneous-equations-graphically-q03","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":3,"marks":1,"tier":"H","calc":false,"text":"

The diagram shows the graphs of y = x + 1 and y = 5 - x

Use the diagram to solve the simultaneous equations

y = x + 1

y = 5 - x

","answerType":"exact","answer":"x = 2, y = 3","answerDisplay":null,"accept":["x = 2 and y = 3","(2, 3)","y = 3, x = 2"],"parts":[],"solution":["Read the coordinates of the point where the two lines cross: (2, 3)","So x = 2 and y = 3","Check: 2 + 1 = 3 and 5 - 2 = 3 ✓"],"diagram":{"type":"axes","x":{"min":0,"max":6,"step":1},"y":{"min":0,"max":6,"step":1},"lines":[{"m":1,"c":1,"label":"y = x + 1","labelAt":[3,4.5]},{"m":-1,"c":5,"label":"y = 5 − x","labelAt":[4,1.5]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"5df4b91443a5"},{"id":"g5-solving-simultaneous-equations-graphically-q04","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":4,"marks":2,"tier":"FH","calc":false,"text":"
  1. Show that the point (4, 3) lies on the line 2x + y = 11 and also on the line y = x - 1
  2. Write down the solution of the simultaneous equations 2x + y = 11 and y = x - 1
","answerType":"multi","answer":"a) 2(4) + 3 = 11 and 3 = 4 - 1, so (4, 3) is on both lines; b) x = 4, y = 3","answerDisplay":null,"accept":["b) (4, 3)","b) x = 4 and y = 3"],"parts":[{"label":"a","answer":"2 × 4 + 3 = 11 and 4 - 1 = 3, so the point lies on both lines"},{"label":"b","answer":"x = 4, y = 3"}],"solution":["a) Substitute x = 4, y = 3 into 2x + y: 8 + 3 = 11 ✓","a) Substitute x = 4 into x - 1: 4 - 1 = 3 = y ✓","b) Since (4, 3) lies on both lines, it is their crossing point, so the solution is x = 4, y = 3"],"diagram":null,"draw":false,"revision":"bc2388543e64"},{"id":"g5-solving-simultaneous-equations-graphically-q05","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":5,"marks":2,"tier":"H","calc":false,"text":"

Katie draws the graphs of y = 2x + 3 and y = 2x - 1 on the same grid.

She wants to use her graphs to solve the simultaneous equations y = 2x + 3 and y = 2x - 1

Explain why Katie will not be able to find a solution.

","answerType":"open","answer":"Both lines have gradient 2, so they are parallel and never cross. There is no crossing point, so the equations have no solution.","answerDisplay":null,"accept":["The lines are parallel so they never intersect","Same gradient, different intercepts, so no point of intersection"],"parts":[],"solution":["Both equations have the same gradient, 2, but different y-intercepts","The lines are parallel, so they never cross","A graphical solution is a crossing point, so there is no solution"],"diagram":null,"draw":false,"revision":"de4256051916"},{"id":"g5-solving-simultaneous-equations-graphically-q06","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":6,"marks":3,"tier":"F","calc":true,"text":"

The graph of y = x + 2 is drawn on the grid.

  1. On the same grid, draw the graph of y = 8 - x for values of x from 0 to 6
  2. Use your graphs to solve the simultaneous equations y = x + 2 and y = 8 - x
","answerType":"multi","answer":"a) straight line through (0, 8) and (6, 2); b) x = 3, y = 5","answerDisplay":null,"accept":["b) (3, 5)","b) x = 3 and y = 5"],"parts":[{"label":"a","answer":"correct straight line through (0, 8), (4, 4) and (6, 2)"},{"label":"b","answer":"x = 3, y = 5"}],"solution":["a) Plot points for y = 8 - x, for example (0, 8), (4, 4), (6, 2), and join with a straight line","b) The lines cross at (3, 5), so x = 3 and y = 5","Check: 3 + 2 = 5 and 8 - 3 = 5 ✓"],"diagram":{"type":"axes","x":{"min":0,"max":6,"step":1},"y":{"min":0,"max":9,"step":1},"lines":[{"m":1,"c":2,"label":"y = x + 2","labelAt":[4,6.5]},{"m":-1,"c":8,"answer":true,"from":0,"to":6}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"a449db01904c"},{"id":"g5-solving-simultaneous-equations-graphically-q07","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":7,"marks":3,"tier":"FH","calc":false,"text":"
  1. Complete the table of values for y = 2x - 3
    x012345
    y-315
  2. The line y = x + 2 passes through the points (0, 2), (3, 5) and (5, 7).
    Use the tables of values to find the solution of the simultaneous equations y = 2x - 3 and y = x + 2
","answerType":"multi","answer":"a) y = -1 (x = 1), y = 3 (x = 3), y = 7 (x = 5); b) x = 5, y = 7","answerDisplay":null,"accept":["a) -1, 3, 7 b) (5, 7)"],"parts":[{"label":"a (x = 1)","answer":"-1"},{"label":"a (x = 3)","answer":"3"},{"label":"a (x = 5)","answer":"7"},{"label":"b","answer":"x = 5, y = 7"}],"solution":["a) When x = 1: y = 2 - 3 = -1; when x = 3: y = 6 - 3 = 3; when x = 5: y = 10 - 3 = 7","b) Both lines pass through (5, 7): y = 2 × 5 - 3 = 7 and y = 5 + 2 = 7","b) So the solution is x = 5, y = 7"],"diagram":null,"draw":false,"revision":"791665df3a38"},{"id":"g5-solving-simultaneous-equations-graphically-q08","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

The diagram shows the graphs of y = 3x - 4 and y = 6 - 2x

Use the diagram to estimate the solution of the simultaneous equations

y = 3x - 4

y = 6 - 2x

","answerType":"exact","answer":"x = 2, y = 2","answerDisplay":null,"accept":["x = 2 and y = 2","(2, 2)","y = 2, x = 2"],"parts":[],"solution":["Read the coordinates of the crossing point of the two lines: (2, 2)","So x = 2 and y = 2","Check: 3 × 2 - 4 = 2 and 6 - 2 × 2 = 2 ✓"],"diagram":{"type":"axes","x":{"min":-1,"max":5,"step":1},"y":{"min":-5,"max":7,"step":1},"lines":[{"m":3,"c":-4,"label":"y = 3x − 4","labelAt":[3.5,5.5]},{"m":-2,"c":6,"label":"y = 6 − 2x","labelAt":[0.5,6]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"dfb961b79d22"},{"id":"g5-solving-simultaneous-equations-graphically-q09","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":9,"marks":3,"tier":"H","calc":false,"text":"

Ben says the graphs of x + 2y = 9 and y = x - 3 cross at the point (5, 2).

Without drawing the graphs, decide whether Ben is correct.

You must show your working.

","answerType":"open","answer":"Yes. 5 + 2(2) = 9 and 2 = 5 - 3, so (5, 2) lies on both lines and is their crossing point.","answerDisplay":null,"accept":["Yes, because the point satisfies both equations"],"parts":[],"solution":["Substitute x = 5, y = 2 into x + 2y: 5 + 4 = 9 ✓","Substitute x = 5 into x - 3: 5 - 3 = 2 = y ✓","The point lies on both lines, so Ben is correct: the graphs cross at (5, 2)"],"diagram":null,"draw":false,"revision":"3a0d836d3af1"},{"id":"g5-solving-simultaneous-equations-graphically-q10","topicId":"g5-solving-simultaneous-equations-graphically","topic":"Solving Simultaneous Equations Graphically","grade":"5","topicGrade":"5","strand":"A","difficulty":10,"marks":4,"tier":"H","calc":false,"text":"

The graph of y = x2 - 3x is drawn on the grid.

By drawing a suitable straight line on the grid, solve the equation

x2 - 3x = x - 3

","answerType":"exact","answer":"x = 1 or x = 3","answerDisplay":null,"accept":["x = 3 or x = 1","x = 1, x = 3","1 and 3"],"parts":[],"solution":["The equation x2 - 3x = x - 3 is solved where the curve y = x2 - 3x meets the line y = x - 3","Draw the straight line y = x - 3 on the grid","The line meets the curve where x = 1 and x = 3","Check algebraically: x2 - 4x + 3 = 0 factorises as (x - 1)(x - 3) = 0 ✓"],"diagram":{"type":"axes","x":{"min":-1,"max":4,"step":1},"y":{"min":-3,"max":5,"step":1},"curves":[{"fn":"x*x-3*x"}],"lines":[{"m":1,"c":-3,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"1d05338e4594"},{"id":"g5-speed-density-and-pressure-q01","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

A cyclist rides at an average speed of 24 km/h for 2.5 hours.

Work out the distance the cyclist rides.

","answerType":"exact","answer":"60 km","answerDisplay":null,"accept":["60","60km","60 kilometres"],"parts":[],"solution":["Use distance = speed × time.","Distance = 24 × 2.5 = 60 km."],"diagram":null,"draw":false,"revision":"dc3e525ed0a0"},{"id":"g5-speed-density-and-pressure-q02","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":2,"marks":2,"tier":"FH","calc":true,"text":"

A block of pine wood has a volume of 240 cm3.

The density of the pine wood is 0.65 g/cm3.

Work out the mass of the block.

","answerType":"exact","answer":"156 g","answerDisplay":null,"accept":["156","156g","0.156 kg"],"parts":[],"solution":["Use mass = density × volume.","Mass = 0.65 × 240 = 156 g."],"diagram":null,"draw":false,"revision":"e81e1f76fa5a"},{"id":"g5-speed-density-and-pressure-q03","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":3,"marks":3,"tier":"FH","calc":false,"text":"

A crate rests on horizontal ground on a square base of side 0.5 m.

The crate exerts a force of 180 N on the ground.

Work out the pressure on the ground.

Give your answer in N/m2.

","answerType":"exact","answer":"720 N/m^2","answerDisplay":null,"accept":["720","720 N/m2","720 N/m^2","720 N/m²","720 newtons per square metre"],"parts":[],"solution":["Area of the base = 0.5 × 0.5 = 0.25 m2.","Use pressure = force ÷ area.","Pressure = 180 ÷ 0.25 = 720 N/m2."],"diagram":null,"draw":false,"revision":"b4f01a7433d3"},{"id":"g5-speed-density-and-pressure-q04","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":4,"marks":3,"tier":"F","calc":true,"text":"

Amber runs 15.6 km in 1 hour 30 minutes.

Work out her average speed in km/h.

","answerType":"exact","answer":"10.4 km/h","answerDisplay":null,"accept":["10.4","10.4km/h","10.4 kmh"],"parts":[],"solution":["Convert the time to hours: 1 hour 30 minutes = 1.5 hours.","Use speed = distance ÷ time.","Speed = 15.6 ÷ 1.5 = 10.4 km/h."],"diagram":null,"draw":false,"revision":"b48f96d516b0"},{"id":"g5-speed-density-and-pressure-q05","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":5,"marks":3,"tier":"FH","calc":true,"text":"

A solid metal cylinder has radius 4 cm and height 10 cm.

The mass of the cylinder is 3768 g.

Work out the density of the metal in g/cm3.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"7.50 g/cm^3","answerDisplay":null,"accept":["7.50","7.50 g/cm3","7.50 g/cm³"],"parts":[],"solution":["Volume of the cylinder = π × 42 × 10 = 160π = 502.65... cm3.","Use density = mass ÷ volume.","Density = 3768 ÷ 502.65... = 7.496...","Density = 7.50 g/cm3 (3 s.f.)."],"diagram":null,"draw":false,"revision":"7248714752ad"},{"id":"g5-speed-density-and-pressure-q06","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":6,"marks":3,"tier":"FH","calc":true,"text":"

An empty water tank holds 8640 litres when full.

Water flows into the tank at a constant rate of 4 litres per minute.

Work out how long it takes to fill the tank.

Give your answer in days.

","answerType":"exact","answer":"1.5 days","answerDisplay":null,"accept":["1.5","1.5 days","1 1/2 days","36 hours = 1.5 days"],"parts":[],"solution":["Time = 8640 ÷ 4 = 2160 minutes.","2160 ÷ 60 = 36 hours.","36 ÷ 24 = 1.5 days."],"diagram":null,"draw":false,"revision":"8c6c2e1ec1aa"},{"id":"g5-speed-density-and-pressure-q07","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":7,"marks":3,"tier":"FH","calc":true,"text":"

A paperweight is made of brass with a volume of 60 cm3.

The density of brass is 8.4 g/cm3.

A delivery box can hold a total mass of at most 4.5 kg.

Work out the greatest number of these paperweights the box can hold.

","answerType":"exact","answer":"8","answerDisplay":null,"accept":["8 paperweights"],"parts":[],"solution":["Mass of one paperweight = 8.4 × 60 = 504 g.","Convert the limit: 4.5 kg = 4500 g.","4500 ÷ 504 = 8.92..., so the greatest number is 8."],"diagram":null,"draw":false,"revision":"002521ab34d2"},{"id":"g5-speed-density-and-pressure-q08","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":8,"marks":3,"tier":"H","calc":false,"text":"

Cube A has sides of length 2 cm and a mass of 40 g.

Cube B has sides of length 4 cm and a mass of 480 g.

Find the ratio of the density of cube A to the density of cube B.

Give your answer in its simplest form.

","answerType":"exact","answer":"2 : 3","answerDisplay":null,"accept":["2:3","2 to 3"],"parts":[],"solution":["Volume of cube A = 23 = 8 cm3, so density of A = 40 ÷ 8 = 5 g/cm3.","Volume of cube B = 43 = 64 cm3, so density of B = 480 ÷ 64 = 7.5 g/cm3.","Ratio = 5 : 7.5 = 2 : 3."],"diagram":null,"draw":false,"revision":"b6f14babc3b2"},{"id":"g5-speed-density-and-pressure-q09","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":9,"marks":3,"tier":"H","calc":false,"text":"

A metal box rests on horizontal ground on a rectangular base.

The box exerts a force of 480 N on the ground.

The pressure on the ground is 60 N/m2.

The width of the base is 2.5 m.

Work out the length of the base.

","answerType":"exact","answer":"3.2 m","answerDisplay":null,"accept":["3.2","3.2m","3.2 metres"],"parts":[],"solution":["Rearrange pressure = force ÷ area to give area = force ÷ pressure.","Area = 480 ÷ 60 = 8 m2.","Length = 8 ÷ 2.5 = 3.2 m."],"diagram":null,"draw":false,"revision":"830ca7bfa6bd"},{"id":"g5-speed-density-and-pressure-q10","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":10,"marks":4,"tier":"FH","calc":true,"text":"

Priya drives 90 km on a motorway at an average speed of 60 km/h.

She then drives 50 km on country roads at an average speed of 40 km/h.

Work out her average speed for the whole journey.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"50.9 km/h","answerDisplay":null,"accept":["50.9","50.9km/h"],"parts":[],"solution":["Time for the motorway: 90 ÷ 60 = 1.5 hours.","Time for the country roads: 50 ÷ 40 = 1.25 hours.","Total distance = 90 + 50 = 140 km and total time = 1.5 + 1.25 = 2.75 hours.","Average speed = 140 ÷ 2.75 = 50.90... = 50.9 km/h (1 d.p.)."],"diagram":null,"draw":false,"revision":"aeb17fb1e8b8"},{"id":"g5-speed-density-and-pressure-q11","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":11,"marks":4,"tier":"F","calc":true,"text":"

A coach leaves Leeds at 09:40 and arrives in Newcastle at 12:10 on the same day.

The distance from Leeds to Newcastle by road is 175 km.

The coach company claims the average speed of the coach was more than 65 km/h.

Work out the average speed of the coach and state whether the claim is correct.

","answerType":"exact","answer":"70 km/h, so the claim is correct","answerDisplay":null,"accept":["70 km/h, yes","yes, 70 km/h","70 km/h, so the claim is correct"],"parts":[],"solution":["Time taken = 09:40 to 12:10 = 2 hours 30 minutes = 2.5 hours.","Average speed = 175 ÷ 2.5 = 70 km/h.","70 > 65, so the claim is correct."],"diagram":null,"draw":false,"revision":"c27323db8245"},{"id":"g5-speed-density-and-pressure-q12","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":12,"marks":4,"tier":"H","calc":true,"text":"

Liquid A has a density of 1.2 g/cm3.

Liquid B has a density of 0.9 g/cm3.

Nadia mixes 300 cm3 of liquid A with 200 cm3 of liquid B to make liquid C.

Work out the density of liquid C.

Assume the volumes add when the liquids are mixed and no liquid is lost.

","answerType":"exact","answer":"1.08 g/cm^3","answerDisplay":null,"accept":["1.08","1.08 g/cm3","1.08 g/cm³"],"parts":[],"solution":["Mass of liquid A = 1.2 × 300 = 360 g.","Mass of liquid B = 0.9 × 200 = 180 g.","Total mass = 540 g and total volume = 500 cm3.","Density of liquid C = 540 ÷ 500 = 1.08 g/cm3."],"diagram":null,"draw":false,"revision":"75561c0cc655"},{"id":"g5-speed-density-and-pressure-q13","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":13,"marks":4,"tier":"H","calc":false,"text":"

Marcus cycles the 60 km from his home to the coast at an average speed of 30 km/h.

He cycles back home along the same route at an average speed of 20 km/h.

Marcus says, "My average speed for the whole ride was 25 km/h, because 25 is halfway between 30 and 20."

Show that Marcus is wrong, and work out his actual average speed for the whole ride.

","answerType":"open","answer":"24 km/h (the two legs take different times, so the average is not 25 km/h)","answerDisplay":null,"accept":["24","24 km/h"],"parts":[],"solution":["Time out = 60 ÷ 30 = 2 hours.","Time back = 60 ÷ 20 = 3 hours.","Total distance = 120 km and total time = 5 hours.","Average speed = 120 ÷ 5 = 24 km/h, not 25 km/h, because he spends longer at the slower speed."],"diagram":null,"draw":false,"revision":"93a11db122cc"},{"id":"g5-speed-density-and-pressure-q14","topicId":"g5-speed-density-and-pressure","topic":"Speed, Density and Pressure","grade":"5","topicGrade":"5","strand":"R","difficulty":14,"marks":5,"tier":"H","calc":true,"text":"

A solid cylinder is made of steel with density 7 g/cm3.

The cylinder has radius 3 cm and height 8 cm.

A solid cube is made of titanium with density 5.2 g/cm3.

The cube has the same mass as the cylinder.

Work out the side length of the cube.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"6.73 cm","answerDisplay":null,"accept":["6.73","6.73cm"],"parts":[],"solution":["Volume of the cylinder = π × 32 × 8 = 72π = 226.19... cm3.","Mass of the cylinder = 7 × 226.19... = 1583.3... g.","Volume of the cube = 1583.3... ÷ 5.2 = 304.49... cm3.","Side length = ∛304.49... = 6.726...","Side length = 6.73 cm (3 s.f.)."],"diagram":null,"draw":false,"revision":"16ffac398f22"},{"id":"g5-spheres-and-cones-q01","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":1,"marks":2,"tier":"H","calc":true,"text":"

A solid hemisphere has radius 6 cm.

Work out the volume of the hemisphere.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"452 cm³","answerDisplay":null,"accept":["452"],"parts":[],"solution":["Volume of a sphere = 43 × π × r³, so volume of a hemisphere = 23 × π × r³.","Volume = 23 × π × 6³ = 23 × π × 216 = 144π.","144π = 452.38... = 452 cm³ (3 sf)."],"diagram":null,"draw":false,"revision":"8f3835893d06"},{"id":"g5-spheres-and-cones-q02","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":2,"marks":4,"tier":"H","calc":true,"text":"

A solid is made from a cone and a hemisphere.

The cone has base radius 5 cm and vertical height 9 cm.

The flat face of the hemisphere, of radius 5 cm, is joined to the base of the cone.

Work out the total volume of the solid.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"497 cm³","answerDisplay":null,"accept":["497"],"parts":[],"solution":["Volume of the cone = 13 × π × 5² × 9 = 13 × π × 225 = 75π cm³.","Volume of the hemisphere = 23 × π × 5³ = 250π3 cm³.","Total volume = 75π + 250π3 = 475π3 = 497.41... cm³.","Total volume = 497 cm³ (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"arcs":[{"c":[0,0],"r":3,"from":180,"to":360}],"segments":[{"from":[-3,0],"to":[3,0],"dashed":true},{"from":[0,0],"to":[3,0],"label":"5 cm"},{"from":[-3,0],"to":[0,5]},{"from":[0,5],"to":[3,0]},{"from":[0,0],"to":[0,5],"label":"9 cm","dashed":true}],"texts":[{"at":[0,-4],"text":"Central vertical cross-section"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"94270fb1197f"},{"id":"g5-spheres-and-cones-q03","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":3,"marks":4,"tier":"H","calc":false,"text":"

Three quarters of the surface area of a sphere is 108π cm².

Work out the diameter of the sphere.

","answerType":"exact","answer":"12 cm","answerDisplay":null,"accept":["12"],"parts":[],"solution":["If 34 of the surface area is 108π, the full surface area = 108π ÷ 3 × 4 = 144π cm².","Surface area of a sphere = 4πr², so 4πr² = 144π.","r² = 36, so r = 6 cm.","Diameter = 2 × 6 = 12 cm."],"diagram":null,"draw":false,"revision":"1bee3fd313e2"},{"id":"g5-spheres-and-cones-q04","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":4,"marks":4,"tier":"H","calc":false,"text":"

A solid hemisphere has volume 18π cm³.

Work out the total surface area of the hemisphere.

Give your answer in terms of π.

","answerType":"exact","answer":"27π cm²","answerDisplay":null,"accept":["27pi","27π"],"parts":[],"solution":["Volume of a hemisphere = 23 × π × r³, so 23 × π × r³ = 18π.","r³ = 18 × 32 = 27, so r = 3 cm.","Total surface area = curved surface + flat circular face = 2πr² + πr² = 3πr².","Total surface area = 3 × π × 9 = 27π cm²."],"diagram":null,"draw":false,"revision":"b288c98717a9"},{"id":"g5-spheres-and-cones-q05","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

A solid cone has base radius 6 cm.

The total surface area of the cone is 96π cm².

Work out the vertical height of the cone.

","answerType":"exact","answer":"8 cm","answerDisplay":null,"accept":["8"],"parts":[],"solution":["Total surface area of a cone = πr² + πrl, where l is the slant height.","π × 36 + π × 6 × l = 96π, so 36 + 6l = 96.","6l = 60, so l = 10 cm.","By Pythagoras, height = √(10² − 6²) = √64 = 8 cm."],"diagram":null,"draw":false,"revision":"102b73a3c3a9"},{"id":"g5-spheres-and-cones-q06","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A large cone has base radius 9 cm and vertical height 12 cm.

A smaller, similar cone of base radius 3 cm and vertical height 4 cm is removed from the top of the large cone, leaving a frustum.

Calculate the curved surface area of the frustum.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"377 cm²","answerDisplay":null,"accept":["377"],"parts":[],"solution":["Slant height of the large cone = √(12² + 9²) = √225 = 15 cm.","Slant height of the small cone = √(4² + 3²) = √25 = 5 cm.","Curved surface area of a cone = πrl.","Curved surface area of the frustum = π × 9 × 15 − π × 3 × 5 = 135π − 15π = 120π.","120π = 376.99... = 377 cm² (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"segments":[{"from":[-4,0],"to":[4,0]},{"from":[-4,0],"to":[0,6],"dashed":true},{"from":[0,6],"to":[4,0],"dashed":true},{"from":[-1.3333333333333333,4],"to":[1.3333333333333333,4]},{"from":[0,0],"to":[4,0],"label":"9 cm"},{"from":[0,4],"to":[1.3333333333333333,4],"label":"3 cm"},{"from":[-5,0],"to":[-5,6],"label":"12 cm","arrow":"both"},{"from":[2,4],"to":[2,6],"label":"4 cm","arrow":"both"}],"texts":[{"at":[0,-1],"text":"Central vertical cross-section"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"8a54100d2db1"},{"id":"g5-spheres-and-cones-q07","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":7,"marks":5,"tier":"H","calc":false,"text":"

A solid is a quarter of a sphere. It is made by cutting a sphere in half, then cutting one hemisphere in half again.

The volume of the solid is 243π cm³.

Work out the total surface area of the solid.

Give your answer in terms of π.

Both cuts pass through the centre of the original sphere, and the second cutting plane is perpendicular to the first.

","answerType":"exact","answer":"162π cm²","answerDisplay":null,"accept":["162pi","162π"],"parts":[],"solution":["Volume of a quarter sphere = 14 × 43 × π × r³ = 13 × π × r³.","13 × π × r³ = 243π, so r³ = 729 and r = 9 cm.","Curved part = 14 of the sphere's surface = 14 × 4πr² = πr² = 81π cm².","The two flat faces are semicircles of radius 9, each of area 12πr², together πr² = 81π cm².","Total surface area = 81π + 81π = 162π cm²."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":340,"items":[{"label":"Section perpendicular to both cuts","diagram":{"type":"shape","notAccurate":true,"arcs":[{"c":[0,0],"r":3,"from":0,"to":90,"sector":true,"fill":true}]}},{"label":"Flat faces, shown separately","diagram":{"type":"shape","notAccurate":true,"arcs":[{"c":[0,0],"r":3,"from":0,"to":180,"sector":true,"fill":true},{"c":[0,-4],"r":3,"from":180,"to":360,"sector":true,"fill":true}]}}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"fb89b32d6c2a"},{"id":"g5-spheres-and-cones-q08","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

A solid cone has base radius 5 cm.

The curved surface area of the cone is 65π cm².

Work out the volume of the cone.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"314 cm³","answerDisplay":null,"accept":["314"],"parts":[],"solution":["Curved surface area = πrl, so π × 5 × l = 65π and l = 13 cm.","By Pythagoras, height = √(13² − 5²) = √144 = 12 cm.","Volume = 13 × π × r² × h = 13 × π × 25 × 12 = 100π.","100π = 314.15... = 314 cm³ (3 sf)."],"diagram":null,"draw":false,"revision":"5fd84dcb814f"},{"id":"g5-spheres-and-cones-q09","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

A solid is made from a cone and a hemisphere, both of radius 4 cm. The flat face of the hemisphere is joined to the base of the cone.

The total volume of the solid is 96π cm³.

Work out the total height of the solid.

","answerType":"exact","answer":"14 cm","answerDisplay":null,"accept":["14"],"parts":[],"solution":["Let h be the vertical height of the cone.","Volume of the cone = 13 × π × 16 × h = 16πh3.","Volume of the hemisphere = 23 × π × 64 = 128π3.","16πh3 + 128π3 = 96π, so 16h + 128 = 288.","16h = 160, so h = 10 cm.","Total height = height of cone + radius of hemisphere = 10 + 4 = 14 cm."],"diagram":{"type":"shape","notAccurate":true,"arcs":[{"c":[0,0],"r":3,"from":180,"to":360}],"segments":[{"from":[-3,0],"to":[3,0],"dashed":true},{"from":[0,0],"to":[3,0],"label":"4 cm"},{"from":[-3,0],"to":[0,5]},{"from":[0,5],"to":[3,0]},{"from":[0,0],"to":[0,5],"dashed":true}],"texts":[{"at":[0,-4],"text":"Central vertical cross-section"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"99ef7ca19f98"},{"id":"g5-spheres-and-cones-q10","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

A frustum is made by removing a cone of base radius 3 cm and vertical height 6 cm from a similar cone of base radius 6 cm and vertical height 12 cm.

A solid hemisphere of radius 6 cm is attached to the larger face of the frustum to make a solid.

Calculate the total volume of the solid.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"848 cm³","answerDisplay":null,"accept":["848"],"parts":[],"solution":["Volume of the large cone = 13 × π × 36 × 12 = 144π cm³.","Volume of the removed cone = 13 × π × 9 × 6 = 18π cm³.","Volume of the frustum = 144π − 18π = 126π cm³.","Volume of the hemisphere = 23 × π × 216 = 144π cm³.","Total volume = 126π + 144π = 270π = 848.23... = 848 cm³ (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"arcs":[{"c":[0,0],"r":3,"from":180,"to":360}],"segments":[{"from":[-3,0],"to":[3,0],"dashed":true},{"from":[0,0],"to":[3,0],"label":"6 cm"},{"from":[-3,0],"to":[-1.5,3]},{"from":[-1.5,3],"to":[1.5,3]},{"from":[1.5,3],"to":[3,0]},{"from":[-1.5,3],"to":[0,6],"dashed":true},{"from":[0,6],"to":[1.5,3],"dashed":true},{"from":[0,0],"to":[0,6],"label":"12 cm","dashed":true},{"from":[1.5,3],"to":[1.5,6],"label":"6 cm","dashed":true},{"from":[0,3],"to":[1.5,3],"label":"3 cm"}],"texts":[{"at":[0,-4],"text":"Central vertical cross-section"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"7c01f7f74015"},{"id":"g5-spheres-and-cones-q11","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":11,"marks":5,"tier":"H","calc":true,"text":"

A solid cone has base radius 6 cm and volume 96π cm³.

The curved surface of the cone is made from a sector of a circle.

Work out the angle of the sector.

","answerType":"exact","answer":"216°","answerDisplay":null,"accept":["216","216 degrees"],"parts":[],"solution":["Volume = 13 × π × 36 × h = 96π, so 12h = 96 and h = 8 cm.","Slant height l = √(8² + 6²) = √100 = 10 cm.","The sector has radius l = 10 cm, and its arc length equals the circumference of the base, 2π × 6 = 12π cm.","Angle ÷ 360 = arc ÷ full circumference = 12π ÷ 20π = 35.","Angle = 35 × 360 = 216°."],"diagram":null,"draw":false,"revision":"0d2321052490"},{"id":"g5-spheres-and-cones-q12","topicId":"g5-spheres-and-cones","topic":"Spheres and Cones","grade":"5","topicGrade":"5","strand":"G","difficulty":12,"marks":6,"tier":"H","calc":false,"text":"

A solid sphere has radius r. A solid cone has base radius r and vertical height h.

The volume of the cone is equal to the volume of the sphere.

  1. Show that h = 4r.
  2. Find the ratio of the curved surface area of the cone to the surface area of the sphere. Give your answer in the form √k : m.
","answerType":"multi","answer":"a) shown (see solution) b) √17 : 4","answerDisplay":null,"accept":["root 17 : 4","sqrt(17):4"],"parts":[{"label":"a","answer":"1/3 πr²h = 4/3 πr³ gives h = 4r","answerDisplay":"13 πr²h = 43 πr³ gives h = 4r"},{"label":"b","answer":"√17 : 4"}],"solution":["a) Volume of cone = 13 × π × r² × h and volume of sphere = 43 × π × r³.","Setting them equal: 13 πr²h = 43 πr³, so h = 4r as required.","b) Slant height l = √(r² + h²) = √(r² + 16r²) = √(17r²) = r√17.","Curved surface area of the cone = πrl = √17 × πr².","Surface area of the sphere = 4πr².","Ratio = √17πr² : 4πr² = √17 : 4."],"diagram":null,"draw":false,"revision":"ea7318f8a851"},{"id":"g5-standard-form-q01","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"
  1. Write 8.14 × 104 as an ordinary number.
  2. Write 0.00092 in standard form.
","answerType":"multi","answer":"a) 81,400 b) 9.2 x 10^-4","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"81,400"},{"label":"b","answer":"9.2 x 10^-4"}],"solution":["Standard form is a × 10ⁿ, where 1 ≤ a < 10 for these positive numbers and n is an integer. A negative power means divide by a power of 10.","a) Multiplying by 10^4 moves the digits four places: 8.14 x 10^4 = 81,400.","b) 0.00092 = 9.2 / 10,000 = 9.2 x 10^-4."],"diagram":null,"draw":false,"revision":"5db875b73d5f"},{"id":"g5-standard-form-q02","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":2,"marks":2,"tier":"FH","calc":true,"text":"
  1. Write 507,000 in standard form.
  2. Write 2.6 × 10−3 as an ordinary number.
","answerType":"multi","answer":"a) 5.07 x 10^5 b) 0.0026","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"5.07 x 10^5"},{"label":"b","answer":"0.0026"}],"solution":["a) 507,000 = 5.07 x 100,000 = 5.07 x 10^5.","b) 2.6 x 10^-3 = 2.6 / 1000 = 0.0026."],"diagram":null,"draw":false,"revision":"4630c85c718a"},{"id":"g5-standard-form-q03","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":3,"marks":2,"tier":"FH","calc":false,"text":"

Work out (7 × 104) × (3 × 105)

Give your answer in standard form.

","answerType":"exact","answer":"2.1 x 10^10","answerDisplay":null,"accept":["2.1 x 10^10","2.1×10^10"],"parts":[],"solution":["Multiply the number parts: 7 x 3 = 21. Add the powers: 10^4 x 10^5 = 10^9.","This gives 21 x 10^9, which is not in standard form.","21 = 2.1 x 10, so the answer is 2.1 x 10^10."],"diagram":null,"draw":false,"revision":"c78150add139"},{"id":"g5-standard-form-q04","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":4,"marks":2,"tier":"FH","calc":false,"text":"

Work out (9 × 108) ÷ (3 × 10−2)

Give your answer in standard form.

","answerType":"exact","answer":"3 x 10^10","answerDisplay":null,"accept":["3 x 10^10","3×10^10"],"parts":[],"solution":["Divide the number parts: 9 / 3 = 3.","Subtract the powers: 10^8 / 10^-2 = 10^(8 - (-2)) = 10^10.","So the answer is 3 x 10^10."],"diagram":null,"draw":false,"revision":"92051c940336"},{"id":"g5-standard-form-q05","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

8w = 2 × 105

Find the value of w.

Give your answer in standard form.

","answerType":"exact","answer":"2.5 x 10^4","answerDisplay":null,"accept":["2.5 x 10^4","2.5×10^4"],"parts":[],"solution":["w = (2 x 10^5) / 8 = 0.25 x 10^5.","0.25 x 10^5 is not in standard form: 0.25 = 2.5 x 10^-1.","So w = 2.5 x 10^4 (= 25,000)."],"diagram":null,"draw":false,"revision":"0a6d4508a943"},{"id":"g5-standard-form-q06","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":6,"marks":2,"tier":"H","calc":true,"text":"

Write these numbers in order of size, starting with the smallest.

3.2 × 104    2.9 × 105    8.7 × 103    3.05 × 104

","answerType":"exact","answer":"8.7 x 10^3, 3.05 x 10^4, 3.2 x 10^4, 2.9 x 10^5","answerDisplay":null,"accept":["8.7 x 10^3, 3.05 x 10^4, 3.2 x 10^4, 2.9 x 10^5"],"parts":[],"solution":["As ordinary numbers: 3.2 x 10^4 = 32,000; 2.9 x 10^5 = 290,000; 8.7 x 10^3 = 8,700; 3.05 x 10^4 = 30,500.","Order smallest first: 8,700; 30,500; 32,000; 290,000.","So: 8.7 x 10^3, 3.05 x 10^4, 3.2 x 10^4, 2.9 x 10^5."],"diagram":null,"draw":false,"revision":"e31200815cf1"},{"id":"g5-standard-form-q07","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":7,"marks":3,"tier":"FH","calc":true,"text":"

Work out (4.6 × 105) + (3.1 × 104)

Give your answer in standard form.

","answerType":"exact","answer":"4.91 x 10^5","answerDisplay":null,"accept":["4.91 x 10^5","4.91×10^5"],"parts":[],"solution":["Write both as ordinary numbers: 4.6 x 10^5 = 460,000 and 3.1 x 10^4 = 31,000.","Add them: 460,000 + 31,000 = 491,000.","In standard form: 491,000 = 4.91 x 10^5."],"diagram":null,"draw":false,"revision":"3f30bc3cd658"},{"id":"g5-standard-form-q08","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":8,"marks":3,"tier":"FH","calc":true,"text":"
  1. Write 0.0698 in standard form.
  2. Which is greater, 7.3 × 10−2 or 0.0698?

    You must show your working.

","answerType":"multi","answer":"a) 6.98 x 10^-2 b) 7.3 x 10^-2, because 7.3 x 10^-2 = 0.073 and 0.073 > 0.0698","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6.98 x 10^-2"},{"label":"b","answer":"7.3 x 10^-2 (0.073 > 0.0698)"}],"solution":["a) 0.0698 = 6.98 / 100 = 6.98 x 10^-2.","b) 7.3 x 10^-2 = 0.073. Comparing 0.073 with 0.0698: 0.073 is larger.","b) So 7.3 x 10^-2 is greater."],"diagram":null,"draw":false,"revision":"13e7a5f57896"},{"id":"g5-standard-form-q09","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":9,"marks":3,"tier":"H","calc":true,"text":"

Work out (6.4 × 106) − (5.9 × 105)

Give your answer in standard form.

","answerType":"exact","answer":"5.81 x 10^6","answerDisplay":null,"accept":["5.81 x 10^6","5.81×10^6"],"parts":[],"solution":["Write both as ordinary numbers: 6.4 x 10^6 = 6,400,000 and 5.9 x 10^5 = 590,000.","Subtract: 6,400,000 - 590,000 = 5,810,000.","In standard form: 5,810,000 = 5.81 x 10^6."],"diagram":null,"draw":false,"revision":"881f76f30d5f"},{"id":"g5-standard-form-q10","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":10,"marks":4,"tier":"FH","calc":true,"text":"

The population of country A is 6.85 × 107

The population of country B is 928,000

Work out the total population of the two countries.

Give your answer in standard form.

","answerType":"exact","answer":"6.9428 x 10^7","answerDisplay":null,"accept":["6.9428 x 10^7","6.9428×10^7"],"parts":[],"solution":["Write country A's population as an ordinary number: 6.85 x 10^7 = 68,500,000.","Add country B's population: 68,500,000 + 928,000 = 69,428,000.","In standard form: 69,428,000 = 6.9428 x 10^7."],"diagram":null,"draw":false,"revision":"e26e1c941adb"},{"id":"g5-standard-form-q11","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":11,"marks":4,"tier":"FH","calc":false,"text":"

A grain of sand has a mass of 0.0032 grams.

A bag contains 5 × 106 grains of sand.

Work out the total mass of the sand in the bag.

Give your answer in grams, in standard form.

","answerType":"exact","answer":"1.6 x 10^4 g","answerDisplay":null,"accept":["1.6 x 10^4","1.6×10^4"],"parts":[],"solution":["Write 0.0032 in standard form: 3.2 x 10^-3.","Multiply: (3.2 x 10^-3) x (5 x 10^6) = (3.2 x 5) x 10^(-3+6) = 16 x 10^3.","16 x 10^3 is not in standard form: 16 = 1.6 x 10, so the answer is 1.6 x 10^4 grams."],"diagram":null,"draw":false,"revision":"99fc10f73667"},{"id":"g5-standard-form-q12","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":12,"marks":4,"tier":"H","calc":true,"text":"

A human heart beats about 70 times each minute.

Estimate the total number of times a human heart beats in a lifetime of 80 years.

Give your answer in standard form, correct to 2 significant figures.

(Use 1 year = 365 days.)

","answerType":"open","answer":"About 2.9 × 10^9 beats","answerDisplay":null,"accept":["2.9 x 10^9"],"parts":[],"solution":["Beats per hour: 70 x 60 = 4,200.","Beats per day: 4,200 x 24 = 100,800.","Beats per year: 100,800 x 365 = 36,792,000.","Beats in 80 years: 36,792,000 x 80 = 2,943,360,000, which is about 2.9 x 10^9.","Using the given 365-day year gives 2.94336 × 10^9. To 2 significant figures this is 2.9 × 10^9; 3 × 10^9 only shows 1 significant figure."],"diagram":null,"draw":false,"revision":"fa448ef92f76"},{"id":"g5-standard-form-q13","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":13,"marks":4,"tier":"H","calc":true,"text":"

Data travels along an undersea fibre-optic cable at a speed of 2 × 105 kilometres per second.

The cable is 1.3 × 104 kilometres long.

  1. Work out the time taken for data to travel the full length of the cable.

    Give your answer in seconds, in standard form.

  2. Write your answer to part (a) in milliseconds.
","answerType":"multi","answer":"a) 6.5 x 10^-2 seconds b) 65 milliseconds","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6.5 x 10^-2 s"},{"label":"b","answer":"65 ms"}],"solution":["a) Time = distance / speed = (1.3 x 10^4) / (2 x 10^5).","a) 1.3 / 2 = 0.65 and 10^4 / 10^5 = 10^-1, giving 0.65 x 10^-1 = 6.5 x 10^-2 seconds.","b) 1 second = 1000 milliseconds, so 6.5 x 10^-2 s = 0.065 x 1000 = 65 milliseconds."],"diagram":null,"draw":false,"revision":"3a7fe8f69582"},{"id":"g5-standard-form-q14","topicId":"g5-standard-form","topic":"Standard Form","grade":"5","topicGrade":"5","strand":"N","difficulty":14,"marks":4,"tier":"H","calc":true,"text":"

The kinetic energy E joules of a moving object is given by

E = 12mv2

where m kg is the mass of the object and v m/s is its speed.

  1. Work out the value of E when m = 8 × 10−3 and v = 2 × 103

    Give your answer in standard form.

  2. The speed v is increased by 10% and the mass m stays the same.

    Work out the percentage increase in E.

","answerType":"multi","answer":"a) 1.6 x 10^4 b) 21%","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1.6 x 10^4"},{"label":"b","answer":"21%"}],"solution":["a) v^2 = (2 x 10^3)^2 = 4 x 10^6.","a) E = 0.5 x (8 x 10^-3) x (4 x 10^6) = (0.5 x 8 x 4) x 10^3 = 16 x 10^3 = 1.6 x 10^4 joules.","b) If v increases by 10%, the new speed is 1.1v, so v^2 becomes (1.1)^2 v^2 = 1.21v^2.","b) E is multiplied by 1.21, which is a 21% increase."],"diagram":null,"draw":false,"revision":"6c4570662efe"},{"id":"g5-vectors-q01","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

a = (3, 5) and b = (2, −1).

Work out a + b as a column vector.

","answerType":"exact","answer":"(5, 4)","answerDisplay":null,"accept":["(5,4)","5, 4"],"parts":[],"solution":["Add the top numbers: 3 + 2 = 5.","Add the bottom numbers: 5 + (-1) = 4.","So a + b = (5, 4)."],"diagram":null,"draw":false,"revision":"1c90fd537171"},{"id":"g5-vectors-q02","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

a = (4, −3) and b = (1, 6).

Work out 2a + b as a column vector.

","answerType":"exact","answer":"(9, 0)","answerDisplay":null,"accept":["(9,0)","9, 0"],"parts":[],"solution":["2a = (2 × 4, 2 × -3) = (8, -6).","2a + b = (8 + 1, -6 + 6) = (9, 0)."],"diagram":null,"draw":false,"revision":"21333d3d07a9"},{"id":"g5-vectors-q03","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":3,"marks":2,"tier":"F","calc":true,"text":"

p = (5, 2) and q = (3, −4).

Work out p − 2q as a column vector.

","answerType":"exact","answer":"(-1, 10)","answerDisplay":null,"accept":["(-1,10)","-1, 10"],"parts":[],"solution":["2q = (6, -8).","p - 2q = (5 - 6, 2 - (-8)) = (-1, 10)."],"diagram":null,"draw":false,"revision":"39ca33e1bc41"},{"id":"g5-vectors-q04","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":4,"marks":2,"tier":"FH","calc":false,"text":"

a = (−2, 5) and b = (3, −1).

Work out 3a − 2b as a column vector.

","answerType":"exact","answer":"(-12, 17)","answerDisplay":null,"accept":["(-12,17)","-12, 17"],"parts":[],"solution":["3a = (-6, 15) and 2b = (6, -2).","3a - 2b = (-6 - 6, 15 - (-2)) = (-12, 17)."],"diagram":null,"draw":false,"revision":"ff635ca3c485"},{"id":"g5-vectors-q05","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":5,"marks":3,"tier":"F","calc":true,"text":"

a = (3, −1) and b = (−1, 2).

On the grid, draw and label the vector a + 2b.

","answerType":"exact","answer":"(1, 3)","answerDisplay":null,"accept":["(1,3)","arrow representing (1, 3)"],"parts":[],"solution":["2b = (-2, 4).","a + 2b = (3 - 2, -1 + 4) = (1, 3).","Draw an arrow going 1 square right and 3 squares up, with the arrowhead at the finishing end."],"diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":5,"step":1},"square":true,"vectors":[{"from":[0,0],"to":[1,3],"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"55a7b8c9137e"},{"id":"g5-vectors-q06","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

k(2, −3) + (5, 1) = (13, m)

where k and m are numbers.

Find the value of k and the value of m.

","answerType":"multi","answer":"k = 4, m = -11","answerDisplay":null,"accept":["k=4 m=-11"],"parts":[{"label":"k","answer":"4"},{"label":"m","answer":"-11"}],"solution":["Top numbers: 2k + 5 = 13, so 2k = 8 and k = 4.","Bottom numbers: k × (-3) + 1 = m.","m = 4 × (-3) + 1 = -12 + 1 = -11."],"diagram":null,"draw":false,"revision":"84c1b68146d3"},{"id":"g5-vectors-q07","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":7,"marks":3,"tier":"FH","calc":true,"text":"

A is the point (1, 2) and B is the point (5, 8).

ABC is a straight line and BC = 12AB.

Work out the coordinates of C.

The points are in the order A, B, C, so C lies beyond B.

","answerType":"exact","answer":"(7, 11)","answerDisplay":null,"accept":["(7,11)","7, 11"],"parts":[],"solution":["Vector AB = (5 - 1, 8 - 2) = (4, 6).","BC = 12AB = (2, 3).","C = B + BC = (5 + 2, 8 + 3) = (7, 11)."],"diagram":null,"draw":false,"revision":"9377501847d5"},{"id":"g5-vectors-q08","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

m(2, 1) + n(1, 3) = (4, 7)

where m and n are numbers.

Find the value of m and the value of n.

","answerType":"multi","answer":"m = 1, n = 2","answerDisplay":null,"accept":["m=1 n=2"],"parts":[{"label":"m","answer":"1"},{"label":"n","answer":"2"}],"solution":["Top numbers: 2m + n = 4.","Bottom numbers: m + 3n = 7.","From the first equation n = 4 - 2m. Substitute: m + 3(4 - 2m) = 7, so m + 12 - 6m = 7 and -5m = -5, giving m = 1.","Then n = 4 - 2 = 2.","Check: 1 + 3 × 2 = 7. Correct."],"diagram":null,"draw":false,"revision":"96991fce2de8"},{"id":"g5-vectors-q09","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":9,"marks":3,"tier":"H","calc":true,"text":"

a = (2, 1) and b = (1, −2).

On the grid, draw and label the vector 2a − b.

","answerType":"exact","answer":"(3, 4)","answerDisplay":null,"accept":["(3,4)","arrow representing (3, 4)"],"parts":[],"solution":["2a = (4, 2).","2a - b = (4 - 1, 2 - (-2)) = (3, 4).","Draw an arrow going 3 squares right and 4 squares up, with an arrowhead."],"diagram":{"type":"axes","x":{"min":-4,"max":4,"step":1},"y":{"min":-4,"max":5,"step":1},"square":true,"vectors":[{"from":[0,0],"to":[3,4],"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"1950fd4069be"},{"id":"g5-vectors-q10","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":10,"marks":3,"tier":"H","calc":false,"text":"

OAB is a triangle. OA = a and OB = b.

M is the midpoint of OA and N is the midpoint of OB.

Show that MN is parallel to AB.

","answerType":"open","answer":"MN = (1/2)(b - a) = (1/2)AB, so MN is parallel to AB","answerDisplay":"MN = (12)(b - a) = (12)AB, so MN is parallel to AB","accept":["MN is a multiple of AB"],"parts":[],"solution":["OM = 12a and ON = 12b.","MN = ON - OM = 12b - 12a = 12(b - a).","AB = OB - OA = b - a.","MN = 12AB, so MN is a scalar multiple of AB and is therefore parallel to AB.","A scalar is an ordinary number multiplying a vector. Multiplying a non-zero vector by 1/2 halves its length without changing its direction, which establishes parallel lines."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[2,5],"label":"B","style":"none","labelPos":"above"},{"at":[3.5,0],"label":"M","style":"none","labelPos":"below-right"},{"at":[1,2.5],"label":"N","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"a","arrow":"end"},{"from":[0,0],"to":[2,5],"label":"b","arrow":"end"},{"from":[7,0],"to":[2,5]},{"from":[3.5,0],"to":[1,2.5]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"120b369cc6e4"},{"id":"g5-vectors-q11","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":11,"marks":4,"tier":"H","calc":true,"text":"

OAB is a triangle. OA = a and OB = b.

P is the point on AB such that AP : PB = 2 : 1.

Express OP in terms of a and b. Give your answer in its simplest form.

","answerType":"exact","answer":"(1/3)a + (2/3)b","answerDisplay":"(13)a + (23)b","accept":["(a + 2b)/3","1/3 a + 2/3 b"],"parts":[],"solution":["AB = OB - OA = b - a.","AP : PB = 2 : 1, so AP = (23)AB = (23)(b - a).","OP = OA + AP = a + (23)(b - a).","OP = a - (23)a + (23)b = (13)a + (23)b."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[2,5],"label":"B","style":"none","labelPos":"above"},{"at":[3.666666666666667,3.333333333333333],"label":"P","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"a","arrow":"end"},{"from":[0,0],"to":[2,5],"label":"b","arrow":"end"},{"from":[7,0],"to":[2,5]},{"from":[0,0],"to":[3.666666666666667,3.333333333333333]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a7df326ae4d4"},{"id":"g5-vectors-q12","topicId":"g5-vectors","topic":"Vectors","grade":"5","topicGrade":"5","strand":"G","difficulty":12,"marks":4,"tier":"H","calc":true,"text":"

OABC is a trapezium. OA = a and OC = c.

CB is parallel to OA and CB = 2a.

M is the midpoint of AB.

Find OM in terms of a and c. Give your answer in its simplest form.

","answerType":"exact","answer":"(3/2)a + (1/2)c","answerDisplay":"(32)a + (12)c","accept":["(3a + c)/2","3/2 a + 1/2 c","1.5a + 0.5c"],"parts":[],"solution":["OB = OC + CB = c + 2a.","AB = OB - OA = c + 2a - a = a + c.","AM = 12AB = 12(a + c).","OM = OA + AM = a + 12a + 12c = (32)a + (12)c."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[5,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[1,4],"label":"C","style":"none","labelPos":"above"},{"at":[11,4],"label":"B","style":"none","labelPos":"above"},{"at":[8,2],"label":"M","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[5,0],"label":"a","arrow":"end","parallel":1},{"from":[0,0],"to":[1,4],"label":"c","arrow":"end"},{"from":[1,4],"to":[11,4],"label":"2a","arrow":"end","parallel":1},{"from":[5,0],"to":[11,4]},{"from":[0,0],"to":[8,2]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"25a880980eb5"},{"id":"g5-venn-diagrams-q01","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":1,"marks":2,"tier":"F","calc":false,"text":"

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {even numbers}   B = {multiples of 5}

Kim draws a Venn diagram. She puts:

2, 4, 8 in A only   6, 10 in the intersection of A and B   5 in B only   1, 3, 9 outside both circles.

Write down two mistakes Kim has made.

","answerType":"open","answer":"6 should be in A only, not in the intersection (6 is not a multiple of 5); the number 7 is missing from the region outside both circles.","answerDisplay":null,"accept":[],"parts":[],"solution":["6 is even but not a multiple of 5, so it belongs in A only, not in A intersection B.","Every number in the universal set must appear exactly once: 7 is odd and not a multiple of 5, so it should be outside both circles, but Kim has left it out."],"diagram":{"type":"venn","sets":["A","B"],"regions":{"A":"2, 4, 8","AB":"6, 10","B":"5","none":"1, 3, 9"}},"draw":false,"revision":"a4eb689ae5d9"},{"id":"g5-venn-diagrams-q02","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":2,"marks":3,"tier":"F","calc":false,"text":"

A = {factors of 12}   B = {factors of 18}

  1. List the members of A ∩ B.
  2. Is 9 a member of A ∪ B? Give a reason for your answer.
","answerType":"multi","answer":"a) 1, 2, 3, 6; b) yes, because 9 is a factor of 18 so it is in B","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1, 2, 3, 6"},{"label":"b","answer":"Yes, because 9 is in B (9 is a factor of 18), so it is in the union"}],"solution":["A = {1, 2, 3, 4, 6, 12} and B = {1, 2, 3, 6, 9, 18}.","The intersection is the numbers in both sets: {1, 2, 3, 6}.","The union contains every number in A or B (or both). 9 is a factor of 18, so 9 is in B and therefore in A ∪ B."],"diagram":null,"draw":false,"revision":"87e9e2769420"},{"id":"g5-venn-diagrams-q03","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":3,"marks":3,"tier":"F","calc":false,"text":"

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A = {multiples of 3}   B = {even numbers}

Complete the Venn diagram.

","answerType":"multi","answer":"A only: 3, 9; A and B: 6, 12; B only: 2, 4, 8, 10; outside: 1, 5, 7, 11","answerDisplay":null,"accept":[],"parts":[{"label":"A only","answer":"3, 9"},{"label":"A and B","answer":"6, 12"},{"label":"B only","answer":"2, 4, 8, 10"},{"label":"outside","answer":"1, 5, 7, 11"}],"solution":["A = {3, 6, 9, 12} and B = {2, 4, 6, 8, 10, 12}.","Numbers in both sets: 6 and 12 go in the intersection.","A only: 3 and 9. B only: 2, 4, 8, 10.","The remaining numbers 1, 5, 7, 11 go outside both circles."],"diagram":{"type":"venn","sets":["A","B"],"regions":{},"answer":{"A":"3, 9","AB":"6, 12","B":"2, 4, 8, 10","none":"1, 5, 7, 11"}},"draw":false,"revision":"2265c2f0a463"},{"id":"g5-venn-diagrams-q04","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":4,"marks":3,"tier":"FH","calc":true,"text":"

40 people were asked whether they drink coffee (C) and whether they drink tea (D).

14 drink coffee only, 6 drink both, 11 drink tea only and 9 drink neither.

One of the 40 people is chosen at random.

  1. Find the probability that this person drinks both coffee and tea.
  2. Find the probability that this person drinks coffee or tea or both.
","answerType":"multi","answer":"a) 3/20; b) 31/40","answerDisplay":"a) 320; b) 3140","accept":[],"parts":[{"label":"a","answer":"3/20 (= 6/40)","answerDisplay":"320 (= 640)"},{"label":"b","answer":"31/40","answerDisplay":"3140"}],"solution":["P(both) = 640 = 320.","Number who drink coffee or tea or both = 14 + 6 + 11 = 31.","P(coffee or tea or both) = 3140"],"diagram":{"type":"venn","sets":["C","D"],"regions":{"A":"14","AB":"6","B":"11","none":"9"}},"draw":false,"revision":"060c52d4b817"},{"id":"g5-venn-diagrams-q05","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":5,"marks":4,"tier":"F","calc":true,"text":"

There are 50 students in a year group.

23 study geography (G).

19 study history (H).

8 study both geography and history.

Complete the Venn diagram to show the number of students in each region.

","answerType":"multi","answer":"G only: 15; both: 8; H only: 11; neither: 16","answerDisplay":null,"accept":[],"parts":[{"label":"G only","answer":"15"},{"label":"both","answer":"8"},{"label":"H only","answer":"11"},{"label":"neither","answer":"16"}],"solution":["Put 8 in the intersection first.","Geography only = 23 - 8 = 15.","History only = 19 - 8 = 11.","Neither = 50 - 15 - 8 - 11 = 16.","Check: 15 + 8 + 11 + 16 = 50."],"diagram":{"type":"venn","sets":["G","H"],"regions":{},"answer":{"A":"15","AB":"8","B":"11","none":"16"}},"draw":false,"revision":"61eb4d84a694"},{"id":"g5-venn-diagrams-q06","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":6,"marks":4,"tier":"FH","calc":false,"text":"

ξ = {odd numbers between 0 and 20}

A = {multiples of 3}   B = {numbers less than 10}

Draw a Venn diagram to show the sets A and B, putting every member of ξ in the correct region.

","answerType":"multi","answer":"A only: 15; A and B: 3, 9; B only: 1, 5, 7; outside: 11, 13, 17, 19","answerDisplay":null,"accept":[],"parts":[{"label":"A only","answer":"15"},{"label":"A and B","answer":"3, 9"},{"label":"B only","answer":"1, 5, 7"},{"label":"outside","answer":"11, 13, 17, 19"}],"solution":["ξ = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}.","A = {3, 9, 15} and B = {1, 3, 5, 7, 9}.","3 and 9 are in both sets, so they go in the intersection.","A only: 15. B only: 1, 5, 7. Outside both: 11, 13, 17, 19."],"diagram":{"type":"venn","sets":["A","B"],"regions":{},"answer":{"A":"15","AB":"3, 9","B":"1, 5, 7","none":"11, 13, 17, 19"}},"draw":false,"revision":"d44cb323fbcc"},{"id":"g5-venn-diagrams-q07","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":7,"marks":5,"tier":"F","calc":true,"text":"

A gym has 80 members.

47 members use the treadmill (T).

39 members use the weights (W).

21 members use both the treadmill and the weights.

  1. Complete the Venn diagram to show the number of members in each region.
  2. One of the 80 members is chosen at random. Work out the probability that this member uses neither the treadmill nor the weights.
","answerType":"multi","answer":"a) T only: 26; both: 21; W only: 18; neither: 15; b) 3/16","answerDisplay":"a) T only: 26; both: 21; W only: 18; neither: 15; b) 316","accept":[],"parts":[{"label":"a","answer":"T only = 26, both = 21, W only = 18, neither = 15"},{"label":"b","answer":"3/16","answerDisplay":"316"}],"solution":["Treadmill only = 47 - 21 = 26 and weights only = 39 - 21 = 18.","Neither = 80 - 26 - 21 - 18 = 15.","P(neither) = 1580 = 316.","Check: 26 + 21 + 18 + 15 = 80."],"diagram":{"type":"venn","sets":["T","W"],"regions":{},"answer":{"A":"26","AB":"21","B":"18","none":"15"}},"draw":false,"revision":"cf6bf5812b07"},{"id":"g5-venn-diagrams-q08","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":8,"marks":5,"tier":"FH","calc":true,"text":"

60 people were asked whether they have a dog (D) or a cat (C).

32 people have a dog.

27 people have a cat.

11 people have both a dog and a cat.

  1. Work out the number of people who have a dog but not a cat.
  2. Work out the number of people who have neither a dog nor a cat.
  3. One of the 60 people is chosen at random. Work out the probability that this person has a dog but not a cat.
","answerType":"multi","answer":"a) 21; b) 12; c) 7/20","answerDisplay":"a) 21; b) 12; c) 720","accept":[],"parts":[{"label":"a","answer":"21"},{"label":"b","answer":"12"},{"label":"c","answer":"7/20","answerDisplay":"720"}],"solution":["Dog but not cat = 32 - 11 = 21.","Cat but not dog = 27 - 11 = 16, so neither = 60 - 21 - 11 - 16 = 12.","P(dog but not cat) = 2160 = 720"],"diagram":null,"draw":false,"revision":"8be506413dc3"},{"id":"g5-venn-diagrams-q09","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"

The Venn diagram regions for sets A and B contain these numbers of people:

A only: 7   A ∩ B: 5   B only: 8   outside both: 10

One of the people is chosen at random.

  1. Find P(A ∩ B).
  2. Find P(A′).
  3. Find P(A ∪ B).
","answerType":"multi","answer":"a) 1/6; b) 3/5; c) 2/3","answerDisplay":"a) 16; b) 35; c) 23","accept":[],"parts":[{"label":"a","answer":"1/6 (= 5/30)","answerDisplay":"16 (= 530)"},{"label":"b","answer":"3/5 (= 18/30)","answerDisplay":"35 (= 1830)"},{"label":"c","answer":"2/3 (= 20/30)","answerDisplay":"23 (= 2030)"}],"solution":["Total = 7 + 5 + 8 + 10 = 30.","P(A ∩ B) = 530 = 16.","A' is everyone not in A: 8 + 10 = 18, so P(A') = 1830 = 35.","A ∪ B contains 7 + 5 + 8 = 20 people, so P(A ∪ B) = 2030 = 23."],"diagram":{"type":"venn","sets":["A","B"],"regions":{"A":"7","AB":"5","B":"8","none":"10"}},"draw":false,"revision":"a0db3759193c"},{"id":"g5-venn-diagrams-q10","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

100 people went to a music festival.

58 saw band X.

45 saw band Y.

23 saw both bands.

  1. Work out the number of people who saw neither band.
  2. One person who saw band X is chosen at random. Work out the probability that this person also saw band Y.
","answerType":"multi","answer":"a) 20; b) 23/58","answerDisplay":"a) 20; b) 2358","accept":[],"parts":[{"label":"a","answer":"20"},{"label":"b","answer":"23/58","answerDisplay":"2358"}],"solution":["X only = 58 - 23 = 35 and Y only = 45 - 23 = 22.","Neither = 100 - 35 - 23 - 22 = 20.","58 people saw band X, and 23 of them also saw band Y.","P(saw Y given saw X) = 2358"],"diagram":null,"draw":false,"revision":"1c70a532e8ac"},{"id":"g5-venn-diagrams-q11","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":11,"marks":6,"tier":"H","calc":true,"text":"

80 students were asked which sports they play from football (F), rugby (R) and cricket (C).

4 students play all three sports.

12 students play football and rugby.

10 students play football and cricket.

9 students play rugby and cricket.

36 students play football.

29 students play rugby.

25 students play cricket.

  1. Complete the Venn diagram to show the number of students in each region.
  2. One of the 80 students is chosen at random. Work out the probability that this student plays none of the three sports.
","answerType":"multi","answer":"a) F only: 18, R only: 12, C only: 10, F and R only: 8, F and C only: 6, R and C only: 5, all three: 4, none: 17; b) 17/80","answerDisplay":"a) F only: 18, R only: 12, C only: 10, F and R only: 8, F and C only: 6, R and C only: 5, all three: 4, none: 17; b) 1780","accept":[],"parts":[{"label":"a","answer":"F only 18, R only 12, C only 10, F and R only 8, F and C only 6, R and C only 5, all three 4, none 17"},{"label":"b","answer":"17/80","answerDisplay":"1780"}],"solution":["Put 4 in the centre first.","F and R only = 12 - 4 = 8; F and C only = 10 - 4 = 6; R and C only = 9 - 4 = 5.","F only = 36 - 8 - 6 - 4 = 18; R only = 29 - 8 - 5 - 4 = 12; C only = 25 - 6 - 5 - 4 = 10.","Sum inside the circles = 18 + 12 + 10 + 8 + 6 + 5 + 4 = 63, so none = 80 - 63 = 17.","P(none) = 1780"],"diagram":{"type":"venn","sets":["F","R","C"],"regions":{},"answer":{"A":"18","B":"12","C":"10","AB":"8","AC":"6","BC":"5","ABC":"4","none":"17"}},"draw":false,"revision":"059a31a01dc4"},{"id":"g5-venn-diagrams-q12","topicId":"g5-venn-diagrams","topic":"Venn Diagrams","grade":"5","topicGrade":"5","strand":"P","difficulty":12,"marks":6,"tier":"H","calc":true,"text":"

90 people were asked which streaming services they use from A, B and C.

5 people use all three services.

14 people use A and B.

11 people use A and C.

13 people use B and C.

40 people use A.

35 people use B.

30 people use C.

  1. Complete the Venn diagram to show the number of people in each region.
  2. One person who uses service A is chosen at random. Work out the probability that this person also uses service C.
","answerType":"multi","answer":"a) A only: 20, B only: 13, C only: 11, A and B only: 9, A and C only: 6, B and C only: 8, all three: 5, none: 18; b) 11/40","answerDisplay":"a) A only: 20, B only: 13, C only: 11, A and B only: 9, A and C only: 6, B and C only: 8, all three: 5, none: 18; b) 1140","accept":[],"parts":[{"label":"a","answer":"A only 20, B only 13, C only 11, A and B only 9, A and C only 6, B and C only 8, all three 5, none 18"},{"label":"b","answer":"11/40","answerDisplay":"1140"}],"solution":["Put 5 in the centre.","A and B only = 14 - 5 = 9; A and C only = 11 - 5 = 6; B and C only = 13 - 5 = 8.","A only = 40 - 9 - 6 - 5 = 20; B only = 35 - 9 - 8 - 5 = 13; C only = 30 - 6 - 8 - 5 = 11.","Inside the circles: 20 + 13 + 11 + 9 + 6 + 8 + 5 = 72, so none = 90 - 72 = 18.","40 people use A, and 6 + 5 = 11 of them also use C, so P(uses C given uses A) = 1140."],"diagram":{"type":"venn","sets":["A","B","C"],"regions":{},"answer":{"A":"20","B":"13","C":"11","AB":"9","AC":"6","BC":"8","ABC":"5","none":"18"}},"draw":false,"revision":"0c4d08889158"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q01","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":1,"marks":2,"tier":"F","calc":true,"text":"

A box contains red, blue and green pens in the ratio 2 : 3 : 7.

Find the fraction of the pens that are blue. Give your fraction in its simplest form.

","answerType":"exact","answer":"1/4","answerDisplay":"14","accept":[],"parts":[],"solution":["Total number of parts = 2 + 3 + 7 = 12","Fraction blue = 312 = 14"],"diagram":null,"draw":false,"revision":"aeb24944f04b"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q02","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":2,"marks":2,"tier":"F","calc":true,"text":"

In a drama club, the ratio of the number of boys to the number of girls is 4 : 5.

Write down the fraction of the club members who are boys.

","answerType":"exact","answer":"4/9","answerDisplay":"49","accept":[],"parts":[],"solution":["Total parts = 4 + 5 = 9","Fraction who are boys = 49"],"diagram":null,"draw":false,"revision":"84d6b1452cf3"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q03","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

A bag contains only red, blue and yellow counters.

The numbers of red, blue and yellow counters are in the ratio 2 : 3 : x.

A counter is taken at random from the bag.

The probability that it is yellow is 27.

Work out the value of x.

","answerType":"exact","answer":"2","answerDisplay":null,"accept":["x = 2"],"parts":[],"solution":["Fraction of counters that are yellow = x(2 + 3 + x) = x(5 + x)","So x(5 + x) = 27","7x = 2(5 + x), so 7x = 10 + 2x, giving 5x = 10 and x = 2","Check: ratio 2 : 3 : 2 has total 7 parts and yellow is 27"],"diagram":null,"draw":false,"revision":"d2a07caffab7"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q04","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":4,"marks":3,"tier":"H","calc":false,"text":"

a : b = 2 : 5 and b : c = 3 : 4

Show that a : c = 3 : 10.

","answerType":"open","answer":"a : b : c = 6 : 15 : 20, so a : c = 6 : 20 = 3 : 10","answerDisplay":null,"accept":[],"parts":[],"solution":["Make the b parts equal: multiply 2 : 5 by 3 to get a : b = 6 : 15, and multiply 3 : 4 by 5 to get b : c = 15 : 20","So a : b : c = 6 : 15 : 20","Then a : c = 6 : 20, which simplifies to 3 : 10"],"diagram":null,"draw":false,"revision":"eda79786674d"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q05","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":5,"marks":3,"tier":"H","calc":false,"text":"

A jar contains only red sweets and green sweets.

The ratio of the number of red sweets to the number of green sweets is 4 : 3.

Show that the total number of sweets in the jar must be a multiple of 7.

","answerType":"open","answer":"Red = 4n and green = 3n for some whole number n, so total = 7n, a multiple of 7","answerDisplay":null,"accept":[],"parts":[],"solution":["Let the number of red sweets be 4n and the number of green sweets be 3n, where n is a whole number","Total = 4n + 3n = 7n","7n is always a multiple of 7, so the total must be a multiple of 7"],"diagram":null,"draw":false,"revision":"da4c78e81f7b"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q06","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

Every student in a school walks, cycles or gets the bus to school.

The ratio of the number who walk to the number who cycle is 5 : 2.

The ratio of the number who cycle to the number who get the bus is 4 : 3.

Work out the fraction of the students who walk to school.

","answerType":"exact","answer":"10/17","answerDisplay":"1017","accept":[],"parts":[],"solution":["Make the cycle parts equal: multiply 5 : 2 by 2 to get walk : cycle = 10 : 4","So walk : cycle : bus = 10 : 4 : 3","Total parts = 10 + 4 + 3 = 17","Fraction who walk = 1017"],"diagram":null,"draw":false,"revision":"b9e9ee36deaa"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q07","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

The ratio of the amount of money Amy has to the amount Ben has is 5 : 3.

Amy gives Ben £14.

The ratio of the amount Amy has to the amount Ben has is now 6 : 5.

Work out how much money Amy had at first.

","answerType":"exact","answer":"£110","answerDisplay":null,"accept":["110","£110"],"parts":[],"solution":["Let Amy have 5x pounds and Ben have 3x pounds at first","After the gift: (5x − 14) : (3x + 14) = 6 : 5","So 5(5x − 14) = 6(3x + 14), giving 25x − 70 = 18x + 84","7x = 154, so x = 22 and Amy had 5 × 22 = £110","Check: 110 − 14 = 96 and 66 + 14 = 80, and 96 : 80 = 6 : 5"],"diagram":null,"draw":false,"revision":"29f23a05976d"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q08","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

The cost of a pencil and the cost of a ruler are in the ratio 2 : 5.

Priti buys 3 pencils and 2 rulers for a total of £3.20.

Work out the cost of a ruler.

","answerType":"exact","answer":"£1","answerDisplay":null,"accept":["100p","£1.00","1"],"parts":[],"solution":["Let a pencil cost 2k pence and a ruler cost 5k pence","3 × 2k + 2 × 5k = 16k = 320 pence","So k = 20, and a ruler costs 5 × 20 = 100p = £1","Check: 3 × 40p + 2 × £1 = £1.20 + £2 = £3.20"],"diagram":null,"draw":false,"revision":"718f97c3d8d2"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q09","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

A box contains red counters and blue counters.

If 5 more red counters are put in the box, the ratio of red to blue will be 2 : 3.

If instead 5 more blue counters are put in the box, the ratio of red to blue will be 1 : 2.

Work out how many counters of each colour are in the box.

","answerType":"multi","answer":"25 red and 45 blue","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"25 red"},{"label":"b","answer":"45 blue"}],"solution":["Let there be r red and b blue counters","First condition: (r + 5)b = 23, so 3r + 15 = 2b","Second condition: r(b + 5) = 12, so 2r = b + 5, giving b = 2r − 5","Substitute: 3r + 15 = 2(2r − 5) = 4r − 10, so r = 25 and b = 45","Check: (25 + 5) : 45 = 30 : 45 = 2 : 3, and 25 : (45 + 5) = 25 : 50 = 1 : 2"],"diagram":null,"draw":false,"revision":"f711f1adebe6"},{"id":"g5-writing-a-ratio-as-a-fraction-or-linear-function-q10","topicId":"g5-writing-a-ratio-as-a-fraction-or-linear-function","topic":"Writing a Ratio as a Fraction or Linear Function","grade":"5","topicGrade":"5","strand":"R","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

a and b are positive numbers such that

a2 + 6b2 = 5ab

Find the possible values of the ratio a : b.

","answerType":"exact","answer":"a : b = 2 : 1 or a : b = 3 : 1","answerDisplay":null,"accept":["2:1 or 3:1"],"parts":[],"solution":["Rearrange: a2 − 5ab + 6b2 = 0","Factorise: (a − 2b)(a − 3b) = 0","So a = 2b or a = 3b","Therefore a : b = 2 : 1 or a : b = 3 : 1","Check: if a = 2b then a2 + 6b2 = 4b2 + 6b2 = 10b2 and 5ab = 10b2; if a = 3b then 9b2 + 6b2 = 15b2 and 5ab = 15b2"],"diagram":null,"draw":false,"revision":"9ab689045de7"},{"id":"g6-box-plots-q01","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":1,"marks":2,"tier":"H","calc":true,"text":"

The table gives information about the times, in minutes, that 51 patients waited at a clinic.

Shortest time15
Lower quartile22
Median38
Upper quartile46
Longest time63

Tom draws a box plot for this information. On his box plot:

Write down two things that are wrong with Tom's box plot.

","answerType":"open","answer":"The left whisker should end at 15 (the shortest time), not 12; the median line should be at 38, not 34","answerDisplay":null,"accept":["Whisker at wrong minimum (should be 15); median in wrong place (should be 38)"],"parts":[],"solution":["The whisker must end at the shortest time, 15 minutes, but Tom drew it at 12.","The line inside the box shows the median, 38 minutes, but Tom drew it at 34."],"diagram":{"type":"axes","x":{"min":0,"max":70,"step":10,"minor":5,"label":"Time (minutes)"},"y":{"min":0,"max":1,"hide":true},"xy":false,"height":150,"rows":1,"boxplots":[{"row":0,"min":12,"q1":22,"med":34,"q3":46,"max":63,"label":"Tom's box plot"}]},"draw":false,"revision":"0e185d20c0a7"},{"id":"g6-box-plots-q02","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

The box plot shows the journey times, in minutes, of 15 deliveries made by a courier.

  1. Write down the median journey time.
  2. Work out the interquartile range of the journey times.
  3. Work out the range of the journey times.
","answerType":"multi","answer":"(a) 19 minutes (b) 12 minutes (c) 26 minutes","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"19 minutes"},{"label":"b","answer":"12 minutes"},{"label":"c","answer":"26 minutes"}],"solution":["The line inside the box is at 19, so the median is 19 minutes.","Interquartile range = upper quartile - lower quartile = 26 - 14 = 12 minutes.","Range = longest - shortest = 34 - 8 = 26 minutes."],"diagram":{"type":"axes","x":{"min":0,"max":40,"step":5,"minor":5,"label":"Journey time (minutes)"},"y":{"min":0,"max":1,"hide":true},"xy":false,"height":150,"rows":1,"boxplots":[{"row":0,"min":8,"q1":14,"med":19,"q3":26,"max":34}]},"draw":false,"revision":"f2150bd365e1"},{"id":"g6-box-plots-q03","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

Here are the numbers of emails Priya received on each of 11 days, written in order.

3  5  6  8  9  11  13  14  16  18  21

  1. Find the median.
  2. Find the lower quartile and the upper quartile.
  3. Work out the interquartile range.
","answerType":"multi","answer":"(a) 11 (b) LQ 6, UQ 16 (c) 10","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"11"},{"label":"b","answer":"lower quartile 6, upper quartile 16"},{"label":"c","answer":"10"}],"solution":["There are 11 values, so the median is the 6th value: 11.","Exclude the overall median, then find the median of each remaining half. The lower five values have middle value 6; the upper five have middle value 16. These are the lower and upper quartiles.","The lower quartile is the 3rd value: 6. The upper quartile is the 9th value: 16.","Interquartile range = 16 - 6 = 10."],"diagram":null,"draw":false,"revision":"281c666687ad"},{"id":"g6-box-plots-q04","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":4,"marks":3,"tier":"H","calc":false,"text":"

The stem and leaf diagram shows the marks of 15 students in a spelling test.

StemLeaves
12  5  8
20  3  4  7  9
31  4  6  8
40  2  5

Key: 2 | 3 means 23 marks

Work out the five values needed to draw a box plot for this data: the lowest mark, the lower quartile, the median, the upper quartile and the highest mark.

","answerType":"multi","answer":"lowest 12, LQ 20, median 29, UQ 38, highest 45","answerDisplay":null,"accept":[],"parts":[{"label":"lowest","answer":"12"},{"label":"lower quartile","answer":"20"},{"label":"median","answer":"29"},{"label":"upper quartile","answer":"38"},{"label":"highest","answer":"45"}],"solution":["The ordered data is 12, 15, 18, 20, 23, 24, 27, 29, 31, 34, 36, 38, 40, 42, 45.","With 15 values the median is the 8th value: 29.","The lower quartile is the 4th value: 20. The upper quartile is the 12th value: 38.","Lowest = 12, highest = 45."],"diagram":null,"draw":false,"revision":"98baabac6fe8"},{"id":"g6-box-plots-q05","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

Two shops sell second-hand bikes. The tables summarise the prices, in pounds, of the bikes sold last month.

Shop A

Minimum60
Lower quartile95
Median130
Upper quartile180
Maximum250

Shop B: median £150, interquartile range £45.

  1. Work out the interquartile range of the prices at Shop A.
  2. Compare the prices of bikes at the two shops. Give two comparisons.
  3. Which shop has the more consistent prices? Give a reason.
","answerType":"multi","answer":"(a) £85 (b) Shop B's median (£150) is higher than Shop A's (£130), so bikes at Shop B cost more on average; Shop A's IQR (£85) is larger than Shop B's (£45), so Shop A's prices are more spread out (c) Shop B, because its interquartile range is smaller","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"£85"},{"label":"b","answer":"Shop B has the higher median so its bikes cost more on average; Shop A has the larger IQR so its prices vary more"},{"label":"c","answer":"Shop B - smaller interquartile range means more consistent prices"}],"solution":["Interquartile range for Shop A = 180 - 95 = £85.","Comparing averages: Shop B's median of £150 is higher than Shop A's £130, so on average bikes at Shop B cost more.","Comparing spread: Shop A's IQR of £85 is larger than Shop B's £45, so Shop A's prices are more spread out.","Shop B is more consistent because its interquartile range is smaller."],"diagram":null,"draw":false,"revision":"336005950915"},{"id":"g6-box-plots-q06","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A box plot summarises the times, in minutes, taken by 120 runners to finish a 10 km race.

On the box plot: the whiskers are at 22 and 58, the box goes from 28 to 41, and the median line is at 33.

  1. Work out the interquartile range of the times.
  2. Estimate how many of the 120 runners took longer than 41 minutes.
  3. Explain why your answer to part (b) is an estimate.
","answerType":"multi","answer":"(a) 13 minutes (b) 30 runners (c) The box plot only shows the quartiles; exactly 25% is assumed to lie above the upper quartile, and the individual times are not known","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"13 minutes"},{"label":"b","answer":"30"},{"label":"c","answer":"A box plot gives only a five-number summary, so we assume exactly a quarter of the runners are above the upper quartile"}],"solution":["Interquartile range = 41 - 28 = 13 minutes.","41 minutes is the upper quartile, so about 25% of runners took longer: 25% of 120 = 30 runners.","The box plot does not show individual times, so we can only assume a quarter of the data lies above the upper quartile."],"diagram":{"type":"axes","x":{"min":0,"max":60,"step":10,"minor":5,"label":"Time (minutes)"},"y":{"min":0,"max":1,"hide":true},"xy":false,"height":150,"rows":1,"boxplots":[{"row":0,"min":22,"q1":28,"med":33,"q3":41,"max":58}]},"draw":false,"revision":"d4d0ba69edab"},{"id":"g6-box-plots-q07","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":7,"marks":5,"tier":"H","calc":true,"text":"

The table gives information about the times, in minutes, that the girls in a class spent on homework one evening.

Shortest time5
Lower quartile9
Median12
Upper quartile15
Longest time22
  1. On the grid, draw a box plot for this information.
  2. The box plot for the boys in the class is shown on the same grid. The boys had a median of 14 minutes and an interquartile range of 8 minutes. Compare the times spent by the girls and the boys. Give two comparisons.
","answerType":"multi","answer":"(a) box plot with whiskers at 5 and 22, box from 9 to 15, median line at 12 (b) The girls' median (12) is lower than the boys' (14), so on average the girls spent less time on homework; the girls' IQR (6) is smaller than the boys' (8), so the girls' times are more consistent","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"whiskers at 5 and 22, box 9 to 15, median at 12"},{"label":"b","answer":"Girls' median lower (12 vs 14) so girls spent less time on average; girls' IQR smaller (6 vs 8) so girls' times were more consistent"}],"solution":["Draw whiskers at 5 and 22, a box from 9 to 15 and the median line at 12.","Girls' interquartile range = 15 - 9 = 6 minutes.","The girls' median of 12 minutes is lower than the boys' 14 minutes, so on average girls spent less time.","The girls' IQR of 6 minutes is smaller than the boys' 8 minutes, so the girls' times were less spread out."],"diagram":{"type":"axes","x":{"min":0,"max":25,"step":5,"minor":5,"label":"Time (minutes)"},"y":{"min":0,"max":1,"hide":true},"xy":false,"height":170,"rows":2,"boxplots":[{"row":0,"min":5,"q1":9,"med":12,"q3":15,"max":22,"label":"Girls","answer":true},{"row":1,"min":4,"q1":8,"med":14,"q3":16,"max":25,"label":"Boys"}]},"draw":true,"revision":"5e42caaf4fd7"},{"id":"g6-box-plots-q08","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":8,"marks":5,"tier":"H","calc":false,"text":"

A weather station recorded the daily rainfall, in mm, on each of 60 days.

  1. Use this information to draw a box plot for the rainfall.
  2. Estimate the number of days on which the rainfall was more than 22 mm.
","answerType":"multi","answer":"(a) box plot with whiskers at 2 and 32, box from 10 to 22, median line at 15 (b) 15 days","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"whiskers at 2 and 32, box 10 to 22, median at 15"},{"label":"b","answer":"15"}],"solution":["Highest rainfall = lowest + range = 2 + 30 = 32 mm.","Upper quartile = lower quartile + IQR = 10 + 12 = 22 mm.","Draw whiskers at 2 and 32, the box from 10 to 22 and the median line at 15.","22 mm is the upper quartile, so about a quarter of the days had more rainfall: 60 ÷ 4 = 15 days."],"diagram":{"type":"axes","x":{"min":0,"max":35,"step":5,"minor":5,"label":"Rainfall (mm)"},"y":{"min":0,"max":1,"hide":true},"xy":false,"height":150,"rows":1,"boxplots":[{"row":0,"min":2,"q1":10,"med":15,"q3":22,"max":32,"answer":true}]},"draw":true,"revision":"eb241a6d7270"},{"id":"g6-box-plots-q09","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

Here are the times, in seconds, taken by the 15 swimmers in Team A to swim 50 metres.

41  38  45  36  50  39  43  47  35  44  40  52  37  46  42

  1. Find the median, the lower quartile and the upper quartile of the times for Team A.
  2. Team B's times had a median of 44 seconds and an interquartile range of 13 seconds. Compare the times of the two teams. Give two comparisons.
","answerType":"multi","answer":"(a) median 42, LQ 38, UQ 46 (b) Team A's median (42 s) is lower than Team B's (44 s), so Team A was faster on average; Team A's IQR (8 s) is smaller than Team B's (13 s), so Team A's times are more consistent","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"median 42, lower quartile 38, upper quartile 46"},{"label":"b","answer":"Team A has the lower median so was faster on average; Team A has the smaller IQR so was more consistent"}],"solution":["Ordered times: 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 52.","With 15 values the median is the 8th value: 42 seconds.","The lower quartile is the 4th value: 38. The upper quartile is the 12th value: 46.","Team A's interquartile range = 46 - 38 = 8 seconds.","Team A's median of 42 is lower than Team B's 44, so on average Team A swam faster; Team A's IQR of 8 is smaller than Team B's 13, so Team A's times were more consistent."],"diagram":null,"draw":false,"revision":"9b8b8b85601c"},{"id":"g6-box-plots-q10","topicId":"g6-box-plots","topic":"Box Plots","grade":"6","topicGrade":"6","strand":"S","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

The table summarises the battery life, in hours, of 80 phones of Model X.

Minimum8
Lower quartile13
Median16
Upper quartile20
Maximum27

Jay tries to draw a box plot for Model X. On his diagram the whiskers are at 13 and 20 and the box goes from 8 to 27 with a line at 16.

  1. Write down two mistakes Jay has made.
  2. On the grid, draw a correct box plot for Model X.
  3. The box plot for Model Y is drawn on the same grid. Model Y has a median of 15 hours and an interquartile range of 4 hours. Compare the battery lives of the two models.
","answerType":"multi","answer":"(a) The box should go from the quartiles (13 to 20), not from the minimum and maximum; the whiskers should reach 8 and 27, not 13 and 20 - Jay has swapped them (b) box plot with whiskers at 8 and 27, box from 13 to 20, median line at 16 (c) Model X's median (16) is higher than Model Y's (15), so Model X lasts longer on average; Model X's IQR (7) is larger than Model Y's (4), so Model X's battery life is less consistent","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Box drawn between min and max instead of between the quartiles; whiskers drawn at the quartiles instead of at min and max"},{"label":"b","answer":"whiskers at 8 and 27, box 13 to 20, median at 16"},{"label":"c","answer":"Model X has the higher median so lasts longer on average; Model X has the larger IQR so is less consistent"}],"solution":["Jay swapped the box and the whiskers: the box must span the quartiles 13 to 20 and the whiskers must reach the minimum 8 and maximum 27.","Draw whiskers at 8 and 27, the box from 13 to 20 and the median line at 16.","Model X's interquartile range = 20 - 13 = 7 hours.","Model X's median of 16 hours is higher than Model Y's 15 hours, so on average Model X lasts longer; Model X's IQR of 7 hours is larger than Model Y's 4 hours, so Model X's battery life varies more."],"diagram":{"type":"axes","x":{"min":0,"max":30,"step":5,"minor":5,"label":"Battery life (hours)"},"y":{"min":0,"max":1,"hide":true},"xy":false,"height":170,"rows":2,"boxplots":[{"row":0,"min":8,"q1":13,"med":16,"q3":20,"max":27,"label":"Model X","answer":true},{"row":1,"min":10,"q1":13,"med":15,"q3":17,"max":22,"label":"Model Y"}]},"draw":true,"revision":"1234925b8f69"},{"id":"g6-capture-recapture-q01","topicId":"g6-capture-recapture","topic":"Capture Recapture","grade":"6","topicGrade":"6","strand":"S","difficulty":1,"marks":2,"tier":"H","calc":false,"text":"

A park keeper wants to estimate the number of fish in a lake.

She catches 30 fish, tags them and puts them back in the lake.

The next day she catches 40 fish. 8 of these fish are tagged.

Work out an estimate for the number of fish in the lake.

","answerType":"exact","answer":"150","answerDisplay":null,"accept":["150 fish"],"parts":[],"solution":["Proportion of tagged fish in the second sample = 840 = 15.","Assume the same proportion of the whole population is tagged, so 30 fish is 15 of the population.","Estimate = 30 × 40 ÷ 8 = 150."],"diagram":null,"draw":false,"revision":"0b680b052088"},{"id":"g6-capture-recapture-q02","topicId":"g6-capture-recapture","topic":"Capture Recapture","grade":"6","topicGrade":"6","strand":"S","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

An ecologist wants to estimate the number of blackbirds in a wood.

She catches 25 blackbirds, puts a ring on each one and releases them.

A week later she catches 60 blackbirds. 5 of them have rings.

  1. Work out an estimate for the number of blackbirds in the wood.
  2. State one assumption you made in working out your estimate.
","answerType":"multi","answer":"(a) 300 (b) The population has not changed between the samples (no births, deaths or migration), the ringed birds mixed fully with the rest, and no rings were lost","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"300"},{"label":"b","answer":"Any one of: the population is unchanged between visits; the ringed birds mix fully with the population; rings are not lost; each bird is equally likely to be caught"}],"solution":["Proportion ringed in the second sample = 560 = 112.","Assume 25 ringed birds is 112 of the population.","Estimate = 25 × 60 ÷ 5 = 300.","For the assumption part, one valid answer is that the marked animals mixed fully with the rest before the second sample. Alternatively, assume no marks were lost or the population did not change. Only one is required."],"diagram":null,"draw":false,"revision":"ddeefa129e7e"},{"id":"g6-capture-recapture-q03","topicId":"g6-capture-recapture","topic":"Capture Recapture","grade":"6","topicGrade":"6","strand":"S","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

A gardener wants to estimate the number of snails in her garden.

One evening she collects 45 snails, marks each shell with a dot of paint and releases them.

Two evenings later she collects 36 snails. 9 of them are marked.

  1. Work out an estimate for the number of snails in the garden.
  2. Write down one assumption you have made.
","answerType":"multi","answer":"(a) 180 (b) The marked snails mixed fully with the others and the population did not change between the two evenings","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"180"},{"label":"b","answer":"Any one of: marked snails mix evenly; no marks wash off; population unchanged; every snail equally likely to be collected"}],"solution":["Proportion marked in the second sample = 936 = 14.","Assume 45 marked snails is 14 of the population.","Estimate = 45 × 36 ÷ 9 = 180.","For the assumption part, one valid answer is that the marked animals mixed fully with the rest before the second sample. Alternatively, assume no marks were lost or the population did not change. Only one is required."],"diagram":null,"draw":false,"revision":"3b53802cf4a9"},{"id":"g6-capture-recapture-q04","topicId":"g6-capture-recapture","topic":"Capture Recapture","grade":"6","topicGrade":"6","strand":"S","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

A scientist wants to estimate the number of beetles in a field.

He captures 50 beetles, marks them and releases them.

Later he captures 48 beetles. 6 of them are marked.

  1. Work out an estimate for the number of beetles in the field.
  2. Some of the paint marks rubbed off before the second capture. Explain how this affects your estimate.
","answerType":"multi","answer":"(a) 400 (b) Fewer beetles would be counted as marked, so the estimate would be too big (an overestimate)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"400"},{"label":"b","answer":"The estimate is an overestimate - marked beetles are undercounted, so dividing by a smaller number makes the estimate too large"}],"solution":["Proportion marked in the second sample = 648 = 18.","Estimate = 50 × 48 ÷ 6 = 400.","If marks rubbed off, some recaptured beetles that were really marked are not recognised, so 6 is too small; dividing by a smaller number gives an estimate that is too big."],"diagram":null,"draw":false,"revision":"3f6b58b65fbf"},{"id":"g6-capture-recapture-q05","topicId":"g6-capture-recapture","topic":"Capture Recapture","grade":"6","topicGrade":"6","strand":"S","difficulty":5,"marks":4,"tier":"H","calc":false,"text":"

A warden wants to estimate the number of rabbits in a nature reserve.

She traps 60 rabbits, tags them and releases them.

The following week she traps 45 rabbits. 9 of them are tagged.

  1. Work out an estimate for the number of rabbits in the reserve.
  2. State one assumption needed for your estimate to be valid.
  3. The warden set her second traps close to where she released the tagged rabbits. Explain what effect this could have on your estimate.
","answerType":"multi","answer":"(a) 300 (b) Tagged rabbits mix fully with the population and the population is unchanged (c) Tagged rabbits would be over-represented in the second sample, so the estimate would be too small (an underestimate)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"300"},{"label":"b","answer":"Any one of: tagged rabbits mix fully; population unchanged; tags not lost; all rabbits equally likely to be trapped"},{"label":"c","answer":"Too many tagged rabbits would be recaptured, making the tagged proportion too high, so the estimate would be an underestimate"}],"solution":["Proportion tagged in the second sample = 945 = 15.","Estimate = 60 × 45 ÷ 9 = 300.","Trapping near the release site means tagged rabbits are more likely to be caught than untagged ones, so the fraction 945 is too high and the population estimate is too low.","For the assumption part, one valid answer is that the marked animals mixed fully with the rest before the second sample. Alternatively, assume no marks were lost or the population did not change. Only one is required."],"diagram":null,"draw":false,"revision":"441b6fe19e63"},{"id":"g6-capture-recapture-q06","topicId":"g6-capture-recapture","topic":"Capture Recapture","grade":"6","topicGrade":"6","strand":"S","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A ranger wants to estimate the number of deer in a forest.

He catches 80 deer, fits each one with an ear tag and releases them.

A month later he observes 120 deer. 15 of them have ear tags.

  1. Work out an estimate for the number of deer in the forest.
  2. The ranger believes the forest holds at least 800 deer. Does your estimate support his belief? Give a reason.
  3. State one assumption your estimate relies on.
","answerType":"multi","answer":"(a) 640 (b) No - the estimate of 640 is less than 800, so it does not support the belief (c) The tagged deer mixed fully with the herd and the population did not change during the month","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"640"},{"label":"b","answer":"No; 640 < 800 so the estimate does not support at least 800 deer"},{"label":"c","answer":"Any one of: tagged deer mix fully; population unchanged over the month; tags not lost; every deer equally likely to be observed"}],"solution":["Proportion tagged in the second sample = 15120 = 18.","Estimate = 80 × 120 ÷ 15 = 640.","640 is less than 800, so the estimate does not support the ranger's belief.","The method assumes the tagged deer mixed evenly and the population stayed the same."],"diagram":null,"draw":false,"revision":"100e59c03b57"},{"id":"g6-capture-recapture-q07","topicId":"g6-capture-recapture","topic":"Capture Recapture","grade":"6","topicGrade":"6","strand":"S","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

Using a capture-recapture study, the number of carp in a reservoir was estimated to be 600. In that study, 72 carp were tagged.

  1. A new sample of 50 carp is caught. Work out how many of these carp you would expect to be tagged.
  2. In fact, 10 of the 50 carp in the new sample are tagged. What does this suggest about the number of carp in the reservoir? Show your working.
","answerType":"multi","answer":"(a) 6 (b) The new estimate is 72 x 50 / 10 = 360, so the reservoir probably holds fewer carp than 600","answerDisplay":"(a) 6 (b) The new estimate is 72 x 5010 = 360, so the reservoir probably holds fewer carp than 600","accept":[],"parts":[{"label":"a","answer":"6"},{"label":"b","answer":"New estimate = 360, so the population is likely to be smaller than 600"}],"solution":["If the population is 600, the proportion tagged is 72600 = 0.12.","Expected tagged carp in a sample of 50 = 0.12 × 50 = 6.","Finding 10 tagged carp gives a new estimate of 72 × 50 ÷ 10 = 360.","More tagged carp than expected means the tagged fish make up a larger share of the population, so the population is probably smaller than 600."],"diagram":null,"draw":false,"revision":"8d2892406b41"},{"id":"g6-capture-recapture-q08","topicId":"g6-capture-recapture","topic":"Capture Recapture","grade":"6","topicGrade":"6","strand":"S","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

A marine biologist wants to estimate the number of crabs on a stretch of shore.

She collects 150 crabs, marks their shells and returns them to the shore.

The next day she collects a sample of 200 crabs. 12% of the crabs in this sample are marked.

  1. Work out an estimate for the number of crabs on the shore.
  2. State two assumptions your estimate relies on.
","answerType":"multi","answer":"(a) 1250 (b) Any two of: the marked crabs mixed fully with the population; the population did not change overnight; no marks were lost; every crab was equally likely to be collected","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1250"},{"label":"b","answer":"Two of: full mixing; unchanged population; no lost marks; equal chance of capture"}],"solution":["Marked crabs in the second sample = 12% of 200 = 24.","Proportion marked = 24200 = 0.12.","Assume 150 marked crabs is 12% of the population.","Estimate = 150 ÷ 0.12 = 1250 (equivalently 150 × 200 ÷ 24 = 1250).","b) Two assumptions are that the marked crabs mixed fully with the rest and that no marks were lost. Other valid assumptions include an unchanged population and equal chances of capture."],"diagram":null,"draw":false,"revision":"f5c861de6527"},{"id":"g6-circle-theorems-q01","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

A, B and C are points on a circle with centre O.

AOC is a diameter of the circle.

Angle CAB = 37°.

Work out the size of angle ACB. Give a reason for each stage of your working.

","answerType":"exact","answer":"53°","answerDisplay":null,"accept":["53"],"parts":[],"solution":["Angle ABC = 90° because the angle in a semicircle is 90°.","Angles in a triangle add up to 180°.","Angle ACB = 180° - 90° - 37° = 53°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[-3,3.6739403974420594e-16],"label":"A","style":"none","labelPos":"below-left"},{"at":[3,0],"label":"C","style":"none","labelPos":"below-right"},{"at":[0.8269120674509975,2.8837850878149567],"label":"B","style":"none","labelPos":"above"}],"segments":[{"from":[-3,3.6739403974420594e-16],"to":[3,0]},{"from":[-3,3.6739403974420594e-16],"to":[0.8269120674509975,2.8837850878149567]},{"from":[0.8269120674509975,2.8837850878149567],"to":[3,0]}],"angles":[{"at":[-3,3.6739403974420594e-16],"from":[3,0],"to":[0.8269120674509975,2.8837850878149567],"label":"37°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"1126bbb66457"},{"id":"g6-circle-theorems-q02","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":2,"marks":3,"tier":"H","calc":true,"text":"

A, B and C are points on a circle with centre O.

B lies on the major arc AC.

Angle AOC = 128°.

Work out the size of angle ABC. Give a reason for your answer.

","answerType":"exact","answer":"64°","answerDisplay":null,"accept":["64"],"parts":[],"solution":["The angle at the centre of a circle is twice the angle at the circumference when both stand on the same arc.","Angle ABC = 128° ÷ 2 = 64°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"above"},{"at":[2.6963821388975013,1.3151134403672322],"label":"A","style":"none","labelPos":"above"},{"at":[-2.6963821388975013,1.3151134403672318],"label":"C","style":"none","labelPos":"above"},{"at":[-5.51091059616309e-16,-3],"label":"B","style":"none","labelPos":"below-left"}],"segments":[{"from":[0,0],"to":[2.6963821388975013,1.3151134403672322]},{"from":[0,0],"to":[-2.6963821388975013,1.3151134403672318]},{"from":[2.6963821388975013,1.3151134403672322],"to":[-5.51091059616309e-16,-3]},{"from":[-5.51091059616309e-16,-3],"to":[-2.6963821388975013,1.3151134403672318]}],"angles":[{"at":[0,0],"from":[2.6963821388975013,1.3151134403672322],"to":[-2.6963821388975013,1.3151134403672318],"label":"128°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"8e6e0638beb6"},{"id":"g6-circle-theorems-q03","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

A and B are points on a circle with centre O.

TA and TB are tangents to the circle from the point T.

Angle ATB = 48°.

Work out the size of angle AOB. Give a reason for each stage of your working.

","answerType":"exact","answer":"132°","answerDisplay":null,"accept":["132"],"parts":[],"solution":["Angle OAT = angle OBT = 90° because a tangent to a circle is perpendicular to the radius at the point of contact.","Angles in a quadrilateral add up to 360°.","Angle AOB = 360° - 90° - 90° - 48° = 132°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[1.2202099292274007,2.7406363729278027],"label":"A","style":"none","labelPos":"above"},{"at":[1.2202099292274007,-2.7406363729278027],"label":"B","style":"none","labelPos":"below-left"},{"at":[7.375780006722715,0],"label":"T","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[1.2202099292274007,2.7406363729278027]},{"from":[0,0],"to":[1.2202099292274007,-2.7406363729278027]},{"from":[7.375780006722715,0],"to":[1.2202099292274007,2.7406363729278027]},{"from":[7.375780006722715,0],"to":[1.2202099292274007,-2.7406363729278027]}],"angles":[{"at":[7.375780006722715,0],"from":[1.2202099292274007,2.7406363729278027],"to":[1.2202099292274007,-2.7406363729278027],"label":"48°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"160c87452968"},{"id":"g6-circle-theorems-q04","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

A, B, C and D are points on a circle, in that order.

Angle ABD = 29° and angle DBC = 46°.

Work out the size of

  1. angle ACD,
  2. angle ADC.

Give a reason for each stage of your working.

","answerType":"multi","answer":"a) 29°, b) 105°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"29°"},{"label":"b","answer":"105°"}],"solution":["a) Angle ACD = angle ABD = 29° because angles in the same segment, standing on chord AD, are equal.","b) Angle ABC = 29° + 46° = 75°.","Opposite angles of a cyclic quadrilateral add up to 180°.","Angle ADC = 180° - 75° = 105°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"above"},{"at":[3,0],"label":"A","style":"none","labelPos":"above"},{"at":[0.5209445330007912,2.954423259036624],"label":"B","style":"none","labelPos":"above"},{"at":[-2.598076211353316,-1.5000000000000004],"label":"C","style":"none","labelPos":"below-left"},{"at":[1.589757792699614,-2.544144288469278],"label":"D","style":"none","labelPos":"below-right"}],"segments":[{"from":[3,0],"to":[0.5209445330007912,2.954423259036624]},{"from":[0.5209445330007912,2.954423259036624],"to":[-2.598076211353316,-1.5000000000000004]},{"from":[-2.598076211353316,-1.5000000000000004],"to":[1.589757792699614,-2.544144288469278]},{"from":[1.589757792699614,-2.544144288469278],"to":[3,0]},{"from":[0.5209445330007912,2.954423259036624],"to":[1.589757792699614,-2.544144288469278]},{"from":[3,0],"to":[-2.598076211353316,-1.5000000000000004]}],"angles":[{"at":[0.5209445330007912,2.954423259036624],"from":[3,0],"to":[1.589757792699614,-2.544144288469278],"label":"29°"},{"at":[0.5209445330007912,2.954423259036624],"from":[1.589757792699614,-2.544144288469278],"to":[-2.598076211353316,-1.5000000000000004],"label":"46°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a27865a3069d"},{"id":"g6-circle-theorems-q05","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

A, B and C are points on a circle with centre O.

C lies on the major arc AB.

Angle OAB = 28°.

Work out the size of angle ACB. Give a reason for each stage of your working.

","answerType":"exact","answer":"62°","answerDisplay":null,"accept":["62"],"parts":[],"solution":["OA = OB because both are radii, so triangle OAB is isosceles and angle OBA = angle OAB = 28°.","Angles in a triangle add up to 180°, so angle AOB = 180° - 28° - 28° = 124°.","The angle at the centre is twice the angle at the circumference standing on the same arc.","Angle ACB = 124° ÷ 2 = 62°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"above"},{"at":[2.648842778576781,1.4084146883576725],"label":"A","style":"none","labelPos":"above"},{"at":[-2.64884277857678,1.4084146883576731],"label":"B","style":"none","labelPos":"above"},{"at":[-5.51091059616309e-16,-3],"label":"C","style":"none","labelPos":"below-left"}],"segments":[{"from":[0,0],"to":[2.648842778576781,1.4084146883576725]},{"from":[0,0],"to":[-2.64884277857678,1.4084146883576731]},{"from":[2.648842778576781,1.4084146883576725],"to":[-2.64884277857678,1.4084146883576731]},{"from":[2.648842778576781,1.4084146883576725],"to":[-5.51091059616309e-16,-3]},{"from":[-2.64884277857678,1.4084146883576731],"to":[-5.51091059616309e-16,-3]}],"angles":[{"at":[2.648842778576781,1.4084146883576725],"from":[0,0],"to":[-2.64884277857678,1.4084146883576731],"label":"28°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"7fd5a31594f5"},{"id":"g6-circle-theorems-q06","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A and C are points on a circle with centre O.

PC is a tangent to the circle at C.

Angle PCA = 58°.

Work out the size of angle OAC. Give a reason for each stage of your working.

","answerType":"exact","answer":"32°","answerDisplay":null,"accept":["32"],"parts":[],"solution":["Angle OCP = 90° because a tangent to a circle is perpendicular to the radius at the point of contact.","Angle OCA = 90° - 58° = 32°.","OA = OC because both are radii, so triangle OAC is isosceles.","Angle OAC = angle OCA = 32°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[3,0],"label":"C","style":"none","labelPos":"below-right"},{"at":[-1.3151134403672327,2.696382138897501],"label":"A","style":"none","labelPos":"above"},{"at":[3,5],"label":"P","style":"none","labelPos":"above"}],"segments":[{"from":[3,5],"to":[3,0]},{"from":[3,0],"to":[-1.3151134403672327,2.696382138897501]},{"from":[0,0],"to":[3,0]},{"from":[0,0],"to":[-1.3151134403672327,2.696382138897501]}],"angles":[{"at":[3,0],"from":[3,5],"to":[-1.3151134403672327,2.696382138897501],"label":"58°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ccb61998f45d"},{"id":"g6-circle-theorems-q07","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

A, B and C are points on a circle with centre O.

B lies on the minor arc AC.

Angle AOC = 114°.

Work out the size of

  1. angle ABC,
  2. angle OAC.

Give a reason for each stage of your working.

","answerType":"multi","answer":"a) 123°, b) 33°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"123°"},{"label":"b","answer":"33°"}],"solution":["a) The reflex angle AOC = 360° - 114° = 246° because angles around a point add up to 360°.","The angle at the centre is twice the angle at the circumference standing on the same arc. Angle ABC stands on the major arc, so it pairs with the reflex angle at O.","Angle ABC = 246° ÷ 2 = 123°.","b) OA = OC because both are radii, so triangle OAC is isosceles.","Angle OAC = (180° - 114°) ÷ 2 = 33°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-right"},{"at":[2.516011703836272,1.6339171050450814],"label":"A","style":"none","labelPos":"above"},{"at":[-2.5160117038362726,1.6339171050450805],"label":"C","style":"none","labelPos":"above"},{"at":[1.8369701987210297e-16,3],"label":"B","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[2.516011703836272,1.6339171050450814]},{"from":[0,0],"to":[-2.5160117038362726,1.6339171050450805]},{"from":[2.516011703836272,1.6339171050450814],"to":[1.8369701987210297e-16,3]},{"from":[1.8369701987210297e-16,3],"to":[-2.5160117038362726,1.6339171050450805]},{"from":[2.516011703836272,1.6339171050450814],"to":[-2.5160117038362726,1.6339171050450805]}],"angles":[{"at":[0,0],"from":[2.516011703836272,1.6339171050450814],"to":[-2.5160117038362726,1.6339171050450805],"label":"114°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"65712ee7da07"},{"id":"g6-circle-theorems-q08","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

A, B and C are points on a circle.

DAE is a tangent to the circle at A.

B is on the opposite side of the chord AC from E.

Angle CAE = 63° and angle BAC = 44°.

Work out the size of

  1. angle ABC,
  2. angle ACB.

Give a reason for each stage of your working.

","answerType":"multi","answer":"a) 63°, b) 73°","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"63°"},{"label":"b","answer":"73°"}],"solution":["a) Angle ABC = angle CAE = 63° because the angle between a tangent and a chord is equal to the angle in the alternate segment.","b) Angles in a triangle add up to 180°.","Angle ACB = 180° - 63° - 44° = 73°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"above"},{"at":[-5.51091059616309e-16,-3],"label":"A","style":"none","labelPos":"below-left"},{"at":[2.4270509831248424,1.7633557568774194],"label":"C","style":"none","labelPos":"above"},{"at":[-1.6775787104122402,2.4871127176651253],"label":"B","style":"none","labelPos":"above"},{"at":[-5,-3],"label":"D","style":"none","labelPos":"below-left"},{"at":[5,-3],"label":"E","style":"none","labelPos":"below-right"}],"segments":[{"from":[-5,-3],"to":[5,-3]},{"from":[-5.51091059616309e-16,-3],"to":[-1.6775787104122402,2.4871127176651253]},{"from":[-1.6775787104122402,2.4871127176651253],"to":[2.4270509831248424,1.7633557568774194]},{"from":[-5.51091059616309e-16,-3],"to":[2.4270509831248424,1.7633557568774194]}],"angles":[{"at":[-5.51091059616309e-16,-3],"from":[2.4270509831248424,1.7633557568774194],"to":[5,-3],"label":"63°"},{"at":[-5.51091059616309e-16,-3],"from":[-1.6775787104122402,2.4871127176651253],"to":[2.4270509831248424,1.7633557568774194],"label":"44°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"720b03220fad"},{"id":"g6-circle-theorems-q09","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":9,"marks":4,"tier":"H","calc":false,"text":"

A, B and C are points on a circle with centre O.

PA and PB are tangents to the circle from the point P.

C lies on the major arc AB.

Angle APB = 42°.

Work out the size of angle ACB. Give a reason for each stage of your working.

","answerType":"exact","answer":"69°","answerDisplay":null,"accept":["69"],"parts":[],"solution":["Angle OAP = angle OBP = 90° because a tangent is perpendicular to the radius at the point of contact.","Angles in a quadrilateral add up to 360°, so angle AOB = 360° - 90° - 90° - 42° = 138°.","The angle at the centre is twice the angle at the circumference standing on the same arc.","Angle ACB = 138° ÷ 2 = 69°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[1.075103848635901,2.8007412794916053],"label":"A","style":"none","labelPos":"above"},{"at":[1.075103848635901,-2.8007412794916053],"label":"B","style":"none","labelPos":"below-left"},{"at":[8.371284328876005,0],"label":"P","style":"none","labelPos":"below-right"},{"at":[-3,3.6739403974420594e-16],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[1.075103848635901,2.8007412794916053]},{"from":[0,0],"to":[1.075103848635901,-2.8007412794916053]},{"from":[8.371284328876005,0],"to":[1.075103848635901,2.8007412794916053]},{"from":[8.371284328876005,0],"to":[1.075103848635901,-2.8007412794916053]},{"from":[1.075103848635901,2.8007412794916053],"to":[-3,3.6739403974420594e-16]},{"from":[1.075103848635901,-2.8007412794916053],"to":[-3,3.6739403974420594e-16]}],"angles":[{"at":[8.371284328876005,0],"from":[1.075103848635901,2.8007412794916053],"to":[1.075103848635901,-2.8007412794916053],"label":"42°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"52c3e384c6a1"},{"id":"g6-circle-theorems-q10","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":10,"marks":4,"tier":"H","calc":false,"text":"

A, B, C and D are points on a circle, in that order.

The chords AC and BD intersect at the point E inside the circle.

Prove that triangle ABE is similar to triangle DCE.

","answerType":"open","answer":"Proof using angles in the same segment (twice) and vertically opposite angles, so the triangles are similar (AA)","answerDisplay":null,"accept":[],"parts":[],"solution":["Angle BAE = angle BAC and angle CDE = angle CDB. Angle BAC = angle BDC because angles in the same segment, standing on chord BC, are equal. So angle BAE = angle CDE.","Angle ABE = angle ABD and angle DCE = angle ACD. Angle ABD = angle ACD because angles in the same segment, standing on chord AD, are equal. So angle ABE = angle DCE.","Angle AEB = angle DEC because vertically opposite angles are equal.","The triangles have equal angles, so triangle ABE is similar to triangle DCE.","Similar triangles have the same shape, though their sizes can differ. AA means that two matching angles are sufficient: the third also matches because each triangle’s angles total 180°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[-2.598076211353316,1.4999999999999998],"label":"A","style":"none","labelPos":"above"},{"at":[2.598076211353316,1.4999999999999998],"label":"B","style":"none","labelPos":"above"},{"at":[2.598076211353316,-1.4999999999999998],"label":"C","style":"none","labelPos":"below-right"},{"at":[-2.598076211353316,-1.4999999999999998],"label":"D","style":"none","labelPos":"below-left"},{"at":[0,0],"label":"E","style":"dot","labelPos":"below-right"}],"segments":[{"from":[-2.598076211353316,1.4999999999999998],"to":[2.598076211353316,1.4999999999999998]},{"from":[2.598076211353316,1.4999999999999998],"to":[2.598076211353316,-1.4999999999999998]},{"from":[2.598076211353316,-1.4999999999999998],"to":[-2.598076211353316,-1.4999999999999998]},{"from":[-2.598076211353316,-1.4999999999999998],"to":[-2.598076211353316,1.4999999999999998]},{"from":[-2.598076211353316,1.4999999999999998],"to":[2.598076211353316,-1.4999999999999998]},{"from":[2.598076211353316,1.4999999999999998],"to":[-2.598076211353316,-1.4999999999999998]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"d57d54c42593"},{"id":"g6-circle-theorems-q11","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":11,"marks":5,"tier":"H","calc":true,"text":"

A, B and C are points on a circle with centre O.

PA and PB are tangents to the circle from the point P.

C lies on the major arc AB.

Angle APB = x°.

Find the size of angle ACB in terms of x. Give your answer in its simplest form and give a reason for each stage of your working.

","answerType":"exact","answer":"(90 - x/2)°","answerDisplay":"(90 - x2)°","accept":["90 - x/2","(180 - x)/2"],"parts":[],"solution":["Angle OAP = angle OBP = 90° because a tangent is perpendicular to the radius at the point of contact.","Angles in a quadrilateral add up to 360°, so angle AOB = 360° - 90° - 90° - x° = (180 - x)°.","The angle at the centre is twice the angle at the circumference standing on the same arc.","Angle ACB = (180 - x)° ÷ 2 = (90 - x2)°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[1.2678547852220983,2.7189233611099497],"label":"A","style":"none","labelPos":"above"},{"at":[1.2678547852220983,-2.7189233611099497],"label":"B","style":"none","labelPos":"below-left"},{"at":[7.098604749457495,0],"label":"P","style":"none","labelPos":"below-right"},{"at":[-3,3.6739403974420594e-16],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[1.2678547852220983,2.7189233611099497]},{"from":[0,0],"to":[1.2678547852220983,-2.7189233611099497]},{"from":[7.098604749457495,0],"to":[1.2678547852220983,2.7189233611099497]},{"from":[7.098604749457495,0],"to":[1.2678547852220983,-2.7189233611099497]},{"from":[1.2678547852220983,2.7189233611099497],"to":[-3,3.6739403974420594e-16]},{"from":[1.2678547852220983,-2.7189233611099497],"to":[-3,3.6739403974420594e-16]}],"angles":[{"at":[7.098604749457495,0],"from":[1.2678547852220983,2.7189233611099497],"to":[1.2678547852220983,-2.7189233611099497],"label":"x°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"08dbbdbc7496"},{"id":"g6-circle-theorems-q12","topicId":"g6-circle-theorems","topic":"Circle Theorems","grade":"6","topicGrade":"6","strand":"G","difficulty":12,"marks":5,"tier":"H","calc":false,"text":"

ABCD is a cyclic quadrilateral.

angle BAD : angle BCD = 4 : 5

angle ABC = angle BAD + 30°

Work out the size of angle ADC. Give a reason for each stage of your working.

","answerType":"exact","answer":"70°","answerDisplay":null,"accept":["70"],"parts":[],"solution":["Opposite angles of a cyclic quadrilateral add up to 180°, so angle BAD + angle BCD = 180°.","The ratio 4 : 5 has 9 parts, so one part = 180° ÷ 9 = 20°.","Angle BAD = 4 × 20° = 80° and angle BCD = 5 × 20° = 100°.","Angle ABC = 80° + 30° = 110°.","Opposite angles of a cyclic quadrilateral add up to 180°, so angle ADC = 180° - 110° = 70°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[-2.954423259036624,0.5209445330007908],"label":"A","style":"none","labelPos":"above"},{"at":[1.8369701987210297e-16,3],"label":"B","style":"none","labelPos":"above"},{"at":[2.954423259036624,0.520944533000791],"label":"C","style":"none","labelPos":"above"},{"at":[1.0260604299770064,-2.819077862357725],"label":"D","style":"none","labelPos":"below-right"}],"segments":[{"from":[-2.954423259036624,0.5209445330007908],"to":[1.8369701987210297e-16,3]},{"from":[1.8369701987210297e-16,3],"to":[2.954423259036624,0.520944533000791]},{"from":[2.954423259036624,0.520944533000791],"to":[1.0260604299770064,-2.819077862357725]},{"from":[1.0260604299770064,-2.819077862357725],"to":[-2.954423259036624,0.5209445330007908]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a83f94e91054"},{"id":"g6-cumulative-frequency-q01","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":1,"marks":1,"tier":"H","calc":false,"text":"

The frequency table shows the times, t minutes, taken by 35 students to finish a puzzle.

Time (t minutes)Frequency
0 < t ≤ 58
5 < t ≤ 1012
10 < t ≤ 159
15 < t ≤ 206

Noah makes a cumulative frequency table for this data. He writes the cumulative frequencies 8, 20, 29, 34.

Explain what is wrong with Noah's cumulative frequencies.

","answerType":"open","answer":"The final cumulative frequency should be 35 (the total frequency), not 34 - the last value is wrong (29 + 6 = 35)","answerDisplay":null,"accept":["The last cumulative frequency must equal the total 35","29 + 6 = 35 not 34"],"parts":[],"solution":["The cumulative frequencies should be 8, 8 + 12 = 20, 20 + 9 = 29, 29 + 6 = 35.","Noah's final value of 34 cannot be right because the cumulative frequencies must end at the total frequency, 35."],"diagram":null,"draw":false,"revision":"9eb55d2c1cb2"},{"id":"g6-cumulative-frequency-q02","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":2,"marks":2,"tier":"H","calc":false,"text":"

The table shows the masses, m grams, of 40 tomatoes.

Mass (m grams)Frequency
50 < m ≤ 605
60 < m ≤ 7011
70 < m ≤ 8014
80 < m ≤ 907
90 < m ≤ 1003

Complete the cumulative frequency table.

Mass (m grams)Cumulative frequency
50 < m ≤ 60
50 < m ≤ 70
50 < m ≤ 80
50 < m ≤ 90
50 < m ≤ 100
","answerType":"multi","answer":"5, 16, 30, 37, 40","answerDisplay":null,"accept":[],"parts":[{"label":"up to 60","answer":"5"},{"label":"up to 70","answer":"16"},{"label":"up to 80","answer":"30"},{"label":"up to 90","answer":"37"},{"label":"up to 100","answer":"40"}],"solution":["Add the frequencies as you go down the table: 5, 5 + 11 = 16, 16 + 14 = 30, 30 + 7 = 37, 37 + 3 = 40.","The final value 40 matches the total number of tomatoes."],"diagram":null,"draw":false,"revision":"b1caf61f5b48"},{"id":"g6-cumulative-frequency-q03","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

The table shows the ages, a years, of the 60 members of a running club.

Age (a years)Frequency
10 < a ≤ 209
20 < a ≤ 3015
30 < a ≤ 4018
40 < a ≤ 5012
50 < a ≤ 606
  1. Complete a cumulative frequency table for this data.
  2. Write down the class interval that contains the median age.
","answerType":"multi","answer":"(a) 9, 24, 42, 54, 60 (b) 30 < a <= 40","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"9, 24, 42, 54, 60"},{"label":"b","answer":"30 < a <= 40"}],"solution":["Cumulative frequencies: 9, 9 + 15 = 24, 24 + 18 = 42, 42 + 12 = 54, 54 + 6 = 60.","For 60 members, the median lies between the 30th and 31st ordered values.","The first 24 members are aged 30 or under and the first 42 are aged 40 or under. Both the 30th and 31st values therefore lie in 30 < a ≤ 40, so the median lies in this class."],"diagram":null,"draw":false,"revision":"f8f53e6405c8"},{"id":"g6-cumulative-frequency-q04","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":4,"marks":3,"tier":"H","calc":false,"text":"

40 students took a typing test. Salma drew a cumulative frequency graph for their times.

Salma plotted each cumulative frequency against the midpoint of its class interval.

  1. Explain the mistake Salma made.
  2. Salma redraws the graph correctly. The correct graph shows a cumulative frequency of 34 at a time of 40 seconds. How many students took longer than 40 seconds?
  3. What percentage of the 40 students took longer than 40 seconds?
","answerType":"multi","answer":"(a) Cumulative frequencies must be plotted at the upper class boundaries, not the midpoints (b) 6 (c) 15%","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"She should plot each cumulative frequency at the upper end (upper class boundary) of the interval, not the midpoint"},{"label":"b","answer":"6"},{"label":"c","answer":"15%"}],"solution":["A cumulative frequency tells you how many values are less than or equal to the top of the class, so it must be plotted at the upper class boundary.","Students taking longer than 40 seconds = 40 - 34 = 6.","6 out of 40 = 640 = 15%."],"diagram":null,"draw":false,"revision":"f28e3688a441"},{"id":"g6-cumulative-frequency-q05","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":5,"marks":6,"tier":"H","calc":true,"text":"

The table shows the heights, h cm, of 80 sunflower seedlings.

Height (h cm)Frequency
0 < h ≤ 106
10 < h ≤ 2014
20 < h ≤ 3026
30 < h ≤ 4022
40 < h ≤ 5012
  1. Complete the cumulative frequency table.
  2. On the grid, draw a cumulative frequency graph for this information.
  3. Use your graph to find an estimate for the median height.
  4. Use your graph to find an estimate for the interquartile range of the heights.
","answerType":"multi","answer":"(a) 6, 20, 46, 68, 80 (b) points plotted at upper boundaries, joined with a smooth increasing curve (c) about 28 cm (accept 26-29) (d) about 16 cm (accept 14-19)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6, 20, 46, 68, 80"},{"label":"b","answer":"points at (10,6),(20,20),(30,46),(40,68),(50,80) joined by a smooth curve through the origin"},{"label":"c","answer":"approximately 28 cm (accept 26-29)"},{"label":"d","answer":"approximately 16 cm (accept 14-19)"}],"solution":["Cumulative frequencies: 6, 20, 46, 68, 80.","Plot each cumulative frequency at the upper class boundary and join with a smooth curve.","Median: read across from 40 on the cumulative frequency axis; this gives roughly 28 cm.","Lower quartile: read across from 20, giving about 20 cm. Upper quartile: read across from 60, giving about 36 cm.","Interquartile range is approximately 36 - 20 = 16 cm."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":50,"step":10,"minor":2,"label":"Height (cm)"},"y":{"min":0,"max":80,"step":10,"minor":5,"label":"Cumulative frequency"},"curves":[{"pts":[[0,0],[10,6],[20,20],[30,46],[40,68],[50,80]],"smooth":true,"marks":true,"answer":true}],"lines":[{"y":40,"from":0,"to":27.7,"dashed":true,"answer":true},{"x":27.7,"from":0,"to":40,"dashed":true,"answer":true},{"y":20,"from":0,"to":20,"dashed":true,"answer":true},{"x":20,"from":0,"to":20,"dashed":true,"answer":true},{"y":60,"from":0,"to":36.4,"dashed":true,"answer":true},{"x":36.4,"from":0,"to":60,"dashed":true,"answer":true}]},"draw":true,"revision":"6229cd1e7279"},{"id":"g6-cumulative-frequency-q06","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":6,"marks":6,"tier":"H","calc":true,"text":"

The table shows the journey times, t minutes, of 60 commuters.

Time (t minutes)Frequency
0 < t ≤ 104
10 < t ≤ 2010
20 < t ≤ 3017
30 < t ≤ 4015
40 < t ≤ 509
50 < t ≤ 605
  1. Complete the cumulative frequency table.
  2. On the grid, draw a cumulative frequency graph for this information.
  3. Use your graph to find an estimate for the median journey time.
  4. Use your graph to estimate the percentage of commuters whose journey took more than 45 minutes.
","answerType":"multi","answer":"(a) 4, 14, 31, 46, 55, 60 (b) points plotted at upper boundaries, joined with a smooth increasing curve (c) about 29 minutes (accept 28-31) (d) about 16% (accept 13% to 20%)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"4, 14, 31, 46, 55, 60"},{"label":"b","answer":"points at (10,4),(20,14),(30,31),(40,46),(50,55),(60,60) joined by a smooth curve"},{"label":"c","answer":"approximately 29 minutes (accept 28-31)"},{"label":"d","answer":"approximately 16% (accept 13-20%)"}],"solution":["Cumulative frequencies: 4, 14, 31, 46, 55, 60.","Plot at the upper class boundaries and join with a smooth curve.","Median: read across from 30 on the cumulative frequency axis, giving roughly 29 minutes.","At 45 minutes the cumulative frequency is roughly 50 or 51, so about 60 - 50.5 = 9.5 commuters took longer.","9.5 out of 60 is roughly 16%."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":60,"step":10,"minor":2,"label":"Journey time (minutes)"},"y":{"min":0,"max":60,"step":10,"minor":5,"label":"Cumulative frequency"},"curves":[{"pts":[[0,0],[10,4],[20,14],[30,31],[40,46],[50,55],[60,60]],"smooth":true,"marks":true,"answer":true}],"lines":[{"y":30,"from":0,"to":29.4,"dashed":true,"answer":true},{"x":29.4,"from":0,"to":30,"dashed":true,"answer":true},{"x":45,"from":0,"to":50.5,"dashed":true,"answer":true},{"y":50.5,"from":0,"to":45,"dashed":true,"answer":true}]},"draw":true,"revision":"39da7af0cbce"},{"id":"g6-cumulative-frequency-q07","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":7,"marks":6,"tier":"H","calc":false,"text":"

The cumulative frequency graph shows the test scores of the 60 students in Class A.

  1. Use the graph to find an estimate for the median score.
  2. Use the graph to find an estimate for the interquartile range of the scores.
  3. The 60 students in Class B had a median score of 45 and an interquartile range of 12. Compare the scores of the two classes.
","answerType":"multi","answer":"(a) about 32 (accept 31-34) (b) about 18 (accept 15-21) (c) Class B's median is higher, so Class B scored better on average; Class B's IQR is smaller, so Class B's scores are more consistent","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"approximately 32 (accept 31-34)"},{"label":"b","answer":"approximately 18 (accept 15-21)"},{"label":"c","answer":"Class B has the higher median (45 vs about 32) so did better on average; Class B has the smaller IQR (12 vs about 18) so its scores are less spread out"}],"solution":["Median: read across from 30 (half of 60) to the curve and down, giving roughly 32 marks.","Lower quartile: read across from 15, giving about 23. Upper quartile: read across from 45, giving about 41.","Interquartile range is approximately 41 - 23 = 18.","Class B's median of 45 is higher than Class A's, so on average Class B scored better.","Class B's interquartile range of 12 is smaller than Class A's, so Class B's scores are more consistent."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":60,"step":10,"minor":5,"label":"Score"},"y":{"min":0,"max":60,"step":10,"minor":5,"label":"Cumulative frequency"},"curves":[{"pts":[[0,0],[10,3],[20,10],[30,26],[40,44],[50,55],[60,60]],"smooth":true,"marks":true}],"lines":[{"y":30,"from":0,"to":32.2,"dashed":true,"answer":true},{"x":32.2,"from":0,"to":30,"dashed":true,"answer":true},{"y":15,"from":0,"to":23.1,"dashed":true,"answer":true},{"x":23.1,"from":0,"to":15,"dashed":true,"answer":true},{"y":45,"from":0,"to":40.9,"dashed":true,"answer":true},{"x":40.9,"from":0,"to":45,"dashed":true,"answer":true}]},"draw":false,"revision":"3b5f2d3e5c2a"},{"id":"g6-cumulative-frequency-q08","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":8,"marks":6,"tier":"H","calc":false,"text":"

The cumulative frequency graph shows the weights, in kg, of 80 parcels sent from a depot on Monday.

The lightest parcel weighed 0.5 kg and the heaviest weighed 9.5 kg.

  1. Use the graph to draw a box plot for the weights of Monday's parcels.
  2. Use the graph to estimate the number of parcels that weighed more than 7 kg.
  3. On Tuesday the parcels had a median weight of 4.4 kg and an interquartile range of 2.1 kg. Compare the weights of Monday's and Tuesday's parcels.
","answerType":"multi","answer":"(a) box plot with min 0.5, LQ about 3.75, median about 5.2, UQ about 6.8, max 9.5 (b) about 18 parcels (accept 15-21) (c) Monday's median (about 5.2 kg) is higher than Tuesday's (4.4 kg), so Monday's parcels were heavier on average; Monday's IQR (about 3 kg) is larger than Tuesday's (2.1 kg), so Monday's weights were more spread out","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"min 0.5, LQ approximately 3.75 (accept 3.5-4), median approximately 5.2 (accept 5-5.4), UQ approximately 6.8 (accept 6.5-7.1), max 9.5"},{"label":"b","answer":"approximately 18 (accept 15-21)"},{"label":"c","answer":"Monday's median is higher so Monday's parcels were heavier on average; Monday's IQR is larger so Monday's weights varied more"}],"solution":["Median: read across from 40, giving about 5.2 kg.","Lower quartile: read across from 20, giving about 3.75 kg. Upper quartile: read across from 60, giving about 6.8 kg.","Draw the box from the lower quartile to the upper quartile with a line at the median, and whiskers to 0.5 kg and 9.5 kg.","At 7 kg the cumulative frequency is about 62, so about 80 - 62 = 18 parcels weighed more than 7 kg.","Monday's median (about 5.2 kg) is higher than Tuesday's (4.4 kg): Monday's parcels were heavier on average. Monday's IQR (about 6.8 - 3.75 = 3.05 kg) is larger than Tuesday's (2.1 kg): Monday's weights were more spread out."],"diagram":{"type":"panels","cols":1,"panelWidth":440,"panelHeight":312,"items":[{"label":"","diagram":{"type":"axes","xy":false,"height":240,"x":{"min":0,"max":10,"step":1,"minor":2,"label":"Weight (kg)"},"y":{"min":0,"max":80,"step":10,"minor":5,"label":"Cumulative frequency"},"curves":[{"pts":[[0.5,0],[2,6],[4,22],[6,52],[8,72],[9.5,80]],"smooth":true,"marks":true}],"lines":[{"y":20,"from":0,"to":3.75,"dashed":true,"answer":true},{"x":3.75,"from":0,"to":20,"dashed":true,"answer":true},{"y":40,"from":0,"to":5.2,"dashed":true,"answer":true},{"x":5.2,"from":0,"to":40,"dashed":true,"answer":true},{"y":60,"from":0,"to":6.8,"dashed":true,"answer":true},{"x":6.8,"from":0,"to":60,"dashed":true,"answer":true},{"x":7,"from":0,"to":62,"dashed":true,"answer":true},{"y":62,"from":0,"to":7,"dashed":true,"answer":true}]}},{"label":"","diagram":{"type":"axes","x":{"min":0,"max":10,"step":1,"minor":2,"label":"Weight (kg)"},"y":{"min":0,"max":1,"hide":true},"xy":false,"height":240,"rows":1,"boxplots":[{"row":0,"min":0.5,"q1":3.75,"med":5.2,"q3":6.8,"max":9.5,"answer":true}]}}]},"draw":true,"revision":"7a52b5e8eb80"},{"id":"g6-cumulative-frequency-q09","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":9,"marks":6,"tier":"H","calc":true,"text":"

The table shows the commute times, t minutes, of 100 office workers.

Time (t minutes)Frequency
0 < t ≤ 1512
15 < t ≤ 3030
30 < t ≤ 4534
45 < t ≤ 6018
60 < t ≤ 756
  1. How many workers have a commute of 30 minutes or less?
  2. Write down the class interval that contains the median commute time.
  3. Write down the class interval that contains the upper quartile.
  4. A local newspaper claims that more than a third of the workers commute for over 45 minutes. Is the claim correct? Show your working.
","answerType":"multi","answer":"(a) 42 (b) 30 < t <= 45 (c) 30 < t <= 45 (d) No - 18 + 6 = 24 workers commute for over 45 minutes, which is 24% and less than a third","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"42"},{"label":"b","answer":"30 < t <= 45"},{"label":"c","answer":"30 < t <= 45"},{"label":"d","answer":"No; 24 out of 100 (24%) commute over 45 minutes, and 24% is less than a third"}],"solution":["Cumulative frequencies: 12, 42, 76, 94, 100.","Workers with a commute of 30 minutes or less = 42.","The median is between the 50th and 51st values; 42 workers take 30 minutes or less and 76 take 45 minutes or less, so the median is in 30 < t ≤ 45.","The upper quartile is the 75th value, which also lies in 30 < t ≤ 45 (cumulative frequency reaches 76 at 45 minutes).","Over 45 minutes: 18 + 6 = 24 workers, which is 24% of 100. A third of 100 is about 33.3, so the claim is not correct."],"diagram":null,"draw":false,"revision":"0c2493cf3827"},{"id":"g6-cumulative-frequency-q10","topicId":"g6-cumulative-frequency","topic":"Cumulative Frequency","grade":"6","topicGrade":"6","strand":"S","difficulty":10,"marks":7,"tier":"H","calc":true,"text":"

80 students sat an exam marked out of 100. The table shows their scores, s marks.

Score (s marks)Frequency
0 < s ≤ 206
20 < s ≤ 4018
40 < s ≤ 6028
60 < s ≤ 8020
80 < s ≤ 1008
  1. Complete the cumulative frequency table.
  2. Write down the class interval that contains the median score.
  3. A cumulative frequency graph of the data gives these estimates: median 51, lower quartile 37, upper quartile 66. Work out an estimate for the interquartile range.
  4. The pass mark was 60. Work out the percentage of students who scored more than 60.
  5. Last year the same exam had a median of 46 and an interquartile range of 24. Compare this year's scores with last year's.
","answerType":"multi","answer":"(a) 6, 24, 52, 72, 80 (b) 40 < s <= 60 (c) 29 (d) 35% (e) This year's median (51) is higher than last year's (46), so students scored better on average; this year's IQR (29) is larger than last year's (24), so scores were more spread out","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"6, 24, 52, 72, 80"},{"label":"b","answer":"40 < s <= 60"},{"label":"c","answer":"29"},{"label":"d","answer":"35%"},{"label":"e","answer":"Higher median this year (51 vs 46) so better on average; larger IQR this year (29 vs 24) so more spread out"}],"solution":["Cumulative frequencies: 6, 24, 52, 72, 80.","The median is between the 40th and 41st values; 24 students scored 40 or less and 52 scored 60 or less, so the median is in 40 < s ≤ 60.","Interquartile range = 66 - 37 = 29.","Students scoring more than 60 = 20 + 8 = 28, and 2880 = 35%.","This year's median of 51 is higher than last year's 46, so on average students did better this year; this year's IQR of 29 is bigger than 24, so this year's scores were more varied."],"diagram":null,"draw":false,"revision":"ff0d6bf3ed78"},{"id":"g6-enlarging-with-negative-scale-factors-q01","topicId":"g6-enlarging-with-negative-scale-factors","topic":"Enlarging with Negative Scale Factors","grade":"6","topicGrade":"6","strand":"G","difficulty":1,"marks":2,"tier":"H","calc":true,"text":"

Triangle P has vertices at (2, 1), (4, 1) and (4, 2).

Triangle P is enlarged by scale factor −1, centre (0, 0), to give triangle Q.

Write down the coordinates of the vertices of triangle Q.

","answerType":"exact","answer":"(-2, -1), (-4, -1), (-4, -2)","answerDisplay":null,"accept":["(-2,-1) (-4,-1) (-4,-2)"],"parts":[],"solution":["Scale factor -1 with centre (0, 0) sends each point (x, y) to (-x, -y).","(2, 1) → (-2, -1); (4, 1) → (-4, -1); (4, 2) → (-4, -2)."],"diagram":null,"draw":false,"revision":"f9d0d8e2de00"},{"id":"g6-enlarging-with-negative-scale-factors-q02","topicId":"g6-enlarging-with-negative-scale-factors","topic":"Enlarging with Negative Scale Factors","grade":"6","topicGrade":"6","strand":"G","difficulty":2,"marks":2,"tier":"H","calc":true,"text":"

Triangle A has vertices at (1, 1), (3, 1) and (1, 2).

Triangle A is enlarged by scale factor −2, centre (0, 0), to give triangle B.

Write down the coordinates of the vertices of triangle B.

","answerType":"exact","answer":"(-2, -2), (-6, -2), (-2, -4)","answerDisplay":null,"accept":["(-2,-2) (-6,-2) (-2,-4)"],"parts":[],"solution":["Scale factor -2 with centre (0, 0) sends each point (x, y) to (-2x, -2y).","(1, 1) → (-2, -2); (3, 1) → (-6, -2); (1, 2) → (-2, -4).","The image is on the opposite side of the centre and twice the size, turned through 180°."],"diagram":null,"draw":false,"revision":"24c2e988ed1b"},{"id":"g6-enlarging-with-negative-scale-factors-q03","topicId":"g6-enlarging-with-negative-scale-factors","topic":"Enlarging with Negative Scale Factors","grade":"6","topicGrade":"6","strand":"G","difficulty":3,"marks":2,"tier":"H","calc":true,"text":"

Triangle T has vertices at (4, 2), (6, 2) and (4, 6).

Triangle T is enlarged by scale factor −12, centre (0, 0), to give triangle U.

Write down the coordinates of the vertices of triangle U.

","answerType":"exact","answer":"(-2, -1), (-3, -1), (-2, -3)","answerDisplay":null,"accept":["(-2,-1) (-3,-1) (-2,-3)"],"parts":[],"solution":["Scale factor -12 with centre (0, 0) sends each point (x, y) to (-x2, -y2).","(4, 2) → (-2, -1); (6, 2) → (-3, -1); (4, 6) → (-2, -3).","The image is half the size and on the opposite side of the centre."],"diagram":null,"draw":false,"revision":"ffda41fc0d46"},{"id":"g6-enlarging-with-negative-scale-factors-q04","topicId":"g6-enlarging-with-negative-scale-factors","topic":"Enlarging with Negative Scale Factors","grade":"6","topicGrade":"6","strand":"G","difficulty":4,"marks":2,"tier":"H","calc":true,"text":"

Triangle M has vertices at (2, 1), (3, 1) and (2, 3).

Triangle M is enlarged by scale factor −3, centre (1, 0), to give triangle N.

Work out the coordinates of the vertices of triangle N.

","answerType":"exact","answer":"(-2, -3), (-5, -3), (-2, -9)","answerDisplay":null,"accept":["(-2,-3) (-5,-3) (-2,-9)"],"parts":[],"solution":["For each vertex, find the vector from the centre (1, 0) to the vertex, multiply it by -3, then add it to the centre.","(2, 1): vector (1, 1) → (-3, -3), image (1, 0) + (-3, -3) = (-2, -3).","(3, 1): vector (2, 1) → (-6, -3), image (-5, -3).","(2, 3): vector (1, 3) → (-3, -9), image (-2, -9)."],"diagram":null,"draw":false,"revision":"7b3a6ceb8c3d"},{"id":"g6-enlarging-with-negative-scale-factors-q05","topicId":"g6-enlarging-with-negative-scale-factors","topic":"Enlarging with Negative Scale Factors","grade":"6","topicGrade":"6","strand":"G","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

Triangle R is drawn on the grid with vertices at (2, 3), (4, 3) and (2, 4).

On the grid, enlarge triangle R by scale factor −2, centre (1, 2).

","answerType":"exact","answer":"Triangle with vertices (-1, 0), (-5, 0), (-1, -2)","answerDisplay":null,"accept":["(-1,0) (-5,0) (-1,-2)"],"parts":[],"solution":["From the centre (1, 2): (2, 3) is (1, 1) away, so its image is (1, 2) + (-2, -2) = (-1, 0).","(4, 3) is (3, 1) away, so its image is (1, 2) + (-6, -2) = (-5, 0).","(2, 4) is (1, 2) away, so its image is (1, 2) + (-2, -4) = (-1, -2).","Join the three image points to draw the enlarged triangle."],"diagram":{"type":"axes","x":{"min":-6,"max":7,"step":1},"y":{"min":-4,"max":6,"step":1},"square":true,"polygons":[{"pts":[[2,3],[4,3],[2,4]],"label":"R"},{"pts":[[-1,0],[-5,0],[-1,-2]],"label":"B","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"df19ac410dea"},{"id":"g6-enlarging-with-negative-scale-factors-q06","topicId":"g6-enlarging-with-negative-scale-factors","topic":"Enlarging with Negative Scale Factors","grade":"6","topicGrade":"6","strand":"G","difficulty":6,"marks":2,"tier":"H","calc":true,"text":"

Triangle A has vertices at (2, 2), (6, 2) and (2, 4).

Triangle B has vertices at (−1, −1), (−3, −1) and (−1, −2).

Describe fully the single transformation that maps triangle A onto triangle B.

","answerType":"open","answer":"Enlargement, scale factor -1/2, centre (0, 0)","answerDisplay":"Enlargement, scale factor -12, centre (0, 0)","accept":["enlargement scale factor -0.5 centre the origin"],"parts":[],"solution":["The sides of B are half the length of the corresponding sides of A, and B is on the opposite side of the origin, so the scale factor is -12.","Check with one vertex: (2, 2) × -12 = (-1, -1). Correct.","Lines joining corresponding points, for example (6, 2) to (-3, -1), all pass through (0, 0), so the centre is (0, 0).","The single transformation is an enlargement, scale factor -12, centre (0, 0)."],"diagram":{"type":"axes","x":{"min":-6,"max":7,"step":1},"y":{"min":-4,"max":6,"step":1},"square":true,"polygons":[{"pts":[[2,2],[6,2],[2,4]],"label":"A"},{"pts":[[-1,-1],[-3,-1],[-1,-2]],"label":"B","answer":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"0405b54b04a8"},{"id":"g6-enlarging-with-negative-scale-factors-q07","topicId":"g6-enlarging-with-negative-scale-factors","topic":"Enlarging with Negative Scale Factors","grade":"6","topicGrade":"6","strand":"G","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

Triangle P has vertices at (3, 1), (5, 1) and (3, 2).

Triangle Q has vertices at (0, 1), (−4, 1) and (0, −1).

Describe fully the single transformation that maps triangle P onto triangle Q.

","answerType":"open","answer":"Enlargement, scale factor -2, centre (2, 1)","answerDisplay":null,"accept":["enlargement scale factor -2 centre (2,1)"],"parts":[],"solution":["The sides of Q are twice the length of the corresponding sides of P and Q is rotated through 180° relative to P, so the scale factor is -2.","The centre lies on every line joining a point to its image. (3, 1) maps to (0, 1): the centre is on the line y = 1... it divides the join so that centre-to-image = 2 × centre-to-object on the opposite side, giving x = 2, so the centre is (2, 1).","Check with (5, 1): vector from (2, 1) is (3, 0); (3, 0) × -2 = (-6, 0); (2, 1) + (-6, 0) = (-4, 1). Correct.","Check with (3, 2): vector (1, 1) × -2 = (-2, -2); image (0, -1). Correct.","The single transformation is an enlargement, scale factor -2, centre (2, 1)."],"diagram":{"type":"axes","x":{"min":-6,"max":7,"step":1},"y":{"min":-4,"max":6,"step":1},"square":true,"polygons":[{"pts":[[3,1],[5,1],[3,2]],"label":"P"},{"pts":[[0,1],[-4,1],[0,-1]],"label":"Q","answer":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"cfefac5d2ead"},{"id":"g6-enlarging-with-negative-scale-factors-q08","topicId":"g6-enlarging-with-negative-scale-factors","topic":"Enlarging with Negative Scale Factors","grade":"6","topicGrade":"6","strand":"G","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

Triangle A has vertices at (2, 2), (6, 2) and (6, 4).

  1. Triangle A is enlarged by scale factor −12, centre (0, 0), to give triangle B. Write down the coordinates of the vertices of triangle B.
  2. Describe fully the single transformation that maps triangle B back onto triangle A.
","answerType":"multi","answer":"a) (-1, -1), (-3, -1), (-3, -2); b) enlargement, scale factor -2, centre (0, 0)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(-1, -1), (-3, -1), (-3, -2)"},{"label":"b","answer":"Enlargement, scale factor -2, centre (0, 0)"}],"solution":["a) Scale factor -12 with centre (0, 0) sends (x, y) to (-x2, -y2).","(2, 2) → (-1, -1); (6, 2) → (-3, -1); (6, 4) → (-3, -2).","b) The inverse of an enlargement scale factor -12 about (0, 0) is an enlargement scale factor -2 about the same centre.","Check: (-1, -1) × -2 = (2, 2). Correct.","So the single transformation is an enlargement, scale factor -2, centre (0, 0)."],"diagram":{"type":"axes","x":{"min":-6,"max":7,"step":1},"y":{"min":-4,"max":6,"step":1},"square":true,"polygons":[{"pts":[[2,2],[6,2],[6,4]],"label":"A"},{"pts":[[-1,-1],[-3,-1],[-3,-2]],"label":"B","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"93b1e22cd2d2"},{"id":"g6-expanding-triple-brackets-q01","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

Expand and simplify (x + 2)(x + 3)(x + 4)

","answerType":"exact","answer":"x^3 + 9x^2 + 26x + 24","answerDisplay":null,"accept":["x^3 + 9x^2 + 26x + 24","x³ + 9x² + 26x + 24"],"parts":[],"solution":["Expand the first two brackets: (x + 2)(x + 3) = x2 + 5x + 6.","Multiply the result by the third bracket: (x2 + 5x + 6)(x + 4).","x3 + 4x2 + 5x2 + 20x + 6x + 24.","Collect like terms: x3 + 9x2 + 26x + 24."],"diagram":null,"draw":false,"revision":"0467863d9281"},{"id":"g6-expanding-triple-brackets-q02","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":true,"text":"

Expand and simplify (x + 5)(x - 2)(x + 1)

","answerType":"exact","answer":"x^3 + 4x^2 - 7x - 10","answerDisplay":null,"accept":["x^3 + 4x^2 - 7x - 10","x³ + 4x² - 7x - 10"],"parts":[],"solution":["Expand the first two brackets: (x + 5)(x - 2) = x2 + 3x - 10.","Multiply by the third bracket: (x2 + 3x - 10)(x + 1).","x3 + x2 + 3x2 + 3x - 10x - 10.","Collect like terms: x3 + 4x2 - 7x - 10."],"diagram":null,"draw":false,"revision":"36eb7e23bcc6"},{"id":"g6-expanding-triple-brackets-q03","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

Expand and simplify (x - 3)(x + 4)(x - 1)

","answerType":"exact","answer":"x^3 - 13x + 12","answerDisplay":null,"accept":["x^3 - 13x + 12","x³ - 13x + 12","x^3 + 0x^2 - 13x + 12"],"parts":[],"solution":["Expand the first two brackets: (x - 3)(x + 4) = x2 + x - 12.","Multiply by the third bracket: (x2 + x - 12)(x - 1).","x3 - x2 + x2 - x - 12x + 12.","Collect like terms: the x2 terms cancel, leaving x3 - 13x + 12."],"diagram":null,"draw":false,"revision":"60c29d7ef25f"},{"id":"g6-expanding-triple-brackets-q04","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

Expand and simplify (x - 2)(x - 5)(x + 3)

","answerType":"exact","answer":"x^3 - 4x^2 - 11x + 30","answerDisplay":null,"accept":["x^3 - 4x^2 - 11x + 30","x³ - 4x² - 11x + 30"],"parts":[],"solution":["Expand the first two brackets: (x - 2)(x - 5) = x2 - 7x + 10.","Multiply by the third bracket: (x2 - 7x + 10)(x + 3).","x3 + 3x2 - 7x2 - 21x + 10x + 30.","Collect like terms: x3 - 4x2 - 11x + 30."],"diagram":null,"draw":false,"revision":"fcdb837f374c"},{"id":"g6-expanding-triple-brackets-q05","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

Expand and simplify (2x + 1)(x + 3)(x - 2)

","answerType":"exact","answer":"2x^3 + 3x^2 - 11x - 6","answerDisplay":null,"accept":["2x^3 + 3x^2 - 11x - 6","2x³ + 3x² - 11x - 6"],"parts":[],"solution":["Expand the last two brackets first: (x + 3)(x - 2) = x2 + x - 6.","Multiply by (2x + 1): (2x + 1)(x2 + x - 6).","2x3 + 2x2 - 12x + x2 + x - 6.","Collect like terms: 2x3 + 3x2 - 11x - 6."],"diagram":null,"draw":false,"revision":"7840d40f2f5c"},{"id":"g6-expanding-triple-brackets-q06","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

Expand and simplify (3x - 1)(x + 2)(x - 4)

","answerType":"exact","answer":"3x^3 - 7x^2 - 22x + 8","answerDisplay":null,"accept":["3x^3 - 7x^2 - 22x + 8","3x³ - 7x² - 22x + 8"],"parts":[],"solution":["Expand the last two brackets first: (x + 2)(x - 4) = x2 - 2x - 8.","Multiply by (3x - 1): (3x - 1)(x2 - 2x - 8).","3x3 - 6x2 - 24x - x2 + 2x + 8.","Collect like terms: 3x3 - 7x2 - 22x + 8."],"diagram":null,"draw":false,"revision":"6047dfd628bd"},{"id":"g6-expanding-triple-brackets-q07","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":7,"marks":3,"tier":"H","calc":false,"text":"

Show that (x + 1)(x - 4)(x + 6) = x3 + 3x2 - 22x - 24

","answerType":"open","answer":"Full expansion shown: (x+1)(x-4) = x^2 - 3x - 4, then (x^2 - 3x - 4)(x+6) = x^3 + 3x^2 - 22x - 24","answerDisplay":null,"accept":[],"parts":[],"solution":["Expand the first two brackets: (x + 1)(x - 4) = x2 - 3x - 4.","Multiply by the third bracket: (x2 - 3x - 4)(x + 6).","x3 + 6x2 - 3x2 - 18x - 4x - 24.","Collect like terms: x3 + 3x2 - 22x - 24, as required."],"diagram":null,"draw":false,"revision":"4f16ba6845ac"},{"id":"g6-expanding-triple-brackets-q08","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"
  1. Factorise y2 - 25
  2. Hence, or otherwise, expand and simplify (y + 5)(y - 5)(2y + 3)
","answerType":"multi","answer":"a) (y + 5)(y - 5) b) 2y^3 + 3y^2 - 50y - 75","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(y + 5)(y - 5)"},{"label":"b","answer":"2y^3 + 3y^2 - 50y - 75"}],"solution":["y2 - 25 is a difference of two squares: (y + 5)(y - 5).","So (y + 5)(y - 5)(2y + 3) = (y2 - 25)(2y + 3).","2y3 + 3y2 - 50y - 75."],"diagram":null,"draw":false,"revision":"ef259035a054"},{"id":"g6-expanding-triple-brackets-q09","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"
  1. Simplify (2m3)4
  2. Expand and simplify (m + 2)(m - 3)(m + 7)
","answerType":"multi","answer":"a) 16m^12 b) m^3 + 6m^2 - 13m - 42","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"16m^12"},{"label":"b","answer":"m^3 + 6m^2 - 13m - 42"}],"solution":["(2m3)4 = 24 × m12 = 16m12.","Expand the first two brackets: (m + 2)(m - 3) = m2 - m - 6.","Multiply by (m + 7): m3 + 7m2 - m2 - 7m - 6m - 42.","Collect like terms: m3 + 6m2 - 13m - 42."],"diagram":null,"draw":false,"revision":"32138aa18bcb"},{"id":"g6-expanding-triple-brackets-q10","topicId":"g6-expanding-triple-brackets","topic":"Expanding Triple Brackets","grade":"6","topicGrade":"6","strand":"A","difficulty":10,"marks":6,"tier":"H","calc":true,"text":"
  1. Show that (x + 2)(x + 5)(x - 2) = x3 + 5x2 - 4x - 20
  2. Hence solve x3 + 5x2 - 4x - 20 = 0
","answerType":"multi","answer":"a) shown b) x = -5, x = -2, x = 2","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Expansion shown: (x + 2)(x - 2) = x^2 - 4, then (x^2 - 4)(x + 5) = x^3 + 5x^2 - 4x - 20"},{"label":"b","answer":"x = -5, x = -2, x = 2"}],"solution":["Use the difference of two squares: (x + 2)(x - 2) = x2 - 4.","Multiply by (x + 5): (x2 - 4)(x + 5) = x3 + 5x2 - 4x - 20, as required.","For part b, use the factorised form: (x + 2)(x + 5)(x - 2) = 0.","A product is zero when any factor is zero, so x = -2, x = -5 or x = 2."],"diagram":null,"draw":false,"revision":"f86294f93e7d"},{"id":"g6-fractional-and-negative-indices-q01","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":1,"marks":1,"tier":"H","calc":false,"text":"

Write down the value of 12112

","answerType":"exact","answer":"11","answerDisplay":null,"accept":["11"],"parts":[],"solution":["A power of 12 means the positive square root.","√121 = 11."],"diagram":null,"draw":false,"revision":"ac72452b7f32"},{"id":"g6-fractional-and-negative-indices-q02","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":2,"marks":1,"tier":"H","calc":true,"text":"

Priya says that 2713 = 9 because 27 ÷ 3 = 9

Explain why Priya is wrong.

","answerType":"open","answer":"A power of 1/3 means the cube root, not dividing by 3. The cube root of 27 is 3, because 3 x 3 x 3 = 27.","answerDisplay":"A power of 13 means the cube root, not dividing by 3. The cube root of 27 is 3, because 3 x 3 x 3 = 27.","accept":[],"parts":[],"solution":["2713 means the cube root of 27, not 27 divided by 3.","Since 3 × 3 × 3 = 27, the correct value is 3."],"diagram":null,"draw":false,"revision":"0590154bef3b"},{"id":"g6-fractional-and-negative-indices-q03","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":3,"marks":2,"tier":"H","calc":false,"text":"
  1. Work out the value of 90
  2. Work out the value of 2−4
","answerType":"multi","answer":"a) 1 b) 1/16","answerDisplay":"a) 1 b) 116","accept":[],"parts":[{"label":"a","answer":"1"},{"label":"b","answer":"1/16","answerDisplay":"116"}],"solution":["(a) Any non-zero number to the power 0 is 1, so 90 = 1.","(b) A negative index means the reciprocal: 2−4 = 124 = 116.","The reciprocal rule keeps the index laws consistent: 2^4 × 2^-4 = 2^0 = 1. Since 2^4 = 16, the other factor must be 1/16. A negative power does not make the answer negative."],"diagram":null,"draw":false,"revision":"ce103df38f3e"},{"id":"g6-fractional-and-negative-indices-q04","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":4,"marks":2,"tier":"H","calc":false,"text":"

Work out the value of (3625)12

","answerType":"exact","answer":"6/5","answerDisplay":"65","accept":["6/5","1.2","1 1/5"],"parts":[],"solution":["A power of 12 means the square root of the numerator and of the denominator.","√36 = 6 and √25 = 5, so the answer is 65."],"diagram":null,"draw":false,"revision":"b24997c91bfe"},{"id":"g6-fractional-and-negative-indices-q05","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":5,"marks":3,"tier":"H","calc":false,"text":"
  1. Work out the value of 823
  2. Work out the value of 25−12
  3. Work out the value of 10−2
","answerType":"multi","answer":"a) 4 b) 1/5 c) 1/100","answerDisplay":"a) 4 b) 15 c) 1100","accept":[],"parts":[{"label":"a","answer":"4"},{"label":"b","answer":"1/5","answerDisplay":"15"},{"label":"c","answer":"1/100","answerDisplay":"1100"}],"solution":["(a) 823 = (cube root of 8)² = 2² = 4.","(b) 25−12 = 12512 = 1√25 = 15.","(c) 10−2 = 110² = 1100 = 0.01."],"diagram":null,"draw":false,"revision":"1fac7ea01d87"},{"id":"g6-fractional-and-negative-indices-q06","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":6,"marks":2,"tier":"H","calc":false,"text":"

2n = 18

Find the value of n.

","answerType":"exact","answer":"-3","answerDisplay":null,"accept":["-3","n = -3"],"parts":[],"solution":["18 = 12³ = 2−3.","So 2n = 2−3, giving n = −3."],"diagram":null,"draw":false,"revision":"a8b3fbecf8ba"},{"id":"g6-fractional-and-negative-indices-q07","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":7,"marks":3,"tier":"H","calc":false,"text":"

Simplify fully (27a9b6)23

","answerType":"exact","answer":"9a^6b^4","answerDisplay":null,"accept":["9a^6b^4","9a6b4"],"parts":[],"solution":["Apply the power 23 to each factor.","2723 = (cube root of 27)² = 3² = 9.","(a9)23 = a6 and (b6)23 = b4.","So the expression simplifies to 9a6b4."],"diagram":null,"draw":false,"revision":"008a19c22827"},{"id":"g6-fractional-and-negative-indices-q08","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":8,"marks":4,"tier":"H","calc":false,"text":"
  1. Work out the value of 64−23
  2. Work out the value of (8116)−34
","answerType":"multi","answer":"a) 1/16 b) 8/27","answerDisplay":"a) 116 b) 827","accept":[],"parts":[{"label":"a","answer":"1/16","answerDisplay":"116"},{"label":"b","answer":"8/27","answerDisplay":"827"}],"solution":["(a) 6423 = (cube root of 64)² = 4² = 16, so 64−23 = 116.","(b) The negative index turns the fraction over: (8116)−34 = (1681)34.","1634 = (fourth root of 16)³ = 2³ = 8 and 8134 = 3³ = 27.","So the answer is 827."],"diagram":null,"draw":false,"revision":"53355f997889"},{"id":"g6-fractional-and-negative-indices-q09","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":9,"marks":3,"tier":"H","calc":false,"text":"

9x = 127

Find the value of x.

","answerType":"exact","answer":"-3/2","answerDisplay":"-32","accept":["-3/2","-1.5","x = -3/2"],"parts":[],"solution":["Write both sides as powers of 3: 9x = (3²)x = 32x.","127 = 3−3.","So 2x = −3, giving x = −32."],"diagram":null,"draw":false,"revision":"ce6b01873f7d"},{"id":"g6-fractional-and-negative-indices-q10","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"
  1. 8n × 25 = 47
    Find the value of n.
  2. Write x√x as a single power of x.
","answerType":"multi","answer":"a) 3 b) x^(3/2)","answerDisplay":"a) 3 b) x32","accept":[],"parts":[{"label":"a","answer":"3"},{"label":"b","answer":"x^(3/2)","answerDisplay":"x32"}],"solution":["(a) Write everything as powers of 2: 8n = 23n and 47 = 214.","So 23n × 25 = 23n+5 = 214.","3n + 5 = 14, so n = 3.","(b) √x = x12, so x√x = x1 × x12 = x32."],"diagram":null,"draw":false,"revision":"546166508a74"},{"id":"g6-fractional-and-negative-indices-q11","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":11,"marks":4,"tier":"H","calc":false,"text":"
  1. Work out the value of 27−23
  2. 4x = 32
    Find the value of x.
","answerType":"multi","answer":"a) 1/9 b) 5/2","answerDisplay":"a) 19 b) 52","accept":[],"parts":[{"label":"a","answer":"1/9","answerDisplay":"19"},{"label":"b","answer":"5/2","answerDisplay":"52"}],"solution":["(a) 2723 = (cube root of 27)² = 3² = 9, so 27−23 = 19.","(b) Write both sides as powers of 2: 4x = 22x and 32 = 25.","So 2x = 5, giving x = 52."],"diagram":null,"draw":false,"revision":"d9e76b00cd0b"},{"id":"g6-fractional-and-negative-indices-q12","topicId":"g6-fractional-and-negative-indices","topic":"Fractional and Negative Indices","grade":"6","topicGrade":"6","strand":"N","difficulty":12,"marks":6,"tier":"H","calc":false,"text":"
  1. Work out the value of (√3)6
  2. Work out the value of (64125)−23
  3. 32n = 1√8
    Find the value of n.
","answerType":"multi","answer":"a) 27 b) 25/16 c) -3/10","answerDisplay":"a) 27 b) 2516 c) -310","accept":[],"parts":[{"label":"a","answer":"27"},{"label":"b","answer":"25/16","answerDisplay":"2516"},{"label":"c","answer":"-3/10","answerDisplay":"-310"}],"solution":["(a) (√3)6 = (312)6 = 3³ = 27.","(b) The negative index turns the fraction over: (12564)23.","12523 = 5² = 25 and 6423 = 4² = 16, so the answer is 2516.","(c) Write both sides as powers of 2: 32n = 25n.","1√8 = 8−12 = (2³)−12 = 2−32.","So 5n = −32, giving n = −310."],"diagram":null,"draw":false,"revision":"c27ee07ffb0b"},{"id":"g6-inequalities-on-graphs-q01","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":1,"marks":3,"tier":"H","calc":true,"text":"

On the grid, shade the region that satisfies all three of these inequalities.

x ≥ 1

y > 2

x + y < 6

Label the region R.

","answerType":"open","answer":"Triangle region with vertices (1, 2), (4, 2), (1, 5): solid boundary on x = 1, dashed boundaries on y = 2 and x + y = 6","answerDisplay":null,"accept":[],"parts":[],"solution":["Draw the line x = 1 (solid, because the inequality includes equals) and shade to the right of it.","Draw the line y = 2 (dashed, because y > 2 is strict) and keep the region above it.","Draw the line x + y = 6 (dashed) through (0, 6) and (6, 0) and keep the region below it.","R is the triangle with vertices (1, 2), (4, 2) and (1, 5).","The stated corner coordinates are where the boundary lines meet. A dashed boundary is excluded from R, including any corner lying on that dashed line."],"diagram":{"type":"axes","x":{"min":-2,"max":7,"step":1},"y":{"min":-2,"max":7,"step":1},"square":true,"lines":[{"x":1,"answer":true},{"y":2,"dashed":true,"answer":true},{"m":-1,"c":6,"dashed":true,"answer":true}],"regions":[{"pts":[[1,2],[4,2],[1,5]],"label":"R","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"c64aa32bcab7"},{"id":"g6-inequalities-on-graphs-q02","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

On the grid, shade the region that satisfies all three of these inequalities.

y ≤ 3

y ≥ x - 1

x > 0

Label the region R.

","answerType":"open","answer":"Region bounded by y = 3 (solid, below), y = x - 1 (solid, above) and x = 0 (dashed, right of); vertices (0, -1), (0, 3), (4, 3)","answerDisplay":null,"accept":[],"parts":[],"solution":["Draw the horizontal line y = 3 (solid) and keep the region below it.","Draw the line y = x - 1 (solid) through (0, -1) and (4, 3) and keep the region above it.","Draw the line x = 0, the y-axis, dashed because x > 0 is strict, and keep the region to the right.","R is the triangle with vertices (0, -1), (0, 3) and (4, 3).","The stated corner coordinates are where the boundary lines meet. A dashed boundary is excluded from R, including any corner lying on that dashed line."],"diagram":{"type":"axes","x":{"min":-2,"max":6,"step":1},"y":{"min":-2,"max":6,"step":1},"square":true,"lines":[{"y":3,"answer":true},{"m":1,"c":-1,"answer":true},{"x":0,"dashed":true,"answer":true}],"regions":[{"pts":[[0,-1],[0,3],[4,3]],"label":"R","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"541495a0dd64"},{"id":"g6-inequalities-on-graphs-q03","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

The region R satisfies all three of these inequalities.

x ≥ 1

y ≥ 2

x + y ≤ 5

Find all the points in R whose coordinates are both integers.

","answerType":"exact","answer":"(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (3, 2)","answerDisplay":null,"accept":["(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (3, 2)"],"parts":[],"solution":["x must be an integer at least 1 and y an integer at least 2, with x + y at most 5.","If x = 1, then y can be 2, 3 or 4, giving (1, 2), (1, 3), (1, 4).","If x = 2, then y can be 2 or 3, giving (2, 2), (2, 3).","If x = 3, then y can only be 2, giving (3, 2). If x = 4 or more, x + y would exceed 5.","The six points are (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (3, 2)."],"diagram":null,"draw":false,"revision":"45fae33aa141"},{"id":"g6-inequalities-on-graphs-q04","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

The region R satisfies all three of these inequalities.

y > x

x ≥ 0

y ≤ 4

Amy says the point (2, 2) is in R.

Is Amy correct? You must explain your answer.

","answerType":"open","answer":"No: at (2, 2) the inequality y > x is not satisfied because 2 is not greater than 2; the point is on the boundary line y = x, which is not included","answerDisplay":null,"accept":[],"parts":[],"solution":["Test (2, 2) in each inequality.","x ≥ 0 is satisfied since 2 ≥ 0, and y ≤ 4 is satisfied since 2 ≤ 4.","But y > x requires 2 > 2, which is false.","The point lies on the boundary line y = x, and a strict inequality does not include its boundary, so Amy is wrong."],"diagram":null,"draw":false,"revision":"cffcf7673306"},{"id":"g6-inequalities-on-graphs-q05","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

A region R is drawn on a coordinate grid.

R is the triangle with vertices (2, 1), (6, 1) and (2, 5), including its boundary.

The three sides of the triangle lie on the lines x = 2, y = 1 and x + y = 7

Find the three inequalities that define R.

","answerType":"multi","answer":"x >= 2, y >= 1, x + y <= 7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x >= 2"},{"label":"b","answer":"y >= 1"},{"label":"c","answer":"x + y <= 7"}],"solution":["The region lies to the right of the vertical line x = 2, so x ≥ 2.","The region lies above the horizontal line y = 1, so y ≥ 1.","The region lies below the line x + y = 7. Check with a point inside, such as (3, 2): 3 + 2 = 5 ≤ 7, correct. So x + y ≤ 7.","All boundaries are included, so every inequality uses ≥ or ≤."],"diagram":{"type":"axes","x":{"min":-2,"max":7,"step":1},"y":{"min":-2,"max":7,"step":1},"square":true,"lines":[{"x":2,"answer":false},{"y":1,"answer":false},{"m":-1,"c":7,"answer":false}],"regions":[{"pts":[[2,1],[6,1],[2,5]],"label":"R","answer":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"62e44b8f9186"},{"id":"g6-inequalities-on-graphs-q06","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

On the grid, shade the region that satisfies all four of these inequalities.

x > 0

y > 1

y < 2x

x + y < 6

Label the region R.

","answerType":"open","answer":"The interior of the triangle with boundary intersections (1/2, 1), (5, 1), (2, 4); all three edges are dashed and excluded. x > 0 holds automatically and x = 0 is not an edge of R.","answerDisplay":"The interior of the triangle with boundary intersections (1/2, 1), (5, 1), (2, 4); all three edges are dashed and excluded. x > 0 holds automatically and x = 0 is not an edge of R.","accept":[],"parts":[],"solution":["Draw y = 1 (dashed) and keep the region above it.","Draw y = 2x (dashed) through (0, 0), (1, 2), (2, 4) and keep the region below it, that is to the right of the line.","Draw x + y = 6 (dashed) through (0, 6) and (6, 0) and keep the region below it.","x > 0 holds automatically in this triangle, since y < 2x with y > 1 forces x > 12.","R is the triangle with vertices (12, 1), (5, 1) and (2, 4), with every boundary dashed.","The stated corner coordinates are where the boundary lines meet. A dashed boundary is excluded from R, including any corner lying on that dashed line."],"diagram":{"type":"axes","x":{"min":-2,"max":7,"step":1},"y":{"min":-2,"max":7,"step":1},"square":true,"lines":[{"x":0,"dashed":true,"answer":true},{"y":1,"dashed":true,"answer":true},{"m":2,"c":0,"dashed":true,"answer":true},{"m":-1,"c":6,"dashed":true,"answer":true}],"regions":[{"pts":[[0.5,1],[5,1],[2,4]],"label":"R","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"964f17a69279"},{"id":"g6-inequalities-on-graphs-q07","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

A region R is drawn on a coordinate grid.

R is the triangle with vertices (0, 1), (4, 1) and (4, 5), including its boundary.

Find the three inequalities that define R.

","answerType":"multi","answer":"y >= 1, x <= 4, y <= x + 1","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"y >= 1"},{"label":"b","answer":"x <= 4"},{"label":"c","answer":"y <= x + 1"}],"solution":["The side from (0, 1) to (4, 1) lies on y = 1, and the region is above it, so y ≥ 1.","The side from (4, 1) to (4, 5) lies on x = 4, and the region is to the left, so x ≤ 4.","The side from (0, 1) to (4, 5) has gradient (5 - 1) ÷ (4 - 0) = 1 and y-intercept 1, so it lies on y = x + 1.","Check with a point inside, such as (3, 2): 2 ≤ 3 + 1, correct, so y ≤ x + 1."],"diagram":{"type":"axes","x":{"min":-2,"max":7,"step":1},"y":{"min":-2,"max":7,"step":1},"square":true,"lines":[{"y":1,"answer":false},{"x":4,"answer":false},{"m":1,"c":1,"answer":false}],"regions":[{"pts":[[0,1],[4,1],[4,5]],"label":"R","answer":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"97dc89ef96a3"},{"id":"g6-inequalities-on-graphs-q08","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"
  1. On the grid, shade the region R that satisfies all three of these inequalities.

    y ≥ 1

    x + y ≤ 8

    y ≤ 3x - 1

  2. Priya says the point (3, 8) is in R. Explain why Priya is wrong.
","answerType":"multi","answer":"a) triangle with vertices (2/3, 1), (7, 1), (9/4, 23/4), all solid boundaries b) 3 + 8 = 11 > 8, so (3, 8) breaks x + y <= 8","answerDisplay":"a) triangle with vertices (23, 1), (7, 1), (94, 234), all solid boundaries b) 3 + 8 = 11 > 8, so (3, 8) breaks x + y <= 8","accept":[],"parts":[{"label":"a","answer":"Shaded triangle bounded by y = 1, x + y = 8 and y = 3x - 1, all solid; vertices (2/3, 1), (7, 1), (9/4, 23/4)","answerDisplay":"Shaded triangle bounded by y = 1, x + y = 8 and y = 3x - 1, all solid; vertices (23, 1), (7, 1), (94, 234)"},{"label":"b","answer":"At (3, 8), x + y = 11, which is greater than 8, so the inequality x + y <= 8 fails"}],"solution":["Draw y = 1 (solid) and keep the region above it.","Draw x + y = 8 (solid) through (0, 8) and (8, 0) and keep the region below it.","Draw y = 3x - 1 (solid) through (0, -1), (1, 2), (2, 5) and keep the region below it, that is to the right of the line.","The vertices of R are where the lines cross: y = 1 meets y = 3x - 1 at (23, 1); y = 1 meets x + y = 8 at (7, 1); y = 3x - 1 meets x + y = 8 where x + 3x - 1 = 8, giving (94, 234).","For part b, test (3, 8) in x + y ≤ 8: 3 + 8 = 11, which is greater than 8, so the point is not in R."],"diagram":{"type":"axes","x":{"min":-2,"max":9,"step":1},"y":{"min":-2,"max":9,"step":1},"square":true,"lines":[{"y":1,"answer":true},{"m":-1,"c":8,"answer":true},{"m":3,"c":-1,"answer":true}],"regions":[{"pts":[[0.6666666666666666,1],[7,1],[2.25,5.75]],"label":"R","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"4da0a0440c22"},{"id":"g6-inequalities-on-graphs-q09","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

The region R satisfies all four of these inequalities.

x ≥ 0

y ≥ 0

2x + y ≤ 10

y ≤ x + 2

The point (x, y) is in R and both coordinates are integers.

Find the greatest possible value of x + y

","answerType":"exact","answer":"7","answerDisplay":null,"accept":["7","x + y = 7"],"parts":[],"solution":["To make x + y large, take points near where 2x + y = 10 meets y = x + 2, which is at x = 83, y = 143, not an integer point.","Try integer x-values near 83. If x = 3: 2x + y ≤ 10 gives y ≤ 4, and y ≤ x + 2 gives y ≤ 5, so the best is y = 4 and x + y = 7.","If x = 2: y ≤ 6 and y ≤ 4, so the best is y = 4 and x + y = 6.","If x = 4: y ≤ 2 and y ≤ 6, so the best is y = 2 and x + y = 6.","To check every possibility: x ≥ 0 and 2x + y ≤ 10 with y ≥ 0 give integer x = 0, 1, 2, 3, 4 or 5. Their greatest allowed integer y-values are 2, 3, 4, 4, 2 and 0 respectively. The corresponding totals x + y are 2, 4, 6, 7, 6 and 5, so no point has a larger total.","The greatest possible value is 7, at the point (3, 4)."],"diagram":null,"draw":false,"revision":"c809ac13c77d"},{"id":"g6-inequalities-on-graphs-q10","topicId":"g6-inequalities-on-graphs","topic":"Inequalities on Graphs","grade":"6","topicGrade":"6","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

A region R is drawn on a coordinate grid.

R is the triangle with vertices (0, 1), (6, 1) and (2, 5), including its boundary.

Find the three inequalities that define R.

","answerType":"multi","answer":"y >= 1, y <= 2x + 1, x + y <= 7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"y >= 1"},{"label":"b","answer":"y <= 2x + 1"},{"label":"c","answer":"x + y <= 7"}],"solution":["The side from (0, 1) to (6, 1) lies on y = 1, and the region is above it, so y ≥ 1.","The side from (0, 1) to (2, 5) has gradient (5 - 1) ÷ (2 - 0) = 2 and y-intercept 1, so it lies on y = 2x + 1. The region is below this line, check with (2, 2): 2 ≤ 5, correct, so y ≤ 2x + 1.","The side from (6, 1) to (2, 5) has gradient (5 - 1) ÷ (2 - 6) = -1, so it lies on y = -x + 7, that is x + y = 7. The region is below it, check with (2, 2): 2 + 2 = 4 ≤ 7, correct, so x + y ≤ 7.","All boundaries are included, so every inequality uses ≥ or ≤."],"diagram":{"type":"axes","x":{"min":-2,"max":7,"step":1},"y":{"min":-2,"max":7,"step":1},"square":true,"lines":[{"y":1,"answer":false},{"m":2,"c":1,"answer":false},{"m":-1,"c":7,"answer":false}],"regions":[{"pts":[[0,1],[6,1],[2,5]],"label":"R","answer":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"e2a78528c450"},{"id":"g6-parallel-and-perpendicular-lines-q01","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":1,"marks":1,"tier":"FH","calc":true,"text":"

Here are the equations of four straight lines.

A: y = 2x - 5

B: y = x + 1

C: y = 3 - 2x

D: 2y = x + 4

Write down the letter of the line that is parallel to y = 2x + 1

","answerType":"exact","answer":"A","answerDisplay":null,"accept":["A","y = 2x - 5"],"parts":[],"solution":["Parallel lines have the same gradient.","The line y = 2x + 1 has gradient 2.","Line A has gradient 2, line B has gradient 1, line C has gradient -2, and line D rearranges to y = x2 + 2 with gradient 12.","So line A is parallel to y = 2x + 1."],"diagram":null,"draw":false,"revision":"cbfdc18002f3"},{"id":"g6-parallel-and-perpendicular-lines-q02","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":2,"marks":1,"tier":"H","calc":false,"text":"

A straight line has equation y = 4x + 3

Write down the gradient of a line perpendicular to this line.

","answerType":"exact","answer":"-1/4","answerDisplay":"-14","accept":["-1/4","-0.25"],"parts":[],"solution":["Perpendicular gradients multiply to give -1.","The gradient of the given line is 4.","The perpendicular gradient is -14, since 4 × (-14) = -1."],"diagram":null,"draw":false,"revision":"3b4447c36eb0"},{"id":"g6-parallel-and-perpendicular-lines-q03","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":3,"marks":2,"tier":"H","calc":false,"text":"

Show that the straight lines with equations y = 3x + 2 and 6x - 2y = 5 are parallel.

","answerType":"open","answer":"Rearranging 6x - 2y = 5 gives y = 3x - 5/2; both gradients are 3, so the lines are parallel","answerDisplay":"Rearranging 6x - 2y = 5 gives y = 3x - 52; both gradients are 3, so the lines are parallel","accept":[],"parts":[],"solution":["Rearrange 6x - 2y = 5 into the form y = mx + c.","2y = 6x - 5, so y = 3x - 52.","Both lines have gradient 3, and parallel lines have equal gradients, so the lines are parallel."],"diagram":null,"draw":false,"revision":"a32af9270495"},{"id":"g6-parallel-and-perpendicular-lines-q04","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":4,"marks":2,"tier":"H","calc":false,"text":"

Sam says that the straight lines with equations y = 5x - 1 and 5y = x + 10 are parallel.

Is Sam correct? You must explain your answer.

","answerType":"open","answer":"No: 5y = x + 10 rearranges to y = x/5 + 2, so the gradients are 5 and 1/5, which are not equal","answerDisplay":"No: 5y = x + 10 rearranges to y = x5 + 2, so the gradients are 5 and 15, which are not equal","accept":[],"parts":[],"solution":["Rearrange 5y = x + 10 into the form y = mx + c: y = x5 + 2.","The gradients are 5 and 15.","The gradients are not equal, so the lines are not parallel and Sam is wrong. In fact 5 × 15 = 1, not -1, so they are not perpendicular either."],"diagram":null,"draw":false,"revision":"06879d45e2cb"},{"id":"g6-parallel-and-perpendicular-lines-q05","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

Find the equation of the straight line that is perpendicular to y = 2x - 3 and passes through the point (4, 1).

","answerType":"exact","answer":"y = -x/2 + 3","answerDisplay":"y = -x2 + 3","accept":["y = -x/2 + 3","y = -0.5x + 3","y = -(1/2)x + 3","y = 3 - x/2","2y = -x + 6","x + 2y = 6"],"parts":[],"solution":["The gradient of y = 2x - 3 is 2, so the perpendicular gradient is -12, since 2 × (-12) = -1.","The new line has the form y = -x2 + c.","Substitute (4, 1): 1 = -2 + c, so c = 3.","The equation is y = -x2 + 3."],"diagram":null,"draw":false,"revision":"05ade2a3a349"},{"id":"g6-parallel-and-perpendicular-lines-q06","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":false,"text":"

The straight line L has equation y = -x3 + 5

Find the equation of the straight line that is perpendicular to L and passes through the point (2, -1).

","answerType":"exact","answer":"y = 3x - 7","answerDisplay":null,"accept":["y = 3x - 7","y = -7 + 3x"],"parts":[],"solution":["The gradient of L is -13, so the perpendicular gradient is 3, since -13 × 3 = -1.","The new line has the form y = 3x + c.","Substitute (2, -1): -1 = 6 + c, so c = -7.","The equation is y = 3x - 7."],"diagram":null,"draw":false,"revision":"f324aa2c93a1"},{"id":"g6-parallel-and-perpendicular-lines-q07","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

The straight line L1 has equation y = px + 4, where p is a constant.

The straight line L2 has equation 2x + 6y = 9

L1 is perpendicular to L2

Find the value of p.

","answerType":"exact","answer":"p = 3","answerDisplay":null,"accept":["3","p = 3"],"parts":[],"solution":["Rearrange 2x + 6y = 9: 6y = -2x + 9, so y = -x3 + 32.","The gradient of L2 is -13.","Perpendicular gradients multiply to -1, so p × (-13) = -1.","p = 3. Check: 3 × (-13) = -1, correct."],"diagram":null,"draw":false,"revision":"5fc8839cadb5"},{"id":"g6-parallel-and-perpendicular-lines-q08","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

The straight line L has equation 3x + 4y = 8

Find the equation of the straight line that is parallel to L and passes through the point (8, 1).

","answerType":"exact","answer":"y = -3x/4 + 7","answerDisplay":"y = -3x4 + 7","accept":["y = -3x/4 + 7","y = -0.75x + 7","y = -(3/4)x + 7","4y = -3x + 28","3x + 4y = 28"],"parts":[],"solution":["Rearrange 3x + 4y = 8: 4y = -3x + 8, so y = -3x4 + 2. The gradient of L is -34.","Parallel lines have equal gradients, so the new line has the form y = -3x4 + c.","Substitute (8, 1): 1 = -6 + c, so c = 7.","The equation is y = -3x4 + 7."],"diagram":null,"draw":false,"revision":"dbcd15f79b5e"},{"id":"g6-parallel-and-perpendicular-lines-q09","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

The straight line L1 has equation y = 2x + 6

L1 crosses the x-axis at the point A and crosses the y-axis at the point B.

The straight line L2 is perpendicular to L1 and passes through B.

L2 crosses the x-axis at the point C.

Work out the area of triangle ABC.

","answerType":"exact","answer":"45","answerDisplay":null,"accept":["45","45 square units","45 units^2"],"parts":[],"solution":["A is where y = 2x + 6 meets y = 0: 2x + 6 = 0, so A is (-3, 0). B is (0, 6).","L2 has gradient -12, since 2 × (-12) = -1, and passes through B, so L2 is y = -x2 + 6.","C is where y = -x2 + 6 meets y = 0: x2 = 6, so C is (12, 0).","Triangle ABC has base AC = 12 - (-3) = 15 on the x-axis and height 6 (the y-coordinate of B).","Area = 12 × 15 × 6 = 45."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[-2,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[0,4],"label":"B","style":"none","labelPos":"above"},{"at":[8,0],"label":"C","style":"none","labelPos":"below-right"}],"segments":[{"from":[-3,0],"to":[9,0],"label":"x","arrow":"end"},{"from":[0,-1],"to":[0,6],"label":"y","arrow":"end"},{"from":[-2,0],"to":[0,4],"label":"L₁"},{"from":[0,4],"to":[8,0],"label":"L₂"}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"8b48aa949f61"},{"id":"g6-parallel-and-perpendicular-lines-q10","topicId":"g6-parallel-and-perpendicular-lines","topic":"Parallel and Perpendicular Lines","grade":"6","topicGrade":"6","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

A is the point (0, 2) and B is the point (8, 6).

The point M lies on AB so that AM : MB = 1 : 3

Find the equation of the straight line that passes through M and is perpendicular to AB.

","answerType":"exact","answer":"y = -2x + 7","answerDisplay":null,"accept":["y = -2x + 7","y = 7 - 2x"],"parts":[],"solution":["AM : MB = 1 : 3 means M is 14 of the way from A to B.","M = (0 + 14 × 8, 2 + 14 × 4) = (2, 3).","The gradient of AB is 6 - 28 - 0 = 12, so the perpendicular gradient is -2, since 12 × (-2) = -1.","The line has the form y = -2x + c. Substitute (2, 3): 3 = -4 + c, so c = 7.","The equation is y = -2x + 7."],"diagram":null,"draw":false,"revision":"5ba5e1090cbb"},{"id":"g6-recurring-decimals-to-fractions-q01","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":1,"marks":1,"tier":"H","calc":false,"text":"

Amelia says that 0.2̇9̇ = 29100

Explain why Amelia is wrong.

","answerType":"open","answer":"0.2̇9̇ means the digits 29 repeat forever, so it equals 0.292929... = 29/99, not 29/100. 29/100 is the terminating decimal 0.29.","answerDisplay":"0.2̇9̇ means the digits 29 repeat forever, so it equals 0.292929... = 2999, not 29100. 29100 is the terminating decimal 0.29.","accept":[],"parts":[],"solution":["0.2̇9̇ means 0.292929... where the digits 29 repeat forever.","29100 = 0.29 exactly, which stops after two decimal places, so it cannot equal the recurring decimal.","The correct fraction is 2999."],"diagram":null,"draw":false,"revision":"6241cc868d8d"},{"id":"g6-recurring-decimals-to-fractions-q02","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":2,"marks":2,"tier":"H","calc":false,"text":"

Express 0.7̇ as a fraction.

","answerType":"exact","answer":"7/9","answerDisplay":"79","accept":["7/9"],"parts":[],"solution":["Let x = 0.777...","Then 10x = 7.777...","Multiplying by 10 shifts the decimal one place, so the repeating tails now match. Subtracting removes that entire matching infinite tail, leaving a whole number.","Subtracting: 10x − x = 7.777... − 0.777..., so 9x = 7.","Therefore x = 79."],"diagram":null,"draw":false,"revision":"c83801da73a6"},{"id":"g6-recurring-decimals-to-fractions-q03","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

Express 0.5̇4̇ as a fraction in its simplest form.

","answerType":"exact","answer":"6/11","answerDisplay":"611","accept":["6/11"],"parts":[],"solution":["Let x = 0.545454...","Then 100x = 54.545454...","Subtracting: 100x − x = 54, so 99x = 54.","x = 5499 = 611 after dividing top and bottom by 9."],"diagram":null,"draw":false,"revision":"9bcb49a308d7"},{"id":"g6-recurring-decimals-to-fractions-q04","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

Prove algebraically that the recurring decimal 0.4̇5̇ can be written as 511

","answerType":"open","answer":"Let x = 0.454545..., 100x = 45.454545..., 99x = 45, x = 45/99 = 5/11","answerDisplay":"Let x = 0.454545..., 100x = 45.454545..., 99x = 45, x = 4599 = 511","accept":[],"parts":[],"solution":["Let x = 0.454545...","Multiply by 100: 100x = 45.454545...","Subtract: 100x − x = 45.454545... − 0.454545..., so 99x = 45.","x = 4599. Dividing numerator and denominator by 9 gives x = 511, as required."],"diagram":null,"draw":false,"revision":"81e2f714af52"},{"id":"g6-recurring-decimals-to-fractions-q05","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

Express the recurring decimal 0.2̇13̇ as a fraction in its simplest form.

(The digits 2, 1 and 3 all recur.)

","answerType":"exact","answer":"71/333","answerDisplay":"71333","accept":["71/333"],"parts":[],"solution":["Let x = 0.213213213...","Multiply by 1000: 1000x = 213.213213...","Subtract: 999x = 213, so x = 213999.","Divide top and bottom by 3: x = 71333."],"diagram":null,"draw":false,"revision":"ca5f9bddf305"},{"id":"g6-recurring-decimals-to-fractions-q06","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

Prove algebraically that the recurring decimal 0.28̇ can be written as 1345

(Only the digit 8 recurs.)

","answerType":"open","answer":"Let x = 0.2888..., 10x = 2.888..., 100x = 28.888..., 90x = 26, x = 26/90 = 13/45","answerDisplay":"Let x = 0.2888..., 10x = 2.888..., 100x = 28.888..., 90x = 26, x = 2690 = 1345","accept":[],"parts":[],"solution":["Let x = 0.28888...","10x = 2.8888... and 100x = 28.8888...","Subtract: 100x − 10x = 28.8888... − 2.8888..., so 90x = 26.","x = 2690 = 1345 after dividing top and bottom by 2, as required."],"diagram":null,"draw":false,"revision":"3075df7bde5b"},{"id":"g6-recurring-decimals-to-fractions-q07","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":7,"marks":3,"tier":"H","calc":false,"text":"

Express the recurring decimal 0.35̇1̇ as a fraction in its simplest form.

(Only the digits 5 and 1 recur.)

","answerType":"exact","answer":"58/165","answerDisplay":"58165","accept":["58/165"],"parts":[],"solution":["Let x = 0.3515151...","10x = 3.515151... and 1000x = 351.515151...","Subtract: 1000x − 10x = 348, so 990x = 348.","x = 348990 = 58165 after dividing top and bottom by 6."],"diagram":null,"draw":false,"revision":"a8ac80b67d1a"},{"id":"g6-recurring-decimals-to-fractions-q08","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":8,"marks":3,"tier":"H","calc":false,"text":"

Work out 0.1̇3̇ + 0.5̇

Give your answer as a fraction in its simplest form.

","answerType":"exact","answer":"68/99","answerDisplay":"6899","accept":["68/99"],"parts":[],"solution":["0.1̇3̇ = 1399 (let x = 0.1313..., 100x − x = 13, so x = 1399).","0.5̇ = 59 (let y = 0.555..., 10y − y = 5, so y = 59).","59 = 5599, so the sum is 1399 + 5599 = 6899.","68 = 4 × 17 and 99 = 9 × 11 share no common factor, so 6899 is in simplest form."],"diagram":null,"draw":false,"revision":"23565546b650"},{"id":"g6-recurring-decimals-to-fractions-q09","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":9,"marks":3,"tier":"H","calc":true,"text":"

Show algebraically that 0.7̇2̇ + 0.2̇7̇ = 1

","answerType":"open","answer":"0.727272... = 72/99 and 0.272727... = 27/99; 72/99 + 27/99 = 99/99 = 1","answerDisplay":"0.727272... = 7299 and 0.272727... = 2799; 7299 + 2799 = 9999 = 1","accept":[],"parts":[],"solution":["Let x = 0.727272..., then 100x = 72.7272..., so 99x = 72 and x = 7299.","Let y = 0.272727..., then 100y = 27.2727..., so 99y = 27 and y = 2799.","x + y = 7299 + 2799 = 9999 = 1, as required."],"diagram":null,"draw":false,"revision":"1288fd028e65"},{"id":"g6-recurring-decimals-to-fractions-q10","topicId":"g6-recurring-decimals-to-fractions","topic":"Recurring Decimals to Fractions","grade":"6","topicGrade":"6","strand":"N","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

Work out 0.1̇8̇ × 0.4̇5̇

Give your answer as a fraction in its simplest form.

You must show all your working.

","answerType":"exact","answer":"10/121","answerDisplay":"10121","accept":["10/121"],"parts":[],"solution":["Let x = 0.181818..., then 100x = 18.1818..., so 99x = 18 and x = 1899 = 211.","Let y = 0.454545..., then 100y = 45.4545..., so 99y = 45 and y = 4599 = 511.","Multiply: 211 × 511 = 10121.","10 = 2 × 5 and 121 = 11² share no common factor, so 10121 is in simplest form."],"diagram":null,"draw":false,"revision":"de49f55a3e6b"},{"id":"g6-repeated-percentage-change-q01","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":1,"marks":3,"tier":"FH","calc":true,"text":"

Hollie puts £8000 into a savings account.

The value of the account increases by 5% each year.

Work out the value of the account after 2 years.

","answerType":"exact","answer":"£8820","answerDisplay":null,"accept":["8820","£8820","£8,820","£8820.00"],"parts":[],"solution":["The multiplier for a 5% increase is 1.05.","After 2 years the value is 8000 × 1.052.","8000 × 1.1025 = £8820."],"diagram":null,"draw":false,"revision":"8ef5f8b27022"},{"id":"g6-repeated-percentage-change-q02","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":2,"marks":3,"tier":"FH","calc":true,"text":"

Dev buys a van for £15 000.

The value of the van depreciates by 12% each year.

Calculate the value of the van at the end of 3 years.

Give your answer to the nearest penny.

","answerType":"exact","answer":"£10222.08","answerDisplay":null,"accept":["10222.08","£10222.08","£10,222.08"],"parts":[],"solution":["The multiplier for a 12% decrease is 0.88.","After 3 years the value is 15000 × 0.883.","0.883 = 0.681472, so the value is 15000 × 0.681472 = £10222.08."],"diagram":null,"draw":false,"revision":"3f73dbebf473"},{"id":"g6-repeated-percentage-change-q03","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

The number of visitors to a website increases by 50% one month and then increases by 50% again the next month.

Petra says, "Two increases of 50% make an increase of 100%."

Petra is wrong.

Find the overall percentage increase over the two months.

","answerType":"exact","answer":"125%","answerDisplay":null,"accept":["125","125 percent","125 per cent"],"parts":[],"solution":["The multiplier for a 50% increase is 1.5, or 32.","Two increases give an overall multiplier of 1.5 × 1.5 = 2.25.","A multiplier of 2.25 is an increase of 125%, not 100%."],"diagram":null,"draw":false,"revision":"35cfb889070f"},{"id":"g6-repeated-percentage-change-q04","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

The population of a colony of seabirds falls by the same percentage each year.

Three years ago the population was 64 000.

The population is now 46 656.

Find the annual percentage decrease.

","answerType":"exact","answer":"10%","answerDisplay":null,"accept":["10","10 percent","10 per cent"],"parts":[],"solution":["If the annual multiplier is m, then 64000 × m3 = 46656.","m3 = 46656 ÷ 64000 = 0.729.","m = ∛0.729 = 0.9, which is a 10% decrease each year."],"diagram":null,"draw":false,"revision":"cd146e2002ea"},{"id":"g6-repeated-percentage-change-q05","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":5,"marks":4,"tier":"FH","calc":true,"text":"

Ellis invests £4000 in a savings account for 3 years.

The account pays compound interest at an annual rate of

3% for the first year

2.5% for each of the second and third years.

Work out the value of the investment at the end of 3 years.

Give your answer to the nearest penny.

","answerType":"exact","answer":"£4328.58","answerDisplay":null,"accept":["4328.58","£4328.58","£4,328.58"],"parts":[],"solution":["After the first year: 4000 × 1.03 = £4120.","After the second year: 4120 × 1.025 = £4223.","After the third year: 4223 × 1.025 = £4328.575.","The value is £4328.58 to the nearest penny."],"diagram":null,"draw":false,"revision":"f4643be5fd0b"},{"id":"g6-repeated-percentage-change-q06","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

At the start of an experiment there are 20 000 bacteria in a dish.

The number of bacteria increases by 8% each hour.

A scientist claims there will be more than 25 000 bacteria after 3 hours.

Work out whether the scientist is correct.

You must show your working.

","answerType":"open","answer":"Yes - 20000 x 1.08^3 = 25194.24, which is more than 25000","answerDisplay":null,"accept":["yes","correct, 25194","25194.24 > 25000"],"parts":[],"solution":["The multiplier for an 8% increase is 1.08.","After 3 hours: 20000 × 1.083 = 20000 × 1.259712 = 25194.24.","25194 > 25000, so the scientist is correct."],"diagram":null,"draw":false,"revision":"2c0da19e7a7b"},{"id":"g6-repeated-percentage-change-q07","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

A running club has 800 members.

The number of members increases by 15% each year.

Find the number of complete years it takes for the number of members to first exceed 1600.

","answerType":"exact","answer":"5 years","answerDisplay":null,"accept":["5","5 years","after 5 years"],"parts":[],"solution":["The multiplier for a 15% increase is 1.15.","After 4 years: 800 × 1.154 = 800 × 1.7490... = 1399.2, which is below 1600.","After 5 years: 800 × 1.155 = 800 × 2.0113... = 1609.1, which exceeds 1600.","So it takes 5 years."],"diagram":null,"draw":false,"revision":"9c1be3652b34"},{"id":"g6-repeated-percentage-change-q08","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

The population of a town is growing by 12% each year.

Find the number of complete years it will take for the population to become more than double its current size.

","answerType":"exact","answer":"7 years","answerDisplay":null,"accept":["7","7 years","after 7 years"],"parts":[],"solution":["The population is multiplied by 1.12 each year, so we need the smallest n with 1.12n > 2.","1.126 = 1.9738..., which is less than 2.","1.127 = 2.2106..., which is more than 2.","So it takes 7 years."],"diagram":null,"draw":false,"revision":"68a9e8f78cf3"},{"id":"g6-repeated-percentage-change-q09","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

Town A has a population of 150 000. Its population is falling by 4% each year.

Town B has a population of 120 000. Its population is rising by 5% each year.

Work out which town will have the larger population at the end of 3 years.

You must show your working.

","answerType":"open","answer":"Town B (138915 compared with about 132710)","answerDisplay":null,"accept":["Town B","B"],"parts":[],"solution":["Town A: 150000 × 0.963 = 150000 × 0.884736 = 132710.4, so about 132 710 people.","Town B: 120000 × 1.053 = 120000 × 1.157625 = 138915 people.","138915 > 132710, so Town B will have the larger population."],"diagram":null,"draw":false,"revision":"be607ae20bab"},{"id":"g6-repeated-percentage-change-q10","topicId":"g6-repeated-percentage-change","topic":"Repeated Percentage Change","grade":"6","topicGrade":"6","strand":"R","difficulty":10,"marks":4,"tier":"H","calc":true,"text":"

The value of a painting increased by 10% each year for two years.

The painting is now worth £9680.

Find the value of the painting two years ago.

","answerType":"exact","answer":"£8000","answerDisplay":null,"accept":["8000","£8000","£8,000"],"parts":[],"solution":["Two increases of 10% give an overall multiplier of 1.12 = 1.21.","If the original value was V, then V × 1.21 = 9680.","V = 9680 ÷ 1.21 = £8000.","Check: 8000 × 1.1 = 8800 and 8800 × 1.1 = 9680."],"diagram":null,"draw":false,"revision":"8b313e886449"},{"id":"g6-similar-shapes-area-and-volume-q01","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

Two solid ornaments, P and Q, are mathematically similar.

P has height 4 cm and Q has height 12 cm.

The volume of P is 20 cm3.

Work out the volume of Q.

","answerType":"exact","answer":"540 cm³","answerDisplay":null,"accept":["540","540 cm^3"],"parts":[],"solution":["Length scale factor = 12 ÷ 4 = 3.","Every length is multiplied by 3. Volume scales in three dimensions, so the multiplier is 3 × 3 × 3 = 27; an area would instead scale by 3 × 3 = 9.","Volume scale factor = 33 = 27.","Volume of Q = 20 × 27 = 540 cm3."],"diagram":null,"draw":false,"revision":"15182b242efb"},{"id":"g6-similar-shapes-area-and-volume-q02","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

Two mathematically similar signs are made for a shop.

The small sign is 5 cm wide and has area 50 cm2.

The large sign is 8 cm wide.

Work out the area of the large sign.

","answerType":"exact","answer":"128 cm²","answerDisplay":null,"accept":["128","128 cm^2"],"parts":[],"solution":["Length scale factor = 8 ÷ 5 = 1.6.","Area scale factor = 1.62 = 2.56.","Area of the large sign = 50 × 2.56 = 128 cm2."],"diagram":null,"draw":false,"revision":"295de61b0ff7"},{"id":"g6-similar-shapes-area-and-volume-q03","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

Two mathematically similar solids, A and B, have volumes in the ratio 64 : 343.

Find the ratio of the surface area of A to the surface area of B.

Give your answer in its simplest form.

","answerType":"exact","answer":"16 : 49","answerDisplay":null,"accept":["16:49"],"parts":[],"solution":["Cube root the volume ratio to get the length ratio: ∛64 : ∛343 = 4 : 7.","Square the length ratio to get the area ratio: 42 : 72 = 16 : 49."],"diagram":null,"draw":false,"revision":"1031990933d0"},{"id":"g6-similar-shapes-area-and-volume-q04","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

Two mathematically similar candles, C and D, are made from the same wax.

C has surface area 36 cm2 and D has surface area 81 cm2.

The volume of C is 24 cm3.

Work out the volume of D.

","answerType":"exact","answer":"81 cm³","answerDisplay":null,"accept":["81","81 cm^3"],"parts":[],"solution":["Area ratio C : D = 36 : 81 = 4 : 9.","Length ratio = √4 : √9 = 2 : 3.","Volume ratio = 23 : 33 = 8 : 27.","Volume of D = 24 × 27 ÷ 8 = 81 cm3."],"diagram":null,"draw":false,"revision":"fd72ddeb98e1"},{"id":"g6-similar-shapes-area-and-volume-q05","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

Two mathematically similar statues are to be painted.

The small statue has height 10 cm and surface area 80 cm2.

The large statue has height 25 cm.

Work out the surface area of the large statue.

","answerType":"exact","answer":"500 cm²","answerDisplay":null,"accept":["500","500 cm^2"],"parts":[],"solution":["Length scale factor = 25 ÷ 10 = 2.5.","Area scale factor = 2.52 = 6.25.","Surface area of the large statue = 80 × 6.25 = 500 cm2."],"diagram":null,"draw":false,"revision":"59e43e2a143e"},{"id":"g6-similar-shapes-area-and-volume-q06","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

Two solid trophies are mathematically similar and are made from the same metal.

The small trophy has height 6 cm and mass 120 g.

The large trophy has height 9 cm.

Work out the mass of the large trophy.

","answerType":"exact","answer":"405 g","answerDisplay":null,"accept":["405","405 grams"],"parts":[],"solution":["Length scale factor = 9 ÷ 6 = 1.5.","Mass is proportional to volume, so the mass scale factor = 1.53 = 3.375.","Mass of the large trophy = 120 × 3.375 = 405 g."],"diagram":null,"draw":false,"revision":"fe23ed369643"},{"id":"g6-similar-shapes-area-and-volume-q07","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

Two mathematically similar bottles hold 250 ml and 686 ml of juice when full.

The small bottle has height 15 cm.

Work out the height of the large bottle.

","answerType":"exact","answer":"21 cm","answerDisplay":null,"accept":["21"],"parts":[],"solution":["Volume ratio = 250 : 686 = 125 : 343.","Length ratio = ∛125 : ∛343 = 5 : 7.","Height of the large bottle = 15 × 7 ÷ 5 = 21 cm."],"diagram":null,"draw":false,"revision":"875c61fc89cb"},{"id":"g6-similar-shapes-area-and-volume-q08","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

Two mathematically similar solids, S and T, have volumes 135 cm3 and 320 cm3.

The surface area of S is 180 cm2.

Work out the surface area of T.

","answerType":"exact","answer":"320 cm²","answerDisplay":null,"accept":["320","320 cm^2"],"parts":[],"solution":["The solids are similar. Their volume ratio is 135 : 320 = 27 : 64.","Take cube roots to find the corresponding length ratio: 3 : 4.","Square the length ratio to find the surface-area ratio: 9 : 16.","Surface area of T = 180 × 16 ÷ 9 = 320 cm^2."],"diagram":null,"draw":false,"revision":"f70ea58bc723"},{"id":"g6-similar-shapes-area-and-volume-q09","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":9,"marks":4,"tier":"H","calc":false,"text":"

In triangle ABC, D lies on AB and E lies on AC so that DE is parallel to BC.

AD : DB = 2 : 1.

The area of triangle ADE is 24 cm2.

Work out the area of the trapezium DBCE.

","answerType":"exact","answer":"30 cm²","answerDisplay":null,"accept":["30","30 cm^2"],"parts":[],"solution":["DE is parallel to BC, so triangles ADE and ABC are similar.","AD : AB = 2 : 3, so the length scale factor from ADE to ABC is 32.","Area scale factor = (32)2 = 94.","Area of ABC = 24 × 94 = 54 cm2.","Area of trapezium DBCE = 54 - 24 = 30 cm2."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,5],"label":"A","style":"none","labelPos":"above"},{"at":[-4,0],"label":"B","style":"none","labelPos":"below-left"},{"at":[4,0],"label":"C","style":"none","labelPos":"below-right"},{"at":[-2.6666666666666665,1.666666666666667],"label":"D","style":"none","labelPos":"above"},{"at":[2.6666666666666665,1.666666666666667],"label":"E","style":"none","labelPos":"above"}],"segments":[{"from":[0,5],"to":[-2.6666666666666665,1.666666666666667]},{"from":[-2.6666666666666665,1.666666666666667],"to":[-4,0]},{"from":[0,5],"to":[4,0]},{"from":[-4,0],"to":[4,0],"parallel":1},{"from":[-2.6666666666666665,1.666666666666667],"to":[2.6666666666666665,1.666666666666667],"parallel":1}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"60b6579d12b7"},{"id":"g6-similar-shapes-area-and-volume-q10","topicId":"g6-similar-shapes-area-and-volume","topic":"Similar Shapes (Area and Volume)","grade":"6","topicGrade":"6","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

Two mathematically similar containers, A and B, have volumes 400 cm3 and 1350 cm3.

  1. Container A has height 12 cm. Work out the height of container B.
  2. Container B has surface area 468 cm2. Work out the surface area of container A.
","answerType":"multi","answer":"a) 18 cm, b) 208 cm²","answerDisplay":null,"accept":["18 cm and 208 cm^2"],"parts":[{"label":"a","answer":"18 cm"},{"label":"b","answer":"208 cm²"}],"solution":["Volume ratio A : B = 400 : 1350 = 8 : 27.","Length ratio = ∛8 : ∛27 = 2 : 3.","a) Height of B = 12 × 3 ÷ 2 = 18 cm.","Area ratio = 22 : 32 = 4 : 9.","b) Surface area of A = 468 × 4 ÷ 9 = 208 cm2."],"diagram":null,"draw":false,"revision":"bbda0801ed16"},{"id":"g6-the-product-rule-for-counting-q01","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":1,"marks":2,"tier":"H","calc":true,"text":"

An ice cream van sells 7 flavours of ice cream and 4 different toppings.

Nadia chooses one flavour and one topping.

Work out the number of different combinations Nadia can choose.

","answerType":"exact","answer":"28","answerDisplay":null,"accept":["28"],"parts":[],"solution":["For each of the 7 flavours there are 4 possible toppings.","7 × 4 = 28 different combinations."],"diagram":null,"draw":false,"revision":"7ea513582c7b"},{"id":"g6-the-product-rule-for-counting-q02","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":2,"marks":2,"tier":"H","calc":true,"text":"

A bistro menu has 5 starters, 8 main courses and 4 desserts.

Callum chooses one starter, one main course and one dessert.

Work out the number of different three-course meals Callum can choose.

","answerType":"exact","answer":"160","answerDisplay":null,"accept":["160"],"parts":[],"solution":["Multiply the number of choices at each stage.","5 × 8 × 4 = 160 different meals."],"diagram":null,"draw":false,"revision":"64a6d5213615"},{"id":"g6-the-product-rule-for-counting-q03","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":3,"marks":2,"tier":"H","calc":true,"text":"

There are 9 boys and 12 girls in a drama club.

One boy and one girl are to be chosen to present the end of term show.

Work out the number of different ways the pair can be chosen.

","answerType":"exact","answer":"108","answerDisplay":null,"accept":["108"],"parts":[],"solution":["Each of the 9 boys can be paired with any of the 12 girls.","9 × 12 = 108 different pairs."],"diagram":null,"draw":false,"revision":"c4e0619a8de5"},{"id":"g6-the-product-rule-for-counting-q04","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":4,"marks":2,"tier":"H","calc":false,"text":"

A pizzeria offers x different pizzas and 6 different side orders.

There are 54 different ways to choose one pizza and one side order.

Work out the value of x.

","answerType":"exact","answer":"9","answerDisplay":null,"accept":["9","x = 9"],"parts":[],"solution":["The number of combinations is x × 6.","x × 6 = 54, so x = 54 ÷ 6 = 9."],"diagram":null,"draw":false,"revision":"e9bcb15dd1d4"},{"id":"g6-the-product-rule-for-counting-q05","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

A locker code is made up of one letter followed by two digits.

The letter is chosen from the 26 letters A to Z.

Each digit is chosen from 0 to 9, and digits may repeat.

Work out the number of different codes that are possible.

","answerType":"exact","answer":"2600","answerDisplay":null,"accept":["2600","2,600"],"parts":[],"solution":["There are 26 choices for the letter, 10 for the first digit and 10 for the second digit.","26 × 10 × 10 = 2600 different codes."],"diagram":null,"draw":false,"revision":"94bcd431cdf3"},{"id":"g6-the-product-rule-for-counting-q06","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":6,"marks":2,"tier":"H","calc":true,"text":"

There are 14 netball teams in a league.

Each team plays every other team twice, once at home and once away.

Work out the total number of games played in the league.

","answerType":"exact","answer":"182","answerDisplay":null,"accept":["182"],"parts":[],"solution":["Each of the 14 teams plays a home game against each of the other 13 teams.","14 × 13 = 182 games. This already counts each pairing twice, once for each team at home."],"diagram":null,"draw":false,"revision":"a5125bb8ccaa"},{"id":"g6-the-product-rule-for-counting-q07","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":7,"marks":2,"tier":"H","calc":true,"text":"

There are 20 students in a chess club.

Two of the students are to be chosen to enter a doubles tournament as a pair.

Work out the number of different pairs that can be chosen.

","answerType":"exact","answer":"190","answerDisplay":null,"accept":["190"],"parts":[],"solution":["There are 20 choices for the first student and 19 for the second, giving 20 × 19 = 380 ordered selections.","Each pair has been counted twice (once in each order), so divide by 2.","380 ÷ 2 = 190 different pairs."],"diagram":null,"draw":false,"revision":"88886b7ba08f"},{"id":"g6-the-product-rule-for-counting-q08","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

A sandwich shop offers a meal deal.

Customers choose one of 8 sandwiches, one of 5 packets of crisps and one of 3 drinks.

Kai says, "There are more than 100 different meal deals."

Show that Kai is correct.

","answerType":"open","answer":"8 x 5 x 3 = 120 and 120 > 100, so Kai is correct","answerDisplay":null,"accept":[],"parts":[],"solution":["Multiply the number of choices at each stage: 8 × 5 × 3.","8 × 5 = 40 and 40 × 3 = 120.","120 > 100, so Kai is correct."],"diagram":null,"draw":false,"revision":"c4affd7a1353"},{"id":"g6-the-product-rule-for-counting-q09","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":9,"marks":3,"tier":"H","calc":true,"text":"

The passcode for a tablet is a 4 digit number.

The first digit cannot be 0.

Digits may repeat.

  1. Work out the number of different passcodes that are possible.
  2. Ruby does not know the passcode. She types one 4 digit number at random, with a non-zero first digit. Write down the probability that Ruby types the correct passcode.
","answerType":"multi","answer":"a) 9000 b) 1/9000","answerDisplay":"a) 9000 b) 19000","accept":[],"parts":[{"label":"a","answer":"9000"},{"label":"b","answer":"1/9000","answerDisplay":"19000"}],"solution":["(a) There are 9 choices for the first digit (1 to 9) and 10 choices for each of the other three digits.","9 × 10 × 10 × 10 = 9000 passcodes.","(b) Exactly one of the 9000 equally likely guesses is correct, so the probability is 19000."],"diagram":null,"draw":false,"revision":"5c800394fc94"},{"id":"g6-the-product-rule-for-counting-q10","topicId":"g6-the-product-rule-for-counting","topic":"The Product Rule for Counting","grade":"6","topicGrade":"6","strand":"N","difficulty":10,"marks":4,"tier":"H","calc":false,"text":"

The code for a padlock is made up of 3 digits.

Each digit is chosen from 1 to 9.

All three digits must be different.

The code must be an odd number.

Work out the number of different codes that are possible.

","answerType":"exact","answer":"280","answerDisplay":null,"accept":["280"],"parts":[],"solution":["The code is odd, so the last digit must be odd: 5 choices (1, 3, 5, 7 or 9).","The first digit can be any of the 8 remaining digits.","The second digit can be any of the 7 digits still left.","5 × 8 × 7 = 280 different codes."],"diagram":null,"draw":false,"revision":"2f342126ae41"},{"id":"g7-3d-pythagoras-and-trigonometry-q01","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":1,"marks":2,"tier":"H","calc":false,"text":"

ABCDEFGH is a cuboid with base ABCD and top face EFGH, where E is directly above A, F is directly above B, G is directly above C and H is directly above D.

AB = 3 cm, BC = 4 cm and CG = 12 cm.

Work out the length of AG.

","answerType":"exact","answer":"13 cm","answerDisplay":null,"accept":["13","13cm"],"parts":[],"solution":["First find the diagonal AC of the base using Pythagoras in triangle ABC: AC² = 3² + 4² = 9 + 16 = 25, so AC = 5 cm.","AG is the space diagonal. Triangle ACG has a right angle at C because CG is vertical.","AG² = AC² + CG² = 25 + 144 = 169.","AG = √169 = 13 cm.","Alternatively, in one step: AG = √(3² + 4² + 12²) = √169 = 13 cm."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[8.8,1.2],"label":"C","style":"none","labelPos":"below-right"},{"at":[1.8,1.2],"label":"D","style":"none","labelPos":"below-left"},{"at":[0,4],"label":"E","style":"none","labelPos":"above"},{"at":[7,4],"label":"F","style":"none","labelPos":"above"},{"at":[8.8,5.2],"label":"G","style":"none","labelPos":"above"},{"at":[1.8,5.2],"label":"H","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"3 cm"},{"from":[7,0],"to":[8.8,1.2],"label":"4 cm"},{"from":[8.8,1.2],"to":[1.8,1.2]},{"from":[1.8,1.2],"to":[0,0]},{"from":[0,4],"to":[7,4]},{"from":[7,4],"to":[8.8,5.2]},{"from":[8.8,5.2],"to":[1.8,5.2]},{"from":[1.8,5.2],"to":[0,4]},{"from":[0,0],"to":[0,4]},{"from":[7,0],"to":[7,4]},{"from":[8.8,1.2],"to":[8.8,5.2],"label":"12 cm"},{"from":[1.8,1.2],"to":[1.8,5.2]},{"from":[0,0],"to":[8.8,5.2],"dashed":true}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"6e1626184ad4"},{"id":"g7-3d-pythagoras-and-trigonometry-q02","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":2,"marks":3,"tier":"H","calc":true,"text":"

A cuboid measures 5 cm by 7 cm by 9 cm.

Calculate the length of the longest straight rod that will fit exactly inside the cuboid.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"12.4 cm","answerDisplay":null,"accept":["12.4"],"parts":[],"solution":["The longest rod that fits inside a cuboid lies along the space diagonal.","Space diagonal = √(5² + 7² + 9²) = √(25 + 49 + 81) = √155.","√155 = 12.4499...","The longest rod is 12.4 cm (1 decimal place)."],"diagram":null,"draw":false,"revision":"9468cf8a2756"},{"id":"g7-3d-pythagoras-and-trigonometry-q03","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

ABCDEFGH is a cuboid with base ABCD and top face EFGH, where E is directly above A, F is directly above B, G is directly above C and H is directly above D.

AB = 8 cm, BC = 6 cm and CG = 5 cm.

Calculate the size of the angle between the line AG and the plane ABCD.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"26.6°","answerDisplay":null,"accept":["26.6"],"parts":[],"solution":["A projection is the line’s shadow on a plane, found by dropping perpendiculars to that plane. The angle between a line and a plane is measured between the line and this shadow. For two planes, use lines in each plane perpendicular to their common edge.","The projection of AG onto the plane ABCD is AC, so the required angle is angle GAC.","Find AC using Pythagoras in the base: AC² = 8² + 6² = 64 + 36 = 100, so AC = 10 cm.","Triangle ACG has a right angle at C because CG is vertical.","tan(angle GAC) = CG ÷ AC = 5 ÷ 10 = 0.5","angle GAC = tan−1(0.5) = 26.565...°","The angle between AG and the plane ABCD is 26.6° (1 decimal place)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[8.8,1.2],"label":"C","style":"none","labelPos":"below-right"},{"at":[1.8,1.2],"label":"D","style":"none","labelPos":"below-left"},{"at":[0,4],"label":"E","style":"none","labelPos":"above"},{"at":[7,4],"label":"F","style":"none","labelPos":"above"},{"at":[8.8,5.2],"label":"G","style":"none","labelPos":"above"},{"at":[1.8,5.2],"label":"H","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"8 cm"},{"from":[7,0],"to":[8.8,1.2],"label":"6 cm"},{"from":[8.8,1.2],"to":[1.8,1.2]},{"from":[1.8,1.2],"to":[0,0]},{"from":[0,4],"to":[7,4]},{"from":[7,4],"to":[8.8,5.2]},{"from":[8.8,5.2],"to":[1.8,5.2]},{"from":[1.8,5.2],"to":[0,4]},{"from":[0,0],"to":[0,4]},{"from":[7,0],"to":[7,4]},{"from":[8.8,1.2],"to":[8.8,5.2],"label":"5 cm"},{"from":[1.8,1.2],"to":[1.8,5.2]},{"from":[0,0],"to":[8.8,5.2],"dashed":true},{"from":[0,0],"to":[8.8,1.2],"answer":true,"dashed":true}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"315244a660e5"},{"id":"g7-3d-pythagoras-and-trigonometry-q04","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

A storage box is a cuboid PQRSTUVW with base PQRS and top face TUVW, where T is directly above P, U is directly above Q, V is directly above R and W is directly above S.

PQ = 9 cm, QR = 4 cm and RV = 7 cm.

Work out the size of the angle that the line PV makes with the base PQRS.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"35.4°","answerDisplay":null,"accept":["35.4"],"parts":[],"solution":["A projection is the line’s shadow on a plane, found by dropping perpendiculars to that plane. The angle between a line and a plane is measured between the line and this shadow. For two planes, use lines in each plane perpendicular to their common edge.","The projection of PV onto the base is PR, so the required angle is angle VPR.","Find PR using Pythagoras in the base: PR² = 9² + 4² = 81 + 16 = 97, so PR = √97 = 9.8488... cm.","Triangle PRV has a right angle at R because RV is vertical.","tan(angle VPR) = RV ÷ PR = 7 ÷ √97 = 0.7107...","angle VPR = tan−1(0.7107...) = 35.40...°","The angle is 35.4° (1 decimal place)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"Q","style":"none","labelPos":"below-right"},{"at":[8.8,1.2],"label":"R","style":"none","labelPos":"below-right"},{"at":[1.8,1.2],"label":"S","style":"none","labelPos":"below-left"},{"at":[0,4],"label":"T","style":"none","labelPos":"above"},{"at":[7,4],"label":"U","style":"none","labelPos":"above"},{"at":[8.8,5.2],"label":"V","style":"none","labelPos":"above"},{"at":[1.8,5.2],"label":"W","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"9 cm"},{"from":[7,0],"to":[8.8,1.2],"label":"4 cm"},{"from":[8.8,1.2],"to":[1.8,1.2]},{"from":[1.8,1.2],"to":[0,0]},{"from":[0,4],"to":[7,4]},{"from":[7,4],"to":[8.8,5.2]},{"from":[8.8,5.2],"to":[1.8,5.2]},{"from":[1.8,5.2],"to":[0,4]},{"from":[0,0],"to":[0,4]},{"from":[7,0],"to":[7,4]},{"from":[8.8,1.2],"to":[8.8,5.2],"label":"7 cm"},{"from":[1.8,1.2],"to":[1.8,5.2]},{"from":[0,0],"to":[8.8,5.2],"dashed":true},{"from":[0,0],"to":[8.8,1.2],"answer":true,"dashed":true}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a822f26d0e6d"},{"id":"g7-3d-pythagoras-and-trigonometry-q05","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

A pyramid has a horizontal square base ABCD with sides of length 12 cm.

The vertex V is directly above the centre O of the base.

The height VO of the pyramid is 8 cm.

Calculate the size of the angle that the edge VA makes with the base ABCD.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"43.3°","answerDisplay":null,"accept":["43.3"],"parts":[],"solution":["A projection is the line’s shadow on a plane, found by dropping perpendiculars to that plane. The angle between a line and a plane is measured between the line and this shadow. For two planes, use lines in each plane perpendicular to their common edge.","The projection of VA onto the base is AO, so the required angle is angle VAO.","AO is half the diagonal of the square base. Diagonal AC = √(12² + 12²) = √288 = 12√2 cm.","So AO = 6√2 = 8.4852... cm.","Triangle VAO has a right angle at O because VO is vertical.","tan(angle VAO) = VO ÷ AO = 8 ÷ 6√2 = 0.9428...","angle VAO = tan−1(0.9428...) = 43.31...°","The angle is 43.3° (1 decimal place)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[8.7,1.7999999999999998],"label":"C","style":"none","labelPos":"above"},{"at":[2.7,1.7999999999999998],"label":"D","style":"none","labelPos":"above"},{"at":[4.35,5.9],"label":"V","style":"none","labelPos":"above"},{"at":[4.35,0.8999999999999999],"label":"O","style":"none","labelPos":"below-left"}],"segments":[{"from":[0,0],"to":[6,0],"label":"12 cm"},{"from":[6,0],"to":[8.7,1.7999999999999998]},{"from":[8.7,1.7999999999999998],"to":[2.7,1.7999999999999998]},{"from":[2.7,1.7999999999999998],"to":[0,0]},{"from":[4.35,5.9],"to":[0,0]},{"from":[4.35,5.9],"to":[6,0]},{"from":[4.35,5.9],"to":[8.7,1.7999999999999998]},{"from":[4.35,5.9],"to":[2.7,1.7999999999999998]},{"from":[4.35,5.9],"to":[4.35,0.8999999999999999],"label":"8 cm","dashed":true},{"from":[0,0],"to":[4.35,0.8999999999999999],"answer":true,"dashed":true}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"54c777f6c109"},{"id":"g7-3d-pythagoras-and-trigonometry-q06","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

The diagram shows a wedge ABCDEF.

The base ABCD is a horizontal rectangle with AB = 24 cm and BC = 10 cm.

E is vertically above D and F is vertically above C, with DE = CF = 6 cm.

ABFE is the sloping face of the wedge.

Work out the size of the angle between the line AF and the base ABCD.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"13.0°","answerDisplay":null,"accept":["13.0"],"parts":[],"solution":["A projection is the line’s shadow on a plane, found by dropping perpendiculars to that plane. The angle between a line and a plane is measured between the line and this shadow. For two planes, use lines in each plane perpendicular to their common edge.","F is vertically above C, so the projection of AF onto the base is AC and the required angle is angle FAC.","Find AC using Pythagoras in the base rectangle: AC² = 24² + 10² = 576 + 100 = 676, so AC = 26 cm.","Triangle ACF has a right angle at C because CF is vertical.","tan(angle FAC) = CF ÷ AC = 6 ÷ 26 = 0.2307...","angle FAC = tan−1(0.2307...) = 12.99...°","The angle is 13.0° (1 decimal place)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[8,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[10.25,1.5],"label":"C","style":"none","labelPos":"below-right"},{"at":[2.25,1.5],"label":"D","style":"none","labelPos":"below-left"},{"at":[2.25,5.5],"label":"E","style":"none","labelPos":"above"},{"at":[10.25,5.5],"label":"F","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[8,0],"label":"24 cm"},{"from":[8,0],"to":[10.25,1.5],"label":"10 cm"},{"from":[10.25,1.5],"to":[2.25,1.5]},{"from":[2.25,1.5],"to":[0,0]},{"from":[2.25,1.5],"to":[2.25,5.5],"label":"6 cm"},{"from":[10.25,1.5],"to":[10.25,5.5],"label":"6 cm"},{"from":[2.25,5.5],"to":[10.25,5.5]},{"from":[0,0],"to":[2.25,5.5]},{"from":[8,0],"to":[10.25,5.5]},{"from":[0,0],"to":[10.25,5.5],"dashed":true},{"from":[0,0],"to":[10.25,1.5],"answer":true,"dashed":true}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"e0d6d4609ee5"},{"id":"g7-3d-pythagoras-and-trigonometry-q07","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

ABCDEFGH is a cuboid with base ABCD and top face EFGH, where E is directly above A, F is directly above B, G is directly above C and H is directly above D.

AB = 5 cm, BC = 12 cm and CG = 9 cm.

Calculate the size of the angle between the line AG and the plane BCGF.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"18.4°","answerDisplay":null,"accept":["18.4"],"parts":[],"solution":["A projection is the line’s shadow on a plane, found by dropping perpendiculars to that plane. The angle between a line and a plane is measured between the line and this shadow. For two planes, use lines in each plane perpendicular to their common edge.","AB is perpendicular to the plane BCGF, because AB is perpendicular to both BC and BF.","So the projection of A onto the plane BCGF is B, the projection of AG onto the plane is BG, and the required angle is angle AGB.","BG is a diagonal of the face BCGF: BG² = BC² + CG² = 12² + 9² = 144 + 81 = 225, so BG = 15 cm.","Triangle ABG has a right angle at B because AB is perpendicular to the plane BCGF.","tan(angle AGB) = AB ÷ BG = 5 ÷ 15 = 0.3333...","angle AGB = tan−1(13) = 18.43...°","The angle is 18.4° (1 decimal place)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[8.8,1.2],"label":"C","style":"none","labelPos":"below-right"},{"at":[1.8,1.2],"label":"D","style":"none","labelPos":"below-left"},{"at":[0,4],"label":"E","style":"none","labelPos":"above"},{"at":[7,4],"label":"F","style":"none","labelPos":"above"},{"at":[8.8,5.2],"label":"G","style":"none","labelPos":"above"},{"at":[1.8,5.2],"label":"H","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"5 cm"},{"from":[7,0],"to":[8.8,1.2],"label":"12 cm"},{"from":[8.8,1.2],"to":[1.8,1.2]},{"from":[1.8,1.2],"to":[0,0]},{"from":[0,4],"to":[7,4]},{"from":[7,4],"to":[8.8,5.2]},{"from":[8.8,5.2],"to":[1.8,5.2]},{"from":[1.8,5.2],"to":[0,4]},{"from":[0,0],"to":[0,4]},{"from":[7,0],"to":[7,4]},{"from":[8.8,1.2],"to":[8.8,5.2],"label":"9 cm"},{"from":[1.8,1.2],"to":[1.8,5.2]},{"from":[0,0],"to":[8.8,5.2],"dashed":true},{"from":[7,0],"to":[8.8,5.2],"answer":true,"dashed":true}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"4ed77651bc87"},{"id":"g7-3d-pythagoras-and-trigonometry-q08","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

A pyramid has a horizontal square base ABCD with sides of length 16 cm.

The vertex V is directly above the centre O of the base.

The height VO of the pyramid is 15 cm.

M is the midpoint of BC.

Calculate the size of the angle between the face VBC and the base ABCD.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"61.9°","answerDisplay":null,"accept":["61.9"],"parts":[],"solution":["A projection is the line’s shadow on a plane, found by dropping perpendiculars to that plane. The angle between a line and a plane is measured between the line and this shadow. For two planes, use lines in each plane perpendicular to their common edge.","The angle between the face VBC and the base is the angle VMO, where M is the midpoint of BC.","VM is perpendicular to BC (VBC is isosceles) and OM is perpendicular to BC, so angle VMO measures the angle between the two planes.","OM is the distance from the centre of the square to the midpoint of a side, so OM = 16 ÷ 2 = 8 cm.","Triangle VOM has a right angle at O because VO is vertical.","tan(angle VMO) = VO ÷ OM = 15 ÷ 8 = 1.875","angle VMO = tan−1(1.875) = 61.92...°","The angle between the face VBC and the base is 61.9° (1 decimal place)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[8.7,1.7999999999999998],"label":"C","style":"none","labelPos":"above"},{"at":[2.7,1.7999999999999998],"label":"D","style":"none","labelPos":"above"},{"at":[4.35,5.9],"label":"V","style":"none","labelPos":"above"},{"at":[4.35,0.8999999999999999],"label":"O","style":"none","labelPos":"below-left"},{"at":[7.35,0.8999999999999999],"label":"M","style":"dot","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[6,0],"label":"16 cm"},{"from":[6,0],"to":[8.7,1.7999999999999998]},{"from":[8.7,1.7999999999999998],"to":[2.7,1.7999999999999998]},{"from":[2.7,1.7999999999999998],"to":[0,0]},{"from":[4.35,5.9],"to":[0,0]},{"from":[4.35,5.9],"to":[6,0]},{"from":[4.35,5.9],"to":[8.7,1.7999999999999998]},{"from":[4.35,5.9],"to":[2.7,1.7999999999999998]},{"from":[4.35,5.9],"to":[4.35,0.8999999999999999],"label":"15 cm","dashed":true},{"from":[4.35,5.9],"to":[7.35,0.8999999999999999],"answer":true,"dashed":true},{"from":[7.35,0.8999999999999999],"to":[4.35,0.8999999999999999],"answer":true,"dashed":true}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"50486881c9c2"},{"id":"g7-3d-pythagoras-and-trigonometry-q09","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

A cuboid has a square base and a height of 8 cm.

The length of the longest diagonal of the cuboid, from one corner of the base to the opposite corner of the top face, is 12 cm.

Work out the volume of the cuboid.

","answerType":"exact","answer":"320 cm³","answerDisplay":null,"accept":["320","320cm^3","320 cm^3"],"parts":[],"solution":["Let the side of the square base be x cm.","The space diagonal of a cuboid satisfies d² = x² + x² + h².","So 12² = x² + x² + 8², which gives 144 = 2x² + 64.","2x² = 80, so x² = 40.","The volume is base area × height = x² × 8 = 40 × 8 = 320.","Volume = 320 cm³. Note that the value of x itself is not needed, only x²."],"diagram":null,"draw":false,"revision":"a1a0d907c690"},{"id":"g7-3d-pythagoras-and-trigonometry-q10","topicId":"g7-3d-pythagoras-and-trigonometry","topic":"3D Pythagoras and Trigonometry","grade":"7","topicGrade":"7","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

The diagram shows a tent in the shape of a prism.

The base ABCD is a horizontal rectangle with AB = 16 m and BC = 6 m.

The ridge EF is horizontal and runs parallel to AB, directly above the middle of the base, with E at the end above AD and F at the end above BC.

The ridge is 4 m above the base.

Work out the size of the angle between the line AF and the base ABCD.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"13.8°","answerDisplay":null,"accept":["13.8"],"parts":[],"solution":["A projection is the line’s shadow on a plane, found by dropping perpendiculars to that plane. The angle between a line and a plane is measured between the line and this shadow. For two planes, use lines in each plane perpendicular to their common edge.","Set up the projection of F onto the base. F is above the point N that lies on the base, on the line BC, midway between B and C, so N is 16 m from the line AD and 3 m from the line AB.","The projection of AF onto the base is AN, so the required angle is angle FAN.","Find AN using Pythagoras in the base: AN² = 16² + 3² = 256 + 9 = 265, so AN = √265 = 16.2788... m.","Triangle ANF has a right angle at N because FN is vertical, with FN = 4 m.","tan(angle FAN) = FN ÷ AN = 4 ÷ √265 = 0.2457...","angle FAN = tan−1(0.2457...) = 13.80...°","The angle between AF and the base is 13.8° (1 decimal place)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[8,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[10.25,1.5],"label":"C","style":"none","labelPos":"below-right"},{"at":[2.25,1.5],"label":"D","style":"none","labelPos":"below-left"},{"at":[1.125,4.75],"label":"E","style":"none","labelPos":"above"},{"at":[9.125,4.75],"label":"F","style":"none","labelPos":"above"},{"at":[9.125,0.75],"label":"N","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[8,0],"label":"16 m"},{"from":[8,0],"to":[10.25,1.5],"label":"6 m"},{"from":[10.25,1.5],"to":[2.25,1.5]},{"from":[2.25,1.5],"to":[0,0]},{"from":[0,0],"to":[1.125,4.75]},{"from":[2.25,1.5],"to":[1.125,4.75]},{"from":[8,0],"to":[9.125,4.75]},{"from":[10.25,1.5],"to":[9.125,4.75]},{"from":[1.125,4.75],"to":[9.125,4.75]},{"from":[9.125,4.75],"to":[9.125,0.75],"label":"4 m","dashed":true},{"from":[0,0],"to":[9.125,4.75],"dashed":true},{"from":[0,0],"to":[9.125,0.75],"answer":true,"dashed":true}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"b272e92fdb85"},{"id":"g7-algebraic-fractions-q01","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

Simplify fully

x2 + 5xx2 + 8x + 15

","answerType":"exact","answer":"x/(x+3)","answerDisplay":"xx+3","accept":["x/(x + 3)"],"parts":[],"solution":["The original expression is defined only when x ≠ -5 and x ≠ -3. A denominator cannot be zero. These restrictions still apply after simplifying.","Factorise the numerator: x^2 + 5x = x(x + 5).","Factorise the denominator: x^2 + 8x + 15 = (x + 3)(x + 5).","Cancel the common factor (x + 5) to get xx + 3.","Cancel common factors only: dividing the whole numerator and denominator by the same nonzero factor leaves the fraction unchanged. You cannot cancel individual terms joined by + or -."],"diagram":null,"draw":false,"revision":"4e4e45381c5f"},{"id":"g7-algebraic-fractions-q02","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

Simplify fully

x2 − 92x2 + 5x − 3

","answerType":"exact","answer":"(x-3)/(2x-1)","answerDisplay":"x-32x-1","accept":["(x - 3)/(2x - 1)"],"parts":[],"solution":["The original expression is defined only when x ≠ -3 and x ≠ 1/2. A denominator cannot be zero. These restrictions still apply after simplifying.","Factorise the numerator using the difference of two squares: x^2 - 9 = (x - 3)(x + 3).","Factorise the denominator: 2x^2 + 5x - 3 = (2x - 1)(x + 3).","Cancel the common factor (x + 3) to get x - 32x - 1.","Cancel common factors only: dividing the whole numerator and denominator by the same nonzero factor leaves the fraction unchanged. You cannot cancel individual terms joined by + or -."],"diagram":null,"draw":false,"revision":"3ee565ac79fe"},{"id":"g7-algebraic-fractions-q03","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

Write

3x + 2 + 5x − 1

as a single fraction in its simplest form.

","answerType":"exact","answer":"(8x+7)/((x+2)(x-1))","answerDisplay":"8x+7(x+2)(x-1)","accept":["(8x + 7)/((x + 2)(x - 1))","(8x+7)/(x^2+x-2)"],"parts":[],"solution":["The original expression is defined only when x ≠ -2 and x ≠ 1. A denominator cannot be zero. These restrictions still apply after simplifying.","Make the denominators match by multiplying each fraction’s numerator and denominator by the same missing factor. This multiplies each fraction by 1, so its value stays the same.","Use the common denominator (x + 2)(x - 1).","3x + 2 + 5x - 1 = 3(x - 1) + 5(x + 2)(x + 2)(x - 1).","Expand the numerator: 3x - 3 + 5x + 10 = 8x + 7.","So the single fraction is 8x + 7(x + 2)(x - 1)."],"diagram":null,"draw":false,"revision":"37fe62ab3c2a"},{"id":"g7-algebraic-fractions-q04","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":false,"text":"

Show that

4x + 3 − 2x + 5

can be written as 2(x + 7)(x + 3)(x + 5)

","answerType":"open","answer":"2(x+7)/((x+3)(x+5))","answerDisplay":"2(x+7)(x+3)(x+5)","accept":[],"parts":[],"solution":["The original expression is defined only when x ≠ -3 and x ≠ -5. A denominator cannot be zero. These restrictions still apply after simplifying.","Make the denominators match by multiplying each fraction’s numerator and denominator by the same missing factor. This multiplies each fraction by 1, so its value stays the same.","Use the common denominator (x + 3)(x + 5).","4x + 3 - 2x + 5 = 4(x + 5) - 2(x + 3)(x + 3)(x + 5).","Expand the numerator: 4x + 20 - 2x - 6 = 2x + 14.","Factorise: 2x + 14 = 2(x + 7), giving 2(x + 7)(x + 3)(x + 5) as required."],"diagram":null,"draw":false,"revision":"838980148275"},{"id":"g7-algebraic-fractions-q05","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

Simplify fully

[x2 + 7x + 10x2 − 4] ÷ [x2 + 10x + 253x − 6]

","answerType":"exact","answer":"3/(x+5)","answerDisplay":"3x+5","accept":["3/(x + 5)"],"parts":[],"solution":["The original expression is defined only when x ≠ 2, -2 or -5. A denominator cannot be zero, and the fraction we divide by cannot be zero. These restrictions still apply after simplifying.","Factorise each expression: x^2 + 7x + 10 = (x + 2)(x + 5), x^2 - 4 = (x - 2)(x + 2), x^2 + 10x + 25 = (x + 5)^2, 3x - 6 = 3(x - 2).","Dividing by a fraction is multiplying by its reciprocal: [(x + 2)(x + 5)(x - 2)(x + 2)] x [3(x - 2)(x + 5)^2].","Cancel (x + 2): gives x + 5x - 2 x 3(x - 2)(x + 5)^2.","Cancel (x - 2) and one factor of (x + 5) to leave 3x + 5.","Cancel common factors only: dividing the whole numerator and denominator by the same nonzero factor leaves the fraction unchanged. You cannot cancel individual terms joined by + or -."],"diagram":null,"draw":false,"revision":"05c66b232754"},{"id":"g7-algebraic-fractions-q06","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

Simplify fully

[2x2 + 7x + 3x2 − 9] × [x2 − 3x4x2 − 1]

","answerType":"exact","answer":"x/(2x-1)","answerDisplay":"x2x-1","accept":["x/(2x - 1)"],"parts":[],"solution":["The original expression is defined only when x ≠ 3, -3, 1/2 or -1/2. A denominator cannot be zero. These restrictions still apply after simplifying.","Factorise each expression: 2x^2 + 7x + 3 = (2x + 1)(x + 3), x^2 - 9 = (x - 3)(x + 3), x^2 - 3x = x(x - 3), 4x^2 - 1 = (2x - 1)(2x + 1).","The product is (2x + 1)(x + 3) x x(x - 3)(x - 3)(x + 3)(2x - 1)(2x + 1).","Cancel (2x + 1), (x + 3) and (x - 3).","This leaves x2x - 1.","Cancel common factors only: dividing the whole numerator and denominator by the same nonzero factor leaves the fraction unchanged. You cannot cancel individual terms joined by + or -."],"diagram":null,"draw":false,"revision":"6fa7207a96a9"},{"id":"g7-algebraic-fractions-q07","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":7,"marks":3,"tier":"H","calc":false,"text":"

Solve

x + 43 + 2x − 15 = 7

","answerType":"exact","answer":"x = 8","answerDisplay":null,"accept":["8"],"parts":[],"solution":["Multiply every term by 15: 5(x + 4) + 3(2x - 1) = 105.","Expand: 5x + 20 + 6x - 3 = 105.","Simplify: 11x + 17 = 105, so 11x = 88.","x = 8. Check: 123 + 155 = 4 + 3 = 7."],"diagram":null,"draw":false,"revision":"611f6bb8c3e3"},{"id":"g7-algebraic-fractions-q08","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

Solve

4x − 1 − 2x + 2 = 1

","answerType":"exact","answer":"x = 4 or x = -3","answerDisplay":null,"accept":["x = -3 or x = 4","4, -3","-3, 4"],"parts":[],"solution":["The original expression is defined only when x ≠ 1 and x ≠ -2. A denominator cannot be zero. These restrictions still apply after simplifying.","Multiply both sides by (x - 1)(x + 2): 4(x + 2) - 2(x - 1) = (x - 1)(x + 2).","Expand: 4x + 8 - 2x + 2 = x^2 + x - 2, so 2x + 10 = x^2 + x - 2.","Rearrange: x^2 - x - 12 = 0.","Factorise: (x - 4)(x + 3) = 0, so x = 4 or x = -3.","Check x = 4: 43 - 26 = 43 - 13 = 1. Check x = -3: 4-4 - 2-1 = -1 + 2 = 1."],"diagram":null,"draw":false,"revision":"de8d4667ffff"},{"id":"g7-algebraic-fractions-q09","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

Simplify fully

(1 + 2x) ÷ (1 − 4x2)

","answerType":"exact","answer":"x/(x-2)","answerDisplay":"xx-2","accept":["x/(x - 2)"],"parts":[],"solution":["The original expression is defined only when x ≠ 0, 2 or -2. A denominator cannot be zero, and the fraction we divide by cannot be zero. These restrictions still apply after simplifying.","Write each bracket as a single fraction: 1 + 2x = x + 2x and 1 - 4x^2 = x^2 - 4x^2.","Dividing gives x + 2x x x^2x^2 - 4.","Factorise x^2 - 4 = (x - 2)(x + 2).","Cancel (x + 2) and one factor of x to leave xx - 2.","Cancel common factors only: dividing the whole numerator and denominator by the same nonzero factor leaves the fraction unchanged. You cannot cancel individual terms joined by + or -."],"diagram":null,"draw":false,"revision":"cf9acd46a613"},{"id":"g7-algebraic-fractions-q10","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

Solve

1x − 2 + 2x + 1 = 3

Give your answers in the form a ± √b where a and b are integers.

","answerType":"exact","answer":"x = 1 + sqrt(2) or x = 1 - sqrt(2)","answerDisplay":null,"accept":["1 ± √2","x = 1 ± √2","1 + √2 and 1 - √2"],"parts":[],"solution":["The original expression is defined only when x ≠ 2 and x ≠ -1. A denominator cannot be zero. These restrictions still apply after simplifying.","Multiply both sides by (x - 2)(x + 1): (x + 1) + 2(x - 2) = 3(x - 2)(x + 1).","Expand: 3x - 3 = 3(x^2 - x - 2).","Divide by 3: x - 1 = x^2 - x - 2, so x^2 - 2x - 1 = 0.","Complete the square: (x - 1)^2 = 2.","x = 1 ± √2. Check x = 1 + √2: 1√2 - 1 + 22 + √2 = (√2 + 1) + (2 - √2) = 3."],"diagram":null,"draw":false,"revision":"cab0e974443d"},{"id":"g7-algebraic-fractions-q11","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":11,"marks":6,"tier":"H","calc":true,"text":"

Show that

x2 + 3xx2 − x − 12 − 2x + 1

can be written as x2 − x + 8(x − 4)(x + 1)

","answerType":"open","answer":"(x^2-x+8)/((x-4)(x+1))","answerDisplay":"x^2-x+8(x-4)(x+1)","accept":[],"parts":[],"solution":["The original expression is defined only when x ≠ -3, 4 or -1. A denominator cannot be zero. These restrictions still apply after simplifying.","Factorise the first fraction: x^2 + 3x = x(x + 3) and x^2 - x - 12 = (x - 4)(x + 3).","Cancel (x + 3): the first fraction simplifies to xx - 4.","Combine over the common denominator (x - 4)(x + 1): x(x + 1) - 2(x - 4)(x - 4)(x + 1).","Expand the numerator: x^2 + x - 2x + 8 = x^2 - x + 8.","This gives x^2 - x + 8(x - 4)(x + 1) as required.","Cancel common factors only: dividing the whole numerator and denominator by the same nonzero factor leaves the fraction unchanged. You cannot cancel individual terms joined by + or -."],"diagram":null,"draw":false,"revision":"64ceb2f62a81"},{"id":"g7-algebraic-fractions-q12","topicId":"g7-algebraic-fractions","topic":"Algebraic Fractions","grade":"7","topicGrade":"7","strand":"A","difficulty":12,"marks":6,"tier":"H","calc":true,"text":"

  1. Simplify fully 2x2 + 5x − 3x2 − 9

  2. Hence, or otherwise, solve

    2x2 + 5x − 3x2 − 9 = 5

","answerType":"multi","answer":"a) (2x-1)/(x-3) b) x = 14/3","answerDisplay":"a) 2x-1x-3 b) x = 143","accept":[],"parts":[{"label":"a","answer":"(2x - 1)/(x - 3)","answerDisplay":"2x - 1x - 3"},{"label":"b","answer":"x = 14/3","answerDisplay":"x = 143"}],"solution":["The original expression is defined only when x ≠ 3 and x ≠ -3. A denominator cannot be zero. These restrictions still apply after simplifying.","a) Factorise: 2x^2 + 5x - 3 = (2x - 1)(x + 3) and x^2 - 9 = (x - 3)(x + 3).","a) Cancel (x + 3) to get 2x - 1x - 3.","b) Using part a, solve 2x - 1x - 3 = 5.","b) Multiply up: 2x - 1 = 5(x - 3) = 5x - 15.","b) So 14 = 3x, giving x = 143.","b) Check: at x = 143 the original fraction is 57591159 = 5.","Cancel common factors only: dividing the whole numerator and denominator by the same nonzero factor leaves the fraction unchanged. You cannot cancel individual terms joined by + or -."],"diagram":null,"draw":false,"revision":"814f1d48bcc3"},{"id":"g7-bounds-q01","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":1,"marks":3,"tier":"H","calc":true,"text":"

y = 7.8 correct to 1 decimal place.

z = 4.1 correct to 1 decimal place.

  1. Write down the upper bound of y.
  2. Work out the upper bound of y − z.
","answerType":"multi","answer":"a) 7.85 b) 3.8","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"7.85"},{"label":"b","answer":"3.8"}],"solution":["(a) y is 7.8 to 1 dp, so 7.75 ≤ y < 7.85. The upper bound of y is 7.85.","(b) To make y − z as large as possible, use the upper bound of y and the lower bound of z.","The lower bound of z is 4.05.","7.85 − 4.05 = 3.8, so the upper bound of y − z is 3.8."],"diagram":null,"draw":false,"revision":"b41ce0c71982"},{"id":"g7-bounds-q02","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":2,"marks":3,"tier":"H","calc":true,"text":"

a = 18.4 correct to 1 decimal place.

b = 2.7 correct to 1 decimal place.

Calculate the upper bound of ab

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"6.96","answerDisplay":null,"accept":["6.96"],"parts":[],"solution":["To make a ÷ b as large as possible, use the upper bound of a and the lower bound of b.","Upper bound of a = 18.45, lower bound of b = 2.65.","18.45 ÷ 2.65 = 6.96226...","To 3 significant figures the upper bound is 6.96."],"diagram":null,"draw":false,"revision":"9f8eb97a26d9"},{"id":"g7-bounds-q03","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

A tram travels a distance of 260 metres, correct to the nearest 10 metres.

The journey takes 32.5 seconds, correct to 1 decimal place.

Calculate the lower bound of the average speed of the tram in m/s.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"7.83","answerDisplay":null,"accept":["7.83","7.83 m/s"],"parts":[],"solution":["Average speed = distance ÷ time. For the lower bound, use the lower bound of the distance and the upper bound of the time.","Lower bound of distance = 255 m. Upper bound of time = 32.55 s.","255 ÷ 32.55 = 7.8341...","To 3 significant figures the lower bound is 7.83 m/s."],"diagram":null,"draw":false,"revision":"52e8d9db6a37"},{"id":"g7-bounds-q04","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

In a right-angled triangle, the angle at A is x°.

The side opposite angle A has length 5.2 cm, correct to 1 decimal place.

The side adjacent to angle A has length 8.4 cm, correct to 1 decimal place.

Calculate the lower bound of x.

Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"31.4","answerDisplay":null,"accept":["31.4","31.4 degrees"],"parts":[],"solution":["tan x = opposite ÷ adjacent.","To make x as small as possible, make tan x as small as possible: use the lower bound of the opposite side and the upper bound of the adjacent side.","Lower bound of opposite = 5.15 cm. Upper bound of adjacent = 8.45 cm.","tan x = 5.15 ÷ 8.45 = 0.60946...","x = tan−1(0.60946...) = 31.36...°, so the lower bound of x is 31.4° to 1 dp."],"diagram":{"type":"shape","notAccurate":true,"width":380,"polygons":[{"pts":[[0,0],[8.4,0],[8.4,5.2]],"labels":["A",null,null],"sides":["8.4 cm","5.2 cm",null],"fill":false}],"angles":[{"at":[0,0],"from":[8.4,0],"to":[8.4,5.2],"label":"x","italic":true},{"at":[8.4,0],"from":[0,0],"to":[8.4,5.2],"right":true}]},"draw":false,"revision":"624630ee9054"},{"id":"g7-bounds-q05","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

The volume of a cube is 730 cm³, correct to 2 significant figures.

Work out the lower bound of the length of one edge of the cube.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"8.98","answerDisplay":null,"accept":["8.98","8.98 cm"],"parts":[],"solution":["730 to 2 significant figures means 725 ≤ volume < 735.","The lower bound of the volume is 725 cm³.","Edge length = cube root of volume, so the lower bound of the edge is the cube root of 725.","72513 = 8.9835...","To 3 significant figures the lower bound is 8.98 cm."],"diagram":null,"draw":false,"revision":"31141fc5ca97"},{"id":"g7-bounds-q06","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

The formula v = u + at gives the final velocity of an object.

v = 27.3 correct to 1 decimal place.

u = 11.6 correct to 1 decimal place.

t = 4.5 correct to 1 decimal place.

By rearranging the formula, calculate the upper bound of a.

Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"3.55","answerDisplay":null,"accept":["3.55"],"parts":[],"solution":["Rearrange: a = (v − u) ÷ t.","For the upper bound of a, make the numerator as large as possible and the denominator as small as possible.","Use the upper bound of v (27.35), the lower bound of u (11.55) and the lower bound of t (4.45).","a = (27.35 − 11.55) ÷ 4.45 = 15.8 ÷ 4.45 = 3.5505...","To 3 significant figures the upper bound of a is 3.55."],"diagram":null,"draw":false,"revision":"e787d1995750"},{"id":"g7-bounds-q07","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

Last week Ben worked for 7.5 hours, correct to the nearest half hour.

He was paid £84, correct to the nearest pound.

Ben says, "My hourly rate of pay was definitely more than £10.90."

Determine whether Ben is correct.

You must show your working using bounds.

","answerType":"open","answer":"Ben is not correct. Lower bound of rate = 83.50 / 7.75 = £10.77 (2 dp), which is less than £10.90, so his rate could have been below £10.90.","answerDisplay":"Ben is not correct. Lower bound of rate = 83.507.75 = £10.77 (2 dp), which is less than £10.90, so his rate could have been below £10.90.","accept":[],"parts":[],"solution":["Hourly rate = pay ÷ hours. The lower bound of the rate uses the lower bound of the pay and the upper bound of the hours.","Pay to the nearest pound: lower bound = £83.50. Hours to the nearest half hour: upper bound = 7.75 hours.","Lower bound of rate = 83.50 ÷ 7.75 = £10.7741... = £10.77 (2 dp).","£10.77 < £10.90, so the rate was not definitely more than £10.90. Ben is not correct.","The upper endpoint 7.75 hours is excluded, so this lower bound is not attained. For a valid example, £83.50 over 7.7 hours gives about £10.84 per hour, below £10.90; both measurements round as stated."],"diagram":null,"draw":false,"revision":"a85c9c632809"},{"id":"g7-bounds-q08","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

A metal block has mass 485 g, correct to the nearest gram.

The volume of the block is 62.4 cm³, correct to 1 decimal place.

density = mass ÷ volume

Work out the density of the metal in g/cm³, giving your answer to a suitable degree of accuracy.

You must explain why your answer is to a suitable degree of accuracy.

","answerType":"open","answer":"7.8 g/cm3, because the upper bound 485.5/62.35 = 7.7867... and the lower bound 484.5/62.45 = 7.7582... both round to 7.8 to 2 significant figures.","answerDisplay":"7.8 g/cm3, because the upper bound 485.562.35 = 7.7867... and the lower bound 484.562.45 = 7.7582... both round to 7.8 to 2 significant figures.","accept":[],"parts":[],"solution":["Upper bound of density = upper bound of mass ÷ lower bound of volume = 485.5 ÷ 62.35 = 7.7866...","Lower bound of density = lower bound of mass ÷ upper bound of volume = 484.5 ÷ 62.45 = 7.7582...","To 2 significant figures both bounds round to 7.8.","So the density is 7.8 g/cm³, because the upper and lower bounds agree to this degree of accuracy."],"diagram":null,"draw":false,"revision":"598754b6b03f"},{"id":"g7-bounds-q09","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

A cyclist travels 93.6 metres, correct to 1 decimal place.

The ride takes 12.0 seconds, correct to 1 decimal place.

Calculate the average speed of the cyclist in m/s, giving your answer to a suitable degree of accuracy.

You must show your working and explain your answer.

","answerType":"open","answer":"7.8 m/s, because the upper bound 93.65/11.95 = 7.8368... and the lower bound 93.55/12.05 = 7.7634... both round to 7.8 to 2 significant figures.","answerDisplay":"7.8 m/s, because the upper bound 93.6511.95 = 7.8368... and the lower bound 93.5512.05 = 7.7634... both round to 7.8 to 2 significant figures.","accept":[],"parts":[],"solution":["Speed = distance ÷ time.","Upper bound of speed = 93.65 ÷ 11.95 = 7.8368...","Lower bound of speed = 93.55 ÷ 12.05 = 7.7634...","To 2 significant figures both bounds round to 7.8.","So the average speed is 7.8 m/s, because the upper and lower bounds agree to this degree of accuracy."],"diagram":null,"draw":false,"revision":"dec5f242752c"},{"id":"g7-bounds-q10","topicId":"g7-bounds","topic":"Bounds","grade":"7","topicGrade":"7","strand":"N","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

The kinetic energy, E joules, of an object is given by

E = 12mv²

m = 1.2 kg, correct to 1 decimal place.

v = 8.4 m/s, correct to 1 decimal place.

  1. Work out the upper bound of E. Give your answer correct to 3 significant figures.
  2. The lower bound of E is 40.1 joules, correct to 3 significant figures. Write down the value of E to a suitable degree of accuracy. Give a reason for your answer.
","answerType":"multi","answer":"a) 44.6 b) 40 (to 1 sf), because the upper bound 44.6... and the lower bound 40.09... both round to 40 to 1 significant figure","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"44.6"},{"label":"b","answer":"40"}],"solution":["(a) For the upper bound of E use the upper bounds of m and v.","Upper bound of m = 1.25, upper bound of v = 8.45.","E = 0.5 × 1.25 × 8.45² = 0.5 × 1.25 × 71.4025 = 44.6265...","To 3 significant figures the upper bound is 44.6 joules.","(b) The bounds are 40.09... and 44.62... To 1 significant figure both round to 40.","So E = 40 joules to 1 significant figure, because the upper and lower bounds agree at this degree of accuracy."],"diagram":null,"draw":false,"revision":"e36f1436e5cd"},{"id":"g7-conditional-probability-q01","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

A bag contains 4 red counters and 6 blue counters.

Two counters are taken from the bag at random, without replacement.

Work out the probability that the two counters are different colours.

","answerType":"exact","answer":"8/15","answerDisplay":"815","accept":["48/90"],"parts":[],"solution":["Different colours can occur in two separate orders: red then blue, or blue then red. Multiply the conditional probabilities along each order, then add because these outcomes cannot happen together.","P(red then blue) = 410 × 69 = 2490.","P(blue then red) = 610 × 49 = 2490.","P(different colours) = 2490 + 2490 = 4890 = 815"],"diagram":null,"draw":false,"revision":"afca6e8c7bd0"},{"id":"g7-conditional-probability-q02","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

Five cards are numbered 1, 2, 3, 4 and 5.

Two cards are taken at random, without replacement.

The two numbers are multiplied together.

Work out the probability that the product is even.

","answerType":"exact","answer":"7/10","answerDisplay":"710","accept":["0.7","70%"],"parts":[],"solution":["The product is odd only if both cards are odd. The odd cards are 1, 3 and 5.","P(both odd) = 35 × 24 = 620 = 310.","P(product even) = 1 - 310 = 710"],"diagram":null,"draw":false,"revision":"8cdf2b103e3b"},{"id":"g7-conditional-probability-q03","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

A box contains 5 green pegs and 3 white pegs.

Two pegs are taken from the box at random, without replacement.

Work out the probability that both pegs are green.

","answerType":"exact","answer":"5/14","answerDisplay":"514","accept":["20/56"],"parts":[],"solution":["P(first peg green) = 58.","After one green peg is removed, 4 green pegs remain out of 7.","P(both green) = 58 × 47 = 2056 = 514"],"diagram":null,"draw":false,"revision":"03ddf30f23fa"},{"id":"g7-conditional-probability-q04","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

A bag contains 7 red beads and 5 blue beads.

Two beads are taken from the bag at random, without replacement.

Work out the probability that exactly one of the beads is red.

","answerType":"exact","answer":"35/66","answerDisplay":"3566","accept":["70/132"],"parts":[],"solution":["P(red then blue) = 712 × 511 = 35132.","P(blue then red) = 512 × 711 = 35132.","P(exactly one red) = 35132 + 35132 = 70132 = 3566"],"diagram":null,"draw":false,"revision":"34ade40c43f9"},{"id":"g7-conditional-probability-q05","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

The probability that Ravi cycles to work on Monday is 0.7.

If he cycles on Monday, the probability that he cycles on Tuesday is 0.9.

If he does not cycle on Monday, the probability that he cycles on Tuesday is 0.4.

Work out the probability that Ravi cycles to work on Tuesday.

","answerType":"exact","answer":"0.75","answerDisplay":null,"accept":["3/4","75%"],"parts":[],"solution":["P(cycles Monday and Tuesday) = 0.7 × 0.9 = 0.63.","P(does not cycle Monday but cycles Tuesday) = 0.3 × 0.4 = 0.12.","P(cycles Tuesday) = 0.63 + 0.12 = 0.75"],"diagram":null,"draw":false,"revision":"a0233171f73f"},{"id":"g7-conditional-probability-q06","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":6,"marks":4,"tier":"H","calc":false,"text":"

Six cards are numbered 1, 2, 3, 4, 5 and 6.

Two cards are taken at random, without replacement.

The two numbers are added together.

Work out the probability that the sum is even.

","answerType":"exact","answer":"2/5","answerDisplay":"25","accept":["12/30","0.4"],"parts":[],"solution":["The sum is even only when both numbers are odd or both are even.","There are 3 odd cards and 3 even cards.","P(both odd) = 36 × 25 = 630, and P(both even) = 36 × 25 = 630.","P(sum even) = 630 + 630 = 1230 = 25"],"diagram":null,"draw":false,"revision":"b1b4b51ab9d5"},{"id":"g7-conditional-probability-q07","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

A tin contains 12 chocolates. 5 are dark chocolate and 7 are milk chocolate.

Nina takes two chocolates at random, without replacement.

Work out the probability that she takes at least one dark chocolate.

","answerType":"exact","answer":"15/22","answerDisplay":"1522","accept":["90/132"],"parts":[],"solution":["P(no dark chocolates) = P(both milk) = 712 × 611 = 42132 = 722.","P(at least one dark) = 1 - 722 = 1522"],"diagram":null,"draw":false,"revision":"183572aba93d"},{"id":"g7-conditional-probability-q08","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":8,"marks":5,"tier":"H","calc":false,"text":"

A bag contains 5 blue counters and 7 white counters.

Tom takes 3 counters from the bag at random, one at a time, without replacement.

He takes 2 blue counters and 1 white counter.

  1. Write down the number of blue counters left in the bag.
  2. Tom now takes two more counters at random, without replacement. Work out the probability that both of these counters are white.
","answerType":"multi","answer":"a) 3; b) 5/12","answerDisplay":"a) 3; b) 512","accept":[],"parts":[{"label":"a","answer":"3"},{"label":"b","answer":"5/12","answerDisplay":"512"}],"solution":["Blue counters left = 5 - 2 = 3, and white counters left = 7 - 1 = 6, so 9 counters remain.","P(first counter white) = 69.","P(second counter white) = 58.","P(both white) = 69 × 58 = 3072 = 512"],"diagram":null,"draw":false,"revision":"448eed870302"},{"id":"g7-conditional-probability-q09","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

A bag contains 4 red discs and 8 blue discs.

Meg takes two discs from the bag at random, without replacement.

Meg says,

\"The probability that both discs are blue is more than three times the probability that both discs are red.\"

Is Meg correct? You must show your working.

","answerType":"open","answer":"Yes. P(both blue) = 8/12 x 7/11 = 56/132 = 14/33 and P(both red) = 4/12 x 3/11 = 12/132 = 1/11. Three times P(both red) = 36/132, and 56/132 > 36/132, so Meg is correct.","answerDisplay":"Yes. P(both blue) = 812 x 711 = 56132 = 1433 and P(both red) = 412 x 311 = 12132 = 111. Three times P(both red) = 36132, and 56132 > 36132, so Meg is correct.","accept":[],"parts":[],"solution":["P(both blue) = 812 × 711 = 56132.","P(both red) = 412 × 311 = 12132.","Three times P(both red) = 36132.","56132 > 36132, so the claim is true: Meg is correct."],"diagram":null,"draw":false,"revision":"fa2071d4799c"},{"id":"g7-conditional-probability-q10","topicId":"g7-conditional-probability","topic":"Conditional Probability","grade":"7","topicGrade":"7","strand":"P","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

There are 40 students in a year group.

The Venn diagram for drama (D) and music (M) shows:

D only: 10   D ∩ M: 8   M only: 12   neither: 10

Two of the students who do music are chosen at random, without replacement.

Work out the probability that both of these students also do drama.

","answerType":"exact","answer":"14/95","answerDisplay":"1495","accept":["56/380"],"parts":[],"solution":["Number of students who do music = 8 + 12 = 20.","Of these, 8 also do drama.","P(first student also does drama) = 820.","P(second student also does drama) = 719.","P(both do drama) = 820 × 719 = 56380 = 1495"],"diagram":{"type":"venn","sets":["D","M"],"regions":{"A":"10","AB":"8","B":"12","none":"10"}},"draw":false,"revision":"b0ccf283075f"},{"id":"g7-congruent-triangles-q01","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":1,"marks":1,"tier":"F","calc":false,"text":"

Here are the side lengths of four triangles.

Triangle A: 5 cm, 7 cm, 9 cm
Triangle B: 5 cm, 7 cm, 8 cm
Triangle C: 9 cm, 7 cm, 5 cm
Triangle D: 5 cm, 6 cm, 9 cm

Write down the letters of the two triangles that are congruent to each other.

","answerType":"exact","answer":"A and C","answerDisplay":null,"accept":["A, C","C and A","A C","triangles A and C"],"parts":[],"solution":["Congruent triangles have the same shape and size. SSS (side-side-side) says that three matching side lengths are enough to establish this, even if one triangle is rotated or reflected.","Triangle A has sides 5 cm, 7 cm and 9 cm. Triangle C has sides 9 cm, 7 cm and 5 cm, which is the same set of three lengths.","Triangle B (5, 7, 8) and Triangle D (5, 6, 9) each differ from this set, so the congruent pair is A and C."],"diagram":null,"draw":false,"revision":"dcd0c1916a73"},{"id":"g7-congruent-triangles-q02","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":2,"marks":1,"tier":"FH","calc":true,"text":"

In triangles PQR and XYZ,

PQ = XY = 6 cm
angle PQR = angle XYZ = 40°
QR = YZ = 8 cm

The two triangles are congruent.

Write down the congruence condition that shows the triangles are congruent.

","answerType":"exact","answer":"SAS","answerDisplay":null,"accept":["side angle side","side-angle-side","S.A.S."],"parts":[],"solution":["Two pairs of sides are equal: PQ = XY and QR = YZ.","The equal angles, angle PQR and angle XYZ, are between these pairs of sides at Q and at Y, so the angle is the included angle in each triangle.","Two sides and the included angle match, so the condition is SAS."],"diagram":null,"draw":false,"revision":"13aa20c86e8e"},{"id":"g7-congruent-triangles-q03","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

ABCD is a kite in which AB = AD and CB = CD.

The diagonal AC is drawn.

Prove that triangle ABC is congruent to triangle ADC.

","answerType":"open","answer":"Triangle ABC is congruent to triangle ADC by SSS","answerDisplay":null,"accept":[],"parts":[],"solution":["AB = AD because ABCD is a kite (given).","CB = CD because ABCD is a kite (given).","AC is common to both triangles, so AC = AC.","Three pairs of sides are equal, so triangle ABC is congruent to triangle ADC (SSS)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,4],"label":"A","style":"none","labelPos":"above"},{"at":[3,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[0,-2],"label":"C","style":"none","labelPos":"below-right"},{"at":[-3,0],"label":"D","style":"none","labelPos":"below-left"}],"segments":[{"from":[0,4],"to":[3,0],"ticks":1},{"from":[0,4],"to":[-3,0],"ticks":1},{"from":[0,-2],"to":[3,0],"ticks":2},{"from":[0,-2],"to":[-3,0],"ticks":2},{"from":[0,4],"to":[0,-2]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"f0dd94bbeb38"},{"id":"g7-congruent-triangles-q04","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":4,"marks":3,"tier":"H","calc":false,"text":"

In triangle ABC, AB = AC.

The point D lies on BC so that AD bisects angle BAC.

Prove that triangle ABD is congruent to triangle ACD.

","answerType":"open","answer":"Triangle ABD is congruent to triangle ACD by SAS","answerDisplay":null,"accept":[],"parts":[],"solution":["AB = AC (given).","Angle BAD = angle CAD because AD bisects angle BAC (given).","AD is common to both triangles, so AD = AD.","In each triangle the equal angle is the included angle between the two equal sides (AB and AD in triangle ABD, AC and AD in triangle ACD).","So triangle ABD is congruent to triangle ACD (SAS)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,5],"label":"A","style":"none","labelPos":"above"},{"at":[-3,0],"label":"B","style":"none","labelPos":"below-left"},{"at":[3,0],"label":"C","style":"none","labelPos":"below-right"},{"at":[0,0],"label":"D","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,5],"to":[-3,0],"ticks":1},{"from":[0,5],"to":[3,0],"ticks":1},{"from":[-3,0],"to":[3,0]},{"from":[0,5],"to":[0,0]}],"angles":[{"at":[0,5],"from":[-3,0],"to":[0,0],"label":"","double":true},{"at":[0,5],"from":[0,0],"to":[3,0],"label":"","double":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"2c7a5fd955b1"},{"id":"g7-congruent-triangles-q05","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

In quadrilateral ABCD the diagonal BD is drawn.

Angle ABD = angle CDB and angle ADB = angle CBD.

Prove that triangle ABD is congruent to triangle CDB.

","answerType":"open","answer":"Triangle ABD is congruent to triangle CDB by ASA","answerDisplay":null,"accept":[],"parts":[],"solution":["Angle ABD = angle CDB (given).","Angle ADB = angle CBD (given).","BD is common to both triangles, so BD = DB.","In triangle ABD the side BD lies between the angles ABD and ADB; in triangle CDB the side DB lies between the corresponding angles CDB and CBD.","Two pairs of angles and the included side are equal, so triangle ABD is congruent to triangle CDB (ASA)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[8,4],"label":"C","style":"none","labelPos":"above"},{"at":[2,4],"label":"D","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"parallel":0,"ticks":0},{"from":[6,0],"to":[8,4],"parallel":0},{"from":[8,4],"to":[2,4],"parallel":0,"ticks":0},{"from":[2,4],"to":[0,0],"parallel":0},{"from":[6,0],"to":[2,4]}],"angles":[{"at":[6,0],"from":[0,0],"to":[2,4],"label":""},{"at":[2,4],"from":[8,4],"to":[6,0],"label":""},{"at":[2,4],"from":[0,0],"to":[6,0],"label":"","double":true},{"at":[6,0],"from":[8,4],"to":[2,4],"label":"","double":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"c7085b162d81"},{"id":"g7-congruent-triangles-q06","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":6,"marks":3,"tier":"H","calc":false,"text":"

OA and OB are two straight lines that meet at O.

The point P lies between the two lines.

M is the point on OA such that PM is perpendicular to OA.
N is the point on OB such that PN is perpendicular to OB.

PM = PN

Prove that triangle OMP is congruent to triangle ONP.

","answerType":"open","answer":"Triangle OMP is congruent to triangle ONP by RHS","answerDisplay":null,"accept":[],"parts":[],"solution":["Angle OMP = angle ONP = 90° because PM is perpendicular to OA and PN is perpendicular to OB (given).","OP is the hypotenuse of both right-angled triangles and is common, so OP = OP.","PM = PN (given).","A right angle, the hypotenuse and one other side are equal in each triangle, so triangle OMP is congruent to triangle ONP (RHS).","Note: it follows that angle MOP = angle NOP, so OP bisects the angle between the two lines."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[0,7],"label":"B","style":"none","labelPos":"above"},{"at":[3,3],"label":"P","style":"none","labelPos":"above"},{"at":[3,0],"label":"M","style":"none","labelPos":"below-right"},{"at":[0,3],"label":"N","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0]},{"from":[0,0],"to":[0,7]},{"from":[3,3],"to":[3,0],"ticks":1},{"from":[3,3],"to":[0,3],"ticks":1},{"from":[0,0],"to":[3,3]}],"angles":[{"at":[3,0],"from":[0,0],"to":[3,3],"label":"","right":true},{"at":[0,3],"from":[0,0],"to":[3,3],"label":"","right":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"dce61af82900"},{"id":"g7-congruent-triangles-q07","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":7,"marks":3,"tier":"H","calc":false,"text":"

A and B are points on a circle with centre O.

AB is not a diameter. M is the midpoint of the chord AB.

Prove that OM bisects angle AOB.

","answerType":"open","answer":"Triangles OAM and OBM are congruent (SSS), so angle AOM = angle BOM and OM bisects angle AOB","answerDisplay":null,"accept":[],"parts":[],"solution":["OA = OB because both are radii of the circle.","AM = BM because M is the midpoint of AB (given).","OM is common to both triangles, so OM = OM.","So triangle OAM is congruent to triangle OBM (SSS).","Corresponding angles of congruent triangles are equal, so angle AOM = angle BOM.","Therefore OM bisects angle AOB."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[2.298133329356934,1.9283628290596178],"label":"A","style":"none","labelPos":"above"},{"at":[2.298133329356934,-1.9283628290596178],"label":"B","style":"none","labelPos":"below-right"},{"at":[2.298133329356934,0],"label":"M","style":"none","labelPos":"below-right"}],"segments":[{"from":[2.298133329356934,1.9283628290596178],"to":[2.298133329356934,0],"ticks":1},{"from":[2.298133329356934,0],"to":[2.298133329356934,-1.9283628290596178],"ticks":1},{"from":[0,0],"to":[2.298133329356934,1.9283628290596178]},{"from":[0,0],"to":[2.298133329356934,-1.9283628290596178]},{"from":[0,0],"to":[2.298133329356934,0]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"16d81161d3d8"},{"id":"g7-congruent-triangles-q08","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

Neither chord is a diameter.

AB and CD are chords of a circle with centre O.

AB = CD

  1. Prove that triangle OAB is congruent to triangle OCD.
  2. Hence explain why angle AOB = angle COD.
","answerType":"multi","answer":"a) congruent by SSS; b) corresponding angles of congruent triangles are equal","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"OA = OC (radii), OB = OD (radii), AB = CD (given), so triangle OAB is congruent to triangle OCD (SSS)"},{"label":"b","answer":"Angle AOB and angle COD are corresponding angles in congruent triangles, so they are equal"}],"solution":["a) OA = OC because both are radii of the circle.","OB = OD because both are radii of the circle.","AB = CD (given).","Three pairs of sides are equal, so triangle OAB is congruent to triangle OCD (SSS).","b) Angle AOB in triangle OAB corresponds to angle COD in triangle OCD.","Corresponding angles of congruent triangles are equal, so angle AOB = angle COD.","This shows that equal chords subtend equal angles at the centre of a circle."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-right"},{"at":[2.4574561328669753,1.7207293090531381],"label":"A","style":"none","labelPos":"above"},{"at":[2.4574561328669753,-1.7207293090531381],"label":"B","style":"none","labelPos":"below-right"},{"at":[-2.457456132866975,1.720729309053139],"label":"C","style":"none","labelPos":"above"},{"at":[-2.457456132866976,-1.7207293090531375],"label":"D","style":"none","labelPos":"below-left"}],"segments":[{"from":[2.4574561328669753,1.7207293090531381],"to":[2.4574561328669753,-1.7207293090531381],"ticks":1},{"from":[-2.457456132866975,1.720729309053139],"to":[-2.457456132866976,-1.7207293090531375],"ticks":1},{"from":[0,0],"to":[2.4574561328669753,1.7207293090531381]},{"from":[0,0],"to":[2.4574561328669753,-1.7207293090531381]},{"from":[0,0],"to":[-2.457456132866975,1.720729309053139]},{"from":[0,0],"to":[-2.457456132866976,-1.7207293090531375]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"49b61e31f8ef"},{"id":"g7-congruent-triangles-q09","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":9,"marks":4,"tier":"H","calc":false,"text":"

ABCD is a quadrilateral in which AB is parallel to DC and AB = DC.

The diagonal BD is drawn.

  1. Prove that triangle ABD is congruent to triangle CDB.
  2. Hence prove that AD = CB.
","answerType":"multi","answer":"a) congruent by SAS; b) AD and CB are corresponding sides of congruent triangles, so AD = CB","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"AB = CD (given), angle ABD = angle CDB (alternate angles, AB parallel to DC), BD common, so triangle ABD is congruent to triangle CDB (SAS)"},{"label":"b","answer":"AD corresponds to CB in the congruent triangles, so AD = CB"}],"solution":["a) AB = CD (given).","Angle ABD = angle BDC because they are alternate angles between the parallel lines AB and DC with transversal BD.","BD is common to both triangles, so BD = DB.","In triangle ABD the angle ABD is included between the sides AB and BD; in triangle CDB the angle CDB is included between the corresponding sides CD and DB.","So triangle ABD is congruent to triangle CDB (SAS).","b) In the congruence, A corresponds to C, B corresponds to D and D corresponds to B, so the side AD corresponds to the side CB.","Corresponding sides of congruent triangles are equal, so AD = CB.","This proves that a quadrilateral with one pair of opposite sides equal and parallel is a parallelogram."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[8,4],"label":"C","style":"none","labelPos":"above"},{"at":[2,4],"label":"D","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"parallel":1,"ticks":1},{"from":[6,0],"to":[8,4],"parallel":0},{"from":[8,4],"to":[2,4],"parallel":1,"ticks":1},{"from":[2,4],"to":[0,0],"parallel":0},{"from":[6,0],"to":[2,4]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"7d47a807e37c"},{"id":"g7-congruent-triangles-q10","topicId":"g7-congruent-triangles","topic":"Congruent Triangles","grade":"7","topicGrade":"7","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

ABCD is a parallelogram.

The diagonals AC and BD intersect at the point M.

Prove that the diagonals of the parallelogram bisect each other.

","answerType":"open","answer":"Triangles ABM and CDM are congruent (ASA), so AM = CM and BM = DM, which means M is the midpoint of both diagonals","answerDisplay":null,"accept":[],"parts":[],"solution":["AB is parallel to DC because ABCD is a parallelogram.","Angle BAM = angle DCM because they are alternate angles between the parallel lines AB and DC with transversal AC.","Angle ABM = angle CDM because they are alternate angles between the parallel lines AB and DC with transversal BD.","AB = CD because opposite sides of a parallelogram are equal.","In triangle ABM the side AB is included between the angles BAM and ABM; in triangle CDM the side CD is included between the corresponding angles DCM and CDM.","So triangle ABM is congruent to triangle CDM (ASA).","In the congruence, A corresponds to C and B corresponds to D, so AM = CM and BM = DM.","Therefore M is the midpoint of both AC and BD, so the diagonals bisect each other."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[8,4],"label":"C","style":"none","labelPos":"above"},{"at":[2,4],"label":"D","style":"none","labelPos":"above"},{"at":[4,2],"label":"M","style":"dot","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[6,0],"parallel":1,"ticks":0},{"from":[6,0],"to":[8,4],"parallel":2},{"from":[8,4],"to":[2,4],"parallel":1,"ticks":0},{"from":[2,4],"to":[0,0],"parallel":2},{"from":[6,0],"to":[2,4]},{"from":[0,0],"to":[8,4]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"48a0ef1720f3"},{"id":"g7-direct-and-inverse-proportion-algebraic-q01","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

y is directly proportional to the square of x.

When x = 4, y = 48.

Find the value of y when x = 5.

","answerType":"exact","answer":"y = 75","answerDisplay":null,"accept":["75"],"parts":[],"solution":["Directly proportional to x^2 means that y ÷ x^2 is a fixed number, called the constant of proportionality k. So write y = kx^2.","Substitute: 48 = k × 16, so k = 3 and y = 3x2.","When x = 5, y = 3 × 25 = 75."],"diagram":null,"draw":false,"revision":"8303485e3463"},{"id":"g7-direct-and-inverse-proportion-algebraic-q02","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

p is inversely proportional to q.

When q = 5, p = 12.

  1. Find a formula for p in terms of q.
  2. Find the value of p when q = 8.
","answerType":"multi","answer":"a) p = 60/q b) p = 7.5","answerDisplay":"a) p = 60q b) p = 7.5","accept":["p=60/q, 7.5"],"parts":[{"label":"a","answer":"p = 60/q","answerDisplay":"p = 60q"},{"label":"b","answer":"7.5"}],"solution":["Inversely proportional means that the product pq is constant: pq = k. Dividing by q gives p = k/q.","Substitute: 12 = k5, so k = 60 and p = 60q.","When q = 8, p = 60 ÷ 8 = 7.5."],"diagram":null,"draw":false,"revision":"08c366b9cbfe"},{"id":"g7-direct-and-inverse-proportion-algebraic-q03","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

y is inversely proportional to the square of x.

When x = 3, y = 4.

Find the negative value of x when y = 9.

","answerType":"exact","answer":"x = -2","answerDisplay":null,"accept":["-2"],"parts":[],"solution":["Write y = kx2.","Substitute: 4 = k9, so k = 36 and y = 36x2.","When y = 9: 9 = 36x2, so x2 = 4.","x = 2 or x = −2, so the negative value is x = −2."],"diagram":null,"draw":false,"revision":"dfbacf20c056"},{"id":"g7-direct-and-inverse-proportion-algebraic-q04","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

The force F newtons between two magnets is inversely proportional to the square of the distance d cm between them.

When d = 2, F = 60.

  1. Find a formula for F in terms of d.
  2. Find the positive value of d when F = 15.
","answerType":"multi","answer":"a) F = 240/d^2 b) d = 4","answerDisplay":"a) F = 240d2 b) d = 4","accept":["F=240/d^2, d=4"],"parts":[{"label":"a","answer":"F = 240/d^2","answerDisplay":"F = 240d2"},{"label":"b","answer":"4"}],"solution":["Write F = kd2.","Substitute: 60 = k4, so k = 240 and F = 240d2.","When F = 15: 15 = 240d2, so d2 = 16.","The positive value is d = 4."],"diagram":null,"draw":false,"revision":"7a08eab27aed"},{"id":"g7-direct-and-inverse-proportion-algebraic-q05","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

The time T seconds for one swing of a pendulum is directly proportional to the square root of its length L metres.

When L = 2.25, T = 3.

  1. Find the value of T when L = 6.25.
  2. Find the value of L when T = 7.
","answerType":"multi","answer":"a) T = 5 b) L = 12.25","answerDisplay":null,"accept":["T=5, L=12.25"],"parts":[{"label":"a","answer":"5"},{"label":"b","answer":"12.25"}],"solution":["Write T = k√L.","Substitute: 3 = k × √2.25 = 1.5k, so k = 2 and T = 2√L.","When L = 6.25: T = 2 × 2.5 = 5.","When T = 7: √L = 3.5, so L = 3.52 = 12.25."],"diagram":null,"draw":false,"revision":"ae8c2ee007b2"},{"id":"g7-direct-and-inverse-proportion-algebraic-q06","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

a is directly proportional to the square of b.

When b = 2, a = 12.

b is directly proportional to the cube of c.

When c = 2, b = 16.

Find a formula for a in terms of c.

","answerType":"exact","answer":"a = 12c^6","answerDisplay":null,"accept":["a=12c^6","12c^6"],"parts":[],"solution":["From the first relation: a = kb2 and 12 = k × 4, so k = 3 and a = 3b2.","From the second relation: b = mc3 and 16 = m × 8, so m = 2 and b = 2c3.","Substitute: a = 3(2c3)2 = 3 × 4c6 = 12c6.","Check with c = 2: b = 16 and a = 3 × 256 = 768 = 12 × 26."],"diagram":null,"draw":false,"revision":"87af2fb2f162"},{"id":"g7-direct-and-inverse-proportion-algebraic-q07","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

x and y are positive quantities.

y is inversely proportional to the square of x.

x is increased by 25%.

Work out the percentage change in y.

","answerType":"exact","answer":"36% decrease","answerDisplay":null,"accept":["36% decrease","-36%","decrease of 36%","36 percent decrease"],"parts":[],"solution":["Write y = kx2.","The new value of x is 1.25x, so the new y is k(1.25x)2 = k(1.5625x2).","1 ÷ 1.5625 = 0.64, so the new y is 0.64 times the old y.","This is a 36% decrease."],"diagram":null,"draw":false,"revision":"f82ca63acc7f"},{"id":"g7-direct-and-inverse-proportion-algebraic-q08","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

y is directly proportional to the square of x.

When x = 5, y = 50.

x is inversely proportional to z.

When z = 2, x = 6.

Find the value of y when z = 4.

","answerType":"exact","answer":"y = 18","answerDisplay":null,"accept":["18"],"parts":[],"solution":["From the first relation: y = kx2 and 50 = k × 25, so k = 2 and y = 2x2.","From the second relation: x = mz and 6 = m2, so m = 12 and x = 12z.","So y = 2(12z)2 = 288z2.","When z = 4, y = 288 ÷ 16 = 18."],"diagram":null,"draw":false,"revision":"847cc96bb8b2"},{"id":"g7-direct-and-inverse-proportion-algebraic-q09","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

t is directly proportional to the square root of m.

When m = 9, t = 6.

m is inversely proportional to w.

When w = 4, m = 9.

Find the value of w when t = 3.

","answerType":"exact","answer":"w = 16","answerDisplay":null,"accept":["16"],"parts":[],"solution":["From the first relation: t = k√m and 6 = k × 3, so k = 2 and t = 2√m.","From the second relation: m = cw and 9 = c4, so c = 36 and m = 36w.","So t = 2√(36w) = 12√w.","When t = 3: √w = 12 ÷ 3 = 4, so w = 16.","Check: m = 36 ÷ 16 = 2.25 and t = 2 × 1.5 = 3."],"diagram":null,"draw":false,"revision":"f156a5a32f06"},{"id":"g7-direct-and-inverse-proportion-algebraic-q10","topicId":"g7-direct-and-inverse-proportion-algebraic","topic":"Direct and Inverse Proportion (Algebraic)","grade":"7","topicGrade":"7","strand":"R","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

Here are some pairs of values of x and y.

x: 4, 25, 100

y: 6, 15, 30

Leon says, "y is directly proportional to x."

  1. Explain why Leon is wrong.
  2. Given that y is directly proportional to the square root of x, find a formula for y in terms of x.
  3. Find the value of y when x = 64.
","answerType":"multi","answer":"a) 6/4 = 1.5 but 15/25 = 0.6, so y/x is not constant b) y = 3 sqrt(x) c) y = 24","answerDisplay":"a) 64 = 1.5 but 1525 = 0.6, so yx is not constant b) y = 3 sqrt(x) c) y = 24","accept":["y=3sqrt(x), 24"],"parts":[{"label":"a","answer":"y/x is not constant (6/4 = 1.5 but 15/25 = 0.6)","answerDisplay":"yx is not constant (64 = 1.5 but 1525 = 0.6)"},{"label":"b","answer":"y = 3√x"},{"label":"c","answer":"24"}],"solution":["If y were directly proportional to x, then yx would be constant, but 6 ÷ 4 = 1.5 and 15 ÷ 25 = 0.6, so Leon is wrong.","Write y = k√x. Using x = 4: 6 = k × 2, so k = 3 and y = 3√x.","Check: 3√25 = 15 and 3√100 = 30, which matches the table.","When x = 64, y = 3 × 8 = 24."],"diagram":null,"draw":false,"revision":"1e276b53fc26"},{"id":"g7-factorising-harder-quadratics-q01","topicId":"g7-factorising-harder-quadratics","topic":"Factorising Harder Quadratics","grade":"7","topicGrade":"7","strand":"A","difficulty":1,"marks":2,"tier":"H","calc":false,"text":"

Factorise 2x2 + 7x + 3

","answerType":"exact","answer":"(2x + 1)(x + 3)","answerDisplay":null,"accept":["(2x + 1)(x + 3)","(x + 3)(2x + 1)"],"parts":[],"solution":["Look for two numbers that multiply to 2 × 3 = 6 and add to 7: they are 6 and 1.","Split the middle term: 2x2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3).","Both groups now contain the whole factor (x + 3). Taking out that shared factor leaves 2x + 1, giving the two brackets below.","Factorise: (2x + 1)(x + 3).","Check by expanding: 2x2 + 6x + x + 3 = 2x2 + 7x + 3, correct."],"diagram":null,"draw":false,"revision":"46b6bf59522b"},{"id":"g7-factorising-harder-quadratics-q02","topicId":"g7-factorising-harder-quadratics","topic":"Factorising Harder Quadratics","grade":"7","topicGrade":"7","strand":"A","difficulty":2,"marks":2,"tier":"H","calc":true,"text":"

Factorise 3x2 - 10x + 8

","answerType":"exact","answer":"(3x - 4)(x - 2)","answerDisplay":null,"accept":["(3x - 4)(x - 2)","(x - 2)(3x - 4)"],"parts":[],"solution":["Look for two numbers that multiply to 3 × 8 = 24 and add to -10: they are -6 and -4.","Split the middle term: 3x2 - 6x - 4x + 8 = 3x(x - 2) - 4(x - 2).","Factorise: (3x - 4)(x - 2).","Check by expanding: 3x2 - 6x - 4x + 8 = 3x2 - 10x + 8, correct."],"diagram":null,"draw":false,"revision":"11c12a0d4b9f"},{"id":"g7-factorising-harder-quadratics-q03","topicId":"g7-factorising-harder-quadratics","topic":"Factorising Harder Quadratics","grade":"7","topicGrade":"7","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

Solve 2x2 + 5x - 12 = 0

","answerType":"exact","answer":"x = 3/2, x = -4","answerDisplay":"x = 32, x = -4","accept":["x = 3/2, x = -4","x = -4, x = 3/2","x = 1.5, x = -4","3/2 and -4"],"parts":[],"solution":["Look for two numbers that multiply to 2 × (-12) = -24 and add to 5: they are 8 and -3.","Split the middle term: 2x2 + 8x - 3x - 12 = 2x(x + 4) - 3(x + 4) = (2x - 3)(x + 4).","Set each factor to zero: 2x - 3 = 0 gives x = 32, and x + 4 = 0 gives x = -4.","Check x = -4: 2(16) - 20 - 12 = 32 - 32 = 0, correct."],"diagram":null,"draw":false,"revision":"955db0270939"},{"id":"g7-factorising-harder-quadratics-q04","topicId":"g7-factorising-harder-quadratics","topic":"Factorising Harder Quadratics","grade":"7","topicGrade":"7","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

Solve 3x2 - 11x + 6 = 0

","answerType":"exact","answer":"x = 2/3, x = 3","answerDisplay":"x = 23, x = 3","accept":["x = 2/3, x = 3","x = 3, x = 2/3","2/3 and 3"],"parts":[],"solution":["Look for two numbers that multiply to 3 × 6 = 18 and add to -11: they are -9 and -2.","Split the middle term: 3x2 - 9x - 2x + 6 = 3x(x - 3) - 2(x - 3) = (3x - 2)(x - 3).","Set each factor to zero: 3x - 2 = 0 gives x = 23, and x - 3 = 0 gives x = 3.","Check x = 3: 3(9) - 33 + 6 = 27 - 27 = 0, correct."],"diagram":null,"draw":false,"revision":"6840ee31226b"},{"id":"g7-factorising-harder-quadratics-q05","topicId":"g7-factorising-harder-quadratics","topic":"Factorising Harder Quadratics","grade":"7","topicGrade":"7","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":false,"text":"

Solve 5x2 + 13x - 6 = 0

","answerType":"exact","answer":"x = 2/5, x = -3","answerDisplay":"x = 25, x = -3","accept":["x = 2/5, x = -3","x = -3, x = 2/5","x = 0.4, x = -3","2/5 and -3"],"parts":[],"solution":["Look for two numbers that multiply to 5 × (-6) = -30 and add to 13: they are 15 and -2.","Split the middle term: 5x2 + 15x - 2x - 6 = 5x(x + 3) - 2(x + 3) = (5x - 2)(x + 3).","Set each factor to zero: 5x - 2 = 0 gives x = 25, and x + 3 = 0 gives x = -3.","Check x = -3: 5(9) - 39 - 6 = 45 - 45 = 0, correct."],"diagram":null,"draw":false,"revision":"ed6b50e7b464"},{"id":"g7-factorising-harder-quadratics-q06","topicId":"g7-factorising-harder-quadratics","topic":"Factorising Harder Quadratics","grade":"7","topicGrade":"7","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

Factorise 6x2 - 7x - 20

","answerType":"exact","answer":"(3x + 4)(2x - 5)","answerDisplay":null,"accept":["(3x + 4)(2x - 5)","(2x - 5)(3x + 4)"],"parts":[],"solution":["Look for two numbers that multiply to 6 × (-20) = -120 and add to -7: they are -15 and 8.","Split the middle term: 6x2 - 15x + 8x - 20 = 3x(2x - 5) + 4(2x - 5).","Factorise: (3x + 4)(2x - 5).","Check by expanding: 6x2 - 15x + 8x - 20 = 6x2 - 7x - 20, correct."],"diagram":null,"draw":false,"revision":"60c8dc148ad0"},{"id":"g7-factorising-harder-quadratics-q07","topicId":"g7-factorising-harder-quadratics","topic":"Factorising Harder Quadratics","grade":"7","topicGrade":"7","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

Solve 2x2 - x - 15 < 0

","answerType":"exact","answer":"-5/2 < x < 3","answerDisplay":"-52 < x < 3","accept":["-5/2 < x < 3","-2.5 < x < 3"],"parts":[],"solution":["Factorise: look for two numbers that multiply to 2 × (-15) = -30 and add to -1: they are -6 and 5.","2x2 - 6x + 5x - 15 = 2x(x - 3) + 5(x - 3) = (2x + 5)(x - 3).","The roots are x = -52 and x = 3. The graph of y = 2x2 - x - 15 is a U-shaped parabola, which is below zero between its roots.","So the solution is -52 < x < 3. Check x = 0: 2(0) - 0 - 15 = -15 < 0, correct."],"diagram":null,"draw":false,"revision":"d108967e0a6d"},{"id":"g7-factorising-harder-quadratics-q08","topicId":"g7-factorising-harder-quadratics","topic":"Factorising Harder Quadratics","grade":"7","topicGrade":"7","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

Solve 6x2 + x - 12 = 0

You must show clear algebraic working.

","answerType":"exact","answer":"x = 4/3, x = -3/2","answerDisplay":"x = 43, x = -32","accept":["x = 4/3, x = -3/2","x = -3/2, x = 4/3","4/3 and -3/2","x = -1.5, x = 4/3"],"parts":[],"solution":["Look for two numbers that multiply to 6 × (-12) = -72 and add to 1: they are 9 and -8.","Split the middle term: 6x2 + 9x - 8x - 12 = 3x(2x + 3) - 4(2x + 3) = (3x - 4)(2x + 3).","Set each factor to zero: 3x - 4 = 0 gives x = 43, and 2x + 3 = 0 gives x = -32.","Check x = 43: 6(169) + 43 - 12 = 323 + 43 - 363 = 0, correct."],"diagram":null,"draw":false,"revision":"ff7d62a59535"},{"id":"g7-finding-the-area-of-any-triangle-q01","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":1,"marks":2,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 6 cm, AC = 9 cm and angle BAC = 40°.

Work out the area of the triangle. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"17.4 cm²","answerDisplay":null,"accept":["17.4","17.4 cm^2"],"parts":[],"solution":["Area is half × base × perpendicular height. If the sides around the given angle are a and b, the height relative to base a is b sin(angle), so area = half × a × b × sin(angle). Use your calculator in degree mode.","Area = 12ab sin C with the two sides that enclose the given angle.","Area = 12 × 6 × 9 × sin 40°.","Area = 27 × 0.64279... = 17.355...","Area = 17.4 cm2 (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6.894399988070802,5.785088487178854],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"6 cm"},{"from":[6,0],"to":[6.894399988070802,5.785088487178854]},{"from":[6.894399988070802,5.785088487178854],"to":[0,0],"label":"9 cm"}],"angles":[{"at":[0,0],"from":[6,0],"to":[6.894399988070802,5.785088487178854],"label":"40°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"0b6bb0cd47c8"},{"id":"g7-finding-the-area-of-any-triangle-q02","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":2,"marks":2,"tier":"H","calc":true,"text":"

In triangle PQR, PQ = 8 cm, PR = 11 cm and angle QPR = 52°.

Calculate the area of the triangle. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"34.7 cm²","answerDisplay":null,"accept":["34.7","34.7 cm^2"],"parts":[],"solution":["Area is half × base × perpendicular height. If the sides around the given angle are a and b, the height relative to base a is b sin(angle), so area = half × a × b × sin(angle). Use your calculator in degree mode.","Area = 12 × 8 × 11 × sin 52°.","Area = 44 × 0.78801... = 34.672...","Area = 34.7 cm2 (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"Q","style":"none","labelPos":"below-right"},{"at":[5.079207171436681,6.501088717255457],"label":"R","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"8 cm"},{"from":[6,0],"to":[5.079207171436681,6.501088717255457]},{"from":[5.079207171436681,6.501088717255457],"to":[0,0],"label":"11 cm"}],"angles":[{"at":[0,0],"from":[6,0],"to":[5.079207171436681,6.501088717255457],"label":"52°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"4d9af2f08314"},{"id":"g7-finding-the-area-of-any-triangle-q03","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":3,"marks":2,"tier":"H","calc":true,"text":"

In triangle DEF, DE = 9.4 cm, DF = 12.6 cm and angle EDF = 105°.

Calculate the area of the triangle. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"57.2 cm²","answerDisplay":null,"accept":["57.2","57.2 cm^2"],"parts":[],"solution":["Area is half × base × perpendicular height. If the sides around the given angle are a and b, the height relative to base a is b sin(angle), so area = half × a × b × sin(angle). Use your calculator in degree mode.","The formula Area = 12ab sin C also works for obtuse angles.","Area = 12 × 9.4 × 12.6 × sin 105°.","Area = 59.22 × 0.96592... = 57.202...","Area = 57.2 cm2 (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"D","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"E","style":"none","labelPos":"below-right"},{"at":[-2.081565937207508,7.768509836963145],"label":"F","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"9.4 cm"},{"from":[6,0],"to":[-2.081565937207508,7.768509836963145]},{"from":[-2.081565937207508,7.768509836963145],"to":[0,0],"label":"12.6 cm"}],"angles":[{"at":[0,0],"from":[6,0],"to":[-2.081565937207508,7.768509836963145],"label":"105°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"c9a23045a180"},{"id":"g7-finding-the-area-of-any-triangle-q04","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 9 cm and AC = 8 cm.

The area of the triangle is 30 cm2.

Angle BAC is acute.

Work out the size of angle BAC. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"56.4°","answerDisplay":null,"accept":["56.4"],"parts":[],"solution":["Area = 12 × 9 × 8 × sin A = 36 sin A.","36 sin A = 30, so sin A = 3036 = 0.8333...","A = sin-1(0.8333...) = 56.442...°.","Angle BAC = 56.4° (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[2.666666666666667,4.618802153517006],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"9 cm"},{"from":[6,0],"to":[2.666666666666667,4.618802153517006]},{"from":[2.666666666666667,4.618802153517006],"to":[0,0],"label":"8 cm"}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"38bb45c59acc"},{"id":"g7-finding-the-area-of-any-triangle-q05","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

In triangle XYZ, XY = 10 cm and angle YXZ = 35°.

The area of the triangle is 42 cm2.

Work out the length of XZ. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"14.6 cm","answerDisplay":null,"accept":["14.6"],"parts":[],"solution":["Area = 12 × 10 × XZ × sin 35° = 42.","XZ = 84 ÷ (10 × sin 35°) = 84 ÷ 5.7357...","XZ = 14.644...","XZ = 14.6 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"X","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"Y","style":"none","labelPos":"below-right"},{"at":[2.9489473594403703,2.064875170863766],"label":"Z","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"10 cm"},{"from":[6,0],"to":[2.9489473594403703,2.064875170863766]},{"from":[2.9489473594403703,2.064875170863766],"to":[0,0]}],"angles":[{"at":[0,0],"from":[6,0],"to":[2.9489473594403703,2.064875170863766],"label":"35°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"c239447bae7a"},{"id":"g7-finding-the-area-of-any-triangle-q06","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A rectangle measures 9 cm by 4 cm.

A triangle has two sides of length 10 cm and 12 cm, with an acute angle of x° between them.

The area of the triangle is equal to the area of the rectangle.

Work out the value of x. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"36.9","answerDisplay":null,"accept":["36.9°","x = 36.9"],"parts":[],"solution":["Area of the rectangle = 9 × 4 = 36 cm2.","Area of the triangle = 12 × 10 × 12 × sin x° = 60 sin x°.","60 sin x° = 36, so sin x° = 0.6.","x = sin-1(0.6) = 36.869...","x = 36.9 (1 dp)."],"diagram":null,"draw":false,"revision":"c30cdd020650"},{"id":"g7-finding-the-area-of-any-triangle-q07","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

A and B are points on a circle with centre O.

The circumference of the circle is 40 cm.

Angle AOB = 68°.

Calculate the area of triangle AOB. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"18.8 cm²","answerDisplay":null,"accept":["18.8","18.8 cm^2"],"parts":[],"solution":["Circumference = 2πr, so r = 40 ÷ (2π) = 6.3661...","OA = OB = r, so Area = 12 × r × r × sin 68°.","Area = 12 × 6.3661...2 × 0.92718... = 18.788...","Area = 18.8 cm2 (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[1.6775787104122404,2.4871127176651253],"label":"A","style":"none","labelPos":"above"},{"at":[-1.6775787104122402,2.4871127176651253],"label":"B","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[1.6775787104122404,2.4871127176651253]},{"from":[0,0],"to":[-1.6775787104122402,2.4871127176651253]},{"from":[1.6775787104122404,2.4871127176651253],"to":[-1.6775787104122402,2.4871127176651253]}],"angles":[{"at":[0,0],"from":[1.6775787104122404,2.4871127176651253],"to":[-1.6775787104122402,2.4871127176651253],"label":"68°"}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"dde7e3449451"},{"id":"g7-finding-the-area-of-any-triangle-q08","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

ABCDE is a regular pentagon with sides of length 9 cm.

Work out the area of triangle ABC. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"38.5 cm²","answerDisplay":null,"accept":["38.5","38.5 cm^2"],"parts":[],"solution":["Interior angle of a regular pentagon = (5 - 2) × 180° ÷ 5 = 108°.","In triangle ABC, AB = BC = 9 cm and angle ABC = 108°.","Area = 12 × 9 × 9 × sin 108°.","Area = 40.5 × 0.95105... = 38.517...","Area = 38.5 cm2 (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"polygons":[{"pts":[[0,0],[4,0],[5.23606797749979,3.804226065180614],[2.0000000000000004,6.155367074350507],[-1.2360679774997894,3.804226065180615]],"labels":["A","B","C","D","E"],"sides":["9 cm","9 cm"]}],"segments":[{"from":[0,0],"to":[5.23606797749979,3.804226065180614]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"64dd14b3dc5d"},{"id":"g7-finding-the-area-of-any-triangle-q09","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 16 cm, angle BAC = 54° and angle ABC = 71°.

Work out the area of the triangle. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"120 cm²","answerDisplay":null,"accept":["120"],"parts":[],"solution":["Angle ACB = 180° - 54° - 71° = 55°.","Use the sine rule to find BC: BCsin A = ABsin C.","BC = 16 × sin 54° ÷ sin 55° = 15.802...","Area = 12 × AB × BC × sin B = 12 × 16 × 15.802... × sin 71°.","Area = 119.52... = 120 cm2 (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[4.070759842795154,5.6029202502033275],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"16 cm"},{"from":[6,0],"to":[4.070759842795154,5.6029202502033275]},{"from":[4.070759842795154,5.6029202502033275],"to":[0,0]}],"angles":[{"at":[0,0],"from":[6,0],"to":[4.070759842795154,5.6029202502033275],"label":"54°","right":false},{"at":[6,0],"from":[4.070759842795154,5.6029202502033275],"to":[0,0],"label":"71°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"0864b9ce4c74"},{"id":"g7-finding-the-area-of-any-triangle-q10","topicId":"g7-finding-the-area-of-any-triangle","topic":"Finding the Area of Any Triangle","grade":"7","topicGrade":"7","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

ABCD is a quadrilateral. The diagonal BD splits it into two triangles.

In triangle ABD, AB = 9 cm, BD = 13 cm and angle ABD = 42°.

In triangle BCD, BC = 11 cm and angle DBC = 51°.

Work out the area of the quadrilateral ABCD. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"94.7 cm²","answerDisplay":null,"accept":["94.7","94.7 cm^2"],"parts":[],"solution":["Area of triangle ABD = 12 × 9 × 13 × sin 42° = 39.144...","Area of triangle BCD = 12 × 11 × 13 × sin 51° = 55.565...","Total area = 39.144... + 55.565... = 94.710...","Area of ABCD = 94.7 cm2 (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,4],"label":"A","style":"none","labelPos":"above"},{"at":[-3,0],"label":"B","style":"none","labelPos":"below-left"},{"at":[0,-4],"label":"C","style":"none","labelPos":"below-left"},{"at":[5,0],"label":"D","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,4],"to":[-3,0],"label":"9 cm"},{"from":[-3,0],"to":[0,-4],"label":"11 cm"},{"from":[0,-4],"to":[5,0]},{"from":[5,0],"to":[0,4]},{"from":[-3,0],"to":[5,0],"label":"13 cm"}],"angles":[{"at":[-3,0],"from":[0,4],"to":[5,0],"label":"42°"},{"at":[-3,0],"from":[5,0],"to":[0,-4],"label":"51°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"fcb405b83864"},{"id":"g7-histograms-q01","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":1,"marks":3,"tier":"H","calc":true,"text":"

The table shows the frequency densities for the lengths, l cm, of the worms found in a soil survey.

Length (l cm)Frequency density
0 < l ≤ 51.2
5 < l ≤ 152.1
15 < l ≤ 203.0
20 < l ≤ 400.4

Work out the total number of worms found in the survey.

","answerType":"exact","answer":"50","answerDisplay":null,"accept":["50 worms"],"parts":[],"solution":["In a histogram, bar area represents frequency. Width × height must therefore equal frequency: the height (frequency density) is frequency divided by class width.","Frequency = frequency density × class width.","0 < l ≤ 5: 1.2 × 5 = 6. 5 < l ≤ 15: 2.1 × 10 = 21.","15 < l ≤ 20: 3.0 × 5 = 15. 20 < l ≤ 40: 0.4 × 20 = 8.","Total = 6 + 21 + 15 + 8 = 50."],"diagram":null,"draw":false,"revision":"ed9803b04d36"},{"id":"g7-histograms-q02","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

The table shows the times, t minutes, that 53 customers spent in a coffee shop.

Time (t minutes)FrequencyFrequency density
0 < t ≤ 1018
10 < t ≤ 1512
15 < t ≤ 2515
25 < t ≤ 458

Complete the frequency density column.

","answerType":"multi","answer":"1.8, 2.4, 1.5, 0.4","answerDisplay":null,"accept":[],"parts":[{"label":"0-10","answer":"1.8"},{"label":"10-15","answer":"2.4"},{"label":"15-25","answer":"1.5"},{"label":"25-45","answer":"0.4"}],"solution":["Frequency density = frequency ÷ class width.","18 ÷ 10 = 1.8 and 12 ÷ 5 = 2.4.","15 ÷ 10 = 1.5 and 8 ÷ 20 = 0.4."],"diagram":null,"draw":false,"revision":"bde0aef1488d"},{"id":"g7-histograms-q03","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

A histogram is drawn to show the masses, m grams, of some plums. The vertical axis has no scale, so the table records the height of each bar in centimetres.

Mass (m grams)Bar height (cm)
20 < m ≤ 306
30 < m ≤ 504.5
50 < m ≤ 558
55 < m ≤ 702

There are 24 plums with masses in the class 20 < m ≤ 30.

  1. Work out the number of plums in the class 30 < m ≤ 50.
  2. Work out the total number of plums.
","answerType":"multi","answer":"(a) 36 (b) 88","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"36"},{"label":"b","answer":"88"}],"solution":["For 20 < m ≤ 30: frequency 24, width 10, so frequency density = 2.4, meaning each cm of height represents 0.4 units of frequency density.","30 < m ≤ 50: frequency density = 4.5 × 0.4 = 1.8, frequency = 1.8 × 20 = 36.","50 < m ≤ 55: frequency density = 8 × 0.4 = 3.2, frequency = 3.2 × 5 = 16.","55 < m ≤ 70: frequency density = 2 × 0.4 = 0.8, frequency = 0.8 × 15 = 12.","Total = 24 + 36 + 16 + 12 = 88."],"diagram":{"type":"axes","xy":false,"height":200,"x":{"min":0,"max":70,"step":10,"minor":2,"label":"Mass (grams)"},"y":{"min":0,"max":9,"hide":true},"bars":[{"from":20,"to":30,"height":6},{"from":30,"to":50,"height":4.5},{"from":50,"to":55,"height":8},{"from":55,"to":70,"height":2}],"texts":[{"at":[8,8.4],"text":"Frequency density","size":11}]},"draw":false,"revision":"8d4b4990fb49"},{"id":"g7-histograms-q04","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

The table and histogram give information about the distances, d km, that some students travel to college.

Distance (d km)Frequency
0 < d ≤ 416
4 < d ≤ 6
6 < d ≤ 1018
10 < d ≤ 16

The bars for 0 < d ≤ 4 and 4 < d ≤ 6 are already drawn on the histogram. The bar for 4 < d ≤ 6 has frequency density 5.5.

  1. Use the histogram to work out the frequency for 4 < d ≤ 6.
  2. The frequency for 10 < d ≤ 16 is 9. Complete the histogram.
","answerType":"multi","answer":"(a) 11 (b) bars drawn at frequency density 4.5 for 6 < d <= 10 and 1.5 for 10 < d <= 16","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"11"},{"label":"b","answer":"bar height 4.5 over 6-10 and bar height 1.5 over 10-16"}],"solution":["Frequency = frequency density × width, so for 4 < d ≤ 6: 5.5 × 2 = 11.","For 6 < d ≤ 10: frequency density = 18 ÷ 4 = 4.5.","For 10 < d ≤ 16: frequency density = 9 ÷ 6 = 1.5.","Draw the two missing bars at these heights."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":16,"step":2,"minor":2,"label":"Distance (km)"},"y":{"min":0,"max":6,"step":1,"minor":2,"label":"Frequency density"},"bars":[{"from":0,"to":4,"height":4},{"from":4,"to":6,"height":5.5},{"from":6,"to":10,"height":4.5,"answer":true},{"from":10,"to":16,"height":1.5,"answer":true}]},"draw":true,"revision":"9f9720c4cba6"},{"id":"g7-histograms-q05","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

The table shows the frequency densities from a histogram of the waiting times, w minutes, of 40 callers to a helpline.

Waiting time (w minutes)Frequency density
0 < w ≤ 42
4 < w ≤ 103
10 < w ≤ 125
12 < w ≤ 200.5
  1. Show that the histogram represents 40 callers.
  2. Calculate an estimate for the mean waiting time.
","answerType":"multi","answer":"(a) frequencies 8, 18, 10, 4 which sum to 40 (b) 7.9 minutes","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8 + 18 + 10 + 4 = 40"},{"label":"b","answer":"7.9 minutes"}],"solution":["Frequencies: 2 × 4 = 8, 3 × 6 = 18, 5 × 2 = 10, 0.5 × 8 = 4. Total = 40.","The exact waiting times within each group are unknown. Use the midpoint to represent each caller in that group, so the resulting mean is an estimate.","Midpoints: 2, 7, 11, 16.","(2 × 8) + (7 × 18) + (11 × 10) + (16 × 4) = 16 + 126 + 110 + 64 = 316.","Estimated mean = 316 ÷ 40 = 7.9 minutes."],"diagram":null,"draw":false,"revision":"b48254969cda"},{"id":"g7-histograms-q06","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A histogram shows the times, t seconds, that 60 shoppers waited at a checkout. The table gives the frequency densities.

Time (t seconds)Frequency density
0 < t ≤ 200.6
20 < t ≤ 302.0
30 < t ≤ 353.2
35 < t ≤ 500.8
  1. Show that the histogram represents 60 shoppers.
  2. One of the 60 shoppers is chosen at random. Work out an estimate for the probability that this shopper waited more than 40 seconds.
","answerType":"multi","answer":"(a) frequencies 12, 20, 16, 12 which sum to 60 (b) 2/15","answerDisplay":"(a) frequencies 12, 20, 16, 12 which sum to 60 (b) 215","accept":[],"parts":[{"label":"a","answer":"12 + 20 + 16 + 12 = 60"},{"label":"b","answer":"2/15 (8/60)","answerDisplay":"215 860"}],"solution":["For estimates using part of a class interval, assume the values are spread evenly within that interval. The data do not give their exact positions within a class.","Frequencies: 0.6 × 20 = 12, 2.0 × 10 = 20, 3.2 × 5 = 16, 0.8 × 15 = 12. Total = 60.","Shoppers waiting more than 40 seconds lie in part of the last class: width 50 - 40 = 10, so estimated frequency = 0.8 × 10 = 8.","Estimated probability = 860 = 215."],"diagram":null,"draw":false,"revision":"0b876c58abdc"},{"id":"g7-histograms-q07","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

The table shows the distances, d km, cycled by 64 members of a cycling club one Sunday.

Distance (d km)Frequency
0 < d ≤ 510
5 < d ≤ 1530
15 < d ≤ 208
20 < d ≤ 4016
  1. On the grid, draw a histogram for this information.
  2. Explain why 5 < d ≤ 15 is the modal class even though 20 < d ≤ 40 has a wider bar.
","answerType":"multi","answer":"(a) bars at frequency densities 2, 3, 1.6, 0.8 (b) The modal class is the one with the greatest frequency density (tallest bar), and 5 < d <= 15 has the greatest frequency density, 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"bar heights 2, 3, 1.6, 0.8"},{"label":"b","answer":"Frequency density (bar height), not bar width, identifies the modal class; 5 < d <= 15 has the highest frequency density"}],"solution":["Frequency densities: 10 ÷ 5 = 2, 30 ÷ 10 = 3, 8 ÷ 5 = 1.6, 16 ÷ 20 = 0.8.","Draw bars with these heights over the correct intervals.","The modal class is the class with the highest frequency density; 5 < d ≤ 15 has frequency density 3, the tallest bar."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":40,"step":10,"minor":5,"label":"Distance (km)"},"y":{"min":0,"max":3.5,"step":0.5,"minor":2,"label":"Frequency density"},"bars":[{"from":0,"to":5,"height":2,"answer":true},{"from":5,"to":15,"height":3,"answer":true},{"from":15,"to":20,"height":1.6,"answer":true},{"from":20,"to":40,"height":0.8,"answer":true}]},"draw":true,"revision":"e597a66bfee5"},{"id":"g7-histograms-q08","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

The histogram shows the masses, m kg, of the 40 dogs seen at a vet's surgery one week.

  1. Use the histogram to complete a frequency table for the four classes.
  2. Write down the class interval that contains the lower quartile of the masses.
  3. Work out an estimate for the number of dogs with a mass greater than 20 kg.
","answerType":"multi","answer":"(a) frequencies 8, 12, 14, 6 (b) 10 < m <= 15 (c) 13","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"8, 12, 14, 6"},{"label":"b","answer":"10 < m <= 15"},{"label":"c","answer":"13"}],"solution":["For estimates using part of a class interval, assume the values are spread evenly within that interval. The data do not give their exact positions within a class.","Frequencies from the histogram: 0.8 × 10 = 8, 2.4 × 5 = 12, 1.4 × 10 = 14, 0.4 × 15 = 6. Total = 40.","The lower quartile is between the 10th and 11th values; the first class holds 8 dogs and the second reaches 20, so the lower quartile is in 10 < m ≤ 15.","Greater than 20 kg: half of the 15 < m ≤ 25 class is 20 to 25, estimated frequency 1.4 × 5 = 7, plus all 6 dogs in 25 < m ≤ 40.","Estimate = 7 + 6 = 13."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":40,"step":5,"label":"Mass (kg)"},"y":{"min":0,"max":2.5,"step":0.5,"minor":5,"label":"Frequency density"},"bars":[{"from":0,"to":10,"height":0.8},{"from":10,"to":15,"height":2.4},{"from":15,"to":25,"height":1.4},{"from":25,"to":40,"height":0.4}]},"draw":false,"revision":"fd767b2a90c2"},{"id":"g7-histograms-q09","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

The table shows the speeds, s mph, of 60 vehicles passing a speed camera.

Speed (s mph)Frequency
0 < s ≤ 208
20 < s ≤ 3018
30 < s ≤ 4024
40 < s ≤ 6010
  1. On the grid, draw a histogram for this information.
  2. Work out an estimate for the fraction of the vehicles travelling at more than 35 mph. Give your fraction in its simplest form.
","answerType":"multi","answer":"(a) bars at frequency densities 0.4, 1.8, 2.4, 0.5 (b) 11/30","answerDisplay":"(a) bars at frequency densities 0.4, 1.8, 2.4, 0.5 (b) 1130","accept":[],"parts":[{"label":"a","answer":"bar heights 0.4, 1.8, 2.4, 0.5"},{"label":"b","answer":"11/30","answerDisplay":"1130"}],"solution":["For estimates using part of a class interval, assume the values are spread evenly within that interval. The data do not give their exact positions within a class.","Frequency densities: 8 ÷ 20 = 0.4, 18 ÷ 10 = 1.8, 24 ÷ 10 = 2.4, 10 ÷ 20 = 0.5.","Draw bars with these heights.","More than 35 mph: half of the 30 < s ≤ 40 class is estimated as 24 ÷ 2 = 12 vehicles, plus all 10 vehicles above 40 mph.","Estimate = 22 vehicles out of 60 = 2260 = 1130."],"diagram":{"type":"axes","xy":false,"x":{"min":0,"max":60,"step":10,"minor":2,"label":"Speed (mph)"},"y":{"min":0,"max":2.5,"step":0.5,"minor":5,"label":"Frequency density"},"bars":[{"from":0,"to":20,"height":0.4,"answer":true},{"from":20,"to":30,"height":1.8,"answer":true},{"from":30,"to":40,"height":2.4,"answer":true},{"from":40,"to":60,"height":0.5,"answer":true}]},"draw":true,"revision":"4f9c53eca627"},{"id":"g7-histograms-q10","topicId":"g7-histograms","topic":"Histograms","grade":"7","topicGrade":"7","strand":"S","difficulty":10,"marks":6,"tier":"H","calc":true,"text":"

A gym has 200 members. A histogram shows the ages, a years, of the members. Some of the information is given in the table.

Age (a years)FrequencyFrequency density
0 < a ≤ 10303
10 < a ≤ 256
25 < a ≤ 45
45 < a ≤ 601
  1. Work out the frequency for the class 10 < a ≤ 25 and the frequency for the class 45 < a ≤ 60.
  2. Work out the frequency and the frequency density for the class 25 < a ≤ 45.
  3. Write down the class interval that contains the median age.
  4. Work out an estimate for the number of members older than 50.
","answerType":"multi","answer":"(a) 90 and 15 (b) frequency 65, frequency density 3.25 (c) 10 < a <= 25 (d) 10","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"10 < a <= 25: 90; 45 < a <= 60: 15"},{"label":"b","answer":"frequency 65, frequency density 3.25"},{"label":"c","answer":"10 < a <= 25"},{"label":"d","answer":"10"}],"solution":["For estimates using part of a class interval, assume the values are spread evenly within that interval. The data do not give their exact positions within a class.","10 < a ≤ 25: frequency = 6 × 15 = 90. 45 < a ≤ 60: frequency = 1 × 15 = 15.","25 < a ≤ 45: frequency = 200 - 30 - 90 - 15 = 65, and frequency density = 65 ÷ 20 = 3.25.","Cumulative frequencies: 30, 120, 185, 200. The median is between the 100th and 101st values, which lie in 10 < a ≤ 25.","Older than 50: part of the last class from 50 to 60, estimated frequency = 1 × 10 = 10."],"diagram":null,"draw":false,"revision":"054e112b6670"},{"id":"g7-inverse-and-composite-functions-q01","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

f(x) = 3x − 4    g(x) = x2

  1. Find f(5)

  2. Find gf(2)

","answerType":"multi","answer":"a) 11 b) 4","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"11"},{"label":"b","answer":"4"}],"solution":["gf(x) means g(f(x)): apply f first, then put its output into g. The function nearest the input is done first; gf does not mean multiply g by f.","a) f(5) = 3(5) - 4 = 11.","b) f(2) = 3(2) - 4 = 2, then g(2) = 2^2 = 4."],"diagram":null,"draw":false,"revision":"4945a56639dd"},{"id":"g7-inverse-and-composite-functions-q02","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

f(x) = 2x + 5

  1. Find f−1(x)

  2. Find f−1(11)

","answerType":"multi","answer":"a) (x-5)/2 b) 3","answerDisplay":"a) x-52 b) 3","accept":[],"parts":[{"label":"a","answer":"f^-1(x) = (x - 5)/2","answerDisplay":"f^-1(x) = x - 52"},{"label":"b","answer":"3"}],"solution":["The inverse function reverses f: undo +5 by subtracting 5, then undo ×2 by dividing by 2. f^-1 means inverse function, not 1/f.","a) Write y = 2x + 5, swap to make x the subject: x = y - 52, so f^-1(x) = x - 52.","b) f^-1(11) = 11 - 52 = 3. Check: f(3) = 11."],"diagram":null,"draw":false,"revision":"2aaab0400348"},{"id":"g7-inverse-and-composite-functions-q03","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

f(x) = x + 3    g(x) = 4x

  1. Find fg(x)

  2. Find (fg)−1(x)

  3. Find (fg)−1(11)

","answerType":"multi","answer":"a) 4x + 3 b) (x-3)/4 c) 2","answerDisplay":"a) 4x + 3 b) x-34 c) 2","accept":[],"parts":[{"label":"a","answer":"4x + 3"},{"label":"b","answer":"(x - 3)/4","answerDisplay":"x - 34"},{"label":"c","answer":"2"}],"solution":["a) fg(x) = f(4x) = 4x + 3.","b) Write y = 4x + 3 and make x the subject: x = y - 34, so (fg)^-1(x) = x - 34.","c) (fg)^-1(11) = 11 - 34 = 2. Check: fg(2) = 8 + 3 = 11."],"diagram":null,"draw":false,"revision":"af8da561266c"},{"id":"g7-inverse-and-composite-functions-q04","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":4,"marks":4,"tier":"H","calc":false,"text":"

f(x) = x + 23    g(x) = x2 − 1

  1. Find fg(3)

  2. Find f−1(x)

","answerType":"multi","answer":"a) 10/3 b) 3x - 2","answerDisplay":"a) 103 b) 3x - 2","accept":[],"parts":[{"label":"a","answer":"10/3","answerDisplay":"103"},{"label":"b","answer":"f^-1(x) = 3x - 2"}],"solution":["a) g(3) = 9 - 1 = 8, then f(8) = 8 + 23 = 103.","b) Write y = x + 23, so 3y = x + 2 and x = 3y - 2, giving f^-1(x) = 3x - 2.","b) Check: f(3x - 2) = 3x - 2 + 23 = x."],"diagram":null,"draw":false,"revision":"f42d5b25a555"},{"id":"g7-inverse-and-composite-functions-q05","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

f(x) = 2x − 1    g(x) = x2 + 3

  1. Show that gf(x) = 4x2 − 4x + 4

  2. Hence find gf(3)

","answerType":"multi","answer":"a) shown b) 28","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"gf(x) = (2x - 1)^2 + 3 = 4x^2 - 4x + 4"},{"label":"b","answer":"28"}],"solution":["a) gf(x) = g(2x - 1) = (2x - 1)^2 + 3.","a) Expand: (2x - 1)^2 = 4x^2 - 4x + 1, so gf(x) = 4x^2 - 4x + 4 as required.","b) gf(3) = 4(9) - 4(3) + 4 = 36 - 12 + 4 = 28.","b) Check directly: f(3) = 5, g(5) = 25 + 3 = 28."],"diagram":null,"draw":false,"revision":"fc718fa17d18"},{"id":"g7-inverse-and-composite-functions-q06","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":false,"text":"

f(x) = x2 + 2x

Express f(x + 3) in the form x2 + bx + c where b and c are integers.

","answerType":"exact","answer":"x^2 + 8x + 15","answerDisplay":null,"accept":["x^2+8x+15"],"parts":[],"solution":["f(x + 3) = (x + 3)^2 + 2(x + 3).","Expand: x^2 + 6x + 9 + 2x + 6.","Collect terms: x^2 + 8x + 15."],"diagram":null,"draw":false,"revision":"0d426ecd7069"},{"id":"g7-inverse-and-composite-functions-q07","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

f(x) = 5x − 3    g(x) = x + 12

  1. Find gf(1)

  2. Find g−1f(2)

","answerType":"multi","answer":"a) 3/2 b) 13","answerDisplay":"a) 32 b) 13","accept":[],"parts":[{"label":"a","answer":"3/2","answerDisplay":"32"},{"label":"b","answer":"13"}],"solution":["a) f(1) = 5 - 3 = 2, then g(2) = 2 + 12 = 32.","b) g^-1(x) = 2x - 1, since y = x + 12 gives x = 2y - 1.","b) f(2) = 10 - 3 = 7, then g^-1(7) = 14 - 1 = 13."],"diagram":null,"draw":false,"revision":"1b3cf8c6a61d"},{"id":"g7-inverse-and-composite-functions-q08","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

f(x) = 3x + 2    g(x) = x2 − 4

  1. Find f−1(x)

  2. Solve gf(x) = 0

","answerType":"multi","answer":"a) (x-2)/3 b) x = 0 or x = -4/3","answerDisplay":"a) x-23 b) x = 0 or x = -43","accept":[],"parts":[{"label":"a","answer":"f^-1(x) = (x - 2)/3","answerDisplay":"f^-1(x) = x - 23"},{"label":"b","answer":"x = 0 or x = -4/3","answerDisplay":"x = 0 or x = -43"}],"solution":["a) Write y = 3x + 2, so x = y - 23 and f^-1(x) = x - 23.","b) gf(x) = (3x + 2)^2 - 4 = 9x^2 + 12x + 4 - 4 = 9x^2 + 12x.","b) Factorise: 3x(3x + 4) = 0, so x = 0 or x = -43.","b) Check x = 0: f(0) = 2, g(2) = 0. Check x = -43: f(-43) = -2, g(-2) = 0."],"diagram":null,"draw":false,"revision":"7473a02af492"},{"id":"g7-inverse-and-composite-functions-q09","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"

f(x) = 4x − 1    g(x) = x2 + 2

  1. Find fg(−3)

  2. Find gf(x), giving your answer in the form ax2 + bx + c

  3. Find f−1(x)

","answerType":"multi","answer":"a) 43 b) 16x^2 - 8x + 3 c) (x+1)/4","answerDisplay":"a) 43 b) 16x^2 - 8x + 3 c) x+14","accept":[],"parts":[{"label":"a","answer":"43"},{"label":"b","answer":"16x^2 - 8x + 3"},{"label":"c","answer":"f^-1(x) = (x + 1)/4","answerDisplay":"f^-1(x) = x + 14"}],"solution":["a) g(-3) = 9 + 2 = 11, then f(11) = 44 - 1 = 43.","b) gf(x) = (4x - 1)^2 + 2 = 16x^2 - 8x + 1 + 2 = 16x^2 - 8x + 3.","c) Write y = 4x - 1, so x = y + 14 and f^-1(x) = x + 14."],"diagram":null,"draw":false,"revision":"c70b5b0fce71"},{"id":"g7-inverse-and-composite-functions-q10","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

f(x) = x2    g(x) = x + 2

  1. Find fg(x), giving your answer in the form x2 + bx + c

  2. Solve fg(x) = gf(x)

","answerType":"multi","answer":"a) x^2 + 4x + 4 b) x = -1/2","answerDisplay":"a) x^2 + 4x + 4 b) x = -12","accept":[],"parts":[{"label":"a","answer":"x^2 + 4x + 4"},{"label":"b","answer":"x = -1/2","answerDisplay":"x = -12"}],"solution":["a) fg(x) = (x + 2)^2 = x^2 + 4x + 4.","b) gf(x) = x^2 + 2.","b) Set them equal: x^2 + 4x + 4 = x^2 + 2, so 4x = -2 and x = -12.","b) Check: fg(-12) = (32)^2 = 94 and gf(-12) = 14 + 2 = 94."],"diagram":null,"draw":false,"revision":"9558a566bd1d"},{"id":"g7-inverse-and-composite-functions-q11","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":11,"marks":6,"tier":"H","calc":false,"text":"

f(x) = 2x + 1x − 3, x ≠ 3

  1. Show that f−1(x) = 3x + 1x − 2

  2. g(x) = x + 4

    Solve fg(x) = 5

","answerType":"multi","answer":"a) shown b) x = 4/3","answerDisplay":"a) shown b) x = 43","accept":[],"parts":[{"label":"a","answer":"Rearranging y = (2x + 1)/(x - 3) gives x = (3y + 1)/(y - 2), so f^-1(x) = (3x + 1)/(x - 2)","answerDisplay":"Rearranging y = 2x + 1x - 3 gives x = 3y + 1y - 2, so f^-1(x) = 3x + 1x - 2"},{"label":"b","answer":"x = 4/3","answerDisplay":"x = 43"}],"solution":["a) Write y = 2x + 1x - 3 and multiply up: y(x - 3) = 2x + 1.","a) Expand and collect x terms: xy - 2x = 3y + 1, so x(y - 2) = 3y + 1.","a) Divide: x = 3y + 1y - 2, so f^-1(x) = 3x + 1x - 2 as required.","The inverse is defined for x ≠ 2: f never outputs 2, since (2x + 1)/(x - 3) = 2 would imply 1 = -6. In part (b), x ≠ -1 because g(x) must not equal 3.","b) fg(x) = f(x + 4) = 2(x + 4) + 1(x + 4) - 3 = 2x + 9x + 1.","b) Solve 2x + 9x + 1 = 5: 2x + 9 = 5x + 5, so 3x = 4 and x = 43.","b) Check: g(43) = 163 and f(163) = 35373 = 5."],"diagram":null,"draw":false,"revision":"6fafd7a87062"},{"id":"g7-inverse-and-composite-functions-q12","topicId":"g7-inverse-and-composite-functions","topic":"Inverse and Composite Functions","grade":"7","topicGrade":"7","strand":"A","difficulty":12,"marks":7,"tier":"H","calc":true,"text":"

f(x) = x + 2    g(x) = x2 − 3x

  1. Find fg(1)

  2. Find f−1(x)

  3. Solve gf(x) = 10

","answerType":"multi","answer":"a) 0 b) x - 2 c) x = -4 or x = 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"0"},{"label":"b","answer":"f^-1(x) = x - 2"},{"label":"c","answer":"x = -4 or x = 3"}],"solution":["a) g(1) = 1 - 3 = -2, then f(-2) = 0.","b) y = x + 2 gives x = y - 2, so f^-1(x) = x - 2.","c) gf(x) = (x + 2)^2 - 3(x + 2) = x^2 + 4x + 4 - 3x - 6 = x^2 + x - 2.","c) Solve x^2 + x - 2 = 10, so x^2 + x - 12 = 0.","c) Factorise: (x + 4)(x - 3) = 0, so x = -4 or x = 3.","c) Check x = 3: f(3) = 5, g(5) = 25 - 15 = 10. Check x = -4: f(-4) = -2, g(-2) = 4 + 6 = 10."],"diagram":null,"draw":false,"revision":"58d6f2383577"},{"id":"g7-iteration-q01","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":1,"marks":2,"tier":"H","calc":true,"text":"

The number of members of a gym is modelled by the rule

un+1 = 1.2un − 50

where un is the number of members after n years.

There are 500 members now, so u0 = 500.

Work out the number of members the model predicts after 2 years.

","answerType":"exact","answer":"610","answerDisplay":null,"accept":["610 members"],"parts":[],"solution":["u0 is the starting value. The subscript n + 1 means the next year: put the current value un into the rule to get the next value. Repeat the rule twice for two years.","u1 = 1.2 x 500 - 50 = 600 - 50 = 550.","u2 = 1.2 x 550 - 50 = 660 - 50 = 610."],"diagram":null,"draw":false,"revision":"772abb1180ad"},{"id":"g7-iteration-q02","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":true,"text":"

A ball is dropped from a height of 5 m onto a hard floor.

After each bounce the ball rises to 60% of the height of the previous bounce, so

hn+1 = 0.6hn, where h0 = 5

  1. Work out the height the ball rises to after the third bounce.

  2. After which bounce does the ball first rise to less than 1 m?

","answerType":"multi","answer":"a) 1.08 m b) the 4th bounce","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1.08 m"},{"label":"b","answer":"4th bounce"}],"solution":["a) h1 = 0.6 x 5 = 3, h2 = 0.6 x 3 = 1.8, h3 = 0.6 x 1.8 = 1.08 m.","b) h4 = 0.6 x 1.08 = 0.648 m, which is the first height below 1 m, so the 4th bounce."],"diagram":null,"draw":false,"revision":"5807a59c6085"},{"id":"g7-iteration-q03","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

Amara opens a savings account with £1000.

The account pays 3% interest at the end of each year, and Amara then pays in another £200.

The amount in the account after n years is given by

an+1 = 1.03an + 200, where a0 = 1000

Work out the amount in the account after 3 years. Give your answer to the nearest penny.

","answerType":"exact","answer":"£1710.91","answerDisplay":null,"accept":["1710.91"],"parts":[],"solution":["a1 = 1.03 x 1000 + 200 = 1230.","a2 = 1.03 x 1230 + 200 = 1466.90.","a3 = 1.03 x 1466.90 + 200 = 1710.907.","To the nearest penny this is £1710.91."],"diagram":null,"draw":false,"revision":"07b682f558c3"},{"id":"g7-iteration-q04","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

A sequence is given by the iteration formula

xn+1 = 3√(5xn + 2), with x1 = 2

  1. Work out the values of x2, x3 and x4. Give each value correct to 3 decimal places.

  2. The values of xn approach a root of an equation. Write down this equation in the form x3 + ax + b = 0

","answerType":"multi","answer":"a) x2 = 2.289, x3 = 2.378, x4 = 2.404 b) x^3 - 5x - 2 = 0","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x2 = 2.289, x3 = 2.378, x4 = 2.404"},{"label":"b","answer":"x^3 - 5x - 2 = 0"}],"solution":["Keep the full calculator value for each next iteration (use the Ans key). Round only the values you write down, otherwise rounding errors can build up.","a) x2 = cube root of (5 x 2 + 2) = cube root of 12 = 2.289428... = 2.289 (3 dp).","a) x3 = cube root of (5 x 2.289428 + 2) = 2.377990... = 2.378 (3 dp).","a) x4 = cube root of (5 x 2.377990 + 2) = 2.403810... = 2.404 (3 dp).","b) At the limit x = cube root of (5x + 2), so x^3 = 5x + 2, which is x^3 - 5x - 2 = 0."],"diagram":null,"draw":false,"revision":"fd5ce3f830d5"},{"id":"g7-iteration-q05","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

A sequence is defined by

un+1 = kun + 3

where k is a constant.

u1 = 5 and u2 = 13

  1. Find the value of k

  2. Find the value of u4

","answerType":"multi","answer":"a) k = 2 b) u4 = 61","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"k = 2"},{"label":"b","answer":"61"}],"solution":["a) u2 = k x u1 + 3, so 13 = 5k + 3, giving 5k = 10 and k = 2.","b) u3 = 2 x 13 + 3 = 29.","b) u4 = 2 x 29 + 3 = 61."],"diagram":null,"draw":false,"revision":"fff48f1882ab"},{"id":"g7-iteration-q06","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

The number of deer in a woodland is modelled by

pn+1 = 0.9pn + 40

where pn is the number of deer n years from now.

There are 300 deer now, so p0 = 300.

  1. Work out the number of deer the model predicts in 2 years' time.

  2. The model predicts that in the long term the number of deer approaches a fixed value L. Work out the value of L.

","answerType":"multi","answer":"a) 319 b) L = 400","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"319"},{"label":"b","answer":"400"}],"solution":["a) p1 = 0.9 x 300 + 40 = 310.","a) p2 = 0.9 x 310 + 40 = 319.","b) At the fixed value, L = 0.9L + 40.","b) So 0.1L = 40, giving L = 400."],"diagram":null,"draw":false,"revision":"7553f4e69c75"},{"id":"g7-iteration-q07","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

  1. Show that the equation x3 − 7x + 3 = 0 can be rearranged to give

    x = 3√(7x − 3)

  2. Using the iteration formula xn+1 = 3√(7xn − 3) with x1 = 2, work out the values of x2, x3 and x4. Give each value correct to 3 decimal places.

","answerType":"multi","answer":"a) shown b) x2 = 2.224, x3 = 2.325, x4 = 2.368","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x^3 = 7x - 3, so x = cube root of (7x - 3)"},{"label":"b","answer":"x2 = 2.224, x3 = 2.325, x4 = 2.368"}],"solution":["Keep the full calculator value for each next iteration (use the Ans key). Round only the values you write down, otherwise rounding errors can build up.","a) Add 7x and subtract 3: x^3 = 7x - 3.","a) Take the cube root of both sides: x = cube root of (7x - 3), as required.","b) x2 = cube root of (7 x 2 - 3) = cube root of 11 = 2.223980... = 2.224 (3 dp).","b) x3 = cube root of (7 x 2.223980 - 3) = 2.324987... = 2.325 (3 dp).","b) x4 = cube root of (7 x 2.324987 - 3) = 2.367793... = 2.368 (3 dp)."],"diagram":null,"draw":false,"revision":"b39942a2fbeb"},{"id":"g7-iteration-q08","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":false,"text":"

f(x) = x2 − 6x + 2

  1. Show that the equation f(x) = 0 has a root between x = 0 and x = 1

  2. Show that the equation x2 − 6x + 2 = 0 can be rearranged to give

    x = x2 + 26

","answerType":"open","answer":"a) f(0) = 2 > 0 and f(1) = -3 < 0, sign change so a root lies between 0 and 1 b) 6x = x^2 + 2 so x = (x^2 + 2)/6","answerDisplay":null,"accept":[],"parts":[],"solution":["This polynomial has a continuous graph: it has no jumps or breaks. If its value changes from positive to negative, the graph must pass through zero somewhere in between.","a) f(0) = 0 - 0 + 2 = 2, which is positive.","a) f(1) = 1 - 6 + 2 = -3, which is negative.","a) f is continuous and changes sign between 0 and 1, so there is a root between x = 0 and x = 1.","b) Rearrange x^2 - 6x + 2 = 0 to 6x = x^2 + 2.","b) Divide both sides by 6: x = x^2 + 26, as required."],"diagram":null,"draw":false,"revision":"790914b20158"},{"id":"g7-iteration-q09","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":9,"marks":6,"tier":"H","calc":true,"text":"

f(x) = x3 + 4x − 9

  1. Show that the equation f(x) = 0 has a root between x = 1 and x = 2

  2. Show that the equation x3 + 4x − 9 = 0 can be rearranged to give

    x = 3√(9 − 4x)

  3. Starting with x0 = 1.5, use the iteration formula xn+1 = 3√(9 − 4xn) to work out the values of x1, x2 and x3. Give each value correct to 4 decimal places.

","answerType":"multi","answer":"a) f(1) = -4 < 0, f(2) = 7 > 0, sign change b) x^3 = 9 - 4x so x = cbrt(9 - 4x) c) x1 = 1.4423, x2 = 1.4784, x3 = 1.4560","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"f(1) = -4 and f(2) = 7; the sign change means a root lies between 1 and 2"},{"label":"b","answer":"x^3 = 9 - 4x, so x = cube root of (9 - 4x)"},{"label":"c","answer":"x1 = 1.4423, x2 = 1.4784, x3 = 1.4560"}],"solution":["This polynomial has a continuous graph: it has no jumps or breaks. If its value changes from positive to negative, the graph must pass through zero somewhere in between.","Keep the full calculator value for each next iteration (use the Ans key). Round only the values you write down, otherwise rounding errors can build up.","a) f(1) = 1 + 4 - 9 = -4, which is negative.","a) f(2) = 8 + 8 - 9 = 7, which is positive.","a) f is continuous and changes sign, so there is a root between x = 1 and x = 2.","b) Rearrange to x^3 = 9 - 4x, then take the cube root: x = cube root of (9 - 4x), as required.","c) x1 = cube root of (9 - 4 x 1.5) = cube root of 3 = 1.442250... = 1.4423 (4 dp).","c) x2 = cube root of (9 - 4 x 1.442250) = 1.478356... = 1.4784 (4 dp).","c) x3 = cube root of (9 - 4 x 1.478356) = 1.455992... = 1.4560 (4 dp)."],"diagram":null,"draw":false,"revision":"b880c2b2a09a"},{"id":"g7-iteration-q10","topicId":"g7-iteration","topic":"Iteration","grade":"7","topicGrade":"7","strand":"A","difficulty":10,"marks":6,"tier":"H","calc":true,"text":"

f(x) = x2 + 2x − 5

  1. Show that the equation f(x) = 0 has a root between x = 1 and x = 2

  2. Show that the equation x2 + 2x − 5 = 0 can be rearranged to give

    x = 5x + 2

  3. Starting with x0 = 1.5, use the iteration formula xn+1 = 5xn + 2 to work out the values of x1, x2 and x3. Give each value correct to 4 decimal places.

  4. Write down an estimate of this root of f(x) = 0 correct to 2 decimal places.

","answerType":"multi","answer":"a) f(1) = -2 < 0, f(2) = 3 > 0 b) x(x + 2) = 5 so x = 5/(x + 2) c) x1 = 1.4286, x2 = 1.4583, x3 = 1.4458 d) 1.45","answerDisplay":"a) f(1) = -2 < 0, f(2) = 3 > 0 b) x(x + 2) = 5 so x = 5x + 2 c) x1 = 1.4286, x2 = 1.4583, x3 = 1.4458 d) 1.45","accept":[],"parts":[{"label":"a","answer":"f(1) = -2 and f(2) = 3; the sign change means a root lies between 1 and 2"},{"label":"b","answer":"x^2 + 2x = 5, so x(x + 2) = 5 and x = 5/(x + 2)","answerDisplay":"x^2 + 2x = 5, so x(x + 2) = 5 and x = 5x + 2"},{"label":"c","answer":"x1 = 1.4286, x2 = 1.4583, x3 = 1.4458"},{"label":"d","answer":"1.45"}],"solution":["This polynomial has a continuous graph: it has no jumps or breaks. If its value changes from positive to negative, the graph must pass through zero somewhere in between.","Keep the full calculator value for each next iteration (use the Ans key). Round only the values you write down, otherwise rounding errors can build up.","a) f(1) = 1 + 2 - 5 = -2, which is negative, and f(2) = 4 + 4 - 5 = 3, which is positive.","a) The sign change means there is a root between x = 1 and x = 2.","b) Rearrange to x^2 + 2x = 5, factorise the left side as x(x + 2) = 5, then divide by (x + 2): x = 5x + 2, as required.","c) x1 = 51.5 + 2 = 1.428571... = 1.4286 (4 dp).","c) x2 = 51.428571 + 2 = 1.458333... = 1.4583 (4 dp).","c) x3 = 51.458333 + 2 = 1.445783... = 1.4458 (4 dp).","d) The iterates settle around 1.45; the exact root is -1 + √6 = 1.449489..., so 1.45 to 2 dp.","To check the 2-decimal-place rounding, f(1.445) = -0.021975 and f(1.455) = 0.027025. The positive root lies between these rounding boundaries, so it rounds to 1.45."],"diagram":null,"draw":false,"revision":"b7213e0760e9"},{"id":"g7-quadratic-formula-q01","topicId":"g7-quadratic-formula","topic":"Quadratic Formula","grade":"7","topicGrade":"7","strand":"A","difficulty":1,"marks":3,"tier":"H","calc":true,"text":"

Solve x2 + 5x + 3 = 0

Give your solutions correct to 2 decimal places.

","answerType":"exact","answer":"x = -0.70, x = -4.30","answerDisplay":null,"accept":["x = -0.70, x = -4.30","x = -4.30, x = -0.70","-0.70 and -4.30"],"parts":[],"solution":["For ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a). Match a, b and c including their signs. ± means calculate once with + and once with -, and divide the entire numerator by 2a.","Use the quadratic formula with a = 1, b = 5, c = 3.","x = -5 ± √(25 - 12)2 = -5 ± √132.","√13 = 3.60555..., so x = -5 + 3.60555...2 = -0.69722... or x = -5 - 3.60555...2 = -4.30277...","To 2 decimal places, x = -0.70 or x = -4.30."],"diagram":null,"draw":false,"revision":"97c44d653ec6"},{"id":"g7-quadratic-formula-q02","topicId":"g7-quadratic-formula","topic":"Quadratic Formula","grade":"7","topicGrade":"7","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":true,"text":"

Solve x2 - 4x - 9 = 0

Give your solutions correct to 2 decimal places.

","answerType":"exact","answer":"x = 5.61, x = -1.61","answerDisplay":null,"accept":["x = 5.61, x = -1.61","x = -1.61, x = 5.61","5.61 and -1.61"],"parts":[],"solution":["For ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a). Match a, b and c including their signs. ± means calculate once with + and once with -, and divide the entire numerator by 2a.","Use the quadratic formula with a = 1, b = -4, c = -9.","x = 4 ± √(16 + 36)2 = 4 ± √522.","√52 = 7.21110..., so x = 4 + 7.21110...2 = 5.60555... or x = 4 - 7.21110...2 = -1.60555...","To 2 decimal places, x = 5.61 or x = -1.61. Take care with the signs: -b = 4 and -4ac = +36."],"diagram":null,"draw":false,"revision":"f566ca850d0d"},{"id":"g7-quadratic-formula-q03","topicId":"g7-quadratic-formula","topic":"Quadratic Formula","grade":"7","topicGrade":"7","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

Solve 2x2 + 7x - 3 = 0

Give your solutions correct to 2 decimal places.

","answerType":"exact","answer":"x = 0.39, x = -3.89","answerDisplay":null,"accept":["x = 0.39, x = -3.89","x = -3.89, x = 0.39","0.39 and -3.89"],"parts":[],"solution":["For ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a). Match a, b and c including their signs. ± means calculate once with + and once with -, and divide the entire numerator by 2a.","Use the quadratic formula with a = 2, b = 7, c = -3.","x = -7 ± √(49 + 24)4 = -7 ± √734.","√73 = 8.54400..., so x = -7 + 8.54400...4 = 0.38600... or x = -7 - 8.54400...4 = -3.88600...","To 2 decimal places, x = 0.39 or x = -3.89."],"diagram":null,"draw":false,"revision":"9d167dd20d7c"},{"id":"g7-quadratic-formula-q04","topicId":"g7-quadratic-formula","topic":"Quadratic Formula","grade":"7","topicGrade":"7","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

Solve 3x2 - 5x - 7 = 0

Give your solutions correct to 2 decimal places.

","answerType":"exact","answer":"x = 2.57, x = -0.91","answerDisplay":null,"accept":["x = 2.57, x = -0.91","x = -0.91, x = 2.57","2.57 and -0.91"],"parts":[],"solution":["For ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a). Match a, b and c including their signs. ± means calculate once with + and once with -, and divide the entire numerator by 2a.","Use the quadratic formula with a = 3, b = -5, c = -7.","x = 5 ± √(25 + 84)6 = 5 ± √1096.","√109 = 10.44030..., so x = 5 + 10.44030...6 = 2.57338... or x = 5 - 10.44030...6 = -0.90671...","To 2 decimal places, x = 2.57 or x = -0.91."],"diagram":null,"draw":false,"revision":"3aec7d698ca6"},{"id":"g7-quadratic-formula-q05","topicId":"g7-quadratic-formula","topic":"Quadratic Formula","grade":"7","topicGrade":"7","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":false,"text":"

Solve x2 + 6x + 4 = 0

Give your solutions in the form a ± √b, where a and b are integers.

","answerType":"exact","answer":"x = -3 + sqrt(5), x = -3 - sqrt(5)","answerDisplay":null,"accept":["x = -3 ± √5","-3 ± √5","x = -3 + √5 and x = -3 - √5"],"parts":[],"solution":["For ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a). Match a, b and c including their signs. ± means calculate once with + and once with -, and divide the entire numerator by 2a.","Use the quadratic formula with a = 1, b = 6, c = 4.","x = -6 ± √(36 - 16)2 = -6 ± √202.","Simplify the surd: √20 = 2√5.","x = -6 ± 2√52 = -3 ± √5."],"diagram":null,"draw":false,"revision":"7d6e0817f3e5"},{"id":"g7-quadratic-formula-q06","topicId":"g7-quadratic-formula","topic":"Quadratic Formula","grade":"7","topicGrade":"7","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":false,"text":"

Solve 2x2 - 8x + 5 = 0

Give your solutions in the form p ± √q2, where p and q are integers.

","answerType":"exact","answer":"x = (4 + sqrt(6))/2, x = (4 - sqrt(6))/2","answerDisplay":"x = 4 + sqrt(6)2, x = 4 - sqrt(6)2","accept":["x = (4 ± √6)/2","(4 ± √6)/2","x = 2 ± (√6)/2"],"parts":[],"solution":["For ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a). Match a, b and c including their signs. ± means calculate once with + and once with -, and divide the entire numerator by 2a.","Use the quadratic formula with a = 2, b = -8, c = 5.","x = 8 ± √(64 - 40)4 = 8 ± √244.","Simplify the surd: √24 = 2√6, so x = 8 ± 2√64.","Divide top and bottom by 2: x = 4 ± √62.","Check by substitution: if x = 4 + √62 then 2x2 = 11 + 4√6 and 8x = 16 + 4√6, so 2x2 - 8x + 5 = 11 + 4√6 - 16 - 4√6 + 5 = 0."],"diagram":null,"draw":false,"revision":"de82ff5c476e"},{"id":"g7-quadratic-formula-q07","topicId":"g7-quadratic-formula","topic":"Quadratic Formula","grade":"7","topicGrade":"7","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"
  1. Using the discriminant, explain why the equation x2 + 3x + 7 = 0 has no real solutions.
  2. Write down the number of real solutions of the equation x2 - 6x + 9 = 0. Give a reason for your answer.
","answerType":"multi","answer":"a) discriminant = 9 - 28 = -19 < 0, so no real solutions b) one (repeated) real solution because the discriminant is 0","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"b^2 - 4ac = 3^2 - 4(1)(7) = -19, which is negative, so there are no real solutions"},{"label":"b","answer":"One repeated real solution, because b^2 - 4ac = 36 - 36 = 0"}],"solution":["For ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a). Match a, b and c including their signs. ± means calculate once with + and once with -, and divide the entire numerator by 2a.","The discriminant of ax2 + bx + c = 0 is b2 - 4ac.","For x2 + 3x + 7 = 0: discriminant = 9 - 28 = -19. A negative number has no real square root, so the formula gives no real solutions.","For x2 - 6x + 9 = 0: discriminant = 36 - 36 = 0, so there is exactly one repeated real solution.","In fact x2 - 6x + 9 = (x - 3)2, so the repeated solution is x = 3."],"diagram":null,"draw":false,"revision":"155b04a1117f"},{"id":"g7-quadratic-formula-q08","topicId":"g7-quadratic-formula","topic":"Quadratic Formula","grade":"7","topicGrade":"7","strand":"A","difficulty":8,"marks":6,"tier":"H","calc":true,"text":"

A rectangular lawn has length (x + 3) metres and width x metres.

The area of the lawn is 20 m2

  1. Show that x2 + 3x - 20 = 0
  2. Find the perimeter of the lawn. Give your answer correct to 3 significant figures.
","answerType":"multi","answer":"a) x(x + 3) = 20 expands to x^2 + 3x - 20 = 0 b) 18.9 m","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Area = x(x + 3) = 20, so x^2 + 3x = 20 and x^2 + 3x - 20 = 0"},{"label":"b","answer":"18.9 m"}],"solution":["For ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a). Match a, b and c including their signs. ± means calculate once with + and once with -, and divide the entire numerator by 2a.","Area of a rectangle = length × width, so x(x + 3) = 20.","Expanding gives x2 + 3x = 20, so x2 + 3x - 20 = 0, as required.","Solve with the quadratic formula, a = 1, b = 3, c = -20: x = -3 ± √(9 + 80)2 = -3 ± √892.","√89 = 9.43398..., so x = 3.21699... or x = -6.21699... A length cannot be negative, so x = 3.21699...","Perimeter = 2(x + x + 3) = 4x + 6 = 4 × 3.21699... + 6 = 18.86796...","To 3 significant figures the perimeter is 18.9 m."],"diagram":null,"draw":false,"revision":"531bcc11e1b7"},{"id":"g7-rearranging-harder-formulae-q01","topicId":"g7-rearranging-harder-formulae","topic":"Rearranging Harder Formulae","grade":"7","topicGrade":"7","strand":"A","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

Assume a ≠ b.

Make x the subject of

ax = bx + c

","answerType":"exact","answer":"x = c/(a-b)","answerDisplay":"x = ca-b","accept":["x = c/(a - b)","x = -c/(b - a)"],"parts":[],"solution":["Collect the x terms on one side: ax - bx = c.","Factorise: x(a - b) = c.","Divide by (a - b): x = ca - b.","Assume a ≠ b. Dividing by zero is not allowed."],"diagram":null,"draw":false,"revision":"8e744a5f487c"},{"id":"g7-rearranging-harder-formulae-q02","topicId":"g7-rearranging-harder-formulae","topic":"Rearranging Harder Formulae","grade":"7","topicGrade":"7","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

Make t the subject of

w = 5(t + 3)t

","answerType":"exact","answer":"t = 15/(w-5)","answerDisplay":"t = 15w-5","accept":["t = 15/(w - 5)"],"parts":[],"solution":["Multiply both sides by t: wt = 5(t + 3) = 5t + 15.","Collect the t terms: wt - 5t = 15.","Factorise: t(w - 5) = 15.","Divide by (w - 5): t = 15w - 5.","Here t ≠ 0 and w ≠ 5. Dividing by zero is not allowed."],"diagram":null,"draw":false,"revision":"a0281eb67848"},{"id":"g7-rearranging-harder-formulae-q03","topicId":"g7-rearranging-harder-formulae","topic":"Rearranging Harder Formulae","grade":"7","topicGrade":"7","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

Make x the subject of

y = 3√(x2) + 5

","answerType":"exact","answer":"x = 2(y-5)^3","answerDisplay":null,"accept":["x = 2(y - 5)^3"],"parts":[],"solution":["Subtract 5 from both sides: y - 5 = cube root of (x2).","Cube both sides: (y - 5)^3 = x2.","Multiply by 2: x = 2(y - 5)^3."],"diagram":null,"draw":false,"revision":"d2e21c3c113a"},{"id":"g7-rearranging-harder-formulae-q04","topicId":"g7-rearranging-harder-formulae","topic":"Rearranging Harder Formulae","grade":"7","topicGrade":"7","strand":"A","difficulty":4,"marks":4,"tier":"H","calc":false,"text":"

Make m the subject of

k = 3m − 2m + 4

","answerType":"exact","answer":"m = (4k+2)/(3-k)","answerDisplay":"m = 4k+23-k","accept":["m = (4k + 2)/(3 - k)","m = (2 + 4k)/(3 - k)","m = -(4k + 2)/(k - 3)"],"parts":[],"solution":["Multiply both sides by (m + 4): k(m + 4) = 3m - 2.","Expand: km + 4k = 3m - 2.","Collect the m terms on one side: km - 3m = -2 - 4k, so 4k + 2 = 3m - km.","Factorise: 4k + 2 = m(3 - k).","Divide by (3 - k): m = 4k + 23 - k.","Here m ≠ -4 and k ≠ 3. Dividing by zero is not allowed."],"diagram":null,"draw":false,"revision":"e2853cb23004"},{"id":"g7-rearranging-harder-formulae-q05","topicId":"g7-rearranging-harder-formulae","topic":"Rearranging Harder Formulae","grade":"7","topicGrade":"7","strand":"A","difficulty":5,"marks":4,"tier":"H","calc":false,"text":"

Assume a ≠ 5.

Make p the subject of

5(p − q) = ap + 7

","answerType":"exact","answer":"p = (5q+7)/(5-a)","answerDisplay":"p = 5q+75-a","accept":["p = (5q + 7)/(5 - a)","p = (7 + 5q)/(5 - a)","p = -(5q + 7)/(a - 5)"],"parts":[],"solution":["Expand the bracket: 5p - 5q = ap + 7.","Collect the p terms on one side: 5p - ap = 7 + 5q.","Factorise: p(5 - a) = 5q + 7.","Divide by (5 - a): p = 5q + 75 - a.","Assume a ≠ 5. Dividing by zero is not allowed."],"diagram":null,"draw":false,"revision":"732784f38238"},{"id":"g7-rearranging-harder-formulae-q06","topicId":"g7-rearranging-harder-formulae","topic":"Rearranging Harder Formulae","grade":"7","topicGrade":"7","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

Make v the subject of

1f = 1u + 1v

","answerType":"exact","answer":"v = fu/(u-f)","answerDisplay":"v = fuu-f","accept":["v = fu/(u - f)","v = uf/(u - f)","v = -fu/(f - u)"],"parts":[],"solution":["Subtract 1u from both sides: 1v = 1f - 1u.","Write the right hand side as a single fraction: 1v = u - ffu.","Take the reciprocal of both sides: v = fuu - f.","Check with f = 2, u = 3: v = 61 = 6, and 13 + 16 = 12.","Here f and u are nonzero and u ≠ f. Dividing by zero is not allowed."],"diagram":null,"draw":false,"revision":"b25448d1eea1"},{"id":"g7-rearranging-harder-formulae-q07","topicId":"g7-rearranging-harder-formulae","topic":"Rearranging Harder Formulae","grade":"7","topicGrade":"7","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

Make x the subject of

y = √[x + 1x − 1]

","answerType":"exact","answer":"x = (y^2+1)/(y^2-1)","answerDisplay":"x = y^2+1y^2-1","accept":["x = (y^2 + 1)/(y^2 - 1)"],"parts":[],"solution":["Square both sides: y^2 = x + 1x - 1.","Multiply by (x - 1): y^2(x - 1) = x + 1, so y^2 x - y^2 = x + 1.","Collect the x terms: y^2 x - x = y^2 + 1.","Factorise: x(y^2 - 1) = y^2 + 1.","Divide by (y^2 - 1): x = y^2 + 1y^2 - 1.","Check with x = 3: 3 + 13 - 1 = 2, y = √2, and 2 + 12 - 1 = 3.","Because y is a square root, y ≥ 0. Also y ≠ 1, so y^2 - 1 is nonzero. Dividing by zero is not allowed."],"diagram":null,"draw":false,"revision":"4aa6a4a4ff09"},{"id":"g7-rearranging-harder-formulae-q08","topicId":"g7-rearranging-harder-formulae","topic":"Rearranging Harder Formulae","grade":"7","topicGrade":"7","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

Make v the subject of

T = √[2vv − 3]

","answerType":"exact","answer":"v = 3T^2/(T^2-2)","answerDisplay":"v = 3T^2T^2-2","accept":["v = 3T^2/(T^2 - 2)"],"parts":[],"solution":["Square both sides: T^2 = 2vv - 3.","Multiply by (v - 3): T^2 v - 3T^2 = 2v.","Collect the v terms: T^2 v - 2v = 3T^2.","Factorise: v(T^2 - 2) = 3T^2.","Divide by (T^2 - 2): v = 3T^2T^2 - 2.","Check with v = 6: 2(6)6 - 3 = 4, T = 2, and 3(4)4 - 2 = 6.","Because T is a square root, T ≥ 0. Also T ≠ √2, so T^2 - 2 is nonzero. Dividing by zero is not allowed."],"diagram":null,"draw":false,"revision":"c3808e3a801b"},{"id":"g7-surds-q01","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":1,"marks":2,"tier":"H","calc":false,"text":"

Rationalise the denominator of 12√3

Give your answer in its simplest form.

","answerType":"exact","answer":"4√3","answerDisplay":null,"accept":["4√3","4√3","4root3"],"parts":[],"solution":["Rationalise means remove the square root from the denominator. Multiplying top and bottom by √3 multiplies the fraction by √3/√3 = 1, so it keeps the same value; √3 × √3 = 3.","Multiply the numerator and denominator by √3.","12√3 = 12√33.","Divide top and bottom by 3: 12√33 = 4√3."],"diagram":null,"draw":false,"revision":"a51ff3661c95"},{"id":"g7-surds-q02","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":2,"marks":2,"tier":"H","calc":true,"text":"

Erin is asked to rationalise the denominator of 8√6

Here is her working.

8√6 = 8 × √6√6 × √6 = 8√636

Erin has made one mistake.

Explain the mistake Erin has made.

","answerType":"open","answer":"√6 x √6 = 6, not 36. The denominator should be 6, giving 8√6/6 = 4√6/3.","answerDisplay":"√6 x √6 = 6, not 36. The denominator should be 6, giving 8√66 = 4√63.","accept":[],"parts":[],"solution":["Erin has squared 6 instead of squaring √6.","√6 × √6 = 6, not 36.","The correct answer is 8√66, which simplifies to 4√63."],"diagram":null,"draw":false,"revision":"7fcdeb48c2b2"},{"id":"g7-surds-q03","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

Expand and simplify (2 + √5)(3 + √5)

","answerType":"exact","answer":"11 + 5√5","answerDisplay":null,"accept":["11 + 5√5","11+5√5","5√5 + 11"],"parts":[],"solution":["Expand the brackets: 2 × 3 + 2√5 + 3√5 + √5 × √5.","√5 × √5 = 5, so this is 6 + 2√5 + 3√5 + 5.","Collect terms: 11 + 5√5."],"diagram":null,"draw":false,"revision":"7f21cb944719"},{"id":"g7-surds-q04","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":4,"marks":3,"tier":"H","calc":false,"text":"

Express √80 + √45 in the form k√5, where k is an integer.

","answerType":"exact","answer":"7√5","answerDisplay":null,"accept":["7√5","7√5"],"parts":[],"solution":["√80 = √(16 × 5) = 4√5.","√45 = √(9 × 5) = 3√5.","4√5 + 3√5 = 7√5, so k = 7."],"diagram":null,"draw":false,"revision":"1b94c26232c4"},{"id":"g7-surds-q05","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":5,"marks":3,"tier":"H","calc":false,"text":"

Expand and simplify (4 − √3)2

Give your answer in the form a + b√3, where a and b are integers.

","answerType":"exact","answer":"19 - 8√3","answerDisplay":null,"accept":["19 - 8√3","19-8√3","a = 19, b = -8"],"parts":[],"solution":["(4 − √3)² = 16 − 4√3 − 4√3 + (√3)².","(√3)² = 3, so this is 16 − 8√3 + 3.","Collect terms: 19 − 8√3, so a = 19 and b = −8."],"diagram":null,"draw":false,"revision":"bd99ae637789"},{"id":"g7-surds-q06","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":6,"marks":4,"tier":"H","calc":false,"text":"

Write 6 + √12√3 in the form a + b√3, where a and b are integers.

","answerType":"exact","answer":"2 + 2√3","answerDisplay":null,"accept":["2 + 2√3","2+2√3","a = 2, b = 2"],"parts":[],"solution":["√12 = 2√3, so the numerator is 6 + 2√3.","Split the fraction: 6√3 + 2√3√3.","6√3 = 6√33 = 2√3, and 2√3√3 = 2.","So the expression equals 2 + 2√3, giving a = 2 and b = 2."],"diagram":null,"draw":false,"revision":"1fb3a14a9c98"},{"id":"g7-surds-q07","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

Rationalise the denominator of 224 + √5

Give your answer in the form a + b√5, where a and b are integers.

","answerType":"exact","answer":"8 - 2√5","answerDisplay":null,"accept":["8 - 2√5","8-2√5","a = 8, b = -2"],"parts":[],"solution":["Use the same two terms with the sign between them reversed (the conjugate). When the brackets multiply, the two square-root middle terms cancel, leaving a rational denominator. Multiply the numerator by the same bracket to keep the value unchanged.","Multiply the numerator and denominator by (4 − √5).","Denominator: (4 + √5)(4 − √5) = 16 − 5 = 11.","Numerator: 22(4 − √5) = 88 − 22√5.","So the fraction is 88 − 22√511 = 8 − 2√5, giving a = 8 and b = −2."],"diagram":null,"draw":false,"revision":"5aac482a3e1d"},{"id":"g7-surds-q08","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":8,"marks":4,"tier":"H","calc":false,"text":"

The first two terms of a geometric sequence are

√5 + 1    5 + √5

Find the common ratio of the sequence.

Give your answer in its simplest form.

","answerType":"exact","answer":"√5","answerDisplay":null,"accept":["√5","√5","root 5"],"parts":[],"solution":["The common ratio is the second term divided by the first: 5 + √5√5 + 1.","Factorise the numerator: 5 + √5 = √5(√5 + 1).","So the ratio is √5(√5 + 1)√5 + 1 = √5."],"diagram":null,"draw":false,"revision":"5f8c2511b461"},{"id":"g7-surds-q09","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"

Show that √50 + 18√2 = 14√2

","answerType":"open","answer":"√50 = 5√2 and 18/√2 = 9√2, so the sum is 14√2","answerDisplay":"√50 = 5√2 and 18√2 = 9√2, so the sum is 14√2","accept":[],"parts":[],"solution":["√50 = √(25 × 2) = 5√2.","Rationalise: 18√2 = 18√22 = 9√2.","5√2 + 9√2 = 14√2, as required."],"diagram":null,"draw":false,"revision":"c0ac802e27d8"},{"id":"g7-surds-q10","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":10,"marks":5,"tier":"H","calc":false,"text":"

Show that 7 + √53 − √5 can be written as 13 + 5√52

","answerType":"open","answer":"Multiply top and bottom by (3 + √5): numerator (7+√5)(3+√5) = 26 + 10√5, denominator 9 - 5 = 4, so the fraction is (26+10√5)/4 = (13+5√5)/2","answerDisplay":"Multiply top and bottom by (3 + √5): numerator (7+√5)(3+√5) = 26 + 10√5, denominator 9 - 5 = 4, so the fraction is 26+10√54 = 13+5√52","accept":[],"parts":[],"solution":["Use the same two terms with the sign between them reversed (the conjugate). When the brackets multiply, the two square-root middle terms cancel, leaving a rational denominator. Multiply the numerator by the same bracket to keep the value unchanged.","Multiply the numerator and denominator by (3 + √5).","Numerator: (7 + √5)(3 + √5) = 21 + 7√5 + 3√5 + 5 = 26 + 10√5.","Denominator: (3 − √5)(3 + √5) = 9 − 5 = 4.","So the fraction is 26 + 10√54. Dividing top and bottom by 2 gives 13 + 5√52, as required."],"diagram":null,"draw":false,"revision":"64ef5ef53b9d"},{"id":"g7-surds-q11","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":11,"marks":6,"tier":"H","calc":false,"text":"

Express √12 + 4√3 − 1 in the form a + b√3, where a and b are integers.

Show each stage of your working.

","answerType":"exact","answer":"5 + 3√3","answerDisplay":null,"accept":["5 + 3√3","5+3√3","a = 5, b = 3"],"parts":[],"solution":["Use the same two terms with the sign between them reversed (the conjugate). When the brackets multiply, the two square-root middle terms cancel, leaving a rational denominator. Multiply the numerator by the same bracket to keep the value unchanged.","Simplify the numerator: √12 = 2√3, so the numerator is 2√3 + 4.","Multiply the numerator and denominator by (√3 + 1).","Numerator: (2√3 + 4)(√3 + 1) = 6 + 2√3 + 4√3 + 4 = 10 + 6√3.","Denominator: (√3 − 1)(√3 + 1) = 3 − 1 = 2.","So the expression is 10 + 6√32 = 5 + 3√3, giving a = 5 and b = 3."],"diagram":null,"draw":false,"revision":"497f277f8bb5"},{"id":"g7-surds-q12","topicId":"g7-surds","topic":"Surds","grade":"7","topicGrade":"7","strand":"N","difficulty":12,"marks":6,"tier":"H","calc":true,"text":"

A trapezium has parallel sides of length (3 + √2) cm and (5 + 3√2) cm.

The perpendicular distance between the parallel sides is (4 − √2) cm.

Work out the exact area of the trapezium.

Give your answer in the form (a + b√2) cm², where a and b are integers.

","answerType":"exact","answer":"12 + 4√2","answerDisplay":null,"accept":["12 + 4√2","12+4√2","a = 12, b = 4"],"parts":[],"solution":["Area of a trapezium = 12(sum of parallel sides) × height.","Sum of the parallel sides: (3 + √2) + (5 + 3√2) = 8 + 4√2.","Half of this is 4 + 2√2.","Area = (4 + 2√2)(4 − √2) = 16 − 4√2 + 8√2 − 2 × 2.","= 16 + 4√2 − 4 = 12 + 4√2, so the area is (12 + 4√2) cm²."],"diagram":{"type":"shape","notAccurate":true,"width":400,"polygons":[{"pts":[[0,0],[9.2,0],[7.2,2.6],[2.8,2.6]],"fill":false,"sides":["(5 + 3√2) cm",null,"(3 + √2) cm",null]}],"segments":[{"from":[0,0],"to":[9.2,0],"parallel":1},{"from":[2.8,2.6],"to":[7.2,2.6],"parallel":1},{"from":[5,2.6],"to":[5,0],"dashed":true,"thin":true,"label":"(4 − √2) cm","labelPos":"right"}],"angles":[{"at":[5,0],"from":[9.2,0],"to":[5,2.6],"right":true}]},"draw":false,"revision":"3e0176fccca3"},{"id":"g7-the-cosine-rule-q01","topicId":"g7-the-cosine-rule","topic":"The Cosine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":1,"marks":4,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 8 cm, AC = 11 cm and angle BAC = 62°.

Work out the length of BC. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"10.1 cm","answerDisplay":null,"accept":["10.1"],"parts":[],"solution":["We know two sides and the angle between them. The cosine rule finds the opposite side; unlike Pythagoras, it works even when the angle is not 90°. Use degree mode and keep the full value until the final rounding.","Cosine rule: BC2 = AB2 + AC2 - 2 × AB × AC × cos A.","BC2 = 82 + 112 - 2 × 8 × 11 × cos 62°.","BC2 = 64 + 121 - 176 × 0.46947... = 102.37...","BC = √102.37... = 10.117...","BC = 10.1 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[3.8731403929835997,7.2843176410861465],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"8 cm"},{"from":[6,0],"to":[3.8731403929835997,7.2843176410861465]},{"from":[3.8731403929835997,7.2843176410861465],"to":[0,0],"label":"11 cm"}],"angles":[{"at":[0,0],"from":[6,0],"to":[3.8731403929835997,7.2843176410861465],"label":"62°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"bf3a051302a3"},{"id":"g7-the-cosine-rule-q02","topicId":"g7-the-cosine-rule","topic":"The Cosine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":2,"marks":4,"tier":"H","calc":true,"text":"

In triangle PQR, PQ = 9.4 cm, PR = 13.7 cm and angle QPR = 118°.

Calculate the length of QR. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"19.9 cm","answerDisplay":null,"accept":["19.9"],"parts":[],"solution":["Cosine rule: QR2 = PQ2 + PR2 - 2 × PQ × PR × cos P.","QR2 = 9.42 + 13.72 - 2 × 9.4 × 13.7 × cos 118°.","cos 118° is negative, so the last term adds on: QR2 = 88.36 + 187.69 + 120.91... = 396.96...","QR = √396.96... = 19.924...","QR = 19.9 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"Q","style":"none","labelPos":"below-right"},{"at":[-4.105378985212787,7.721094907766361],"label":"R","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"9.4 cm"},{"from":[6,0],"to":[-4.105378985212787,7.721094907766361]},{"from":[-4.105378985212787,7.721094907766361],"to":[0,0],"label":"13.7 cm"}],"angles":[{"at":[0,0],"from":[6,0],"to":[-4.105378985212787,7.721094907766361],"label":"118°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"fe491deed2f0"},{"id":"g7-the-cosine-rule-q03","topicId":"g7-the-cosine-rule","topic":"The Cosine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 7 cm, BC = 9 cm and AC = 11 cm.

Find the size of the largest angle of the triangle. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"85.9°","answerDisplay":null,"accept":["85.9"],"parts":[],"solution":["The largest angle is opposite the longest side, AC = 11 cm, so it is angle ABC.","Cosine rule: cos B = AB2 + BC2 - AC22 × AB × BC.","cos B = 49 + 81 - 1212 × 7 × 9 = 9126 = 0.07142...","B = cos-1(0.07142...) = 85.903...°.","The largest angle is 85.9° (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[5.448979591836735,7.694581248607376],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"7 cm"},{"from":[6,0],"to":[5.448979591836735,7.694581248607376],"label":"9 cm"},{"from":[5.448979591836735,7.694581248607376],"to":[0,0],"label":"11 cm"}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"83f0abb7f63d"},{"id":"g7-the-cosine-rule-q04","topicId":"g7-the-cosine-rule","topic":"The Cosine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":4,"marks":4,"tier":"H","calc":false,"text":"

In triangle ABC, AB = 3 cm, AC = 5 cm and BC = √19 cm.

Find the size of angle BAC.

","answerType":"exact","answer":"60°","answerDisplay":null,"accept":["60"],"parts":[],"solution":["Angle BAC is opposite BC, so use the cosine rule for the angle.","cos A = AB2 + AC2 - BC22 × AB × AC.","cos A = 9 + 25 - 192 × 3 × 5 = 1530 = 12.","cos A = 12, so angle BAC = 60°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[4.999999999999999,8.660254037844387],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"3 cm"},{"from":[6,0],"to":[4.999999999999999,8.660254037844387],"label":"√19 cm"},{"from":[4.999999999999999,8.660254037844387],"to":[0,0],"label":"5 cm"}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"f82e862370fb"},{"id":"g7-the-cosine-rule-q05","topicId":"g7-the-cosine-rule","topic":"The Cosine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":5,"marks":5,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 8 cm, BC = 12 cm and AC = 15 cm.

  1. Work out the size of angle ABC. Give your answer correct to 1 decimal place.
  2. Work out the area of the triangle. Give your answer correct to 3 significant figures.
","answerType":"multi","answer":"a) 95.1°, b) 47.8 cm²","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"95.1°"},{"label":"b","answer":"47.8 cm²"}],"solution":["a) Angle ABC is opposite AC. Cosine rule: cos B = AB2 + BC2 - AC22 × AB × BC.","cos B = 64 + 144 - 2252 × 8 × 12 = -17192 = -0.08854...","B = cos-1(-0.08854...) = 95.079...° = 95.1° (1 dp).","b) Area = 12 × AB × BC × sin B = 12 × 8 × 12 × sin 95.079...°.","Area = 48 × 0.99607... = 47.811... = 47.8 cm2 (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6.796875,8.96465226511185],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"8 cm"},{"from":[6,0],"to":[6.796875,8.96465226511185],"label":"12 cm"},{"from":[6.796875,8.96465226511185],"to":[0,0],"label":"15 cm"}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a3b15e85e20c"},{"id":"g7-the-cosine-rule-q06","topicId":"g7-the-cosine-rule","topic":"The Cosine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":6,"marks":5,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 9 cm, AC = 12 cm and angle BAC = 48°.

Work out the size of angle ABC. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"83.8°","answerDisplay":null,"accept":["83.8"],"parts":[],"solution":["First find BC with the cosine rule: BC2 = 92 + 122 - 2 × 9 × 12 × cos 48°.","BC2 = 225 - 216 × 0.66913... = 80.467..., so BC = 8.9703...","Angle ABC is opposite AC. Use the cosine rule for the angle so there is no acute/obtuse ambiguity:","cos B = AB2 + BC2 - AC22 × AB × BC = 81 + 80.467... - 1442 × 9 × 8.9703... = 0.10818...","B = cos-1(0.10818...) = 83.789...°.","Angle ABC = 83.8° (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[5.353044850870866,5.945158603819154],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"9 cm"},{"from":[6,0],"to":[5.353044850870866,5.945158603819154]},{"from":[5.353044850870866,5.945158603819154],"to":[0,0],"label":"12 cm"}],"angles":[{"at":[0,0],"from":[6,0],"to":[5.353044850870866,5.945158603819154],"label":"48°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"c885d5fbc137"},{"id":"g7-the-cosine-rule-q07","topicId":"g7-the-cosine-rule","topic":"The Cosine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":7,"marks":5,"tier":"H","calc":true,"text":"

The diagram shows quadrilateral ABCD. The diagonal BD splits it into triangles ABD and BCD.

In triangle ABD, AB = 7 cm, AD = 10 cm and angle BAD = 65°.

In triangle BCD, angle DBC = 40° and angle BCD = 68°.

Work out the length of CD. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"6.57 cm","answerDisplay":null,"accept":["6.57"],"parts":[],"solution":["Cosine rule in triangle ABD: BD2 = 72 + 102 - 2 × 7 × 10 × cos 65°.","BD2 = 149 - 140 × 0.42261... = 89.833..., so BD = 9.4780...","Sine rule in triangle BCD: CDsin(DBC) = BDsin(BCD).","CD = 9.4780... × sin 40° ÷ sin 68°.","CD = 9.4780... × 0.64278... ÷ 0.92718... = 6.5708...","CD = 6.57 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,4],"label":"A","style":"none","labelPos":"above"},{"at":[-3,0],"label":"B","style":"none","labelPos":"below-left"},{"at":[0,-4],"label":"C","style":"none","labelPos":"below-left"},{"at":[5,0],"label":"D","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,4],"to":[-3,0],"label":"7 cm"},{"from":[-3,0],"to":[0,-4]},{"from":[0,-4],"to":[5,0]},{"from":[5,0],"to":[0,4],"label":"10 cm"},{"from":[-3,0],"to":[5,0]}],"angles":[{"at":[0,4],"from":[-3,0],"to":[5,0],"label":"65°"},{"at":[-3,0],"from":[5,0],"to":[0,-4],"label":"40°"},{"at":[0,-4],"from":[-3,0],"to":[5,0],"label":"68°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"bd4265cc0537"},{"id":"g7-the-cosine-rule-q08","topicId":"g7-the-cosine-rule","topic":"The Cosine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

Two ships leave a port at the same time.

Ship A sails 24 km on a bearing of 062°.

Ship B sails 31 km on a bearing of 148°.

Work out the distance between the two ships. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"37.9 km","answerDisplay":null,"accept":["37.9"],"parts":[],"solution":["The angle between the two courses at the port = 148° - 62° = 86°.","Cosine rule: d2 = 242 + 312 - 2 × 24 × 31 × cos 86°.","d2 = 576 + 961 - 1488 × 0.06975... = 1433.2...","d = √1433.2... = 37.857...","Distance = 37.9 km (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"above"},{"at":[3.531790371435708,1.8778862511435632],"label":"A","style":"none","labelPos":"above"},{"at":[2.7555801740126658,-4.409850100013416],"label":"B","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[3.531790371435708,1.8778862511435632],"label":"24 km","labelPos":"below-right"},{"from":[0,0],"to":[2.7555801740126658,-4.409850100013416],"label":"31 km"},{"from":[3.531790371435708,1.8778862511435632],"to":[2.7555801740126658,-4.409850100013416]}],"angles":[{"at":[0,0],"from":[0,3],"to":[3.531790371435708,1.8778862511435632],"label":"062°","r":0.7},{"at":[0,0],"from":[0,3],"to":[2.6495963211660243,-4.240240480782131],"label":"","r":1.2}],"north":[{"at":[0,0],"len":3}],"alt":"Question diagram. Use the labels and information in the question.","texts":[{"at":[1.4,-1.1],"text":"148°"}]},"draw":false,"revision":"577c453fa6a7"},{"id":"g7-the-sine-rule-q01","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":1,"marks":3,"tier":"H","calc":true,"text":"

In triangle ABC, angle BAC = 35°, angle ABC = 78° and BC = 9.4 cm.

Work out the length of AC. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"16.0 cm","answerDisplay":null,"accept":["16.0"],"parts":[],"solution":["AC is opposite angle B and BC is opposite angle A.","Sine rule: ACsin B = BCsin A.","AC = 9.4 × sin 78° ÷ sin 35°.","AC = 9.4 × 0.97814... ÷ 0.57357... = 16.030...","AC = 16.0 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[5.222687987481867,3.656965498552101],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0]},{"from":[6,0],"to":[5.222687987481867,3.656965498552101],"label":"9.4 cm"},{"from":[5.222687987481867,3.656965498552101],"to":[0,0]}],"angles":[{"at":[0,0],"from":[6,0],"to":[5.222687987481867,3.656965498552101],"label":"35°","right":false},{"at":[6,0],"from":[5.222687987481867,3.656965498552101],"to":[0,0],"label":"78°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"5d1cf5a51344"},{"id":"g7-the-sine-rule-q02","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":2,"marks":3,"tier":"H","calc":true,"text":"

In triangle ABC, angle BAC = 52°, angle ACB = 63° and AC = 11 cm.

Calculate the length of BC. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"9.56 cm","answerDisplay":null,"accept":["9.56"],"parts":[],"solution":["Angle ABC = 180° - 52° - 63° = 65°.","BC is opposite angle A and AC is opposite angle B.","Sine rule: BCsin A = ACsin B.","BC = 11 × sin 52° ÷ sin 65° = 11 × 0.78801... ÷ 0.90630...","BC = 9.564... = 9.56 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[3.7574054114208986,4.809259615105172],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0]},{"from":[6,0],"to":[3.7574054114208986,4.809259615105172]},{"from":[3.7574054114208986,4.809259615105172],"to":[0,0],"label":"11 cm"}],"angles":[{"at":[0,0],"from":[6,0],"to":[3.7574054114208986,4.809259615105172],"label":"52°","right":false},{"at":[3.7574054114208986,4.809259615105172],"from":[0,0],"to":[6,0],"label":"63°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"1ea38aa91663"},{"id":"g7-the-sine-rule-q03","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 12.5 cm, BC = 8.4 cm and angle BAC = 33°.

Angle ACB is acute.

Work out the size of angle ACB. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"54.1°","answerDisplay":null,"accept":["54.1"],"parts":[],"solution":["AB is opposite angle C and BC is opposite angle A.","Sine rule: sin CAB = sin ABC.","sin C = 12.5 × sin 33° ÷ 8.4 = 0.81047...","C = sin-1(0.81047...) = 54.142...°.","Angle ACB = 54.1° (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6.201014804894486,4.026986099828658],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"12.5 cm"},{"from":[6,0],"to":[6.201014804894486,4.026986099828658],"label":"8.4 cm"},{"from":[6.201014804894486,4.026986099828658],"to":[0,0]}],"angles":[{"at":[0,0],"from":[6,0],"to":[6.201014804894486,4.026986099828658],"label":"33°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"24d17d6ad8f3"},{"id":"g7-the-sine-rule-q04","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

In triangle PQR, QR = 14.2 cm, PR = 9.6 cm and angle QPR = 81°.

Calculate the size of angle PQR. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"41.9°","answerDisplay":null,"accept":["41.9"],"parts":[],"solution":["QR is opposite angle P and PR is opposite angle Q.","Sine rule: sin QPR = sin PQR.","sin Q = 9.6 × sin 81° ÷ 14.2 = 0.66773...","Q = sin-1(0.66773...) = 41.892...°.","Angle PQR = 41.9° (1 dp). The other angle with this sine is 180° - 41.892...° = 138.107...°, but 138.107...° + 81° exceeds 180°, so it cannot fit in this triangle."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"P","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"Q","style":"none","labelPos":"below-right"},{"at":[0.7463906950859497,4.7125253816382715],"label":"R","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0]},{"from":[6,0],"to":[0.7463906950859497,4.7125253816382715],"label":"14.2 cm"},{"from":[0.7463906950859497,4.7125253816382715],"to":[0,0],"label":"9.6 cm"}],"angles":[{"at":[0,0],"from":[6,0],"to":[0.7463906950859497,4.7125253816382715],"label":"81°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"cf86ba99c431"},{"id":"g7-the-sine-rule-q05","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

B is due east of A. AB = 250 m.

The bearing of a mast C from A is 040°.

The bearing of C from B is 300°.

Work out the distance AC. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"127 m","answerDisplay":null,"accept":["127"],"parts":[],"solution":["At A: the bearing of C is 040° and B is due east (bearing 090°), so angle CAB = 90° - 40° = 50°.","At B: A is due west (bearing 270°) and the bearing of C is 300°, so angle CBA = 300° - 270° = 30°.","Angle ACB = 180° - 50° - 30° = 100°.","Sine rule: ACsin(CBA) = ABsin(ACB), so AC = 250 × sin 30° ÷ sin 100°.","AC = 125 ÷ 0.98480... = 126.928... = 127 m (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[8,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[2.5,3],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[8,0],"label":"250 m"},{"from":[0,0],"to":[2.5,3]},{"from":[8,0],"to":[2.5,3]}],"angles":[{"at":[0,0],"from":[0,4],"to":[2.5,3],"label":"040°"},{"at":[8,0],"from":[8,4],"to":[2.5,3],"label":"300°","reflex":true,"r":0.8}],"north":[{"at":[0,0],"len":4},{"at":[8,0],"len":4}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"6f93757b55a9"},{"id":"g7-the-sine-rule-q06","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":false,"text":"

In triangle ABC, angle BAC = 30°, BC = 5 cm and AB = 8 cm.

Work out the exact value of sin C.

","answerType":"exact","answer":"4/5","answerDisplay":"45","accept":["0.8","8/10"],"parts":[],"solution":["BC is opposite angle A and AB is opposite angle C.","Sine rule: sin CAB = sin ABC.","sin C = 8 × sin 30° ÷ 5.","sin 30° = 12, so sin C = 8 × 12 ÷ 5 = 45."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[6.448557158514989,3.7230762113533165],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"8 cm"},{"from":[6,0],"to":[6.448557158514989,3.7230762113533165],"label":"5 cm"},{"from":[6.448557158514989,3.7230762113533165],"to":[0,0]}],"angles":[{"at":[0,0],"from":[6,0],"to":[6.448557158514989,3.7230762113533165],"label":"30°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"729d965f43c6"},{"id":"g7-the-sine-rule-q07","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

A triangular sail has angles of 71° and 44°.

The side opposite the 71° angle has length 6.4 m.

Work out the length of the side opposite the 44° angle. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"4.70 m","answerDisplay":null,"accept":["4.70"],"parts":[],"solution":["Sine rule: xsin 44° = 6.4sin 71°.","x = 6.4 × sin 44° ÷ sin 71°.","x = 6.4 × 0.69465... ÷ 0.94551... = 4.7019...","x = 4.70 m (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[1.497230721505912,4.348273747787889],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0]},{"from":[6,0],"to":[1.497230721505912,4.348273747787889],"label":"6.4 m"},{"from":[1.497230721505912,4.348273747787889],"to":[0,0]}],"angles":[{"at":[0,0],"from":[6,0],"to":[1.497230721505912,4.348273747787889],"label":"71°"},{"at":[6,0],"from":[0,0],"to":[1.497230721505912,4.348273747787889],"label":"44°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"779f5be15f17"},{"id":"g7-the-sine-rule-q08","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

In triangle ABC, angle BAC = 25°, BC = 7.2 cm and AB = 13.5 cm.

Angle ACB is obtuse.

Work out the size of angle ACB. Give your answer correct to 1 decimal place.

","answerType":"exact","answer":"127.6°","answerDisplay":null,"accept":["127.6"],"parts":[],"solution":["BC is opposite angle A and AB is opposite angle C.","Sine rule: sin C = 13.5 × sin 25° ÷ 7.2 = 0.79240...","sin-1(0.79240...) = 52.411...°, but angle ACB is obtuse.","Sine is also positive in the second quadrant: C = 180° - 52.411...° = 127.588...°.","Check: 25° + 127.6° = 152.6° < 180°, so this is a valid triangle.","Angle ACB = 127.6° (1 dp)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[3.159279517823571,1.4731962334133626],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"13.5 cm"},{"from":[6,0],"to":[3.159279517823571,1.4731962334133626],"label":"7.2 cm"},{"from":[3.159279517823571,1.4731962334133626],"to":[0,0]}],"angles":[{"at":[0,0],"from":[6,0],"to":[3.159279517823571,1.4731962334133626],"label":"25°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"06c20116c749"},{"id":"g7-the-sine-rule-q09","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

The diagram shows quadrilateral ABCD. The diagonal BD splits it into triangles ABD and BCD.

In triangle ABD, angle DAB = 58°, angle ABD = 47° and AD = 9.3 cm.

In triangle BCD, angle DBC = 39° and angle BCD = 64°.

Work out the length of CD. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"7.55 cm","answerDisplay":null,"accept":["7.55"],"parts":[],"solution":["In triangle ABD, side AD is opposite angle ABD = 47°. Side BD is opposite angle DAB = 58°. Always pair a side with its opposite angle.","Sine rule: BD/sin 58° = 9.3/sin 47°.","BD = 9.3 × sin 58° ÷ sin 47° = 10.7839028869... cm. Keep this full value for the next step.","In triangle BCD, CD is opposite 39° and BD is opposite 64°, so CD/sin 39° = BD/sin 64°.","CD = 10.7839028869... × sin 39° ÷ sin 64° = 7.5507064268... cm.","Therefore CD = 7.55 cm to 3 significant figures."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,4],"label":"A","style":"none","labelPos":"above"},{"at":[-3,0],"label":"B","style":"none","labelPos":"below-left"},{"at":[0,-4],"label":"C","style":"none","labelPos":"below-left"},{"at":[5,0],"label":"D","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,4],"to":[-3,0]},{"from":[-3,0],"to":[0,-4]},{"from":[0,-4],"to":[5,0]},{"from":[5,0],"to":[0,4],"label":"9.3 cm"},{"from":[-3,0],"to":[5,0]}],"angles":[{"at":[0,4],"from":[-3,0],"to":[5,0],"label":"58°"},{"at":[-3,0],"from":[5,0],"to":[0,-4],"label":"39°"},{"at":[0,-4],"from":[-3,0],"to":[5,0],"label":"64°"},{"at":[-3,0],"from":[0,4],"to":[5,0],"label":"47°"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"0489c95cb0ba"},{"id":"g7-the-sine-rule-q10","topicId":"g7-the-sine-rule","topic":"The Sine Rule","grade":"7","topicGrade":"7","strand":"G","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

In triangle ABC, AB = 12 cm, angle BAC = 40° and angle ABC = 75°.

Work out the perimeter of the triangle. Give your answer correct to 3 significant figures.

","answerType":"exact","answer":"33.3 cm","answerDisplay":null,"accept":["33.3"],"parts":[],"solution":["Angle ACB = 180° - 40° - 75° = 65°.","Sine rule: BCsin A = ABsin C, so BC = 12 × sin 40° ÷ sin 65° = 8.5108...","ACsin B = ABsin C, so AC = 12 × sin 75° ÷ sin 65° = 12.789...","Perimeter = 12 + 8.5108... + 12.789... = 33.300...","Perimeter = 33.3 cm (3 sf)."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"A","style":"none","labelPos":"below-left"},{"at":[6,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[4.898614724120819,4.110425808289372],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[6,0],"label":"12 cm"},{"from":[6,0],"to":[4.898614724120819,4.110425808289372]},{"from":[4.898614724120819,4.110425808289372],"to":[0,0]}],"angles":[{"at":[0,0],"from":[6,0],"to":[4.898614724120819,4.110425808289372],"label":"40°","right":false},{"at":[6,0],"from":[4.898614724120819,4.110425808289372],"to":[0,0],"label":"75°","right":false}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"f22ea4a3e25e"},{"id":"g7-trigonometric-and-exponential-graphs-q01","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":1,"marks":1,"tier":"H","calc":false,"text":"

Write down the coordinates of the point where the graph of y = 4x crosses the y-axis.

","answerType":"exact","answer":"(0, 1)","answerDisplay":null,"accept":["(0,1)"],"parts":[],"solution":["On the y-axis, x = 0.","4^0 = 1, so the graph crosses the y-axis at (0, 1)."],"diagram":null,"draw":false,"revision":"6dd724599df0"},{"id":"g7-trigonometric-and-exponential-graphs-q02","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":2,"marks":2,"tier":"H","calc":false,"text":"

The curve C has equation y = cos x° for 0 ≤ x ≤ 360.

Write down the coordinates of the minimum point of C.

","answerType":"exact","answer":"(180, -1)","answerDisplay":null,"accept":["(180,-1)"],"parts":[],"solution":["The cosine curve starts at (0, 1), falls to its lowest value of -1 at x = 180, and returns to 1 at x = 360.","So the minimum point is (180, -1)."],"diagram":null,"draw":false,"revision":"b55446bdf06e"},{"id":"g7-trigonometric-and-exponential-graphs-q03","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":3,"marks":2,"tier":"H","calc":true,"text":"

Complete the table of values for y = 3x

x−10123
y13

","answerType":"multi","answer":"x=-1: 1/3, x=2: 9, x=3: 27","answerDisplay":"x=-1: 13, x=2: 9, x=3: 27","accept":[],"parts":[{"label":"x = -1","answer":"1/3","answerDisplay":"13"},{"label":"x = 2","answer":"9"},{"label":"x = 3","answer":"27"}],"solution":["3^(-1) = 13.","3^2 = 9.","3^3 = 27."],"diagram":null,"draw":false,"revision":"d859cc92e4fd"},{"id":"g7-trigonometric-and-exponential-graphs-q04","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":4,"marks":2,"tier":"H","calc":true,"text":"

The graph of y = kx, where k is a positive constant, passes through the point (3, 125).

Find the value of k.

","answerType":"exact","answer":"k = 5","answerDisplay":null,"accept":["5"],"parts":[],"solution":["Substitute the point: k^3 = 125.","Take the cube root: k = 5."],"diagram":null,"draw":false,"revision":"077682ae886a"},{"id":"g7-trigonometric-and-exponential-graphs-q05","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

The curve y = abx, where b > 0, passes through the points (0, 7) and (2, 63).

Find the value of a and the value of b.

","answerType":"multi","answer":"a = 7, b = 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"7"},{"label":"b","answer":"3"}],"solution":["At (0, 7): a x b^0 = a = 7.","At (2, 63): 7b^2 = 63, so b^2 = 9.","Since b > 0, b = 3."],"diagram":null,"draw":false,"revision":"4a2be3918a6e"},{"id":"g7-trigonometric-and-exponential-graphs-q06","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":6,"marks":2,"tier":"H","calc":false,"text":"

Sketch the graph of y = tan x° for 0 ≤ x ≤ 360.

Show clearly where the graph crosses the x-axis and mark the values of x where the graph has asymptotes.

","answerType":"open","answer":"Correct tan shape: branches through (0,0), (180,0), (360,0) with vertical asymptotes at x = 90 and x = 270","answerDisplay":null,"accept":[],"parts":[],"solution":["The tangent graph crosses the x-axis at x = 0, 180 and 360.","It has vertical asymptotes at x = 90 and x = 270.","Each branch increases from bottom left to top right, repeating every 180.","A vertical asymptote is a line that the graph approaches while its values grow without bound. Here tan x = sin x ÷ cos x is undefined at 90° and 270° because cos x = 0. Do not join the branches across those lines."],"diagram":{"type":"axes","x":{"min":0,"max":360,"step":90,"minor":3,"label":"x (degrees)"},"y":{"min":-4,"max":4,"step":1},"curves":[{"fn":"tan(x*pi/180)","answer":true}],"lines":[{"x":90,"dashed":true,"answer":true},{"x":270,"dashed":true,"answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"91bd83fc31eb"},{"id":"g7-trigonometric-and-exponential-graphs-q07","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":7,"marks":2,"tier":"H","calc":true,"text":"

You are given that sin 40° = 0.643 correct to 3 decimal places.

Using the symmetry of the graph of y = sin x°, write down the two solutions of

sin x° = 0.643 for 0 ≤ x ≤ 360

Give the angle solutions to the nearest whole degree, using the supplied approximation.

","answerType":"exact","answer":"x = 40 and x = 140","answerDisplay":null,"accept":["40, 140","40 and 140"],"parts":[],"solution":["One solution is x = 40.","The sine graph is symmetric about x = 90, so the second solution is 180 - 40 = 140."],"diagram":{"type":"axes","x":{"min":0,"max":360,"step":90,"minor":3,"label":"x (degrees)"},"y":{"min":-1.5,"max":1.5,"step":1},"curves":[{"fn":"sin(x*pi/180)","answer":false}],"lines":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ff5ba727e869"},{"id":"g7-trigonometric-and-exponential-graphs-q08","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

You are given that cos 60° = 0.5

  1. Find all the solutions of cos x° = 0.5 for 0 ≤ x ≤ 360

  2. Find all the solutions of cos x° = −0.5 for 0 ≤ x ≤ 360

","answerType":"multi","answer":"a) 60 and 300 b) 120 and 240","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x = 60 and x = 300"},{"label":"b","answer":"x = 120 and x = 240"}],"solution":["a) One solution is x = 60. The cosine graph is symmetric about x = 180, so the other is 360 - 60 = 300.","b) cos x = -0.5 when x = 180 - 60 = 120 and x = 180 + 60 = 240."],"diagram":null,"draw":false,"revision":"3ba338dc58c1"},{"id":"g7-trigonometric-and-exponential-graphs-q09","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

  1. Sketch the graph of y = sin x° for 0 ≤ x ≤ 360, marking the values where the graph meets the x-axis.

  2. You are given that sin 35° = 0.574 correct to 3 decimal places.

    Find all the solutions of sin x° = −0.574 for 0 ≤ x ≤ 360

Give the angle solutions to the nearest whole degree, using the supplied approximation.

","answerType":"multi","answer":"a) correct sine wave through (0,0), (180,0), (360,0), max (90,1), min (270,-1) b) x = 215 and x = 325","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"Sine wave through (0,0), (180,0), (360,0) with maximum (90, 1) and minimum (270, -1)"},{"label":"b","answer":"x = 215 and x = 325"}],"solution":["a) The sine curve passes through (0, 0), rises to (90, 1), falls through (180, 0) to (270, -1), then returns to (360, 0).","b) sin x is negative between 180 and 360.","b) The solutions are x = 180 + 35 = 215 and x = 360 - 35 = 325."],"diagram":{"type":"axes","x":{"min":0,"max":360,"step":90,"minor":3,"label":"x (degrees)"},"y":{"min":-1.5,"max":1.5,"step":1},"curves":[{"fn":"sin(x*pi/180)","answer":true}],"lines":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"57873959a0ed"},{"id":"g7-trigonometric-and-exponential-graphs-q10","topicId":"g7-trigonometric-and-exponential-graphs","topic":"Trigonometric and Exponential Graphs","grade":"7","topicGrade":"7","strand":"A","difficulty":10,"marks":4,"tier":"H","calc":false,"text":"

f(x) = cos x° for all real values of x

  1. Write down the coordinates of the minimum point of the graph of y = f(x) + 2

  2. Write down the coordinates of the maximum point of the graph of y = f(x − 30) for 0 ≤ x ≤ 360

  3. The graph of y = f(x) is reflected in the x-axis. Write down the equation of the reflected graph.

","answerType":"multi","answer":"a) (180, 1) b) (30, 1) c) y = -cos x","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(180, 1)"},{"label":"b","answer":"(30, 1)"},{"label":"c","answer":"y = -cos x"}],"solution":["a) y = f(x) + 2 translates the curve up 2, so the minimum (180, -1) moves to (180, 1).","b) y = f(x - 30) translates the curve 30 to the right, so the maximum (0, 1) moves to (30, 1).","c) Reflecting in the x-axis replaces y with -y, giving y = -cos x."],"diagram":null,"draw":false,"revision":"9c619a6f4423"},{"id":"g89-completing-the-square-q01","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":1,"marks":1,"tier":"H","calc":true,"text":"

The curve C has equation y = (x − 3)2 + 5

Write down the coordinates of the turning point of C.

","answerType":"exact","answer":"(3, 5)","answerDisplay":null,"accept":["(3,5)"],"parts":[],"solution":["A squared real number cannot be negative. (x - 3)^2 is smallest when x - 3 = 0, so x = 3.","At x = 3, y = 0 + 5 = 5. The curve therefore has its minimum (turning point) at (3, 5)."],"diagram":null,"draw":false,"revision":"d8e6e5b9806f"},{"id":"g89-completing-the-square-q02","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":2,"marks":2,"tier":"H","calc":false,"text":"

Write x2 + 8x + 3 in the form (x + a)2 + b, where a and b are integers.

","answerType":"exact","answer":"(x + 4)^2 - 13","answerDisplay":null,"accept":["(x+4)^2-13"],"parts":[],"solution":["Expanding (x + a)^2 gives x^2 + 2ax + a^2. To match 8x, we need 2a = 8, so a = 4.","Halve the coefficient of x: (x + 4)^2 = x^2 + 8x + 16.","Adjust the constant: x^2 + 8x + 3 = (x + 4)^2 - 16 + 3 = (x + 4)^2 - 13.","Check by expanding: x^2 + 8x + 16 - 13 = x^2 + 8x + 3."],"diagram":null,"draw":false,"revision":"9de69987097d"},{"id":"g89-completing-the-square-q03","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

  1. Write x2 − 6x + 11 in the form (x − a)2 + b, where a and b are integers.

  2. Hence find the coordinates of the turning point of the curve y = x2 − 6x + 11

","answerType":"multi","answer":"a) (x - 3)^2 + 2 b) (3, 2)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(x - 3)^2 + 2"},{"label":"b","answer":"(3, 2)"}],"solution":["a) (x - 3)^2 = x^2 - 6x + 9, so x^2 - 6x + 11 = (x - 3)^2 + 2.","a) Check by expanding: x^2 - 6x + 9 + 2 = x^2 - 6x + 11.","b) The minimum of (x - 3)^2 + 2 is when x = 3, giving the turning point (3, 2)."],"diagram":null,"draw":false,"revision":"bf0f8d641563"},{"id":"g89-completing-the-square-q04","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

By completing the square, find the coordinates of the minimum point of the curve

y = x2 + 10x − 2

","answerType":"exact","answer":"(-5, -27)","answerDisplay":null,"accept":["(-5,-27)"],"parts":[],"solution":["Complete the square: x^2 + 10x - 2 = (x + 5)^2 - 25 - 2 = (x + 5)^2 - 27.","Check by expanding: x^2 + 10x + 25 - 27 = x^2 + 10x - 2.","The minimum is when x = -5, so the minimum point is (-5, -27)."],"diagram":null,"draw":false,"revision":"4a97a0cfe5eb"},{"id":"g89-completing-the-square-q05","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

By completing the square, solve

x2 + 6x + 4 = 0

Give your answers in the form a ± √b where a and b are integers.

","answerType":"exact","answer":"x = -3 + sqrt(5) or x = -3 - sqrt(5)","answerDisplay":null,"accept":["-3 ± √5","x = -3 ± √5"],"parts":[],"solution":["Complete the square: x^2 + 6x + 4 = (x + 3)^2 - 9 + 4 = (x + 3)^2 - 5.","So (x + 3)^2 = 5.","Both √5 and -√5 square to 5, so x + 3 = ±√5. The ± sign represents these two possibilities.","x = -3 ± √5.","Check by expanding: (x + 3)^2 - 5 = x^2 + 6x + 9 - 5 = x^2 + 6x + 4."],"diagram":null,"draw":false,"revision":"b0cba4be2464"},{"id":"g89-completing-the-square-q06","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

  1. Write 2x2 + 12x + 5 in the form a(x + b)2 + c, where a, b and c are integers.

  2. Hence write down the coordinates of the turning point of the curve y = 2x2 + 12x + 5

","answerType":"multi","answer":"a) 2(x + 3)^2 - 13 b) (-3, -13)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2(x + 3)^2 - 13"},{"label":"b","answer":"(-3, -13)"}],"solution":["a) Take out the factor 2 from the x terms: 2(x^2 + 6x) + 5.","a) Complete the square inside: x^2 + 6x = (x + 3)^2 - 9.","a) So 2x^2 + 12x + 5 = 2(x + 3)^2 - 18 + 5 = 2(x + 3)^2 - 13.","a) Check by expanding: 2(x^2 + 6x + 9) - 13 = 2x^2 + 12x + 5.","b) A square is at least zero, and its coefficient 2 is positive. The smallest value is therefore -13, when x + 3 = 0: the turning point is (-3, -13)."],"diagram":null,"draw":false,"revision":"57b551ee7745"},{"id":"g89-completing-the-square-q07","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":false,"text":"

The curve y = x2 + bx + c passes through the points (0, 7) and (1, 4).

Find the coordinates of the turning point of the curve.

","answerType":"exact","answer":"(2, 3)","answerDisplay":null,"accept":["(2,3)"],"parts":[],"solution":["At (0, 7): c = 7.","At (1, 4): 1 + b + 7 = 4, so b = -4.","The curve is y = x^2 - 4x + 7.","Complete the square: x^2 - 4x + 7 = (x - 2)^2 + 3.","Check by expanding: x^2 - 4x + 4 + 3 = x^2 - 4x + 7.","The turning point is (2, 3)."],"diagram":null,"draw":false,"revision":"760b9b5ea017"},{"id":"g89-completing-the-square-q08","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

The curve C has equation y = x2 − 4x − 5

  1. Write x2 − 4x − 5 in the form (x − a)2 + b

  2. Sketch C, showing clearly the coordinates of the turning point and the coordinates of the points where C crosses the axes.

","answerType":"multi","answer":"a) (x - 2)^2 - 9 b) parabola with minimum (2, -9), x-intercepts (-1, 0) and (5, 0), y-intercept (0, -5)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(x - 2)^2 - 9"},{"label":"b","answer":"U-shaped parabola, turning point (2, -9), crossing the x-axis at (-1, 0) and (5, 0) and the y-axis at (0, -5)"}],"solution":["a) x^2 - 4x - 5 = (x - 2)^2 - 4 - 5 = (x - 2)^2 - 9.","a) Check by expanding: x^2 - 4x + 4 - 9 = x^2 - 4x - 5.","b) The turning point is (2, -9).","b) For the x-intercepts, factorise: x^2 - 4x - 5 = (x - 5)(x + 1), so the curve crosses at (5, 0) and (-1, 0).","b) When x = 0, y = -5, so the y-intercept is (0, -5).","b) Draw a U-shaped parabola through these points with its minimum at (2, -9)."],"diagram":{"type":"axes","x":{"min":-3,"max":7,"step":1},"y":{"min":-12,"max":12,"step":3},"curves":[{"fn":"x*x-4*x-5","answer":true}],"points":[{"at":[-1,0],"label":"(-1, 0)","answer":true},{"at":[5,0],"label":"(5, 0)","answer":true},{"at":[0,-5],"label":"(0, -5)","answer":true},{"at":[2,-9],"label":"(2, -9)","answer":true}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"1f801aae7d2f"},{"id":"g89-completing-the-square-q09","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"

  1. Show that x2 + 5x + 9 can be written as (x + 52)2 + 114

  2. Write down the minimum value of x2 + 5x + 9

  3. Hence explain why the equation x2 + 5x + 9 = 0 has no real solutions.

","answerType":"multi","answer":"a) shown b) 11/4 c) the expression is always at least 11/4 > 0, so it can never equal 0","answerDisplay":"a) shown b) 114 c) the expression is always at least 114 > 0, so it can never equal 0","accept":[],"parts":[{"label":"a","answer":"(x + 5/2)^2 = x^2 + 5x + 25/4, so x^2 + 5x + 9 = (x + 5/2)^2 + 11/4","answerDisplay":"(x + 52)2 = x2 + 5x + 254, so x2 + 5x + 9 = (x + 52)2 + 114"},{"label":"b","answer":"11/4","answerDisplay":"114"},{"label":"c","answer":"Since (x + 5/2)^2 >= 0, the expression is always >= 11/4, which is positive, so it is never 0","answerDisplay":"Since (x + 52)2 >= 0, the expression is always >= 114, which is positive, so it is never 0"}],"solution":["a) Halve the coefficient of x: (x + 52)^2 = x^2 + 5x + 254.","a) So x^2 + 5x + 9 = (x + 52)^2 - 254 + 9 = (x + 52)^2 + 114, since 9 = 364.","b) The square term is at least 0, so the minimum value is 114, when x = -52.","c) The expression is always at least 114, which is greater than 0, so it can never equal 0 and the equation has no real solutions."],"diagram":null,"draw":false,"proofSupport":{"plan":["Expand the proposed square and compare coefficients.","Use the smallest possible value of a square to find the minimum.","Explain what that minimum tells you about real roots."],"checks":["(x + 5/2)² = x² + 5x + 25/4 — Expand the proposed square","x² + 5x + 9 = (x + 5/2)² + 11/4 — 9 − 25/4 = 11/4","Minimum = 11/4 at x = −5/2 — The square term is at least zero","11/4 > 0: no real roots — The expression never reaches zero"],"visuals":[{"caption":"The completed square gives the minimum. The curve never meets the x-axis.","diagram":{"type":"axes","x":{"min":-6,"max":1,"step":1},"y":{"min":0,"max":16,"step":2},"curves":[{"fn":"x*x+5*x+9","from":-6,"to":1}],"points":[{"at":[-2.5,2.75],"label":"Minimum (−5/2, 11/4)","labelPos":"above","answer":true}],"alt":"The quadratic has its minimum above the x-axis, at (−5/2, 11/4)."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"(x + 5/2)² = x² + 5x + 25/4","reason":"Expand the proposed square"},{"math":"x² + 5x + 9 = (x + 5/2)² + 11/4","reason":"9 − 25/4 = 11/4"},{"math":"Minimum = 11/4 at x = −5/2","reason":"The square term is at least zero"},{"math":"11/4 > 0: no real roots","reason":"The expression never reaches zero"}],"alt":"(x + 5/2)² = x² + 5x + 25/4: Expand the proposed square. x² + 5x + 9 = (x + 5/2)² + 11/4: 9 − 25/4 = 11/4. Minimum = 11/4 at x = −5/2: The square term is at least zero. 11/4 > 0: no real roots: The expression never reaches zero"}}]},"revision":"7d127feb0050"},{"id":"g89-completing-the-square-q10","topicId":"g89-completing-the-square","topic":"Completing the Square","grade":"89","topicGrade":"89","strand":"A","difficulty":10,"marks":7,"tier":"H","calc":true,"text":"

f(x) = 2x2 − 8x + 3

  1. Write f(x) in the form a(x − b)2 + c, where a, b and c are integers.

  2. Hence solve f(x) = 0, giving your answers in the form p ± √q2

  3. Priya says that the minimum point of the curve y = f(x) is (−2, −5).

    Explain the mistake Priya has made and write down the correct minimum point.

","answerType":"multi","answer":"a) 2(x - 2)^2 - 5 b) x = 2 ± (√10)/2 c) the x-coordinate is +2 not -2; minimum point (2, -5)","answerDisplay":"a) 2(x - 2)2 - 5 b) x = 2 ± √102 c) the x-coordinate is +2 not -2; minimum point (2, -5)","accept":[],"parts":[{"label":"a","answer":"2(x - 2)^2 - 5"},{"label":"b","answer":"x = 2 + (√10)/2 or x = 2 - (√10)/2","answerDisplay":"x = 2 + √102 or x = 2 - √102"},{"label":"c","answer":"The square is zero at x = 2 and its coefficient is positive, so Priya has the wrong sign; the minimum point is (2, -5)"}],"solution":["a) Take out the factor 2 from the x terms: 2(x^2 - 4x) + 3.","a) Complete the square: x^2 - 4x = (x - 2)^2 - 4.","a) So f(x) = 2(x - 2)^2 - 8 + 3 = 2(x - 2)^2 - 5.","a) Check by expanding: 2(x^2 - 4x + 4) - 5 = 2x^2 - 8x + 3.","b) Solve 2(x - 2)^2 - 5 = 0: (x - 2)^2 = 52.","b) x - 2 = ±√(52) = ±√102, so x = 2 ± √102.","b) Check with the formula: x = 8 ± √(64 - 24)4 = 8 ± √404 = 2 ± √102.","c) Since 2 is positive and (x - 2)^2 is at least zero, the minimum occurs when x - 2 = 0. This means x = 2, not -2. Priya copied the sign inside the bracket instead of finding the x-value that makes it zero.","c) The correct minimum point is (2, -5)."],"diagram":null,"draw":false,"revision":"68436b202746"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q01","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":1,"marks":1,"tier":"H","calc":false,"text":"

A circle has equation x2 + y2 = 81

Write down the radius of the circle.

","answerType":"exact","answer":"9","answerDisplay":null,"accept":["r = 9","9 units"],"parts":[],"solution":["The circle x^2 + y^2 = r^2 has centre the origin and radius r.","Here r^2 = 81, so r = 9."],"diagram":null,"draw":false,"revision":"8cd5983c9b73"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q02","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

The circle C has equation x2 + y2 = 61

  1. Show that the point A (5, −6) lies on C.
  2. Does the point B (4, 7) lie on C? You must justify your answer.

","answerType":"multi","answer":"a) 5^2 + (-6)^2 = 61 so A lies on C; b) no, 4^2 + 7^2 = 65, not 61","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"25 + 36 = 61, so A lies on the circle"},{"label":"b","answer":"No: 16 + 49 = 65, which is not 61, so B does not lie on the circle"}],"solution":["a) Substitute A into the left-hand side: 5^2 + (-6)^2 = 25 + 36 = 61. This equals 61, so A lies on C.","b) Substitute B: 4^2 + 7^2 = 16 + 49 = 65. Since 65 is not equal to 61, B does not lie on C. In fact 65 > 61, so B is outside the circle."],"diagram":null,"draw":false,"proofSupport":{"plan":["Substitute each point’s coordinates into x² + y².","Compare each total with the right-hand side of the circle equation.","State and justify whether each point lies on the circle."],"checks":["A: 5² + (−6)² = 61 — Substitute both coordinates into the circle equation","A lies on x² + y² = 61 — The equation is satisfied","B: 4² + 7² = 65 > 61 — B does not lie on the circle; it is outside"],"visuals":[{"caption":"Substitution is the proof; the positions illustrate why A is on the circle and B is outside.","diagram":{"type":"axes","square":true,"x":{"min":-9,"max":9,"step":3},"y":{"min":-9,"max":9,"step":3},"curves":[{"pts":[[7.810249675906654,0],[7.7805293179109825,0.6807081115483009],[7.6915944337939575,1.3562356233449238],[7.544121871724062,2.021441363130432],[7.339233986945016,2.6712627135628515],[7.078490099974672,3.3007541417925315],[6.7638746292343415,3.9051248379533265],[6.397781988426371,4.4797751761184506],[5.982998363600092,5.020331720231106],[5.522680508593631,5.52268050859363],[5.020331720231107,5.982998363600092],[4.479775176118451,6.39778198842637],[3.905124837953328,6.763874629234341],[3.3007541417925315,7.078490099974672],[2.6712627135628524,7.339233986945015],[2.021441363130434,7.5441218717240615],[1.3562356233449244,7.6915944337939575],[0.6807081115483007,7.7805293179109825],[4.782398633070368e-16,7.810249675906654],[-0.6807081115482997,7.7805293179109825],[-1.3562356233449235,7.6915944337939575],[-2.021441363130433,7.544121871724062],[-2.6712627135628497,7.339233986945016],[-3.3007541417925306,7.0784900999746725],[-3.905124837953325,6.7638746292343415],[-4.479775176118451,6.39778198842637],[-5.020331720231107,5.982998363600092],[-5.52268050859363,5.522680508593631],[-5.982998363600091,5.020331720231107],[-6.397781988426372,4.47977517611845],[-6.76387462923434,3.9051248379533297],[-7.078490099974672,3.300754141792532],[-7.339233986945015,2.671262713562853],[-7.544121871724062,2.021441363130431],[-7.6915944337939575,1.3562356233449233],[-7.7805293179109825,0.6807081115483011],[-7.810249675906654,9.564797266140737e-16],[-7.7805293179109825,-0.6807081115482991],[-7.691594433793958,-1.3562356233449213],[-7.544121871724063,-2.021441363130429],[-7.339233986945016,-2.671262713562851],[-7.078490099974674,-3.300754141792527],[-6.763874629234341,-3.905124837953328],[-6.397781988426373,-4.479775176118449],[-5.982998363600094,-5.020331720231104],[-5.5226805085936315,-5.52268050859363],[-5.020331720231107,-5.982998363600091],[-4.479775176118448,-6.397781988426374],[-3.9051248379533305,-6.76387462923434],[-3.300754141792535,-7.07849009997467],[-2.67126271356285,-7.339233986945016],[-2.0214413631304313,-7.544121871724062],[-1.3562356233449238,-7.6915944337939575],[-0.6807081115483016,-7.7805293179109825],[-1.4347195899211104e-15,-7.810249675906654],[0.6807081115482987,-7.7805293179109825],[1.3562356233449209,-7.691594433793958],[2.0214413631304287,-7.544121871724063],[2.6712627135628537,-7.339233986945015],[3.3007541417925266,-7.078490099974674],[3.9051248379533217,-6.763874629234344],[4.4797751761184506,-6.397781988426371],[5.020331720231106,-5.9829983636000925],[5.522680508593629,-5.5226805085936315],[5.98299836360009,-5.020331720231108],[6.397781988426369,-4.479775176118454],[6.763874629234342,-3.9051248379533243],[7.07849009997467,-3.3007541417925355],[7.339233986945016,-2.6712627135628506],[7.544121871724062,-2.0214413631304318],[7.6915944337939575,-1.3562356233449242],[7.7805293179109825,-0.6807081115483021],[7.810249675906654,-1.9129594532281473e-15]]}],"points":[{"at":[5,-6],"label":"A (5, −6)","answer":true},{"at":[4,7],"label":"B (4, 7)","answer":true}],"alt":"A lies on the circle of radius square root of 61; B is outside."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"A: 5² + (−6)² = 61","reason":"Substitute both coordinates into the circle equation"},{"math":"A lies on x² + y² = 61","reason":"The equation is satisfied"},{"math":"B: 4² + 7² = 65 > 61","reason":"B does not lie on the circle; it is outside"}],"alt":"A: 5² + (−6)² = 61: Substitute both coordinates into the circle equation. A lies on x² + y² = 61: The equation is satisfied. B: 4² + 7² = 65 > 61: B does not lie on the circle; it is outside"}}]},"revision":"f3aba10b0261"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q03","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

The line L is perpendicular to the line with equation y = 2x − 3

L passes through the point (4, 1).

Find an equation of L.

","answerType":"exact","answer":"y = -x/2 + 3","answerDisplay":"y = -x2 + 3","accept":["y = -0.5x + 3","x + 2y = 6","2y = -x + 6","y = 3 - x/2"],"parts":[],"solution":["The gradient of y = 2x - 3 is 2, so a perpendicular line has gradient -12 (the negative reciprocal, since 2 times -12 = -1).","A line with gradient m through (a, b) satisfies y - b = m(x - a): vertical change equals gradient × horizontal change. Here (a, b) = (4, 1).","Using y - 1 = -12 (x - 4):","y - 1 = -x2 + 2, so y = -x2 + 3.","Check: at x = 4, y = -2 + 3 = 1, so the line passes through (4, 1)."],"diagram":null,"draw":false,"revision":"420b98566ee8"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q04","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":4,"marks":4,"tier":"H","calc":false,"text":"

The circle C has equation x2 + y2 = 25

The point P (3, 4) lies on C.

Find an equation of the tangent to C at the point P.

","answerType":"exact","answer":"y = -3x/4 + 25/4","answerDisplay":"y = -3x4 + 254","accept":["3x + 4y = 25","y = -0.75x + 6.25","4y = -3x + 25","y - 4 = -3/4(x - 3)"],"parts":[],"solution":["Check P lies on C: 3^2 + 4^2 = 9 + 16 = 25.","O is the origin (0, 0). The radius gradient is the change in y divided by the change in x from O to P.","The gradient of the radius OP is 43.","The tangent is perpendicular to the radius at P, so its gradient is -34 (since 43 times -34 = -1).","Using y - 4 = -34 (x - 3):","y = -3x4 + 94 + 4 = -3x4 + 254, which can be written 3x + 4y = 25."],"diagram":{"type":"shape","notAccurate":true,"circles":[{"c":[0,0],"r":3}],"segments":[{"from":[-5,0],"to":[6,0],"arrow":"end"},{"from":[0,-4],"to":[0,5],"arrow":"end"},{"from":[0,0],"to":[1.8,2.4],"answer":true,"label":"radius"},{"from":[4.2,0.5999999999999999],"to":[-0.5999999999999999,4.2],"answer":true,"label":"tangent"}],"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[1.8,2.4],"label":"P (3, 4)","style":"dot","labelPos":"above-right"}],"texts":[{"at":[6,-0.5],"text":"x"},{"at":[0.5,5],"text":"y"}],"alt":"Question diagram. Use the labels and information in the question.","angles":[{"at":[1.8,2.4],"from":[0,0],"to":[-0.5999999999999999,4.2],"right":true,"answer":true}]},"draw":false,"revision":"52d0f904f594"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q05","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

The circle C has equation x2 + y2 = 29

The point P (−2, 5) lies on C.

Find an equation of the tangent to C at the point P.

","answerType":"exact","answer":"y = 2x/5 + 29/5","answerDisplay":"y = 2x5 + 295","accept":["5y = 2x + 29","y = 0.4x + 5.8","y - 5 = 2/5(x + 2)","-2x + 5y = 29"],"parts":[],"solution":["Check P lies on C: (-2)^2 + 5^2 = 4 + 25 = 29.","O is the origin (0, 0). The radius gradient is the change in y divided by the change in x from O to P.","The gradient of the radius OP is 5-2 = -52.","The tangent is perpendicular to the radius, so its gradient is 25 (since -52 times 25 = -1).","Using y - 5 = 25 (x + 2):","y = 2x5 + 45 + 5 = 2x5 + 295, which can be written 5y = 2x + 29."],"diagram":{"type":"shape","notAccurate":true,"circles":[{"c":[0,0],"r":3}],"segments":[{"from":[-5,0],"to":[6,0],"arrow":"end"},{"from":[0,-4],"to":[0,5],"arrow":"end"},{"from":[0,0],"to":[-1.1141720290623112,2.785430072655778],"answer":true,"label":"radius"},{"from":[1.671258043593467,3.8996021017180893],"to":[-3.8996021017180893,1.671258043593467],"answer":true,"label":"tangent"}],"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[-1.1141720290623112,2.785430072655778],"label":"P (-2, 5)","style":"dot","labelPos":"above-right"}],"texts":[{"at":[6,-0.5],"text":"x"},{"at":[0.5,5],"text":"y"}],"alt":"Question diagram. Use the labels and information in the question.","angles":[{"at":[-1.1141720290623112,2.785430072655778],"from":[0,0],"to":[-3.8996021017180893,1.671258043593467],"right":true,"answer":true}]},"draw":false,"revision":"35b819d01cf5"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q06","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A circle has centre O, the origin, and passes through the point T (5, 12).

  1. Find an equation of the circle.
  2. The point D has coordinates (10, 9). Decide whether D is inside the circle, on the circle, or outside the circle. You must show your working.

","answerType":"multi","answer":"a) x^2 + y^2 = 169; b) outside, since 10^2 + 9^2 = 181 > 169","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"x^2 + y^2 = 169"},{"label":"b","answer":"outside the circle, because 100 + 81 = 181 > 169"}],"solution":["a) The radius squared is OT^2 = 5^2 + 12^2 = 25 + 144 = 169 (radius 13).","So the circle has equation x^2 + y^2 = 169.","b) For D: 10^2 + 9^2 = 100 + 81 = 181.","181 > 169, so D is further from O than the radius: D is outside the circle."],"diagram":null,"draw":false,"revision":"7a9ada4b7cd8"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q07","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

The circle C has centre O, the origin, and equation x2 + y2 = 100

The tangent to C at the point P (6, 8) crosses the x-axis at the point A.

Find the exact coordinates of A.

","answerType":"exact","answer":"(50/3, 0)","answerDisplay":"(503, 0)","accept":["(16 2/3, 0)"],"parts":[],"solution":["Check P lies on C: 6^2 + 8^2 = 36 + 64 = 100.","The gradient of the radius OP is 86 = 43, so the tangent at P has gradient -34.","Tangent: y - 8 = -34 (x - 6), so y = -3x4 + 92 + 8 = -3x4 + 252.","At A, y = 0: 3x4 = 252, so x = 503.","A is the point (503, 0)."],"diagram":{"type":"shape","notAccurate":true,"circles":[{"c":[0,0],"r":3}],"segments":[{"from":[-5,0],"to":[6,0],"arrow":"end"},{"from":[0,-4],"to":[0,5],"arrow":"end"},{"from":[5,0],"to":[-0.5999999999999999,4.2]}],"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[1.8,2.4],"label":"P (6, 8)","style":"dot","labelPos":"above-right"},{"at":[5,0],"label":"A","style":"none","labelPos":"below-right"}],"texts":[{"at":[6,-0.5],"text":"x"},{"at":[0.5,5],"text":"y"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"439af5830e87"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q08","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":true,"text":"

The circle C has centre O, the origin, and equation x2 + y2 = 45

A tangent to C touches the circle at the point P.

The tangent has gradient −12 and P is in the first quadrant.

Find the coordinates of P.

","answerType":"exact","answer":"(3, 6)","answerDisplay":null,"accept":["P(3, 6)","x = 3, y = 6"],"parts":[],"solution":["The radius OP is perpendicular to the tangent, so the gradient of OP is 2 (the negative reciprocal of -12).","P lies on the line y = 2x through the origin.","Substitute into the circle equation: x^2 + (2x)^2 = 45, so 5x^2 = 45 and x^2 = 9.","x = 3 or x = -3. P is in the first quadrant, so x = 3 and y = 6.","Check: 3^2 + 6^2 = 9 + 36 = 45, so P (3, 6) lies on C."],"diagram":{"type":"shape","notAccurate":true,"circles":[{"c":[0,0],"r":3}],"segments":[{"from":[-5,0],"to":[6,0],"arrow":"end"},{"from":[0,-4],"to":[0,5],"arrow":"end"},{"from":[4.024922359499621,1.3416407864998738],"to":[-1.3416407864998738,4.024922359499621]}],"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[1.3416407864998738,2.6832815729997477],"label":"P","style":"dot","labelPos":"above-right"}],"texts":[{"at":[6,-0.5],"text":"x"},{"at":[0.5,5],"text":"y"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"f1bf828363c8"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q09","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

The line with equation y = x + 2 intersects the circle with equation x2 + y2 = 34 at the points A and B.

Find the coordinates of A and B.

","answerType":"exact","answer":"(-5, -3) and (3, 5)","answerDisplay":null,"accept":["(3, 5) and (-5, -3)","A(-5,-3), B(3,5)"],"parts":[],"solution":["Substitute y = x + 2 into the circle equation: x^2 + (x + 2)^2 = 34.","Expand: x^2 + x^2 + 4x + 4 = 34, so 2x^2 + 4x - 30 = 0.","Divide by 2: x^2 + 2x - 15 = 0, so (x + 5)(x - 3) = 0.","x = -5 gives y = -3; x = 3 gives y = 5.","Check both points: (-5)^2 + (-3)^2 = 25 + 9 = 34 and 3^2 + 5^2 = 9 + 25 = 34.","The points are (-5, -3) and (3, 5)."],"diagram":null,"draw":false,"revision":"b514263e4915"},{"id":"g89-equation-of-a-circle-perpendicular-lines-and-tangents-q10","topicId":"g89-equation-of-a-circle-perpendicular-lines-and-tangents","topic":"Equation of a Circle, Perpendicular Lines and Tangents","grade":"89","topicGrade":"89","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

The line with equation 3x + 4y = 30 is a tangent to a circle with centre O, the origin.

Find an equation of the circle.

","answerType":"exact","answer":"x^2 + y^2 = 36","answerDisplay":null,"accept":["x2+y2=36","x^2+y^2=36"],"parts":[],"solution":["The radius to the point of contact is perpendicular to the tangent.","The tangent 3x + 4y = 30 has gradient -34, so the radius lies on the line y = 4x3 through O.","Substitute y = 4x3 into 3x + 4y = 30: 3x + 16x3 = 30, so 25x3 = 30 and x = 185.","Then y = 43 times 185 = 245. The point of contact is (185, 245).","r^2 = (185)^2 + (245)^2 = 32425 + 57625 = 90025 = 36.","So the circle has equation x^2 + y^2 = 36 (radius 6, which matches the distance 305 = 6 from O to the line)."],"diagram":{"type":"shape","notAccurate":true,"circles":[{"c":[0,0],"r":3}],"segments":[{"from":[-5,0],"to":[6,0],"arrow":"end"},{"from":[0,-4],"to":[0,5],"arrow":"end"},{"from":[4.2,0.5999999999999999],"to":[-0.5999999999999999,4.2],"label":"3x + 4y = 30"}],"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[1.8,2.4],"label":"","style":"dot","labelPos":"above-right"}],"texts":[{"at":[6,-0.5],"text":"x"},{"at":[0.5,5],"text":"y"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"c331e840f71c"},{"id":"g89-probability-equation-questions-q01","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":1,"marks":2,"tier":"H","calc":false,"text":"

A bag contains n counters. 3 of the counters are red.

A counter is taken at random, replaced, and then a second counter is taken at random.

Write down an expression, in terms of n, for the probability that both counters are red.

","answerType":"exact","answer":"9/n^2","answerDisplay":"9n2","accept":["9/n²","(3/n)^2","(3/n)(3/n)"],"parts":[],"solution":["Each pick has P(red) = 3n because the counter is replaced.","P(both red) = 3n × 3n = 9n2"],"diagram":null,"draw":false,"revision":"a5c1ee402914"},{"id":"g89-probability-equation-questions-q02","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":2,"marks":2,"tier":"H","calc":false,"text":"

A box contains n counters. 5 of the counters are green.

A counter is taken at random and not replaced. A second counter is then taken at random.

Write down an expression, in terms of n, for the probability that both counters are green.

","answerType":"exact","answer":"20/(n(n-1))","answerDisplay":"20n(n-1)","accept":["20/(n^2-n)","(5/n)(4/(n-1))"],"parts":[],"solution":["P(first green) = 5n.","After one green counter is removed, 4 green counters remain out of n - 1.","P(both green) = 5n × 4n - 1 = 20n(n - 1)"],"diagram":null,"draw":false,"revision":"dda5fc836181"},{"id":"g89-probability-equation-questions-q03","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":3,"marks":2,"tier":"H","calc":true,"text":"

A biased coin is flipped twice. The flips are independent.

The probability of getting heads on both flips is 0.64.

Work out the probability of getting heads on one flip of the coin.

","answerType":"exact","answer":"0.8","answerDisplay":null,"accept":["4/5","80%"],"parts":[],"solution":["Let P(heads) = p. Then p × p = 0.64.","p = √0.64 = 0.8 (taking the positive root, since a probability cannot be negative)."],"diagram":null,"draw":false,"revision":"9b1fdb20c3f1"},{"id":"g89-probability-equation-questions-q04","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

50 people were asked whether they read fiction (A) or non-fiction (B).

The Venn diagram regions contain these numbers of people:

A only: x   A ∩ B: 5   B only: 2x − 3   neither: x + 8

  1. Work out the value of x.
  2. One of the 50 people is chosen at random. Work out the probability that this person reads non-fiction.
","answerType":"multi","answer":"a) x = 10; b) 11/25","answerDisplay":"a) x = 10; b) 1125","accept":[],"parts":[{"label":"a","answer":"10"},{"label":"b","answer":"11/25","answerDisplay":"1125"}],"solution":["The four regions sum to 50: x + 5 + (2x - 3) + (x + 8) = 50.","4x + 10 = 50, so x = 10.","The regions are 10, 5, 17 and 18 (check: 10 + 5 + 17 + 18 = 50).","Number who read non-fiction = 5 + 17 = 22, so the probability = 2250 = 1125."],"diagram":{"type":"venn","sets":["A","B"],"regions":{"A":"x","AB":"5","B":"2x − 3","none":"x + 8"}},"draw":false,"revision":"6b1f63d349b5"},{"id":"g89-probability-equation-questions-q05","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":5,"marks":4,"tier":"H","calc":false,"text":"

Two independent events A and B are such that

P(B) = 2 × P(A)

P(A and B) = 18

Find P(A).

","answerType":"exact","answer":"1/4","answerDisplay":"14","accept":["0.25"],"parts":[],"solution":["Let P(A) = x, so P(B) = 2x.","A and B are independent, so P(A and B) = x × 2x = 2x².","2x² = 18, so x² = 116.","x = 14 (taking the positive root), so P(A) = 14."],"diagram":null,"draw":false,"revision":"ffb9faed58d0"},{"id":"g89-probability-equation-questions-q06","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":6,"marks":5,"tier":"H","calc":false,"text":"

A jar contains n sweets. 6 of the sweets are orange. The rest are lemon.

Hana takes two sweets at random, without replacement.

The probability that both sweets are orange is 13.

  1. Show that n² − n − 90 = 0.
  2. Find the value of n.
","answerType":"multi","answer":"a) shown; b) n = 10","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"P(both orange) = 6/n × 5/(n-1) = 30/(n(n-1)) = 1/3, so n(n-1) = 90, giving n² - n - 90 = 0","answerDisplay":"P(both orange) = 6n × 5n-1 = 30n(n-1) = 13, so n(n-1) = 90, giving n² - n - 90 = 0"},{"label":"b","answer":"10"}],"solution":["P(both orange) = 6n × 5n - 1 = 30n(n - 1).","Setting this equal to 13 gives 30n(n - 1) = 13, so n(n - 1) = 90.","Expanding: n² - n - 90 = 0, as required.","Factorising: (n - 10)(n + 9) = 0, so n = 10 or n = -9.","n must be a positive number of sweets, so n = 10.","Check: 610 × 59 = 3090 = 13."],"diagram":null,"draw":false,"proofSupport":{"plan":["Write the probabilities for the first and second draws without replacement.","Form an equation using the given probability, then clear the denominators.","Solve for the total, reject impossible counts and complete any remaining probability calculation."],"checks":["P(orange, orange) = (6/n)(5/(n − 1)) — Without replacement: both counts decrease","30/[n(n − 1)] = 1/3 — Use the stated probability","n(n − 1) = 90 ⇒ n² − n − 90 = 0 — Multiply through and expand","(n − 10)(n + 9) = 0 ⇒ n = 10 — Reject the negative number of sweets"],"visuals":[{"caption":"Multiply along the orange–orange path. The second draw has one fewer object in total.","diagram":{"type":"tree","kind":"prob","stage1":{"outcomes":["Orange","Lemon"],"p":["6/n","(n − 6)/n"]},"stage2":{"outcomes":["Orange","Lemon"],"values":[["5/(n − 1)","(n − 6)/(n − 1)"],["6/(n − 1)","(n − 7)/(n − 1)"]]},"alt":"Without-replacement probability tree. Multiply the two orange branches."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"P(orange, orange) = (6/n)(5/(n − 1))","reason":"Without replacement: both counts decrease"},{"math":"30/[n(n − 1)] = 1/3","reason":"Use the stated probability"},{"math":"n(n − 1) = 90 ⇒ n² − n − 90 = 0","reason":"Multiply through and expand"},{"math":"(n − 10)(n + 9) = 0 ⇒ n = 10","reason":"Reject the negative number of sweets"}],"alt":"P(orange, orange) = (6/n)(5/(n − 1)): Without replacement: both counts decrease. 30/[n(n − 1)] = 1/3: Use the stated probability. n(n − 1) = 90 ⇒ n² − n − 90 = 0: Multiply through and expand. (n − 10)(n + 9) = 0 ⇒ n = 10: Reject the negative number of sweets"}}]},"revision":"f339b8add829"},{"id":"g89-probability-equation-questions-q07","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":7,"marks":5,"tier":"H","calc":true,"text":"

A bag contains x green counters and 8 white counters.

Two counters are taken at random, without replacement.

The probability that both counters are white is 1433.

Work out the value of x.

","answerType":"exact","answer":"4","answerDisplay":null,"accept":["x = 4"],"parts":[],"solution":["The bag holds x + 8 counters, so P(both white) = 8x + 8 × 7x + 7 = 56(x + 8)(x + 7).","Setting this equal to 1433: 56(x + 8)(x + 7) = 1433.","(x + 8)(x + 7) = 56 × 3314 = 132.","x² + 15x + 56 = 132, so x² + 15x - 76 = 0.","Factorising: (x + 19)(x - 4) = 0, so x = 4 (x must be positive).","Check: 812 × 711 = 56132 = 1433."],"diagram":null,"draw":false,"revision":"8073a91dfe02"},{"id":"g89-probability-equation-questions-q08","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

A bag contains only red counters and green counters.

The probability of taking a red counter at random is 25.

10 red counters are then removed from the bag.

The probability of taking a red counter at random is now 14.

Work out the number of counters that were in the bag originally.

","answerType":"exact","answer":"50","answerDisplay":null,"accept":["50 counters"],"parts":[],"solution":["Let the original total be t. The number of red counters is 2t5.","After removing 10 red counters there are 2t5 - 10 red counters and t - 10 counters in total.","2t5 - 10t - 10 = 14, so 4(2t5 - 10) = t - 10.","8t5 - 40 = t - 10, so 3t5 = 30 and t = 50.","Check: originally 20 red out of 50 (P = 25); after removal 10 red out of 40 (P = 14)."],"diagram":null,"draw":false,"revision":"943632002c16"},{"id":"g89-probability-equation-questions-q09","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":9,"marks":6,"tier":"H","calc":false,"text":"

A bag contains x black counters and 6 white counters.

Two counters are taken at random, without replacement.

The probability that the two counters are different colours is 12.

Find the possible values of x.

","answerType":"exact","answer":"x = 3 or x = 10","answerDisplay":null,"accept":["3 and 10","3, 10"],"parts":[],"solution":["The bag holds x + 6 counters.","P(black then white) = xx + 6 × 6x + 5 and P(white then black) = 6x + 6 × xx + 5.","P(different colours) = 12x(x + 6)(x + 5) = 12.","24x = (x + 6)(x + 5) = x² + 11x + 30.","x² - 13x + 30 = 0, so (x - 3)(x - 10) = 0.","x = 3 or x = 10.","Check x = 3: 2 × 3 × 69 × 8 = 3672 = 12. Check x = 10: 2 × 10 × 616 × 15 = 120240 = 12."],"diagram":null,"draw":false,"revision":"9f49fb45b1b5"},{"id":"g89-probability-equation-questions-q10","topicId":"g89-probability-equation-questions","topic":"Probability Equation Questions","grade":"89","topicGrade":"89","strand":"P","difficulty":10,"marks":8,"tier":"H","calc":true,"text":"

There are n counters in a box. 8 of the counters are orange. The rest are brown.

Meg takes two counters at random, without replacement.

The probability that both counters are orange is 415.

  1. Show that n² − n − 210 = 0.
  2. Find the value of n.
  3. Work out the probability that both counters Meg takes are brown.
","answerType":"multi","answer":"a) shown; b) n = 15; c) 1/5","answerDisplay":"a) shown; b) n = 15; c) 15","accept":[],"parts":[{"label":"a","answer":"P(both orange) = 8/n × 7/(n-1) = 56/(n(n-1)) = 4/15, so n(n-1) = 210, giving n² - n - 210 = 0","answerDisplay":"P(both orange) = 8n × 7n-1 = 56n(n-1) = 415, so n(n-1) = 210, giving n² - n - 210 = 0"},{"label":"b","answer":"15"},{"label":"c","answer":"1/5","answerDisplay":"15"}],"solution":["P(both orange) = 8n × 7n - 1 = 56n(n - 1).","Setting this equal to 415: 56n(n - 1) = 415, so n(n - 1) = 56 × 154 = 210.","Expanding: n² - n - 210 = 0, as required.","Factorising: (n - 15)(n + 14) = 0, so n = 15 (n must be positive).","Check: 815 × 714 = 56210 = 415.","There are 15 - 8 = 7 brown counters.","P(both brown) = 715 × 614 = 42210 = 15"],"diagram":null,"draw":false,"proofSupport":{"plan":["Write the probabilities for the first and second draws without replacement.","Form an equation using the given probability, then clear the denominators.","Solve for the total, reject impossible counts and complete any remaining probability calculation."],"checks":["P(orange, orange) = (8/n)(7/(n − 1)) — Without replacement: both counts decrease","56/[n(n − 1)] = 4/15 — Use the stated probability","n(n − 1) = 210 ⇒ n² − n − 210 = 0 — Multiply through and expand","(n − 15)(n + 14) = 0 ⇒ n = 15 — Reject the negative root","P(brown, brown) = (7/15)(6/14) = 1/5 — There are 7 brown counters initially"],"visuals":[{"caption":"Multiply along the orange–orange path. The second draw has one fewer object in total.","diagram":{"type":"tree","kind":"prob","stage1":{"outcomes":["Orange","Brown"],"p":["8/n","(n − 8)/n"]},"stage2":{"outcomes":["Orange","Brown"],"values":[["7/(n − 1)","(n − 8)/(n − 1)"],["8/(n − 1)","(n − 9)/(n − 1)"]]},"alt":"Without-replacement probability tree. Multiply the two orange branches."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"P(orange, orange) = (8/n)(7/(n − 1))","reason":"Without replacement: both counts decrease"},{"math":"56/[n(n − 1)] = 4/15","reason":"Use the stated probability"},{"math":"n(n − 1) = 210 ⇒ n² − n − 210 = 0","reason":"Multiply through and expand"},{"math":"(n − 15)(n + 14) = 0 ⇒ n = 15","reason":"Reject the negative root"},{"math":"P(brown, brown) = (7/15)(6/14) = 1/5","reason":"There are 7 brown counters initially"}],"alt":"P(orange, orange) = (8/n)(7/(n − 1)): Without replacement: both counts decrease. 56/[n(n − 1)] = 4/15: Use the stated probability. n(n − 1) = 210 ⇒ n² − n − 210 = 0: Multiply through and expand. (n − 15)(n + 14) = 0 ⇒ n = 15: Reject the negative root. P(brown, brown) = (7/15)(6/14) = 1/5: There are 7 brown counters initially"}}]},"revision":"eac180f7dce5"},{"id":"g89-proof-of-the-circle-theorems-q01","topicId":"g89-proof-of-the-circle-theorems","topic":"Proof of the Circle Theorems","grade":"89","topicGrade":"89","strand":"G","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

AB is a diameter of a circle with centre O.

C is a point on the circle, not at A or B.

Prove that angle ACB = 90°.

","answerType":"open","answer":"Using the isosceles triangles OAC and OBC, angle ACB = x + y where 2x + 2y = 180, so angle ACB = 90 degrees","answerDisplay":null,"accept":[],"parts":[],"solution":["Join O to C. OA = OC and OB = OC because all three are radii of the circle.","Triangle OAC is isosceles, so angle OAC = angle OCA. Call this angle x.","Triangle OBC is isosceles, so angle OBC = angle OCB. Call this angle y.","In triangle ABC the three angles are angle CAB = x, angle CBA = y and angle ACB = x + y.","Angles in a triangle sum to 180°, so x + y + (x + y) = 180°, which gives 2(x + y) = 180°.","So x + y = 90°, and therefore angle ACB = 90°.","This proves that the angle in a semicircle is a right angle."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[-3,3.6739403974420594e-16],"label":"A","style":"none","labelPos":"below-left"},{"at":[3,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[1.2678547852220983,2.7189233611099497],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[-3,3.6739403974420594e-16],"to":[3,0]},{"from":[-3,3.6739403974420594e-16],"to":[1.2678547852220983,2.7189233611099497]},{"from":[3,0],"to":[1.2678547852220983,2.7189233611099497]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Mark the given facts and any extra radii or construction lines you need.","Link equal lengths and angles to named geometric facts.","Connect those facts to the exact statement you need to prove."],"checks":["OA = OC = OB — All are radii: two isosceles triangles","∠CAB = x; ∠CBA = y; ∠ACB = x + y — Base angles in each isosceles triangle are equal","2(x + y) = 180° — Add the three angles of triangle ABC","∠ACB = x + y = 90° — Divide by 2 to reach the required result"],"visuals":[{"caption":"Construction and equalities used in the proof. The written solution explains why each relationship holds.","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-left"},{"at":[-3,3.6739403974420594e-16],"label":"A","style":"none","labelPos":"below-left"},{"at":[3,0],"label":"B","style":"none","labelPos":"below-right"},{"at":[1.2678547852220983,2.7189233611099497],"label":"C","style":"none","labelPos":"above"}],"segments":[{"from":[-3,3.6739403974420594e-16],"to":[3,0]},{"from":[-3,3.6739403974420594e-16],"to":[1.2678547852220983,2.7189233611099497]},{"from":[3,0],"to":[1.2678547852220983,2.7189233611099497]},{"from":[0,0],"to":[-3,3.6739403974420594e-16],"label":"","answer":true,"ticks":1},{"from":[0,0],"to":[3,0],"label":"","answer":true,"ticks":1},{"from":[0,0],"to":[1.2678547852220983,2.7189233611099497],"label":"","answer":true,"ticks":1}],"angles":[{"at":[-3,3.6739403974420594e-16],"from":[0,0],"to":[1.2678547852220983,2.7189233611099497],"label":"x","answer":true,"r":0.45},{"at":[1.2678547852220983,2.7189233611099497],"from":[-3,3.6739403974420594e-16],"to":[0,0],"label":"x","answer":true,"r":0.45},{"at":[3,0],"from":[1.2678547852220983,2.7189233611099497],"to":[0,0],"label":"y","answer":true,"r":0.45},{"at":[1.2678547852220983,2.7189233611099497],"from":[0,0],"to":[3,0],"label":"y","answer":true,"r":0.45}],"circles":[{"c":[0,0],"r":3}],"alt":"Worked construction for proof-of-the-circle-theorems-q01. Green additions correspond to the reasoning below."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"OA = OC = OB","reason":"All are radii: two isosceles triangles"},{"math":"∠CAB = x; ∠CBA = y; ∠ACB = x + y","reason":"Base angles in each isosceles triangle are equal"},{"math":"2(x + y) = 180°","reason":"Add the three angles of triangle ABC"},{"math":"∠ACB = x + y = 90°","reason":"Divide by 2 to reach the required result"}],"alt":"OA = OC = OB: All are radii: two isosceles triangles. ∠CAB = x; ∠CBA = y; ∠ACB = x + y: Base angles in each isosceles triangle are equal. 2(x + y) = 180°: Add the three angles of triangle ABC. ∠ACB = x + y = 90°: Divide by 2 to reach the required result"}}]},"revision":"ad8b33855ef0"},{"id":"g89-proof-of-the-circle-theorems-q02","topicId":"g89-proof-of-the-circle-theorems","topic":"Proof of the Circle Theorems","grade":"89","topicGrade":"89","strand":"G","difficulty":2,"marks":4,"tier":"H","calc":true,"text":"

AB is a chord of a circle with centre O.

M is the point on AB such that OM is perpendicular to AB.

Prove that M is the midpoint of AB.

","answerType":"open","answer":"Triangles OMA and OMB are congruent (RHS), so AM = BM and M is the midpoint of AB","answerDisplay":null,"accept":[],"parts":[],"solution":["If AB is a diameter, the perpendicular meets it at O, so M = O and AM = BM because both are radii. Otherwise, use the two right-angled triangles below.","Join O to A and O to B.","Angle OMA = angle OMB = 90° because OM is perpendicular to AB (given).","OA = OB because both are radii of the circle. These are the hypotenuses of the two right-angled triangles.","OM is common to both triangles, so OM = OM.","So triangle OMA is congruent to triangle OMB: the triangles have the same shape and size. RHS means right angle, equal hypotenuse and one equal side.","Corresponding sides of congruent triangles are equal, so AM = BM.","Therefore M is the midpoint of AB, which proves that the perpendicular from the centre of a circle to a chord bisects the chord."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[2.4574561328669753,1.7207293090531381],"label":"A","style":"none","labelPos":"above"},{"at":[2.4574561328669753,-1.7207293090531381],"label":"B","style":"none","labelPos":"below-right"},{"at":[2.4574561328669753,0],"label":"M","style":"none","labelPos":"below-right"}],"segments":[{"from":[2.4574561328669753,1.7207293090531381],"to":[2.4574561328669753,-1.7207293090531381]},{"from":[0,0],"to":[2.4574561328669753,0]}],"angles":[{"at":[2.4574561328669753,0],"from":[0,0],"to":[2.4574561328669753,1.7207293090531381],"label":"","right":true}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Mark the given facts and any extra radii or construction lines you need.","Link equal lengths and angles to named geometric facts.","Connect those facts to the exact statement you need to prove."],"checks":["OA = OB; OM is common — Equal hypotenuses and a shared side","∠OMA = ∠OMB = 90° — OM is perpendicular to the chord","△OMA ≅ △OMB (RHS) — Right angle, hypotenuse and side","AM = BM — Corresponding sides are equal, so M is the midpoint"],"visuals":[{"caption":"Construction and equalities used in the proof. The written solution explains why each relationship holds.","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[2.4574561328669753,1.7207293090531381],"label":"A","style":"none","labelPos":"above"},{"at":[2.4574561328669753,-1.7207293090531381],"label":"B","style":"none","labelPos":"below-right"},{"at":[2.4574561328669753,0],"label":"M","style":"none","labelPos":"below-right"}],"segments":[{"from":[2.4574561328669753,1.7207293090531381],"to":[2.4574561328669753,-1.7207293090531381]},{"from":[0,0],"to":[2.4574561328669753,0]},{"from":[0,0],"to":[2.4574561328669753,1.7207293090531381],"label":"","answer":true,"ticks":1},{"from":[0,0],"to":[2.4574561328669753,-1.7207293090531381],"label":"","answer":true,"ticks":1},{"from":[0,0],"to":[2.4574561328669753,0],"label":"","answer":true,"ticks":2},{"from":[2.4574561328669753,1.7207293090531381],"to":[2.4574561328669753,0],"label":"","answer":true,"ticks":3},{"from":[2.4574561328669753,0],"to":[2.4574561328669753,-1.7207293090531381],"label":"","answer":true,"ticks":3}],"angles":[{"at":[2.4574561328669753,0],"from":[0,0],"to":[2.4574561328669753,1.7207293090531381],"label":"","right":true}],"circles":[{"c":[0,0],"r":3}],"alt":"Worked construction for proof-of-the-circle-theorems-q02. Green additions correspond to the reasoning below."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"OA = OB; OM is common","reason":"Equal hypotenuses and a shared side"},{"math":"∠OMA = ∠OMB = 90°","reason":"OM is perpendicular to the chord"},{"math":"△OMA ≅ △OMB (RHS)","reason":"Right angle, hypotenuse and side"},{"math":"AM = BM","reason":"Corresponding sides are equal, so M is the midpoint"}],"alt":"OA = OB; OM is common: Equal hypotenuses and a shared side. ∠OMA = ∠OMB = 90°: OM is perpendicular to the chord. △OMA ≅ △OMB (RHS): Right angle, hypotenuse and side. AM = BM: Corresponding sides are equal, so M is the midpoint"}}]},"revision":"9236cd0e85d4"},{"id":"g89-proof-of-the-circle-theorems-q03","topicId":"g89-proof-of-the-circle-theorems","topic":"Proof of the Circle Theorems","grade":"89","topicGrade":"89","strand":"G","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

P is a point outside a circle with centre O.

The two tangents from P touch the circle at A and at B.

Prove that PA = PB.

","answerType":"open","answer":"Triangles OAP and OBP are congruent (RHS), so PA = PB","answerDisplay":null,"accept":[],"parts":[],"solution":["Join O to A, O to B and O to P.","Angle OAP = angle OBP = 90° because a tangent to a circle is perpendicular to the radius at the point of contact.","OA = OB because both are radii of the circle.","OP is the hypotenuse of both right-angled triangles and is common, so OP = OP.","So triangle OAP is congruent to triangle OBP (same shape and size), by RHS: a right angle, equal hypotenuse and one equal side.","Corresponding sides of congruent triangles are equal, so PA = PB.","This proves that the two tangents drawn to a circle from an external point are equal in length."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[1.7207293090531381,2.4574561328669753],"label":"A","style":"none","labelPos":"above"},{"at":[1.7207293090531381,-2.4574561328669753],"label":"B","style":"none","labelPos":"below-left"},{"at":[5.230340386863294,0],"label":"P","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[1.7207293090531381,2.4574561328669753]},{"from":[0,0],"to":[1.7207293090531381,-2.4574561328669753]},{"from":[5.230340386863294,0],"to":[1.7207293090531381,2.4574561328669753]},{"from":[5.230340386863294,0],"to":[1.7207293090531381,-2.4574561328669753]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Mark the given facts and any extra radii or construction lines you need.","Link equal lengths and angles to named geometric facts.","Connect those facts to the exact statement you need to prove."],"checks":["∠OAP = ∠OBP = 90° — A tangent is perpendicular to its radius","OA = OB; OP is common — Equal radii and the same hypotenuse","△OAP ≅ △OBP (RHS) — Right angle, hypotenuse and side","PA = PB — Corresponding sides of congruent triangles"],"visuals":[{"caption":"Construction and equalities used in the proof. The written solution explains why each relationship holds.","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[1.7207293090531381,2.4574561328669753],"label":"A","style":"none","labelPos":"above"},{"at":[1.7207293090531381,-2.4574561328669753],"label":"B","style":"none","labelPos":"below-left"},{"at":[5.230340386863294,0],"label":"P","style":"none","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[1.7207293090531381,2.4574561328669753]},{"from":[0,0],"to":[1.7207293090531381,-2.4574561328669753]},{"from":[5.230340386863294,0],"to":[1.7207293090531381,2.4574561328669753]},{"from":[5.230340386863294,0],"to":[1.7207293090531381,-2.4574561328669753]},{"from":[0,0],"to":[5.230340386863294,0],"label":"","answer":true,"dashed":true,"ticks":2},{"from":[0,0],"to":[1.7207293090531381,2.4574561328669753],"label":"","answer":true,"ticks":1},{"from":[0,0],"to":[1.7207293090531381,-2.4574561328669753],"label":"","answer":true,"ticks":1}],"angles":[{"at":[1.7207293090531381,2.4574561328669753],"from":[0,0],"to":[5.230340386863294,0],"label":"","answer":true,"r":0.45,"right":true},{"at":[1.7207293090531381,-2.4574561328669753],"from":[0,0],"to":[5.230340386863294,0],"label":"","answer":true,"r":0.45,"right":true}],"circles":[{"c":[0,0],"r":3}],"alt":"Worked construction for proof-of-the-circle-theorems-q03. Green additions correspond to the reasoning below."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"∠OAP = ∠OBP = 90°","reason":"A tangent is perpendicular to its radius"},{"math":"OA = OB; OP is common","reason":"Equal radii and the same hypotenuse"},{"math":"△OAP ≅ △OBP (RHS)","reason":"Right angle, hypotenuse and side"},{"math":"PA = PB","reason":"Corresponding sides of congruent triangles"}],"alt":"∠OAP = ∠OBP = 90°: A tangent is perpendicular to its radius. OA = OB; OP is common: Equal radii and the same hypotenuse. △OAP ≅ △OBP (RHS): Right angle, hypotenuse and side. PA = PB: Corresponding sides of congruent triangles"}}]},"revision":"ea8c44dd6997"},{"id":"g89-proof-of-the-circle-theorems-q04","topicId":"g89-proof-of-the-circle-theorems","topic":"Proof of the Circle Theorems","grade":"89","topicGrade":"89","strand":"G","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

A, B and P are points on a circle with centre O.

O lies inside triangle ABP.

Prove that angle AOB = 2 × angle APB.

","answerType":"open","answer":"Extending PO to D and using the exterior angles of the isosceles triangles OAP and OBP gives angle AOB = 2x + 2y = 2 angle APB","answerDisplay":null,"accept":[],"parts":[],"solution":["Join P to O and extend the line to a point D on the other side of O.","OA = OP because both are radii, so triangle OAP is isosceles and angle OPA = angle OAP. Call this angle x.","The exterior angle of a triangle equals the sum of the two interior opposite angles, so angle AOD = angle OPA + angle OAP = 2x.","OB = OP because both are radii, so triangle OBP is isosceles and angle OPB = angle OBP. Call this angle y.","By the same exterior angle argument, angle BOD = 2y.","Angle AOB = angle AOD + angle DOB = 2x + 2y = 2(x + y).","Angle APB = angle APD + angle DPB = x + y.","Therefore angle AOB = 2 × angle APB, which proves that the angle at the centre is twice the angle at the circumference standing on the same arc."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"above"},{"at":[2.8190778623577253,1.0260604299770062],"label":"A","style":"none","labelPos":"above"},{"at":[-2.598076211353316,1.4999999999999998],"label":"B","style":"none","labelPos":"above"},{"at":[-5.51091059616309e-16,-3],"label":"P","style":"none","labelPos":"below-left"}],"segments":[{"from":[2.8190778623577253,1.0260604299770062],"to":[-2.598076211353316,1.4999999999999998]},{"from":[-2.598076211353316,1.4999999999999998],"to":[-5.51091059616309e-16,-3]},{"from":[-5.51091059616309e-16,-3],"to":[2.8190778623577253,1.0260604299770062]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Mark the given facts and any extra radii or construction lines you need.","Link equal lengths and angles to named geometric facts.","Connect those facts to the exact statement you need to prove."],"checks":["∠OPA = ∠OAP = x; ∠OPB = ∠OBP = y — Equal radii give isosceles triangles","∠AOD = 2x; ∠DOB = 2y — Exterior angles equal the opposite interior sum","∠AOB = 2x + 2y = 2(x + y) — Add the adjacent central angles","∠APB = x + y — Hence the central angle is twice the angle at P"],"visuals":[{"caption":"Construction and equalities used in the proof. The written solution explains why each relationship holds.","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"above"},{"at":[2.8190778623577253,1.0260604299770062],"label":"A","style":"none","labelPos":"above"},{"at":[-2.598076211353316,1.4999999999999998],"label":"B","style":"none","labelPos":"above"},{"at":[-5.51091059616309e-16,-3],"label":"P","style":"none","labelPos":"below-left"},{"at":[0,3.6],"label":"D","style":"dot","labelPos":"above"}],"segments":[{"from":[2.8190778623577253,1.0260604299770062],"to":[-2.598076211353316,1.4999999999999998]},{"from":[-2.598076211353316,1.4999999999999998],"to":[-5.51091059616309e-16,-3]},{"from":[-5.51091059616309e-16,-3],"to":[2.8190778623577253,1.0260604299770062]},{"from":[0,0],"to":[2.8190778623577253,1.0260604299770062],"label":"","answer":true,"ticks":1},{"from":[0,0],"to":[-2.598076211353316,1.4999999999999998],"label":"","answer":true,"ticks":1},{"from":[0,0],"to":[-5.51091059616309e-16,-3],"label":"","answer":true,"ticks":1},{"from":[-5.51091059616309e-16,-3],"to":[0,3.6],"label":"","answer":true,"dashed":true}],"angles":[{"at":[-5.51091059616309e-16,-3],"from":[0,0],"to":[2.8190778623577253,1.0260604299770062],"label":"x","answer":true,"r":0.45},{"at":[-5.51091059616309e-16,-3],"from":[-2.598076211353316,1.4999999999999998],"to":[0,0],"label":"y","answer":true,"r":0.45},{"at":[0,0],"from":[2.8190778623577253,1.0260604299770062],"to":[0,3.6],"label":"2x","answer":true,"r":0.65},{"at":[0,0],"from":[0,3.6],"to":[-2.598076211353316,1.4999999999999998],"label":"2y","answer":true,"r":0.8}],"circles":[{"c":[0,0],"r":3}],"alt":"Worked construction for proof-of-the-circle-theorems-q04. Green additions correspond to the reasoning below."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"∠OPA = ∠OAP = x; ∠OPB = ∠OBP = y","reason":"Equal radii give isosceles triangles"},{"math":"∠AOD = 2x; ∠DOB = 2y","reason":"Exterior angles equal the opposite interior sum"},{"math":"∠AOB = 2x + 2y = 2(x + y)","reason":"Add the adjacent central angles"},{"math":"∠APB = x + y","reason":"Hence the central angle is twice the angle at P"}],"alt":"∠OPA = ∠OAP = x; ∠OPB = ∠OBP = y: Equal radii give isosceles triangles. ∠AOD = 2x; ∠DOB = 2y: Exterior angles equal the opposite interior sum. ∠AOB = 2x + 2y = 2(x + y): Add the adjacent central angles. ∠APB = x + y: Hence the central angle is twice the angle at P"}}]},"revision":"50dad3bfa466"},{"id":"g89-proof-of-the-circle-theorems-q05","topicId":"g89-proof-of-the-circle-theorems","topic":"Proof of the Circle Theorems","grade":"89","topicGrade":"89","strand":"G","difficulty":5,"marks":4,"tier":"H","calc":true,"text":"

A, B, P and Q are points on a circle with centre O.

P and Q are on the same side of the chord AB, and both are on the major arc AB.

You may use the theorem that the angle at the centre of a circle is twice the angle at the circumference standing on the same arc.

Prove that angle APB = angle AQB.

","answerType":"open","answer":"Both angles equal half of angle AOB by the angle at the centre theorem, so they are equal","answerDisplay":null,"accept":[],"parts":[],"solution":["Join O to A and O to B.","P is on the major arc AB, so by the angle at the centre theorem, angle AOB = 2 × angle APB.","Q is on the same arc, so by the same theorem, angle AOB = 2 × angle AQB.","Therefore 2 × angle APB = 2 × angle AQB.","Dividing both sides by 2 gives angle APB = angle AQB.","This proves that angles in the same segment of a circle are equal."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"above"},{"at":[2.7189233611099497,1.2678547852220983],"label":"A","style":"none","labelPos":"above"},{"at":[-2.7189233611099497,1.2678547852220985],"label":"B","style":"none","labelPos":"above"},{"at":[-1.5000000000000013,-2.5980762113533156],"label":"P","style":"none","labelPos":"below-left"},{"at":[1.5000000000000004,-2.598076211353316],"label":"Q","style":"none","labelPos":"below-right"}],"segments":[{"from":[2.7189233611099497,1.2678547852220983],"to":[-2.7189233611099497,1.2678547852220985]},{"from":[2.7189233611099497,1.2678547852220983],"to":[-1.5000000000000013,-2.5980762113533156]},{"from":[-2.7189233611099497,1.2678547852220985],"to":[-1.5000000000000013,-2.5980762113533156]},{"from":[2.7189233611099497,1.2678547852220983],"to":[1.5000000000000004,-2.598076211353316]},{"from":[-2.7189233611099497,1.2678547852220985],"to":[1.5000000000000004,-2.598076211353316]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Mark the given facts and any extra radii or construction lines you need.","Link equal lengths and angles to named geometric facts.","Connect those facts to the exact statement you need to prove."],"checks":["∠AOB = 2∠APB — Angle at the centre theorem on the same arc","∠AOB = 2∠AQB — Apply the same theorem for Q","2∠APB = 2∠AQB — Both equal the same central angle","∠APB = ∠AQB — Divide both sides by 2"],"visuals":[{"caption":"Construction and equalities used in the proof. The written solution explains why each relationship holds.","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"above"},{"at":[2.7189233611099497,1.2678547852220983],"label":"A","style":"none","labelPos":"above"},{"at":[-2.7189233611099497,1.2678547852220985],"label":"B","style":"none","labelPos":"above"},{"at":[-1.5000000000000013,-2.5980762113533156],"label":"P","style":"none","labelPos":"below-left"},{"at":[1.5000000000000004,-2.598076211353316],"label":"Q","style":"none","labelPos":"below-right"}],"segments":[{"from":[2.7189233611099497,1.2678547852220983],"to":[-2.7189233611099497,1.2678547852220985]},{"from":[2.7189233611099497,1.2678547852220983],"to":[-1.5000000000000013,-2.5980762113533156]},{"from":[-2.7189233611099497,1.2678547852220985],"to":[-1.5000000000000013,-2.5980762113533156]},{"from":[2.7189233611099497,1.2678547852220983],"to":[1.5000000000000004,-2.598076211353316]},{"from":[-2.7189233611099497,1.2678547852220985],"to":[1.5000000000000004,-2.598076211353316]},{"from":[0,0],"to":[2.7189233611099497,1.2678547852220983],"label":"","answer":true},{"from":[0,0],"to":[-2.7189233611099497,1.2678547852220985],"label":"","answer":true}],"angles":[{"at":[0,0],"from":[2.7189233611099497,1.2678547852220983],"to":[-2.7189233611099497,1.2678547852220985],"label":"θ","answer":true,"r":0.7},{"at":[-1.5000000000000013,-2.5980762113533156],"from":[2.7189233611099497,1.2678547852220983],"to":[-2.7189233611099497,1.2678547852220985],"label":"θ/2","answer":true,"r":0.45},{"at":[1.5000000000000004,-2.598076211353316],"from":[2.7189233611099497,1.2678547852220983],"to":[-2.7189233611099497,1.2678547852220985],"label":"θ/2","answer":true,"r":0.45}],"circles":[{"c":[0,0],"r":3}],"alt":"Worked construction for proof-of-the-circle-theorems-q05. Green additions correspond to the reasoning below."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"∠AOB = 2∠APB","reason":"Angle at the centre theorem on the same arc"},{"math":"∠AOB = 2∠AQB","reason":"Apply the same theorem for Q"},{"math":"2∠APB = 2∠AQB","reason":"Both equal the same central angle"},{"math":"∠APB = ∠AQB","reason":"Divide both sides by 2"}],"alt":"∠AOB = 2∠APB: Angle at the centre theorem on the same arc. ∠AOB = 2∠AQB: Apply the same theorem for Q. 2∠APB = 2∠AQB: Both equal the same central angle. ∠APB = ∠AQB: Divide both sides by 2"}}]},"revision":"e9d139e2c41d"},{"id":"g89-proof-of-the-circle-theorems-q06","topicId":"g89-proof-of-the-circle-theorems","topic":"Proof of the Circle Theorems","grade":"89","topicGrade":"89","strand":"G","difficulty":6,"marks":4,"tier":"H","calc":false,"text":"

ABCD is a cyclic quadrilateral in a circle with centre O.

You may use the theorem that the angle at the centre of a circle is twice the angle at the circumference standing on the same arc.

Prove that angle ABC + angle ADC = 180°.

","answerType":"open","answer":"The two angles at the centre standing on arcs ADC and ABC sum to 360 degrees and equal 2 angle ABC and 2 angle ADC, so angle ABC + angle ADC = 180 degrees","answerDisplay":null,"accept":[],"parts":[],"solution":["A cyclic quadrilateral has all four vertices on a circle. Join O to A and O to C.","There are two arcs from A to C: one through B and one through D. Let α be the central angle following arc ADC, and β the central angle following arc ABC. Together these angles make a full turn, so α + β = 360°.","Angle ABC stands on arc ADC (the arc not containing B), so the angle-at-the-centre theorem gives α = 2 × angle ABC.","Angle ADC stands on the other arc, ABC, so β = 2 × angle ADC.","Substitute into α + β = 360°: 2 × angle ABC + 2 × angle ADC = 360°.","Divide by 2: angle ABC + angle ADC = 180°. This also covers AC being a diameter: both central angles are then 180°."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-right"},{"at":[-2.598076211353316,1.4999999999999998],"label":"A","style":"none","labelPos":"above"},{"at":[1.5000000000000004,2.598076211353316],"label":"B","style":"none","labelPos":"above"},{"at":[2.9544232590366235,-0.5209445330007938],"label":"C","style":"none","labelPos":"below-right"},{"at":[-1.9283628290596184,-2.2981333293569337],"label":"D","style":"none","labelPos":"below-left"}],"segments":[{"from":[-2.598076211353316,1.4999999999999998],"to":[1.5000000000000004,2.598076211353316]},{"from":[1.5000000000000004,2.598076211353316],"to":[2.9544232590366235,-0.5209445330007938]},{"from":[2.9544232590366235,-0.5209445330007938],"to":[-1.9283628290596184,-2.2981333293569337]},{"from":[-1.9283628290596184,-2.2981333293569337],"to":[-2.598076211353316,1.4999999999999998]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Cyclic quadrilateral ABCD with centre O. All four vertices lie on the circle."},"draw":false,"proofSupport":{"plan":["Mark the given facts and any extra radii or construction lines you need.","Link equal lengths and angles to named geometric facts.","Connect those facts to the exact statement you need to prove."],"checks":["α = 2∠ABC — The angle at B stands on arc ADC","β = 2∠ADC — The angle at D stands on arc ABC","2∠ABC + 2∠ADC = 360° — The two central angles make a full turn","∠ABC + ∠ADC = 180° — Divide by 2"],"visuals":[{"caption":"Construction and equalities used in the proof. The written solution explains why each relationship holds.","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"below-right"},{"at":[-2.598076211353316,1.4999999999999998],"label":"A","style":"none","labelPos":"above"},{"at":[1.5000000000000004,2.598076211353316],"label":"B","style":"none","labelPos":"above"},{"at":[2.9544232590366235,-0.5209445330007938],"label":"C","style":"none","labelPos":"below-right"},{"at":[-1.9283628290596184,-2.2981333293569337],"label":"D","style":"none","labelPos":"below-left"}],"segments":[{"from":[-2.598076211353316,1.4999999999999998],"to":[1.5000000000000004,2.598076211353316]},{"from":[1.5000000000000004,2.598076211353316],"to":[2.9544232590366235,-0.5209445330007938]},{"from":[2.9544232590366235,-0.5209445330007938],"to":[-1.9283628290596184,-2.2981333293569337]},{"from":[-1.9283628290596184,-2.2981333293569337],"to":[-2.598076211353316,1.4999999999999998]},{"from":[0,0],"to":[-2.598076211353316,1.4999999999999998],"label":"","answer":true},{"from":[0,0],"to":[2.9544232590366235,-0.5209445330007938],"label":"","answer":true}],"angles":[{"at":[1.5000000000000004,2.598076211353316],"from":[-2.598076211353316,1.4999999999999998],"to":[2.9544232590366235,-0.5209445330007938],"label":"x","answer":true,"r":0.45},{"at":[-1.9283628290596184,-2.2981333293569337],"from":[-2.598076211353316,1.4999999999999998],"to":[2.9544232590366235,-0.5209445330007938],"label":"y","answer":true,"r":0.45},{"at":[0,0],"from":[-2.598076211353316,1.4999999999999998],"to":[2.9544232590366235,-0.5209445330007938],"label":"2y","answer":true,"r":0.6},{"at":[0,0],"from":[-2.598076211353316,1.4999999999999998],"to":[2.9544232590366235,-0.5209445330007938],"label":"2x","answer":true,"r":1.1,"reflex":true}],"circles":[{"c":[0,0],"r":3}],"alt":"Worked construction for proof-of-the-circle-theorems-q06. Green additions correspond to the reasoning below."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"α = 2∠ABC","reason":"The angle at B stands on arc ADC"},{"math":"β = 2∠ADC","reason":"The angle at D stands on arc ABC"},{"math":"2∠ABC + 2∠ADC = 360°","reason":"The two central angles make a full turn"},{"math":"∠ABC + ∠ADC = 180°","reason":"Divide by 2"}],"alt":"α = 2∠ABC: The angle at B stands on arc ADC. β = 2∠ADC: The angle at D stands on arc ABC. 2∠ABC + 2∠ADC = 360°: The two central angles make a full turn. ∠ABC + ∠ADC = 180°: Divide by 2"}}]},"revision":"6d2fa6537249"},{"id":"g89-proof-of-the-circle-theorems-q07","topicId":"g89-proof-of-the-circle-theorems","topic":"Proof of the Circle Theorems","grade":"89","topicGrade":"89","strand":"G","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

The line ST is a tangent to a circle with centre O, touching the circle at A.

AB is a chord of the circle.

C is a point on the major arc AB, so that C is in the alternate segment to angle TAB.

Prove that angle TAB = angle ACB.

","answerType":"open","answer":"Drawing the diameter AD: angle TAB = 90 - angle DAB = angle ADB (angles of right-angled triangle ABD), and angle ADB = angle ACB (same segment), so angle TAB = angle ACB","answerDisplay":null,"accept":[],"parts":[],"solution":["If AB is a diameter, angle TAB is 90° (tangent perpendicular to radius) and angle ACB is 90° (angle in a semicircle), so they are equal. Otherwise use the construction below.","Draw the diameter AD from A through the centre O, and join D to B.","Angle DAT = 90° because a tangent is perpendicular to the radius (and so to the diameter) at the point of contact.","Therefore angle TAB = 90° − angle DAB.","Angle ABD = 90° because it is the angle in a semicircle standing on the diameter AD.","In triangle ABD the angles sum to 180°, so angle ADB = 180° − 90° − angle DAB = 90° − angle DAB.","So angle TAB = angle ADB.","Angle ADB = angle ACB because both are angles in the same segment standing on the chord AB.","Therefore angle TAB = angle ACB, which proves the alternate segment theorem."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"above"},{"at":[-5.51091059616309e-16,-3],"label":"A","style":"none","labelPos":"below-right"},{"at":[2.4574561328669753,1.7207293090531381],"label":"B","style":"none","labelPos":"above"},{"at":[-2.598076211353316,1.4999999999999998],"label":"C","style":"none","labelPos":"above"},{"at":[-5,-3],"label":"S","style":"none","labelPos":"below-left"},{"at":[5,-3],"label":"T","style":"none","labelPos":"below-right"}],"segments":[{"from":[-5,-3],"to":[5,-3]},{"from":[-5.51091059616309e-16,-3],"to":[2.4574561328669753,1.7207293090531381]},{"from":[-5.51091059616309e-16,-3],"to":[-2.598076211353316,1.4999999999999998]},{"from":[2.4574561328669753,1.7207293090531381],"to":[-2.598076211353316,1.4999999999999998]}],"angles":[],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Mark the given facts and any extra radii or construction lines you need.","Link equal lengths and angles to named geometric facts.","Connect those facts to the exact statement you need to prove."],"checks":["∠DAT = 90°; let ∠DAB = x — Draw diameter AD; radius and tangent are perpendicular","∠TAB = 90° − x — Split the right angle at A","∠ABD = 90°; ∠ADB = 90° − x — Angle in a semicircle and triangle angle sum","∠ADB = ∠ACB = ∠TAB — Angles in the same segment complete the proof"],"visuals":[{"caption":"Construction and equalities used in the proof. The written solution explains why each relationship holds.","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"dot","labelPos":"above"},{"at":[-5.51091059616309e-16,-3],"label":"A","style":"none","labelPos":"below-right"},{"at":[2.4574561328669753,1.7207293090531381],"label":"B","style":"none","labelPos":"above"},{"at":[-2.598076211353316,1.4999999999999998],"label":"C","style":"none","labelPos":"above"},{"at":[-5,-3],"label":"S","style":"none","labelPos":"below-left"},{"at":[5,-3],"label":"T","style":"none","labelPos":"below-right"},{"at":[0,3],"label":"D","style":"dot","labelPos":"above"}],"segments":[{"from":[-5,-3],"to":[5,-3]},{"from":[-5.51091059616309e-16,-3],"to":[2.4574561328669753,1.7207293090531381]},{"from":[-5.51091059616309e-16,-3],"to":[-2.598076211353316,1.4999999999999998]},{"from":[2.4574561328669753,1.7207293090531381],"to":[-2.598076211353316,1.4999999999999998]},{"from":[-5.51091059616309e-16,-3],"to":[0,3],"label":"","answer":true,"dashed":true},{"from":[0,3],"to":[2.4574561328669753,1.7207293090531381],"label":"","answer":true}],"angles":[{"at":[-5.51091059616309e-16,-3],"from":[0,3],"to":[5,-3],"label":"","answer":true,"r":0.45,"right":true},{"at":[2.4574561328669753,1.7207293090531381],"from":[-5.51091059616309e-16,-3],"to":[0,3],"label":"","answer":true,"r":0.45,"right":true},{"at":[-5.51091059616309e-16,-3],"from":[0,3],"to":[2.4574561328669753,1.7207293090531381],"label":"x","answer":true,"r":0.45},{"at":[0,3],"from":[-5.51091059616309e-16,-3],"to":[2.4574561328669753,1.7207293090531381],"label":"90°−x","answer":true,"r":0.6},{"at":[-5.51091059616309e-16,-3],"from":[5,-3],"to":[2.4574561328669753,1.7207293090531381],"label":"90°−x","answer":true,"r":1}],"circles":[{"c":[0,0],"r":3}],"alt":"Worked construction for proof-of-the-circle-theorems-q07. Green additions correspond to the reasoning below."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"∠DAT = 90°; let ∠DAB = x","reason":"Draw diameter AD; radius and tangent are perpendicular"},{"math":"∠TAB = 90° − x","reason":"Split the right angle at A"},{"math":"∠ABD = 90°; ∠ADB = 90° − x","reason":"Angle in a semicircle and triangle angle sum"},{"math":"∠ADB = ∠ACB = ∠TAB","reason":"Angles in the same segment complete the proof"}],"alt":"∠DAT = 90°; let ∠DAB = x: Draw diameter AD; radius and tangent are perpendicular. ∠TAB = 90° − x: Split the right angle at A. ∠ABD = 90°; ∠ADB = 90° − x: Angle in a semicircle and triangle angle sum. ∠ADB = ∠ACB = ∠TAB: Angles in the same segment complete the proof"}}]},"revision":"ae30d4673c5a"},{"id":"g89-proof-of-the-circle-theorems-q08","topicId":"g89-proof-of-the-circle-theorems","topic":"Proof of the Circle Theorems","grade":"89","topicGrade":"89","strand":"G","difficulty":8,"marks":5,"tier":"H","calc":false,"text":"

AB and CD are chords of a circle with centre O.

AB = CD

M is the point on AB such that OM is perpendicular to AB.
N is the point on CD such that ON is perpendicular to CD.

Prove that OM = ON.

","answerType":"open","answer":"The perpendicular from the centre bisects each chord, so AM = CN; then Pythagoras (or RHS congruence) in triangles OMA and ONC gives OM = ON","answerDisplay":null,"accept":[],"parts":[],"solution":["If the equal chords are diameters, both perpendicular distances are zero and the result follows. Otherwise use the two right-angled triangles below.","Join O to A and O to C.","The perpendicular from the centre of a circle to a chord bisects the chord, so AM = AB ÷ 2 and CN = CD ÷ 2.","AB = CD (given), so AM = CN.","OA = OC because both are radii of the circle.","Triangle OMA has a right angle at M, so by Pythagoras, OM² = OA² − AM².","Triangle ONC has a right angle at N, so by Pythagoras, ON² = OC² − CN².","Since OA = OC and AM = CN, it follows that OM² = ON².","Distances are nonnegative, so OM = ON. This proves that equal chords of a circle are equidistant from the centre.","Alternatively, triangles OMA and ONC are congruent (RHS: right angles at M and N, hypotenuses OA = OC, and AM = CN), so OM = ON."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-right"},{"at":[2.4574561328669753,1.7207293090531381],"label":"A","style":"none","labelPos":"above"},{"at":[2.4574561328669753,-1.7207293090531381],"label":"B","style":"none","labelPos":"below-right"},{"at":[-2.457456132866975,1.720729309053139],"label":"C","style":"none","labelPos":"above"},{"at":[-2.457456132866976,-1.7207293090531375],"label":"D","style":"none","labelPos":"below-left"},{"at":[2.4574561328669753,0],"label":"M","style":"none","labelPos":"below-right"},{"at":[-2.4574561328669753,0],"label":"N","style":"none","labelPos":"below-left"}],"segments":[{"from":[2.4574561328669753,1.7207293090531381],"to":[2.4574561328669753,-1.7207293090531381],"ticks":1},{"from":[-2.457456132866975,1.720729309053139],"to":[-2.457456132866976,-1.7207293090531375],"ticks":1},{"from":[0,0],"to":[2.4574561328669753,0]},{"from":[0,0],"to":[-2.4574561328669753,0]}],"angles":[{"at":[2.4574561328669753,0],"from":[0,0],"to":[2.4574561328669753,1.7207293090531381],"label":"","right":true},{"at":[-2.4574561328669753,0],"from":[0,0],"to":[-2.457456132866975,1.720729309053139],"label":"","right":true}],"circles":[{"c":[0,0],"r":3}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Mark the given facts and any extra radii or construction lines you need.","Link equal lengths and angles to named geometric facts.","Connect those facts to the exact statement you need to prove."],"checks":["AM = AB/2; CN = CD/2; AB = CD — Perpendiculars from the centre bisect chords","AM = CN; OA = OC — Equal halves and equal radii","△OMA ≅ △ONC (RHS) — Right angles, equal hypotenuses and one equal side","OM = ON — Corresponding sides are equal"],"visuals":[{"caption":"Construction and equalities used in the proof. The written solution explains why each relationship holds.","diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-right"},{"at":[2.4574561328669753,1.7207293090531381],"label":"A","style":"none","labelPos":"above"},{"at":[2.4574561328669753,-1.7207293090531381],"label":"B","style":"none","labelPos":"below-right"},{"at":[-2.457456132866975,1.720729309053139],"label":"C","style":"none","labelPos":"above"},{"at":[-2.457456132866976,-1.7207293090531375],"label":"D","style":"none","labelPos":"below-left"},{"at":[2.4574561328669753,0],"label":"M","style":"none","labelPos":"below-right"},{"at":[-2.4574561328669753,0],"label":"N","style":"none","labelPos":"below-left"}],"segments":[{"from":[2.4574561328669753,1.7207293090531381],"to":[2.4574561328669753,-1.7207293090531381],"ticks":1},{"from":[-2.457456132866975,1.720729309053139],"to":[-2.457456132866976,-1.7207293090531375],"ticks":1},{"from":[0,0],"to":[2.4574561328669753,0]},{"from":[0,0],"to":[-2.4574561328669753,0]},{"from":[0,0],"to":[2.4574561328669753,1.7207293090531381],"label":"","answer":true,"ticks":2},{"from":[0,0],"to":[-2.457456132866975,1.720729309053139],"label":"","answer":true,"ticks":2},{"from":[2.4574561328669753,1.7207293090531381],"to":[2.4574561328669753,0],"label":"","answer":true,"ticks":3},{"from":[-2.457456132866975,1.720729309053139],"to":[-2.4574561328669753,0],"label":"","answer":true,"ticks":3}],"angles":[{"at":[2.4574561328669753,0],"from":[0,0],"to":[2.4574561328669753,1.7207293090531381],"label":"","right":true},{"at":[-2.4574561328669753,0],"from":[0,0],"to":[-2.457456132866975,1.720729309053139],"label":"","right":true}],"circles":[{"c":[0,0],"r":3}],"alt":"Worked construction for proof-of-the-circle-theorems-q08. Green additions correspond to the reasoning below."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"AM = AB/2; CN = CD/2; AB = CD","reason":"Perpendiculars from the centre bisect chords"},{"math":"AM = CN; OA = OC","reason":"Equal halves and equal radii"},{"math":"△OMA ≅ △ONC (RHS)","reason":"Right angles, equal hypotenuses and one equal side"},{"math":"OM = ON","reason":"Corresponding sides are equal"}],"alt":"AM = AB/2; CN = CD/2; AB = CD: Perpendiculars from the centre bisect chords. AM = CN; OA = OC: Equal halves and equal radii. △OMA ≅ △ONC (RHS): Right angles, equal hypotenuses and one equal side. OM = ON: Corresponding sides are equal"}}]},"revision":"dce05f884253"},{"id":"g89-proof-q01","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":1,"marks":2,"tier":"H","calc":false,"text":"

Prove that n2 + n is even for every integer n.

","answerType":"open","answer":"n^2 + n = n(n + 1), the product of consecutive integers, so it is even","answerDisplay":null,"accept":[],"parts":[],"solution":["Factorise: n^2 + n = n(n + 1).","n and n + 1 are consecutive integers, so one of them must be even.","The product of an even number with any integer is even.","Therefore n^2 + n is even for every integer n."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["n² + n = n(n + 1) — Factorise the expression","n or n + 1 is even — Consecutive integers include an even integer","n(n + 1) is even — An even factor makes the product even"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"n² + n = n(n + 1)","reason":"Factorise the expression"},{"math":"n or n + 1 is even","reason":"Consecutive integers include an even integer"},{"math":"n(n + 1) is even","reason":"An even factor makes the product even"}],"alt":"n² + n = n(n + 1): Factorise the expression. n or n + 1 is even: Consecutive integers include an even integer. n(n + 1) is even: An even factor makes the product even"}}]},"revision":"1ccf4c1f9ccc"},{"id":"g89-proof-q02","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":false,"text":"

Prove algebraically that the sum of any three consecutive integers is a multiple of 3.

","answerType":"open","answer":"(n - 1) + n + (n + 1) = 3n, a multiple of 3","answerDisplay":null,"accept":[],"parts":[],"solution":["Let the three consecutive integers be n - 1, n and n + 1, where n is an integer.","Sum = (n - 1) + n + (n + 1) = 3n.","3n is 3 multiplied by an integer, so it is a multiple of 3.","Therefore the sum of any three consecutive integers is a multiple of 3."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["n − 1, n, n + 1; n is an integer — Represent any three consecutive integers","(n − 1) + n + (n + 1) = 3n — Collect like terms","3n is a multiple of 3 — It is 3 times an integer"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"n − 1, n, n + 1; n is an integer","reason":"Represent any three consecutive integers"},{"math":"(n − 1) + n + (n + 1) = 3n","reason":"Collect like terms"},{"math":"3n is a multiple of 3","reason":"It is 3 times an integer"}],"alt":"n − 1, n, n + 1; n is an integer: Represent any three consecutive integers. (n − 1) + n + (n + 1) = 3n: Collect like terms. 3n is a multiple of 3: It is 3 times an integer"}}]},"revision":"ff6153ada3fe"},{"id":"g89-proof-q03","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

Prove algebraically that the sum of the squares of any two consecutive even numbers is a multiple of 4.

","answerType":"open","answer":"(2n)^2 + (2n + 2)^2 = 8n^2 + 8n + 4 = 4(2n^2 + 2n + 1), a multiple of 4","answerDisplay":null,"accept":[],"parts":[],"solution":["Let the two consecutive even numbers be 2n and 2n + 2, where n is an integer.","(2n)^2 + (2n + 2)^2 = 4n^2 + 4n^2 + 8n + 4 = 8n^2 + 8n + 4.","Factorise: 8n^2 + 8n + 4 = 4(2n^2 + 2n + 1).","2n^2 + 2n + 1 is an integer, so the sum is 4 multiplied by an integer: a multiple of 4."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["2n and 2n + 2; n is an integer — Represent consecutive even integers","(2n)² + (2n + 2)² = 8n² + 8n + 4 — Expand both squares","8n² + 8n + 4 = 4(2n² + 2n + 1) — Factor out 4","4 × an integer — Therefore the sum is a multiple of 4"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"2n and 2n + 2; n is an integer","reason":"Represent consecutive even integers"},{"math":"(2n)² + (2n + 2)² = 8n² + 8n + 4","reason":"Expand both squares"},{"math":"8n² + 8n + 4 = 4(2n² + 2n + 1)","reason":"Factor out 4"},{"math":"4 × an integer","reason":"Therefore the sum is a multiple of 4"}],"alt":"2n and 2n + 2; n is an integer: Represent consecutive even integers. (2n)² + (2n + 2)² = 8n² + 8n + 4: Expand both squares. 8n² + 8n + 4 = 4(2n² + 2n + 1): Factor out 4. 4 × an integer: Therefore the sum is a multiple of 4"}}]},"revision":"f84107351e6a"},{"id":"g89-proof-q04","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":false,"text":"

Prove algebraically that (3n + 1)2 − (3n − 1)2 is a multiple of 12 for every integer n.

","answerType":"open","answer":"(3n + 1)^2 - (3n - 1)^2 = 12n, a multiple of 12","answerDisplay":null,"accept":[],"parts":[],"solution":["Expand both squares: (3n + 1)^2 = 9n^2 + 6n + 1 and (3n - 1)^2 = 9n^2 - 6n + 1.","Subtract: (9n^2 + 6n + 1) - (9n^2 - 6n + 1) = 12n.","12n is 12 multiplied by an integer, so it is a multiple of 12.","Therefore (3n + 1)^2 - (3n - 1)^2 is a multiple of 12 for every integer n."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["(3n + 1)² = 9n² + 6n + 1 — Expand the first square","(3n − 1)² = 9n² − 6n + 1 — Expand the second square","Difference = 12n — Subtract every term of the second expression","12 × an integer — Therefore the difference is a multiple of 12"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"(3n + 1)² = 9n² + 6n + 1","reason":"Expand the first square"},{"math":"(3n − 1)² = 9n² − 6n + 1","reason":"Expand the second square"},{"math":"Difference = 12n","reason":"Subtract every term of the second expression"},{"math":"12 × an integer","reason":"Therefore the difference is a multiple of 12"}],"alt":"(3n + 1)² = 9n² + 6n + 1: Expand the first square. (3n − 1)² = 9n² − 6n + 1: Expand the second square. Difference = 12n: Subtract every term of the second expression. 12 × an integer: Therefore the difference is a multiple of 12"}}]},"revision":"1a499999501b"},{"id":"g89-proof-q05","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

Prove algebraically that the product of any two consecutive odd numbers is always 1 less than a multiple of 4.

","answerType":"open","answer":"(2n - 1)(2n + 1) = 4n^2 - 1, which is 1 less than the multiple of 4 given by 4n^2","answerDisplay":null,"accept":[],"parts":[],"solution":["Let the two consecutive odd numbers be 2n - 1 and 2n + 1, where n is an integer.","Product = (2n - 1)(2n + 1) = 4n^2 + 2n - 2n - 1 = 4n^2 - 1. The middle terms cancel; this is the difference-of-two-squares identity.","4n^2 is a multiple of 4, since n^2 is an integer.","So the product is 1 less than a multiple of 4."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["2n − 1 and 2n + 1; n is an integer — Represent consecutive odd integers","(2n − 1)(2n + 1) = 4n² − 1 — Use the difference of two squares","4 × an integer − 1 — The product is 1 less than a multiple of 4"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"2n − 1 and 2n + 1; n is an integer","reason":"Represent consecutive odd integers"},{"math":"(2n − 1)(2n + 1) = 4n² − 1","reason":"Use the difference of two squares"},{"math":"4 × an integer − 1","reason":"The product is 1 less than a multiple of 4"}],"alt":"2n − 1 and 2n + 1; n is an integer: Represent consecutive odd integers. (2n − 1)(2n + 1) = 4n² − 1: Use the difference of two squares. 4 × an integer − 1: The product is 1 less than a multiple of 4"}}]},"revision":"9673321684ed"},{"id":"g89-proof-q06","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

Prove algebraically that the difference between the squares of any two consecutive odd numbers is a multiple of 8.

","answerType":"open","answer":"(2n + 1)^2 - (2n - 1)^2 = 8n, a multiple of 8","answerDisplay":null,"accept":[],"parts":[],"solution":["Let the two consecutive odd numbers be 2n - 1 and 2n + 1, where n is an integer.","(2n + 1)^2 = 4n^2 + 4n + 1 and (2n - 1)^2 = 4n^2 - 4n + 1.","Difference = (4n^2 + 4n + 1) - (4n^2 - 4n + 1) = 8n.","8n is 8 multiplied by an integer, so the difference is a multiple of 8."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["2n − 1 and 2n + 1; n is an integer — Represent consecutive odd integers","(2n + 1)² − (2n − 1)² = 8n — Expand, subtract and collect terms","8 × an integer — Therefore the difference is a multiple of 8"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"2n − 1 and 2n + 1; n is an integer","reason":"Represent consecutive odd integers"},{"math":"(2n + 1)² − (2n − 1)² = 8n","reason":"Expand, subtract and collect terms"},{"math":"8 × an integer","reason":"Therefore the difference is a multiple of 8"}],"alt":"2n − 1 and 2n + 1; n is an integer: Represent consecutive odd integers. (2n + 1)² − (2n − 1)² = 8n: Expand, subtract and collect terms. 8 × an integer: Therefore the difference is a multiple of 8"}}]},"revision":"1132e31ac6cc"},{"id":"g89-proof-q07","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":7,"marks":4,"tier":"H","calc":true,"text":"

  1. Show that (n + 3)2 − (n − 2)2 ≡ 5(2n + 1)
  2. Hence explain why the difference between (n + 3)2 and (n − 2)2 is always an odd multiple of 5, for every integer n.

","answerType":"open","answer":"a) both sides equal 10n + 5; b) 2n + 1 is odd, so 5(2n + 1) is 5 times an odd number","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(n + 3)^2 - (n - 2)^2 = (n^2 + 6n + 9) - (n^2 - 4n + 4) = 10n + 5 = 5(2n + 1)"},{"label":"b","answer":"2n + 1 is odd for every integer n, so the difference is 5 multiplied by an odd number: an odd multiple of 5"}],"solution":["a) Expand: (n + 3)^2 = n^2 + 6n + 9 and (n - 2)^2 = n^2 - 4n + 4.","Subtract: (n^2 + 6n + 9) - (n^2 - 4n + 4) = 10n + 5.","Factorise: 10n + 5 = 5(2n + 1), as required.","b) For every integer n, 2n + 1 is an odd number.","So the difference is 5 multiplied by an odd number, which is an odd multiple of 5."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["(n + 3)² − (n − 2)² = 10n + 5 — Expand both squares and subtract","10n + 5 = 5(2n + 1) — Factor out 5","2n + 1 is odd for integer n — The difference is an odd multiple of 5"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"(n + 3)² − (n − 2)² = 10n + 5","reason":"Expand both squares and subtract"},{"math":"10n + 5 = 5(2n + 1)","reason":"Factor out 5"},{"math":"2n + 1 is odd for integer n","reason":"The difference is an odd multiple of 5"}],"alt":"(n + 3)² − (n − 2)² = 10n + 5: Expand both squares and subtract. 10n + 5 = 5(2n + 1): Factor out 5. 2n + 1 is odd for integer n: The difference is an odd multiple of 5"}}]},"revision":"f47812f51fad"},{"id":"g89-proof-q08","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":8,"marks":4,"tier":"H","calc":false,"text":"

Prove algebraically that the sum of the squares of any two consecutive integers is an odd number.

","answerType":"open","answer":"n^2 + (n + 1)^2 = 2(n^2 + n) + 1, which is odd","answerDisplay":null,"accept":[],"parts":[],"solution":["Let the two consecutive integers be n and n + 1, where n is an integer.","n^2 + (n + 1)^2 = n^2 + n^2 + 2n + 1 = 2n^2 + 2n + 1.","Factorise the first two terms: 2n^2 + 2n + 1 = 2(n^2 + n) + 1.","n^2 + n is an integer, so this is 2 multiplied by an integer plus 1: an odd number.","Therefore the sum of the squares of any two consecutive integers is odd."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["n and n + 1; n is an integer — Represent consecutive integers","n² + (n + 1)² = 2n² + 2n + 1 — Expand and collect terms","2(n² + n) + 1 — This is twice an integer plus 1: odd"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"n and n + 1; n is an integer","reason":"Represent consecutive integers"},{"math":"n² + (n + 1)² = 2n² + 2n + 1","reason":"Expand and collect terms"},{"math":"2(n² + n) + 1","reason":"This is twice an integer plus 1: odd"}],"alt":"n and n + 1; n is an integer: Represent consecutive integers. n² + (n + 1)² = 2n² + 2n + 1: Expand and collect terms. 2(n² + n) + 1: This is twice an integer plus 1: odd"}}]},"revision":"30631401dc92"},{"id":"g89-proof-q09","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

Prove algebraically that the square of any odd number is 1 more than a multiple of 8.

","answerType":"open","answer":"(2n + 1)^2 = 4n(n + 1) + 1; n(n + 1) is even, so 4n(n + 1) is a multiple of 8","answerDisplay":null,"accept":[],"parts":[],"solution":["Let the odd number be 2n + 1, where n is an integer.","(2n + 1)^2 = 4n^2 + 4n + 1 = 4n(n + 1) + 1.","n and n + 1 are consecutive integers, so n(n + 1) is even: write n(n + 1) = 2k for an integer k.","Then (2n + 1)^2 = 4 x 2k + 1 = 8k + 1.","8k is a multiple of 8, so the square of any odd number is 1 more than a multiple of 8."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["(2n + 1)² = 4n(n + 1) + 1 — Represent an odd integer, expand and factorise","n(n + 1) = 2k for an integer k — A product of consecutive integers is even","4(2k) + 1 = 8k + 1 — The square is 1 more than a multiple of 8"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"(2n + 1)² = 4n(n + 1) + 1","reason":"Represent an odd integer, expand and factorise"},{"math":"n(n + 1) = 2k for an integer k","reason":"A product of consecutive integers is even"},{"math":"4(2k) + 1 = 8k + 1","reason":"The square is 1 more than a multiple of 8"}],"alt":"(2n + 1)² = 4n(n + 1) + 1: Represent an odd integer, expand and factorise. n(n + 1) = 2k for an integer k: A product of consecutive integers is even. 4(2k) + 1 = 8k + 1: The square is 1 more than a multiple of 8"}}]},"revision":"b0ad641cfad2"},{"id":"g89-proof-q10","topicId":"g89-proof","topic":"Proof","grade":"89","topicGrade":"89","strand":"A","difficulty":10,"marks":6,"tier":"H","calc":true,"text":"

Prove algebraically that the difference between the squares of any two odd numbers is a multiple of 8.

","answerType":"open","answer":"(2a + 1)^2 - (2b + 1)^2 = 4a(a + 1) - 4b(b + 1) = 8m - 8k, a multiple of 8","answerDisplay":null,"accept":[],"parts":[],"solution":["Let the two odd numbers be 2a + 1 and 2b + 1, where a and b are integers (they need not be consecutive).","(2a + 1)^2 - (2b + 1)^2 = (4a^2 + 4a + 1) - (4b^2 + 4b + 1) = 4a^2 + 4a - 4b^2 - 4b.","Factorise: = 4a(a + 1) - 4b(b + 1).","a(a + 1) is a product of consecutive integers, so it is even: a(a + 1) = 2m for an integer m. Similarly b(b + 1) = 2k.","So the difference = 4 x 2m - 4 x 2k = 8m - 8k = 8(m - k).","m - k is an integer, so the difference between the squares of any two odd numbers is a multiple of 8."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose general variables and state what they represent.","Write the algebraic steps, with a reason for each change.","Explain why the result proves the claim for every allowed value."],"checks":["Odd integers: 2a + 1 and 2b + 1 — a and b are independent integers","Difference = 4a(a + 1) − 4b(b + 1) — Expand both squares and factorise","a(a + 1) = 2m; b(b + 1) = 2k — Each consecutive product is even","Difference = 8(m − k) — m − k is an integer: a multiple of 8"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"Odd integers: 2a + 1 and 2b + 1","reason":"a and b are independent integers"},{"math":"Difference = 4a(a + 1) − 4b(b + 1)","reason":"Expand both squares and factorise"},{"math":"a(a + 1) = 2m; b(b + 1) = 2k","reason":"Each consecutive product is even"},{"math":"Difference = 8(m − k)","reason":"m − k is an integer: a multiple of 8"}],"alt":"Odd integers: 2a + 1 and 2b + 1: a and b are independent integers. Difference = 4a(a + 1) − 4b(b + 1): Expand both squares and factorise. a(a + 1) = 2m; b(b + 1) = 2k: Each consecutive product is even. Difference = 8(m − k): m − k is an integer: a multiple of 8"}}]},"revision":"b38e1b15b539"},{"id":"g89-quadratic-inequalities-q01","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":1,"marks":2,"tier":"H","calc":false,"text":"

Solve (x + 2)(x − 5) < 0

","answerType":"exact","answer":"-2 < x < 5","answerDisplay":null,"accept":["-2 -2 and x < 5","-2 < x and x < 5"],"parts":[],"solution":["The graph of y = (x + 2)(x - 5) is a U-shaped parabola with roots at x = -2 and x = 5.","Between -2 and 5, x + 2 is positive and x - 5 is negative, so their product is negative. Outside that interval the factors have the same sign, giving a positive product. At either endpoint the product is zero, which is excluded by < 0.","The product is negative between the roots.","So -2 < x < 5."],"diagram":null,"draw":false,"revision":"c11d9dcebaf4"},{"id":"g89-quadratic-inequalities-q02","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":2,"marks":2,"tier":"H","calc":true,"text":"

Solve (x − 3)(x + 7) ≥ 0

","answerType":"exact","answer":"x <= -7 or x >= 3","answerDisplay":null,"accept":["x =< -7 or x >= 3","x <= -7, x >= 3","x >= 3 or x <= -7"],"parts":[],"solution":["The roots of (x - 3)(x + 7) = 0 are x = 3 and x = -7.","The parabola is U-shaped, so the product is greater than or equal to zero outside the roots (including the roots themselves because the inequality is not strict).","So x <= -7 or x >= 3."],"diagram":null,"draw":false,"revision":"8fd3130c1df0"},{"id":"g89-quadratic-inequalities-q03","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":false,"text":"

Solve x2 − 9x + 18 < 0

","answerType":"exact","answer":"3 < x < 6","answerDisplay":null,"accept":["3 3 and x < 6"],"parts":[],"solution":["Factorise: x^2 - 9x + 18 = (x - 3)(x - 6).","The roots are x = 3 and x = 6, and the parabola is U-shaped.","The expression is negative between the roots, so 3 < x < 6."],"diagram":null,"draw":false,"revision":"be8558ced2de"},{"id":"g89-quadratic-inequalities-q04","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

Solve x2 + 3x − 28 ≥ 0

","answerType":"exact","answer":"x <= -7 or x >= 4","answerDisplay":null,"accept":["x =< -7 or x >= 4","x >= 4 or x <= -7"],"parts":[],"solution":["Factorise: x^2 + 3x - 28 = (x + 7)(x - 4).","The roots are x = -7 and x = 4, and the parabola is U-shaped.","The expression is greater than or equal to zero outside the roots.","So x <= -7 or x >= 4."],"diagram":null,"draw":false,"revision":"4342cc4684a1"},{"id":"g89-quadratic-inequalities-q05","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":5,"marks":5,"tier":"H","calc":true,"text":"

n is an integer such that

n2 − 2n − 15 ≤ 0  and  3n + 1 > 4

Find all the possible values of n.

","answerType":"exact","answer":"2, 3, 4, 5","answerDisplay":null,"accept":["2,3,4,5","n = 2, 3, 4, 5","2 3 4 5"],"parts":[],"solution":["Factorise the quadratic: n^2 - 2n - 15 = (n - 5)(n + 3), so (n - 5)(n + 3) <= 0.","This gives -3 <= n <= 5.","Solve the linear inequality: 3n + 1 > 4 gives 3n > 3, so n > 1.","The integers satisfying both -3 <= n <= 5 and n > 1 are 2, 3, 4 and 5."],"diagram":null,"draw":false,"revision":"20b4652e309b"},{"id":"g89-quadratic-inequalities-q06","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":6,"marks":5,"tier":"H","calc":true,"text":"

Solve

5x − 4 > 2x + 8  and  x2 − 10x + 21 < 0

Give your answer as a single inequality.

","answerType":"exact","answer":"4 < x < 7","answerDisplay":null,"accept":["4 4 and x < 7"],"parts":[],"solution":["Solve the linear inequality: 5x - 4 > 2x + 8 gives 3x > 12, so x > 4.","Factorise the quadratic: x^2 - 10x + 21 = (x - 3)(x - 7), so (x - 3)(x - 7) < 0.","The expression is negative between the roots, so 3 < x < 7.","Both must hold: x > 4 combined with 3 < x < 7 gives 4 < x < 7."],"diagram":null,"draw":false,"revision":"9e4de888a83e"},{"id":"g89-quadratic-inequalities-q07","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":7,"marks":5,"tier":"H","calc":false,"text":"

Solve 3 < x2 − 6 ≤ 19

","answerType":"exact","answer":"-5 <= x < -3 or 3 < x <= 5","answerDisplay":null,"accept":["-5 =< x < -3 or 3 < x =< 5","3 < x <= 5 or -5 <= x < -3"],"parts":[],"solution":["Add 6 to every part: 9 < x^2 <= 25.","From x^2 > 9: x < -3 or x > 3.","From x^2 <= 25: -5 <= x <= 5.","Combine the two conditions on each side of zero.","Answer: -5 <= x < -3 or 3 < x <= 5."],"diagram":null,"draw":false,"revision":"7f2f0edfc0be"},{"id":"g89-quadratic-inequalities-q08","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

Solve 2x2 + 5x − 3 ≤ 0

","answerType":"exact","answer":"-3 <= x <= 1/2","answerDisplay":"-3 ≤ x ≤ 12","accept":["-3 =< x =< 0.5","-3 <= x <= 0.5","x >= -3 and x <= 1/2"],"parts":[],"solution":["Factorise: 2x^2 + 5x - 3 = (2x - 1)(x + 3).","Check: (2x - 1)(x + 3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3.","The roots are x = 12 and x = -3, and the parabola is U-shaped.","The expression is less than or equal to zero between the roots.","So -3 ≤ x ≤ 12."],"diagram":null,"draw":false,"revision":"1c3872a63ba3"},{"id":"g89-quadratic-inequalities-q09","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":9,"marks":5,"tier":"H","calc":false,"text":"

Solve

x2 − x − 20 ≤ 0  and  x2 − 4 > 0

","answerType":"exact","answer":"-4 <= x < -2 or 2 < x <= 5","answerDisplay":null,"accept":["-4 =< x < -2 or 2 < x =< 5","2 < x <= 5 or -4 <= x < -2"],"parts":[],"solution":["Factorise the first quadratic: x^2 - x - 20 = (x - 5)(x + 4), so (x - 5)(x + 4) <= 0 gives -4 <= x <= 5.","Factorise the second: x^2 - 4 = (x - 2)(x + 2), so (x - 2)(x + 2) > 0 gives x < -2 or x > 2.","Both must hold, so intersect -4 <= x <= 5 with x < -2 or x > 2.","Answer: -4 <= x < -2 or 2 < x <= 5."],"diagram":null,"draw":false,"revision":"5624c9d4a1ba"},{"id":"g89-quadratic-inequalities-q10","topicId":"g89-quadratic-inequalities","topic":"Quadratic Inequalities","grade":"89","topicGrade":"89","strand":"A","difficulty":10,"marks":6,"tier":"H","calc":true,"text":"

A rectangular allotment bed has length (x + 7) metres and width (x − 2) metres, where x > 2.

The area of the bed is less than 90 m2.

  1. Show that x2 + 5x − 104 < 0
  2. Find the range of possible values of x.

","answerType":"multi","answer":"a) proof; b) 2 < x < 8","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(x + 7)(x - 2) < 90 expands to x^2 + 5x - 14 < 90, so x^2 + 5x - 104 < 0"},{"label":"b","answer":"2 < x < 8"}],"solution":["a) Area = (x + 7)(x - 2) = x^2 + 5x - 14. The area is less than 90, so x^2 + 5x - 14 < 90.","Subtract 90 from both sides: x^2 + 5x - 104 < 0, as required.","b) Factorise: x^2 + 5x - 104 = (x + 13)(x - 8), since 13 times 8 = 104 and 13 - 8 = 5.","(x + 13)(x - 8) < 0 gives -13 < x < 8.","The question states x > 2 (the width must be positive), so the range is 2 < x < 8."],"diagram":null,"draw":false,"proofSupport":{"plan":["Write an expression for the area using length × width.","Rearrange the area inequality and find its boundary values.","Combine your solution with the restriction on x in the question."],"checks":["Area = (x + 7)(x − 2) < 90 — Use length × width","x² + 5x − 104 < 0 — Expand and subtract 90","(x + 13)(x − 8) < 0 ⇒ −13 < x < 8 — The upward quadratic is negative between its roots","2 < x < 8 — Also use the condition x > 2"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"Area = (x + 7)(x − 2) < 90","reason":"Use length × width"},{"math":"x² + 5x − 104 < 0","reason":"Expand and subtract 90"},{"math":"(x + 13)(x − 8) < 0 ⇒ −13 < x < 8","reason":"The upward quadratic is negative between its roots"},{"math":"2 < x < 8","reason":"Also use the condition x > 2"}],"alt":"Area = (x + 7)(x − 2) < 90: Use length × width. x² + 5x − 104 < 0: Expand and subtract 90. (x + 13)(x − 8) < 0 ⇒ −13 < x < 8: The upward quadratic is negative between its roots. 2 < x < 8: Also use the condition x > 2"}}]},"revision":"1cf301a02595"},{"id":"g89-quadratic-simultaneous-equations-q01","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":1,"marks":4,"tier":"H","calc":true,"text":"

Solve the simultaneous equations

y = x2

y = x + 6

","answerType":"exact","answer":"x = 3, y = 9 and x = -2, y = 4","answerDisplay":null,"accept":["(3, 9) and (-2, 4)"],"parts":[],"solution":["Substitute y = x^2 into the second equation: x^2 = x + 6.","Rearrange: x^2 - x - 6 = 0.","Factorise: (x - 3)(x + 2) = 0, so x = 3 or x = -2.","When x = 3, y = 9. When x = -2, y = 4.","Check in y = x + 6: 3 + 6 = 9 and -2 + 6 = 4."],"diagram":null,"draw":false,"revision":"ba98d93964b8"},{"id":"g89-quadratic-simultaneous-equations-q02","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":2,"marks":4,"tier":"H","calc":true,"text":"

Solve the simultaneous equations

y = x2 − 3x

y = 2x − 6

","answerType":"exact","answer":"x = 2, y = -2 and x = 3, y = 0","answerDisplay":null,"accept":["(2, -2) and (3, 0)"],"parts":[],"solution":["Set the expressions for y equal: x^2 - 3x = 2x - 6.","Rearrange: x^2 - 5x + 6 = 0.","Factorise: (x - 2)(x - 3) = 0, so x = 2 or x = 3.","When x = 2, y = 2(2) - 6 = -2. When x = 3, y = 2(3) - 6 = 0.","Check in the quadratic: 4 - 6 = -2 and 9 - 9 = 0."],"diagram":null,"draw":false,"revision":"2f6eef3c1d69"},{"id":"g89-quadratic-simultaneous-equations-q03","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

Solve the simultaneous equations

x2 + y2 = 20

y = 2x

","answerType":"exact","answer":"x = 2, y = 4 and x = -2, y = -4","answerDisplay":null,"accept":["(2, 4) and (-2, -4)"],"parts":[],"solution":["Substitute y = 2x into the circle equation: x^2 + 4x^2 = 20.","So 5x^2 = 20, giving x^2 = 4.","x = 2 or x = -2.","When x = 2, y = 4. When x = -2, y = -4.","Check: 4 + 16 = 20 for both pairs."],"diagram":null,"draw":false,"revision":"a2701f1029a4"},{"id":"g89-quadratic-simultaneous-equations-q04","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

Solve the simultaneous equations

y = x2 + 2x − 1

y = 3x + 5

","answerType":"exact","answer":"x = 3, y = 14 and x = -2, y = -1","answerDisplay":null,"accept":["(3, 14) and (-2, -1)"],"parts":[],"solution":["Set the expressions for y equal: x^2 + 2x - 1 = 3x + 5.","Rearrange: x^2 - x - 6 = 0.","Factorise: (x - 3)(x + 2) = 0, so x = 3 or x = -2.","When x = 3, y = 3(3) + 5 = 14. When x = -2, y = 3(-2) + 5 = -1.","Check in the quadratic: 9 + 6 - 1 = 14 and 4 - 4 - 1 = -1."],"diagram":null,"draw":false,"revision":"ab9a388415d8"},{"id":"g89-quadratic-simultaneous-equations-q05","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":5,"marks":5,"tier":"H","calc":true,"text":"

Solve the simultaneous equations

x2 + y2 = 13

x + y = 5

","answerType":"exact","answer":"x = 2, y = 3 and x = 3, y = 2","answerDisplay":null,"accept":["(2, 3) and (3, 2)"],"parts":[],"solution":["From the linear equation, y = 5 - x.","Substitute: x^2 + (5 - x)^2 = 13.","Expand: x^2 + 25 - 10x + x^2 = 13, so 2x^2 - 10x + 12 = 0.","Divide by 2: x^2 - 5x + 6 = 0, so (x - 2)(x - 3) = 0 and x = 2 or x = 3.","When x = 2, y = 3. When x = 3, y = 2.","Check: 4 + 9 = 13 and 9 + 4 = 13."],"diagram":null,"draw":false,"revision":"e1f3346cc9d3"},{"id":"g89-quadratic-simultaneous-equations-q06","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":6,"marks":5,"tier":"H","calc":true,"text":"

Solve the simultaneous equations

y = 2x2 − 3

y = 5x

","answerType":"exact","answer":"x = 3, y = 15 and x = -1/2, y = -5/2","answerDisplay":"x = 3, y = 15 and x = -12, y = -52","accept":["(3, 15) and (-1/2, -5/2)","(3, 15) and (-0.5, -2.5)"],"parts":[],"solution":["Set the expressions for y equal: 2x^2 - 3 = 5x.","Rearrange: 2x^2 - 5x - 3 = 0.","Factorise: (2x + 1)(x - 3) = 0, so x = -12 or x = 3.","When x = 3, y = 5(3) = 15. When x = -12, y = 5(-12) = -52.","Check in the quadratic: 2(9) - 3 = 15 and 2(14) - 3 = -52."],"diagram":null,"draw":false,"revision":"1121a2dd7a6b"},{"id":"g89-quadratic-simultaneous-equations-q07","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":7,"marks":5,"tier":"H","calc":true,"text":"

The circle C has equation x2 + y2 = 10

The line L has equation y = x − 2

Find the coordinates of the points where L intersects C.

","answerType":"exact","answer":"(3, 1) and (-1, -3)","answerDisplay":null,"accept":["x = 3, y = 1 and x = -1, y = -3"],"parts":[],"solution":["Substitute y = x - 2 into the circle equation: x^2 + (x - 2)^2 = 10.","Expand: x^2 + x^2 - 4x + 4 = 10, so 2x^2 - 4x - 6 = 0.","Divide by 2: x^2 - 2x - 3 = 0, so (x - 3)(x + 1) = 0 and x = 3 or x = -1.","When x = 3, y = 1. When x = -1, y = -3.","Check: 9 + 1 = 10 and 1 + 9 = 10."],"diagram":null,"draw":false,"revision":"c0594d4a651b"},{"id":"g89-quadratic-simultaneous-equations-q08","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

Solve the simultaneous equations

y = x2

x + y = 12

","answerType":"exact","answer":"x = 3, y = 9 and x = -4, y = 16","answerDisplay":null,"accept":["(3, 9) and (-4, 16)"],"parts":[],"solution":["Substitute y = x^2 into the linear equation: x + x^2 = 12.","Rearrange: x^2 + x - 12 = 0.","Factorise: (x + 4)(x - 3) = 0, so x = -4 or x = 3.","When x = 3, y = 9. When x = -4, y = 16.","Check in x + y = 12: 3 + 9 = 12 and -4 + 16 = 12."],"diagram":null,"draw":false,"revision":"f9c329cd043d"},{"id":"g89-quadratic-simultaneous-equations-q09","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

The curve C has equation y = x2 − 4x + 7

The line L has equation y = 2x − 2

Show that L touches C at exactly one point, and find the coordinates of this point.

","answerType":"open","answer":"x^2 - 6x + 9 = (x - 3)^2 = 0 has the repeated root x = 3, so exactly one point of intersection, (3, 4)","answerDisplay":null,"accept":["(3, 4)"],"parts":[],"solution":["Set the expressions for y equal: x^2 - 4x + 7 = 2x - 2.","Rearrange: x^2 - 6x + 9 = 0.","Factorise: (x - 3)^2 = 0.","Both factors in (x - 3)(x - 3) are zero at the same value, x = 3; this is called a repeated root. Also, curve minus line = (x - 3)^2 is never negative, so the curve stays on one side of the line and touches it at this one point.","When x = 3, y = 2(3) - 2 = 4, so the point is (3, 4).","Check on the curve: 9 - 12 + 7 = 4."],"diagram":null,"draw":false,"proofSupport":{"plan":["Equate the two expressions for y.","Solve the resulting equation and count the distinct intersections.","Find the coordinates and explain why the line touches the curve."],"checks":["x² − 4x + 7 = 2x − 2 — At an intersection the y-values agree","x² − 6x + 9 = (x − 3)² = 0 — Rearrange and factorise","One repeated root: x = 3 — The line touches the curve at one point","y = 4; point of contact (3, 4) — Substitute into either original equation"],"visuals":[{"caption":"The repeated root proves tangency. The diagram shows the single point of contact.","diagram":{"type":"axes","x":{"min":0,"max":6,"step":1},"y":{"min":-2,"max":20,"step":2},"curves":[{"fn":"x*x-4*x+7","from":0,"to":6}],"lines":[{"m":2,"c":-2,"label":"L: y = 2x − 2"}],"points":[{"at":[3,4],"label":"(3, 4)","answer":true,"labelPos":"below-right"}],"alt":"The line touches the quadratic at (3, 4)."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"x² − 4x + 7 = 2x − 2","reason":"At an intersection the y-values agree"},{"math":"x² − 6x + 9 = (x − 3)² = 0","reason":"Rearrange and factorise"},{"math":"One repeated root: x = 3","reason":"The line touches the curve at one point"},{"math":"y = 4; point of contact (3, 4)","reason":"Substitute into either original equation"}],"alt":"x² − 4x + 7 = 2x − 2: At an intersection the y-values agree. x² − 6x + 9 = (x − 3)² = 0: Rearrange and factorise. One repeated root: x = 3: The line touches the curve at one point. y = 4; point of contact (3, 4): Substitute into either original equation"}}]},"revision":"3c0c64bd784d"},{"id":"g89-quadratic-simultaneous-equations-q10","topicId":"g89-quadratic-simultaneous-equations","topic":"Quadratic Simultaneous Equations","grade":"89","topicGrade":"89","strand":"A","difficulty":10,"marks":5,"tier":"H","calc":true,"text":"

Solve the simultaneous equations

x2 + y2 = 9

x + 2y = 3

","answerType":"exact","answer":"x = 3, y = 0 and x = -9/5, y = 12/5","answerDisplay":"x = 3, y = 0 and x = -95, y = 125","accept":["(3, 0) and (-9/5, 12/5)","(3, 0) and (-1.8, 2.4)"],"parts":[],"solution":["From the linear equation, x = 3 - 2y.","Substitute: (3 - 2y)^2 + y^2 = 9.","Expand: 9 - 12y + 4y^2 + y^2 = 9, so 5y^2 - 12y = 0.","Factorise: y(5y - 12) = 0, so y = 0 or y = 125.","When y = 0, x = 3. When y = 125, x = 3 - 245 = -95.","Check: 9 + 0 = 9 and 8125 + 14425 = 22525 = 9."],"diagram":null,"draw":false,"revision":"61ebb7ae46f4"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q01","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":1,"marks":2,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

2   7   14   23   34

Find an expression, in terms of n, for the nth term of this sequence.

","answerType":"exact","answer":"n^2 + 2n - 1","answerDisplay":null,"accept":["n2+2n-1","n^2+2n-1","(n+1)^2 - 2"],"parts":[],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","First differences: 5, 7, 9, 11. Second difference: 2, so the n^2 coefficient is 2 divided by 2 = 1.","Subtract n^2 (1, 4, 9, 16, 25) from the sequence: 1, 3, 5, 7, 9, which has nth term 2n - 1.","So the nth term is n^2 + 2n - 1.","Check: n = 1 gives 1 + 2 - 1 = 2; n = 2 gives 4 + 4 - 1 = 7; n = 3 gives 9 + 6 - 1 = 14; n = 4 gives 16 + 8 - 1 = 23; n = 5 gives 25 + 10 - 1 = 34."],"diagram":null,"draw":false,"revision":"44df0aa5dc12"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q02","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":2,"marks":2,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

5   11   19   29   41

Find an expression, in terms of n, for the nth term of this sequence.

","answerType":"exact","answer":"n^2 + 3n + 1","answerDisplay":null,"accept":["n2+3n+1","n^2+3n+1"],"parts":[],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","First differences: 6, 8, 10, 12. Second difference: 2, so the n^2 coefficient is 1.","Subtract n^2 (1, 4, 9, 16, 25) from the sequence: 4, 7, 10, 13, 16, which has nth term 3n + 1.","So the nth term is n^2 + 3n + 1.","Check: n = 1 gives 1 + 3 + 1 = 5; n = 2 gives 4 + 6 + 1 = 11; n = 3 gives 9 + 9 + 1 = 19; n = 4 gives 16 + 12 + 1 = 29; n = 5 gives 25 + 15 + 1 = 41."],"diagram":null,"draw":false,"revision":"b506635b6621"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q03","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":3,"marks":3,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

1   10   25   46   73

Find an expression, in terms of n, for the nth term of this sequence.

","answerType":"exact","answer":"3n^2 - 2","answerDisplay":null,"accept":["3n2-2","3n^2-2"],"parts":[],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","First differences: 9, 15, 21, 27. Second difference: 6, so the n^2 coefficient is 6 divided by 2 = 3.","Subtract 3n^2 (3, 12, 27, 48, 75) from the sequence: -2, -2, -2, -2, -2, a constant -2.","So the nth term is 3n^2 - 2.","Check: n = 1 gives 3 - 2 = 1; n = 2 gives 12 - 2 = 10; n = 3 gives 27 - 2 = 25; n = 4 gives 48 - 2 = 46; n = 5 gives 75 - 2 = 73."],"diagram":null,"draw":false,"revision":"bf072c382831"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q04","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":4,"marks":3,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

6   15   28   45   66

Find an expression, in terms of n, for the nth term of this sequence.

","answerType":"exact","answer":"2n^2 + 3n + 1","answerDisplay":null,"accept":["2n2+3n+1","2n^2+3n+1","(2n+1)(n+1)"],"parts":[],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","First differences: 9, 13, 17, 21. Second difference: 4, so the n^2 coefficient is 2.","Subtract 2n^2 (2, 8, 18, 32, 50) from the sequence: 4, 7, 10, 13, 16, which has nth term 3n + 1.","So the nth term is 2n^2 + 3n + 1.","Check: n = 1 gives 2 + 3 + 1 = 6; n = 2 gives 8 + 6 + 1 = 15; n = 3 gives 18 + 9 + 1 = 28; n = 4 gives 32 + 12 + 1 = 45; n = 5 gives 50 + 15 + 1 = 66."],"diagram":null,"draw":false,"revision":"cb0a61fffbe4"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q05","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":5,"marks":3,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

0   7   18   33   52

Find an expression, in terms of n, for the nth term of this sequence.

","answerType":"exact","answer":"2n^2 + n - 3","answerDisplay":null,"accept":["2n2+n-3","2n^2+n-3","(2n+3)(n-1)"],"parts":[],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","First differences: 7, 11, 15, 19. Second difference: 4, so the n^2 coefficient is 2.","Subtract 2n^2 (2, 8, 18, 32, 50) from the sequence: -2, -1, 0, 1, 2, which has nth term n - 3.","So the nth term is 2n^2 + n - 3.","Check: n = 1 gives 2 + 1 - 3 = 0; n = 2 gives 8 + 2 - 3 = 7; n = 3 gives 18 + 3 - 3 = 18; n = 4 gives 32 + 4 - 3 = 33; n = 5 gives 50 + 5 - 3 = 52."],"diagram":null,"draw":false,"revision":"1842059a7003"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q06","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":6,"marks":3,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

4   13   26   43   64

  1. Find an expression, in terms of n, for the nth term of this sequence.
  2. Hence find the 10th term of the sequence.

","answerType":"multi","answer":"a) 2n^2 + 3n - 1; b) 229","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2n^2 + 3n - 1"},{"label":"b","answer":"229"}],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","a) First differences: 9, 13, 17, 21. Second difference: 4, so the n^2 coefficient is 2.","Subtract 2n^2 (2, 8, 18, 32, 50) from the sequence: 2, 5, 8, 11, 14, which has nth term 3n - 1.","So the nth term is 2n^2 + 3n - 1.","Check: n = 1 gives 2 + 3 - 1 = 4; n = 2 gives 8 + 6 - 1 = 13; n = 3 gives 18 + 9 - 1 = 26; n = 4 gives 32 + 12 - 1 = 43; n = 5 gives 50 + 15 - 1 = 64.","b) When n = 10: 2(100) + 30 - 1 = 200 + 30 - 1 = 229."],"diagram":null,"draw":false,"revision":"6177288663f5"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q07","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

20   38   52   62   68

Find an expression, in terms of n, for the nth term of this sequence.

","answerType":"exact","answer":"-2n^2 + 24n - 2","answerDisplay":null,"accept":["-2n2+24n-2","-2n^2+24n-2","24n - 2n^2 - 2"],"parts":[],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","First differences: 18, 14, 10, 6. Second difference: -4, so the n^2 coefficient is -2.","Subtract -2n^2 (-2, -8, -18, -32, -50) from the sequence: 22, 46, 70, 94, 118, which has nth term 24n - 2.","So the nth term is -2n^2 + 24n - 2.","Check: n = 1 gives -2 + 24 - 2 = 20; n = 2 gives -8 + 48 - 2 = 38; n = 3 gives -18 + 72 - 2 = 52; n = 4 gives -32 + 96 - 2 = 62; n = 5 gives -50 + 120 - 2 = 68."],"diagram":null,"draw":false,"revision":"b7505db1123c"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q08","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":8,"marks":3,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

3   8   14   21   29

Find an expression, in terms of n, for the nth term of this sequence.

","answerType":"exact","answer":"(n^2 + 7n - 2)/2","answerDisplay":"n2 + 7n - 22","accept":["0.5n^2 + 3.5n - 1","(1/2)n^2 + (7/2)n - 1","n(n+7)/2 - 1"],"parts":[],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","First differences: 5, 6, 7, 8. Second difference: 1, so the n^2 coefficient is 12.","Subtract n^22 (0.5, 2, 4.5, 8, 12.5) from the sequence: 2.5, 6, 9.5, 13, 16.5, an arithmetic sequence with nth term 3.5n - 1.","So the nth term is 0.5n^2 + 3.5n - 1, which can be written as n^2 + 7n - 22.","Check: n = 1 gives 1 + 7 - 22 = 3; n = 2 gives 4 + 14 - 22 = 8; n = 3 gives 9 + 21 - 22 = 14; n = 4 gives 16 + 28 - 22 = 21; n = 5 gives 25 + 35 - 22 = 29."],"diagram":null,"draw":false,"revision":"a32704d76ee4"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q09","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":9,"marks":6,"tier":"H","calc":true,"text":"

The nth term of a quadratic sequence is an2 + bn, where a and b are constants.

The 2nd term of the sequence is 10.

The 4th term of the sequence is 36.

  1. Find an expression, in terms of n, for the nth term of the sequence.
  2. Find the value of n for which the nth term of the sequence is 105.

","answerType":"multi","answer":"a) 2n^2 + n; b) n = 7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2n^2 + n"},{"label":"b","answer":"7"}],"solution":["a) n = 2 gives 4a + 2b = 10, so 2a + b = 5. n = 4 gives 16a + 4b = 36, so 4a + b = 9.","Subtracting: 2a = 4, so a = 2, and then b = 5 - 4 = 1.","The nth term is 2n^2 + n.","Check: 2nd term = 8 + 2 = 10; 4th term = 32 + 4 = 36.","b) Solve 2n^2 + n = 105, so 2n^2 + n - 105 = 0.","Factorise: (2n + 15)(n - 7) = 2n^2 - 14n + 15n - 105 = 2n^2 + n - 105, so n = 7 or n = -152.","n must be a positive integer, so n = 7.","Check: 2(49) + 7 = 98 + 7 = 105."],"diagram":null,"draw":false,"revision":"8092ec15d573"},{"id":"g89-the-nth-term-of-a-quadratic-sequence-q10","topicId":"g89-the-nth-term-of-a-quadratic-sequence","topic":"The Nth Term of a Quadratic Sequence","grade":"89","topicGrade":"89","strand":"A","difficulty":10,"marks":6,"tier":"H","calc":true,"text":"

Here are the first five terms of a quadratic sequence.

7   16   29   46   67

  1. Find an expression, in terms of n, for the nth term of this sequence.
  2. Show that 500 is not a term of this sequence.

","answerType":"multi","answer":"a) 2n^2 + 3n + 2; b) proof (no integer n gives 500)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"2n^2 + 3n + 2"},{"label":"b","answer":"2n^2 + 3n + 2 = 500 has no positive integer solution"}],"solution":["For a quadratic sequence an^2 + bn + c, the second difference is 2a. The n^2 terms 1, 4, 9, 16, … have first differences 3, 5, 7, … and second difference 2; multiplying by a makes it 2a. The linear part bn + c adds no second difference.","a) First differences: 9, 13, 17, 21. Second difference: 4, so the n^2 coefficient is 2.","Subtract 2n^2 (2, 8, 18, 32, 50) from the sequence: 5, 8, 11, 14, 17, which has nth term 3n + 2.","So the nth term is 2n^2 + 3n + 2.","Check: n = 1 gives 2 + 3 + 2 = 7; n = 2 gives 8 + 6 + 2 = 16; n = 3 gives 18 + 9 + 2 = 29; n = 4 gives 32 + 12 + 2 = 46; n = 5 gives 50 + 15 + 2 = 67.","b) If 500 were a term, then 2n^2 + 3n + 2 = 500, so 2n^2 + 3n - 498 = 0.","The discriminant (the quantity b^2 - 4ac under the square root in the quadratic formula) is 3^2 + 4 x 2 x 498 = 9 + 3984 = 3993.","3993 is not a square number (63^2 = 3969 and 64^2 = 4096), so n is not a whole number.","Alternatively: the 15th term is 2(225) + 45 + 2 = 497 and the 16th term is 2(256) + 48 + 2 = 562, and the sequence is increasing, so it skips 500.","Either way, 500 is not a term of the sequence."],"diagram":null,"draw":false,"proofSupport":{"plan":["Use first and second differences to find the nth term.","Test whether 500 has a positive integer term number, or locate it between consecutive terms.","Explain why your calculation rules out 500."],"checks":["First differences: 9, 13, 17, 21 — Second difference 4 gives coefficient 2 for n²","T(n) = 2n² + 3n + 2 — Subtract 2n² to leave 5, 8, 11, 14, 17","T(15) = 497; T(16) = 562 — 500 lies strictly between consecutive terms","T(n + 1) − T(n) = 4n + 5 > 0 — For positive integer n the sequence increases","500 is not a term — The increasing sequence skips 500"],"visuals":[{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"First differences: 9, 13, 17, 21","reason":"Second difference 4 gives coefficient 2 for n²"},{"math":"T(n) = 2n² + 3n + 2","reason":"Subtract 2n² to leave 5, 8, 11, 14, 17"},{"math":"T(15) = 497; T(16) = 562","reason":"500 lies strictly between consecutive terms"},{"math":"T(n + 1) − T(n) = 4n + 5 > 0","reason":"For positive integer n the sequence increases"},{"math":"500 is not a term","reason":"The increasing sequence skips 500"}],"alt":"First differences: 9, 13, 17, 21: Second difference 4 gives coefficient 2 for n². T(n) = 2n² + 3n + 2: Subtract 2n² to leave 5, 8, 11, 14, 17. T(15) = 497; T(16) = 562: 500 lies strictly between consecutive terms. T(n + 1) − T(n) = 4n + 5 > 0: For positive integer n the sequence increases. 500 is not a term: The increasing sequence skips 500"}}]},"revision":"1d4077b38128"},{"id":"g89-transforming-graphs-y-f-x-q01","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":1,"marks":2,"tier":"H","calc":true,"text":"

The graph of y = f(x) has a turning point at (3, −5).

  1. Write down the coordinates of the turning point of the graph of y = f(x) + 4
  2. Write down the coordinates of the turning point of the graph of y = f(x − 2)

","answerType":"multi","answer":"a) (3, -1); b) (5, -5)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(3, -1)"},{"label":"b","answer":"(5, -5)"}],"solution":["a) y = f(x) + 4 translates the graph 4 units up, so (3, -5) maps to (3, -5 + 4) = (3, -1).","b) y = f(x - 2) translates the graph 2 units to the right, so (3, -5) maps to (3 + 2, -5) = (5, -5).","For part (b), the original turning point occurs when the input of f is 3. In f(x - 2), solve x - 2 = 3, giving x = 5. This explains why subtracting inside the function moves the point to the right."],"diagram":null,"draw":false,"revision":"f92b81329599"},{"id":"g89-transforming-graphs-y-f-x-q02","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":2,"marks":2,"tier":"H","calc":true,"text":"

The point A (4, 7) lies on the graph of y = f(x).

  1. Find the coordinates of the image of A on the graph of y = f(−x)
  2. Find the coordinates of the image of A on the graph of y = −f(x)

","answerType":"multi","answer":"a) (-4, 7); b) (4, -7)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(-4, 7)"},{"label":"b","answer":"(4, -7)"}],"solution":["a) y = f(-x) is a reflection in the y-axis: the x-coordinate changes sign, so A maps to (-4, 7).","b) y = -f(x) is a reflection in the x-axis: the y-coordinate changes sign, so A maps to (4, -7)."],"diagram":null,"draw":false,"revision":"5154180510d2"},{"id":"g89-transforming-graphs-y-f-x-q03","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":3,"marks":2,"tier":"H","calc":false,"text":"

The graph of y = sin x for 0° ≤ x ≤ 360° has a maximum point at (90, 1).

  1. Write down the coordinates of the maximum point of the graph of y = sin x − 3
  2. Write down the coordinates of the maximum point of the graph of y = sin (x − 30°)

","answerType":"multi","answer":"a) (90, -2); b) (120, 1)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(90, -2)"},{"label":"b","answer":"(120, 1)"}],"solution":["a) y = sin x - 3 translates the graph 3 units down, so (90, 1) maps to (90, -2).","b) y = sin(x - 30) translates the graph 30 degrees to the right, so (90, 1) maps to (120, 1)."],"diagram":null,"draw":false,"revision":"d07beafcdd6a"},{"id":"g89-transforming-graphs-y-f-x-q04","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":4,"marks":2,"tier":"H","calc":true,"text":"

The graph of y = f(x) passes through the point (2, 5).

  1. Write down the coordinates of a point that must lie on the graph of y = f(x + 6)
  2. Write down the coordinates of a point that must lie on the graph of y = f(x) − 2

","answerType":"multi","answer":"a) (-4, 5); b) (2, 3)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(-4, 5)"},{"label":"b","answer":"(2, 3)"}],"solution":["a) y = f(x + 6) translates the graph 6 units to the left, so (2, 5) maps to (2 - 6, 5) = (-4, 5).","b) y = f(x) - 2 translates the graph 2 units down, so (2, 5) maps to (2, 3)."],"diagram":null,"draw":false,"revision":"38fb763239a7"},{"id":"g89-transforming-graphs-y-f-x-q05","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":5,"marks":2,"tier":"H","calc":true,"text":"

  1. Describe fully the single transformation that maps the graph of y = x2 onto the graph of y = (x + 4)2
  2. Describe fully the single transformation that maps the graph of y = x2 onto the graph of y = −x2

","answerType":"multi","answer":"a) translation 4 units left (vector (-4, 0)); b) reflection in the x-axis","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"translation by the vector (-4, 0), i.e. 4 units to the left"},{"label":"b","answer":"reflection in the x-axis"}],"solution":["a) Replacing x with x + 4 translates the graph 4 units in the negative x direction, a translation by the vector (-4, 0).","b) Multiplying the whole function by -1 reflects the graph in the x-axis."],"diagram":null,"draw":false,"revision":"551ffeed05a7"},{"id":"g89-transforming-graphs-y-f-x-q06","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":6,"marks":2,"tier":"H","calc":false,"text":"

The graph of y = cos x is drawn for 0° ≤ x ≤ 360°.

The graph of y = cos x + 2 is sketched on the same axes for 0° ≤ x ≤ 360°.

  1. Write down the coordinates of the minimum point of y = cos x + 2
  2. Write down the coordinates of the maximum point of y = cos x + 2 with the smallest x-coordinate

","answerType":"multi","answer":"a) (180, 1); b) (0, 3)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(180, 1)"},{"label":"b","answer":"(0, 3)"}],"solution":["y = cos x + 2 translates the graph of y = cos x by 2 units up.","a) y = cos x has its minimum in this interval at (180, -1), which maps to (180, 1).","b) y = cos x has maximum points at (0, 1) and (360, 1); the one with the smallest x-coordinate maps to (0, 3)."],"diagram":{"type":"axes","x":{"min":0,"max":360,"step":90,"label":"x (degrees)"},"y":{"min":-1,"max":4,"step":1},"curves":[{"fn":"cos(x*pi/180)"},{"fn":"cos(x*pi/180)+2"}],"alt":"Question diagram. Use the labels and information in the question.","texts":[{"at":[60,-0.45],"text":"y = cos x","italic":true,"serif":true,"color":"#2f6be0"},{"at":[60,1.55],"text":"y = cos x + 2","italic":true,"serif":true,"color":"#2f6be0"}]},"draw":false,"revision":"c78076aabfc2"},{"id":"g89-transforming-graphs-y-f-x-q07","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":7,"marks":3,"tier":"H","calc":true,"text":"

The curve C has equation y = f(x) where f(x) = x2 + 2x.

  1. Curve A is the reflection of C in the y-axis. Find the equation of curve A in the form y = x2 + bx + c
  2. Curve B is the translation of C by 3 units in the positive y direction. Find the equation of curve B.

","answerType":"multi","answer":"a) y = x^2 - 2x; b) y = x^2 + 2x + 3","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"y = x^2 - 2x"},{"label":"b","answer":"y = x^2 + 2x + 3"}],"solution":["a) A reflection in the y-axis replaces x with -x: f(-x) = (-x)^2 + 2(-x) = x^2 - 2x. So curve A is y = x^2 - 2x.","b) A translation 3 units up adds 3 to the function: y = f(x) + 3 = x^2 + 2x + 3."],"diagram":null,"draw":false,"revision":"bbeb5a61e6f0"},{"id":"g89-transforming-graphs-y-f-x-q08","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":8,"marks":3,"tier":"H","calc":false,"text":"

The graph of y = f(x) has a maximum point at (−1, 4) and crosses the x-axis at (−3, 0) and (1, 0).

  1. Write down the coordinates of the minimum point of the graph of y = −f(x)
  2. Write down the coordinates of the maximum point of the graph of y = f(x − 2)
  3. Write down the coordinates of the points where the graph of y = f(x − 2) crosses the x-axis.

","answerType":"multi","answer":"a) (-1, -4); b) (1, 4); c) (-1, 0) and (3, 0)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"(-1, -4)"},{"label":"b","answer":"(1, 4)"},{"label":"c","answer":"(-1, 0) and (3, 0)"}],"solution":["a) y = -f(x) reflects the graph in the x-axis, so the maximum (-1, 4) becomes a minimum at (-1, -4).","b) y = f(x - 2) translates the graph 2 units to the right, so the maximum moves to (-1 + 2, 4) = (1, 4).","c) The x-intercepts also move 2 units right: (-3, 0) maps to (-1, 0) and (1, 0) maps to (3, 0)."],"diagram":{"type":"axes","x":{"min":-5,"max":5,"step":1},"y":{"min":-5,"max":6,"step":1},"ticks":false,"grid":false,"curves":[{"fn":"4-(x+1)*(x+1)","label":"y = f(x)","labelAt":[1,3]}],"points":[{"at":[-3,0],"label":"(−3, 0)","labelPos":"below-left"},{"at":[1,0],"label":"(1, 0)","labelPos":"below-right"},{"at":[-1,4],"label":"(−1, 4)","labelPos":"above"}],"texts":[{"at":[0,-4],"text":"Illustrative sketch: use the given points"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"1a9e1d02476b"},{"id":"g89-transforming-graphs-y-f-x-q09","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":9,"marks":4,"tier":"H","calc":true,"text":"

f(x) = x2 − 4x + 7

  1. Write f(x) in the form (x − a)2 + b and hence write down the coordinates of the turning point of y = f(x)
  2. Write down the coordinates of the turning point of y = f(x − 3)
  3. Find the equation of y = f(−x), giving your answer in the form y = x2 + bx + c

","answerType":"multi","answer":"a) (x - 2)^2 + 3, turning point (2, 3); b) (5, 3); c) y = x^2 + 4x + 7","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"f(x) = (x - 2)^2 + 3; turning point (2, 3)"},{"label":"b","answer":"(5, 3)"},{"label":"c","answer":"y = x^2 + 4x + 7"}],"solution":["a) Complete the square: x^2 - 4x + 7 = (x - 2)^2 - 4 + 7 = (x - 2)^2 + 3. The turning point is (2, 3).","b) y = f(x - 3) translates the graph 3 units to the right, so the turning point moves to (5, 3).","c) f(-x) = (-x)^2 - 4(-x) + 7 = x^2 + 4x + 7, so y = x^2 + 4x + 7."],"diagram":null,"draw":false,"revision":"8d6a0f6c7199"},{"id":"g89-transforming-graphs-y-f-x-q10","topicId":"g89-transforming-graphs-y-f-x","topic":"Transforming Graphs y=f(x)","grade":"89","topicGrade":"89","strand":"A","difficulty":10,"marks":4,"tier":"H","calc":false,"text":"

The diagram shows the graph of y = sin x for 0° ≤ x ≤ 360°.

  1. On the same axes, sketch the graph of y = sin x − 2 for 0° ≤ x ≤ 360°
  2. The point P (30, 0.5) lies on the graph of y = sin x. P is mapped to the point Q on the graph of y = sin (x − 60°). Find the coordinates of Q.

","answerType":"multi","answer":"a) sketch of y = sin x translated 2 down; b) (90, 0.5)","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"the sine curve translated 2 units down: through (0, -2), maximum (90, -1), (180, -2), minimum (270, -3), (360, -2)"},{"label":"b","answer":"(90, 0.5)"}],"solution":["a) y = sin x - 2 is a translation of y = sin x by 2 units down. The sketch passes through (0, -2), has a maximum at (90, -1), passes through (180, -2), has a minimum at (270, -3) and ends at (360, -2).","b) y = sin(x - 60) translates the graph 60 degrees to the right, so P (30, 0.5) maps to Q (30 + 60, 0.5) = (90, 0.5)."],"diagram":{"type":"axes","x":{"min":0,"max":360,"step":90,"label":"x (degrees)"},"y":{"min":-3,"max":1.5,"step":1},"curves":[{"fn":"sin(x*pi/180)"},{"fn":"sin(x*pi/180)-2","answer":true}],"points":[{"at":[30,0.5],"label":"P"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":true,"revision":"2fe152b32993"},{"id":"g89-vectors-proof-questions-q01","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":1,"marks":4,"tier":"H","calc":false,"text":"

OAB is a triangle.

Vector OA = a and vector OB = b.

M is the midpoint of OA and N is the midpoint of OB.

Prove that MN is parallel to AB.

","answerType":"open","answer":"MN = (1/2)(b - a) = (1/2)AB, so MN is a scalar multiple of AB and the lines are parallel","answerDisplay":"MN = 12(b - a) = 12AB, so MN is a scalar multiple of AB and the lines are parallel","accept":[],"parts":[],"solution":["Vector OM = 12a because M is the midpoint of OA.","Vector ON = 12b because N is the midpoint of OB.","Vector MN = ON − OM = 12b − 12a = 12(b − a).","Vector AB = OB − OA = b − a.","A scalar is an ordinary number. Multiplying a nonzero vector by a scalar changes its length (and may reverse it), but keeps its direction parallel.","So MN = 12AB, which means MN is a scalar multiple of AB.","Therefore MN is parallel to AB."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[2,5],"label":"B","style":"none","labelPos":"above"},{"at":[3.5,0],"label":"M","style":"none","labelPos":"below-right"},{"at":[1,2.5],"label":"N","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"a","arrow":"end"},{"from":[0,0],"to":[2,5],"label":"b","arrow":"end"},{"from":[7,0],"to":[2,5]},{"from":[3.5,0],"to":[1,2.5]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["OM = a/2; ON = b/2 — M and N are midpoints","MN = ON − OM = (b − a)/2 — Use the route M → O → N","AB = b − a; MN = AB/2 — Compare the two directed vectors","MN is parallel to AB — A scalar multiple gives parallel directions"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"O","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"A","at":[7,0],"style":"dot","labelPos":"above-left"},{"label":"B","at":[2,5],"style":"dot","labelPos":"above-left"},{"label":"M","at":[3.5,0],"style":"dot","labelPos":"above-left"},{"label":"N","at":[1,2.5],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[7,0],"label":"a","answer":true,"arrow":"end"},{"from":[0,0],"to":[2,5],"label":"b","answer":true,"arrow":"end"},{"from":[7,0],"to":[2,5],"label":"b − a","answer":true,"arrow":"end"},{"from":[3.5,0],"to":[1,2.5],"label":"(b − a)/2","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"OM = a/2; ON = b/2","reason":"M and N are midpoints"},{"math":"MN = ON − OM = (b − a)/2","reason":"Use the route M → O → N"},{"math":"AB = b − a; MN = AB/2","reason":"Compare the two directed vectors"},{"math":"MN is parallel to AB","reason":"A scalar multiple gives parallel directions"}],"alt":"OM = a/2; ON = b/2: M and N are midpoints. MN = ON − OM = (b − a)/2: Use the route M → O → N. AB = b − a; MN = AB/2: Compare the two directed vectors. MN is parallel to AB: A scalar multiple gives parallel directions"}}]},"revision":"d4ac3b1d4d80"},{"id":"g89-vectors-proof-questions-q02","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":2,"marks":4,"tier":"H","calc":false,"text":"

ABCD is a quadrilateral.

Vector AB = 4a + 2b
Vector BC = a − 3b
Vector CD = −4a − 2b

Prove that ABCD is a parallelogram.

","answerType":"open","answer":"DC = -CD = 4a + 2b = AB, so AB and DC are equal and parallel, which proves ABCD is a parallelogram","answerDisplay":null,"accept":[],"parts":[],"solution":["Vector DC = −CD = −(−4a − 2b) = 4a + 2b.","So DC = AB.","This means the sides AB and DC are parallel and equal in length.","A quadrilateral with one pair of opposite sides that are equal and parallel is a parallelogram.","Therefore ABCD is a parallelogram.","Check with the remaining side: AD = AB + BC + CD = (4a + 2b) + (a − 3b) + (−4a − 2b) = a − 3b = BC, which confirms AD is equal and parallel to BC as expected."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["DC = −CD = 4a + 2b — Reverse the direction of CD","AB = DC — The vectors are equal","AB and DC are equal and parallel — Interpret equality of directed vectors","ABCD is a parallelogram — One opposite pair is both equal and parallel"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"A","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"B","at":[4,2],"style":"dot","labelPos":"above-left"},{"label":"C","at":[5,-1],"style":"dot","labelPos":"above-left"},{"label":"D","at":[1,-3],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[4,2],"label":"4a + 2b","answer":true,"arrow":"end"},{"from":[1,-3],"to":[5,-1],"label":"4a + 2b","answer":true,"arrow":"end"},{"from":[0,0],"to":[1,-3],"label":"a − 3b","answer":true,"arrow":"end"},{"from":[4,2],"to":[5,-1],"label":"a − 3b","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"DC = −CD = 4a + 2b","reason":"Reverse the direction of CD"},{"math":"AB = DC","reason":"The vectors are equal"},{"math":"AB and DC are equal and parallel","reason":"Interpret equality of directed vectors"},{"math":"ABCD is a parallelogram","reason":"One opposite pair is both equal and parallel"}],"alt":"DC = −CD = 4a + 2b: Reverse the direction of CD. AB = DC: The vectors are equal. AB and DC are equal and parallel: Interpret equality of directed vectors. ABCD is a parallelogram: One opposite pair is both equal and parallel"}}]},"revision":"7d2fd6392cb8"},{"id":"g89-vectors-proof-questions-q03","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

Vector EF = 4a − 2b

Vector GH = 6a − 3b

Prove that EF is parallel to GH.

","answerType":"open","answer":"GH = (3/2)EF, so GH is a scalar multiple of EF and the lines are parallel","answerDisplay":"GH = 32EF, so GH is a scalar multiple of EF and the lines are parallel","accept":[],"parts":[],"solution":["Write both vectors with a common factor taken out.","EF = 4a − 2b = 2(2a − b).","GH = 6a − 3b = 3(2a − b).","So GH = 32 × EF, which means GH is a scalar multiple of EF.","Therefore EF is parallel to GH."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["EF = 2(2a − b) — Factor out 2","GH = 3(2a − b) — Factor out 3","GH = (3/2)EF — The same vector is multiplied by a scalar","EF is parallel to GH — Scalar-multiple vectors have parallel directions"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"E","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"F","at":[4,-2],"style":"dot","labelPos":"above-left"},{"label":"G","at":[0,3],"style":"dot","labelPos":"above-left"},{"label":"H","at":[6,0],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[4,-2],"label":"2(2a − b)","answer":true,"arrow":"end"},{"from":[0,3],"to":[6,0],"label":"3(2a − b)","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"EF = 2(2a − b)","reason":"Factor out 2"},{"math":"GH = 3(2a − b)","reason":"Factor out 3"},{"math":"GH = (3/2)EF","reason":"The same vector is multiplied by a scalar"},{"math":"EF is parallel to GH","reason":"Scalar-multiple vectors have parallel directions"}],"alt":"EF = 2(2a − b): Factor out 2. GH = 3(2a − b): Factor out 3. GH = (3/2)EF: The same vector is multiplied by a scalar. EF is parallel to GH: Scalar-multiple vectors have parallel directions"}}]},"revision":"b784e4b2aa0f"},{"id":"g89-vectors-proof-questions-q04","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":4,"marks":4,"tier":"H","calc":false,"text":"

P, Q and R are three points.

Vector PQ = 2a − 3b

Vector QR = 6a − 9b

Prove that P, Q and R lie on the same straight line.

","answerType":"open","answer":"QR = 3PQ, so PQ and QR are parallel; they share the point Q, so P, Q and R are collinear","answerDisplay":null,"accept":[],"parts":[],"solution":["QR = 6a − 9b = 3(2a − 3b) = 3 × PQ.","So QR is a scalar multiple of PQ, which means PQ and QR are parallel.","PQ and QR both pass through the point Q.","Two parallel line segments through the same point lie on one straight line.","Therefore P, Q and R lie on the same straight line (they are collinear)."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["QR = 6a − 9b = 3(2a − 3b) — Factor out 3","QR = 3PQ — The vectors are parallel","Both segments pass through Q — A shared point is also needed","P, Q and R are collinear — Parallel directions through the same point"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"P","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"Q","at":[2,-3],"style":"dot","labelPos":"above-left"},{"label":"R","at":[8,-12],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[2,-3],"label":"v = 2a − 3b","answer":true,"arrow":"end"},{"from":[2,-3],"to":[8,-12],"label":"3v","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"QR = 6a − 9b = 3(2a − 3b)","reason":"Factor out 3"},{"math":"QR = 3PQ","reason":"The vectors are parallel"},{"math":"Both segments pass through Q","reason":"A shared point is also needed"},{"math":"P, Q and R are collinear","reason":"Parallel directions through the same point"}],"alt":"QR = 6a − 9b = 3(2a − 3b): Factor out 3. QR = 3PQ: The vectors are parallel. Both segments pass through Q: A shared point is also needed. P, Q and R are collinear: Parallel directions through the same point"}}]},"revision":"8bb3aac6110b"},{"id":"g89-vectors-proof-questions-q05","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":5,"marks":5,"tier":"H","calc":true,"text":"

OAB is a triangle.

Vector OA = 2a and vector OB = 3b.

M is the midpoint of OA.

N is the point on OB such that ON : NB = 2 : 1.

  1. Find vector MN in terms of a and b.
  2. Show that MN is not parallel to AB.
","answerType":"multi","answer":"a) MN = 2b - a; b) AB = 3b - 2a is not a scalar multiple of 2b - a, so they are not parallel","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"MN = 2b - a"},{"label":"b","answer":"If 2b - a = k(3b - 2a) then k = 1/2 from the a terms and k = 2/3 from the b terms; these are different, so MN is not parallel to AB","answerDisplay":"If 2b - a = k(3b - 2a) then k = 12 from the a terms and k = 23 from the b terms; these are different, so MN is not parallel to AB"}],"solution":["a) Vector OM = 12 × 2a = a because M is the midpoint of OA.","Vector ON = 23 × 3b = 2b because ON : NB = 2 : 1.","Vector MN = ON − OM = 2b − a.","b) Vector AB = OB − OA = 3b − 2a.","Suppose MN were parallel to AB. Then 2b − a = k(3b − 2a) for some scalar k.","Comparing the a terms: −1 = −2k, so k = 12.","Comparing the b terms: 2 = 3k, so k = 23.","The two values of k are different, so no such scalar exists.","Therefore MN is not parallel to AB."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[2,5],"label":"B","style":"none","labelPos":"above"},{"at":[3.5,0],"label":"M","style":"none","labelPos":"below-right"},{"at":[1.3333333333333333,3.333333333333333],"label":"N","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"2a","arrow":"end"},{"from":[0,0],"to":[2,5],"label":"3b","arrow":"end"},{"from":[7,0],"to":[2,5]},{"from":[3.5,0],"to":[1.3333333333333333,3.333333333333333]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["OM = a; ON = 2b — Use the midpoint and the 2:1 division","MN = 2b − a; AB = 3b − 2a — Subtract position vectors","a terms require k = 1/2 — Test whether MN = kAB","b terms require k = 2/3 — No single k works: MN and AB are not parallel"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"O","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"A","at":[7,0],"style":"dot","labelPos":"above-left"},{"label":"B","at":[2,5],"style":"dot","labelPos":"above-left"},{"label":"M","at":[3.5,0],"style":"dot","labelPos":"above-left"},{"label":"N","at":[1.3333333333333333,3.3333333333333335],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[3.5,0],"label":"a","answer":true,"arrow":"end"},{"from":[0,0],"to":[1.3333333333333333,3.3333333333333335],"label":"2b","answer":true,"arrow":"end"},{"from":[3.5,0],"to":[1.3333333333333333,3.3333333333333335],"label":"2b − a","answer":true,"arrow":"end"},{"from":[7,0],"to":[2,5],"label":"3b − 2a","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"OM = a; ON = 2b","reason":"Use the midpoint and the 2:1 division"},{"math":"MN = 2b − a; AB = 3b − 2a","reason":"Subtract position vectors"},{"math":"a terms require k = 1/2","reason":"Test whether MN = kAB"},{"math":"b terms require k = 2/3","reason":"No single k works: MN and AB are not parallel"}],"alt":"OM = a; ON = 2b: Use the midpoint and the 2:1 division. MN = 2b − a; AB = 3b − 2a: Subtract position vectors. a terms require k = 1/2: Test whether MN = kAB. b terms require k = 2/3: No single k works: MN and AB are not parallel"}}]},"revision":"4502403e78a7"},{"id":"g89-vectors-proof-questions-q06","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":6,"marks":5,"tier":"H","calc":false,"text":"

OACB is a parallelogram.

Vector OA = a and vector OB = b, so vector OC = a + b.

M is the midpoint of OB.
X is the midpoint of OC.
D is the midpoint of AC.

Prove that M, X and D lie on the same straight line.

","answerType":"open","answer":"MX = (1/2)a and XD = (1/2)a, so MX = XD; the segments are parallel and share the point X, so M, X and D are collinear","answerDisplay":"MX = 12a and XD = 12a, so MX = XD; the segments are parallel and share the point X, so M, X and D are collinear","accept":[],"parts":[],"solution":["Vector OM = 12b because M is the midpoint of OB.","Vector OX = 12(a + b) because X is the midpoint of OC.","D is the midpoint of AC, so vector OD = OA + 12AC. Vector AC = OC − OA = b, so OD = a + 12b.","Vector MX = OX − OM = 12a + 12b − 12b = 12a.","Vector XD = OD − OX = a + 12b − 12a − 12b = 12a.","So MX = XD. The two segments are parallel (both are scalar multiples of a) and share the point X.","Therefore M, X and D lie on the same straight line, and X is the midpoint of MD."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[2,5],"label":"B","style":"none","labelPos":"above"},{"at":[9,5],"label":"C","style":"none","labelPos":"above"},{"at":[1,2.5],"label":"M","style":"dot","labelPos":"below-left"},{"at":[4.5,2.5],"label":"X","style":"dot","labelPos":"below-right"},{"at":[8,2.5],"label":"D","style":"dot","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[7,0],"label":"a","arrow":"end"},{"from":[0,0],"to":[2,5],"label":"b","arrow":"end"},{"from":[7,0],"to":[9,5]},{"from":[2,5],"to":[9,5]},{"from":[0,0],"to":[9,5]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["OM = b/2; OX = (a + b)/2 — Use the two midpoints","OD = a + b/2 — D is halfway along AC","MX = a/2; XD = a/2 — Subtract position vectors","MX = XD and both pass through X — M, X and D are collinear; X is the midpoint"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"O","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"A","at":[7,0],"style":"dot","labelPos":"above-left"},{"label":"B","at":[2,5],"style":"dot","labelPos":"above-left"},{"label":"C","at":[9,5],"style":"dot","labelPos":"above-left"},{"label":"M","at":[1,2.5],"style":"dot","labelPos":"above-left"},{"label":"X","at":[4.5,2.5],"style":"dot","labelPos":"above-left"},{"label":"D","at":[8,2.5],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[7,0],"label":"a","answer":true,"arrow":"end"},{"from":[0,0],"to":[2,5],"label":"b","answer":true,"arrow":"end"},{"from":[7,0],"to":[9,5],"label":"b","answer":true,"arrow":"end"},{"from":[2,5],"to":[9,5],"label":"a","answer":true,"arrow":"end"},{"from":[1,2.5],"to":[4.5,2.5],"label":"a/2","answer":true,"arrow":"end"},{"from":[4.5,2.5],"to":[8,2.5],"label":"a/2","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"OM = b/2; OX = (a + b)/2","reason":"Use the two midpoints"},{"math":"OD = a + b/2","reason":"D is halfway along AC"},{"math":"MX = a/2; XD = a/2","reason":"Subtract position vectors"},{"math":"MX = XD and both pass through X","reason":"M, X and D are collinear; X is the midpoint"}],"alt":"OM = b/2; OX = (a + b)/2: Use the two midpoints. OD = a + b/2: D is halfway along AC. MX = a/2; XD = a/2: Subtract position vectors. MX = XD and both pass through X: M, X and D are collinear; X is the midpoint"}}]},"revision":"cea351e4da03"},{"id":"g89-vectors-proof-questions-q07","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":7,"marks":5,"tier":"H","calc":true,"text":"

The points A and B have position vectors

OA = 4a + 3b and OB = 8a + 11b.

  1. Find vector AB in terms of a and b.
  2. The point P lies on AB such that AP : PB = 3 : 1. Work out vector OP in terms of a and b.
  3. The point D has position vector OD = 14a + 18b. Prove that O, P and D lie on the same straight line.
","answerType":"multi","answer":"a) AB = 4a + 8b; b) OP = 7a + 9b; c) OD = 2OP so O, P and D are collinear","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"AB = 4a + 8b"},{"label":"b","answer":"OP = 7a + 9b"},{"label":"c","answer":"OD = 14a + 18b = 2(7a + 9b) = 2OP, so OD is a scalar multiple of OP; both pass through O, so O, P and D are collinear"}],"solution":["a) Vector AB = OB − OA = (8a + 11b) − (4a + 3b) = 4a + 8b.","b) AP : PB = 3 : 1, so AP = 34AB = 34(4a + 8b) = 3a + 6b.","Vector OP = OA + AP = 4a + 3b + 3a + 6b = 7a + 9b.","c) OD = 14a + 18b = 2(7a + 9b) = 2 × OP.","So OD is a scalar multiple of OP, which means OD and OP are parallel.","OD and OP both pass through the point O.","Therefore O, P and D lie on the same straight line, with P the midpoint of OD."],"diagram":null,"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["AB = 4a + 8b; AP = 3a + 6b — P divides AB in the ratio 3:1","OP = OA + AP = 7a + 9b — Follow the route O → A → P","OD = 14a + 18b = 2OP — Compare the position vectors","OP and OD both start at O — O, P and D are collinear"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"O","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"A","at":[4,3],"style":"dot","labelPos":"above-left"},{"label":"B","at":[8,11],"style":"dot","labelPos":"above-left"},{"label":"P","at":[7,9],"style":"dot","labelPos":"above-left"},{"label":"D","at":[14,18],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[4,3],"label":"4a + 3b","answer":true,"arrow":"end"},{"from":[4,3],"to":[8,11],"label":"4a + 8b","answer":true,"arrow":"end"},{"from":[0,0],"to":[7,9],"label":"7a + 9b","answer":true,"arrow":"end"},{"from":[7,9],"to":[14,18],"label":"7a + 9b","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"AB = 4a + 8b; AP = 3a + 6b","reason":"P divides AB in the ratio 3:1"},{"math":"OP = OA + AP = 7a + 9b","reason":"Follow the route O → A → P"},{"math":"OD = 14a + 18b = 2OP","reason":"Compare the position vectors"},{"math":"OP and OD both start at O","reason":"O, P and D are collinear"}],"alt":"AB = 4a + 8b; AP = 3a + 6b: P divides AB in the ratio 3:1. OP = OA + AP = 7a + 9b: Follow the route O → A → P. OD = 14a + 18b = 2OP: Compare the position vectors. OP and OD both start at O: O, P and D are collinear"}}]},"revision":"987702e5f897"},{"id":"g89-vectors-proof-questions-q08","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

The vectors a and b are non-parallel.

The points A, B and C have position vectors

OA = 2a + 5b
OB = 4a + λb
OC = 8a + 23b

where λ is a number.

A, B and C lie on the same straight line.

Find the value of λ.

","answerType":"exact","answer":"11","answerDisplay":null,"accept":["lambda = 11","λ = 11"],"parts":[],"solution":["Vector AB = OB − OA = (4a + λb) − (2a + 5b) = 2a + (λ − 5)b.","Vector AC = OC − OA = (8a + 23b) − (2a + 5b) = 6a + 18b.","A, B and C are collinear, so AB must be a scalar multiple of AC: 2a + (λ − 5)b = k(6a + 18b).","Comparing the a terms: 2 = 6k, so k = 13.","Comparing the b terms: λ − 5 = 18k = 18 × 13 = 6.","So λ = 11.","Check: with λ = 11, AB = 2a + 6b and AC = 6a + 18b = 3AB, so the three points are indeed collinear."],"diagram":null,"draw":false,"revision":"f05c0c340ef9"},{"id":"g89-vectors-proof-questions-q09","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":9,"marks":5,"tier":"H","calc":true,"text":"

OAB is a triangle.

Vector OA = a and vector OB = b.

M is the midpoint of AB.
N is the midpoint of OB.
P is the point on OM such that OP : PM = 2 : 1.

  1. Prove that A, P and N lie on the same straight line.
  2. Find AP : PN.
","answerType":"multi","answer":"a) AN = (3/2)AP, so A, P and N are collinear; b) AP : PN = 2 : 1","answerDisplay":"a) AN = 32AP, so A, P and N are collinear; b) AP : PN = 2 : 1","accept":[],"parts":[{"label":"a","answer":"AP = -(2/3)a + (1/3)b and AN = -a + (1/2)b = (3/2)AP, so AN is a scalar multiple of AP; both pass through A, so A, P and N are collinear","answerDisplay":"AP = -23a + 13b and AN = -a + 12b = 32AP, so AN is a scalar multiple of AP; both pass through A, so A, P and N are collinear"},{"label":"b","answer":"2 : 1"}],"solution":["a) Vector OM = 12(a + b) because M is the midpoint of AB.","OP : PM = 2 : 1, so OP = 23OM = 13(a + b).","Vector AP = OP − OA = 13a + 13b − a = −23a + 13b.","Vector ON = 12b because N is the midpoint of OB.","Vector AN = ON − OA = −a + 12b.","32 × AP = 32(−23a + 13b) = −a + 12b = AN.","So AN = 32AP, which means AP and AN are parallel; both pass through A, so A, P and N lie on the same straight line.","b) PN = AN − AP = 32AP − AP = 12AP.","So AP : PN = 1 : 12 = 2 : 1.","This is the well-known result that the medians of a triangle meet at a point that divides each median in the ratio 2 : 1."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[2,5],"label":"B","style":"none","labelPos":"above"},{"at":[4.5,2.5],"label":"M","style":"none","labelPos":"above"},{"at":[1,2.5],"label":"N","style":"dot","labelPos":"above"},{"at":[3,1.6666666666666665],"label":"P","style":"dot","labelPos":"below-right"}],"segments":[{"from":[0,0],"to":[7,0],"label":"a","arrow":"end"},{"from":[0,0],"to":[2,5],"label":"b","arrow":"end"},{"from":[7,0],"to":[2,5]},{"from":[0,0],"to":[4.5,2.5]}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["OP = (a + b)/3; ON = b/2 — Use the midpoint and the 2:1 division","AP = (−2a + b)/3; AN = (−2a + b)/2 — Subtract OA from each position vector","AN = (3/2)AP; both start at A — A, P and N are collinear","PN = AP/2; AP:PN = 2:1 — Subtract AP from AN to find the remaining segment"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"O","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"A","at":[7,0],"style":"dot","labelPos":"above-left"},{"label":"B","at":[2,5],"style":"dot","labelPos":"above-left"},{"label":"M","at":[4.5,2.5],"style":"dot","labelPos":"above-left"},{"label":"N","at":[1,2.5],"style":"dot","labelPos":"above-left"},{"label":"P","at":[3,1.6666666666666667],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[7,0],"label":"a","answer":true,"arrow":"end"},{"from":[0,0],"to":[2,5],"label":"b","answer":true,"arrow":"end"},{"from":[7,0],"to":[2,5],"label":"","answer":true,"arrow":"end"},{"from":[0,0],"to":[4.5,2.5],"label":"","answer":true,"arrow":"end"},{"from":[7,0],"to":[3,1.6666666666666667],"label":"(−2a + b)/3","answer":true,"arrow":"end"},{"from":[3,1.6666666666666667],"to":[1,2.5],"label":"(−2a + b)/6","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"OP = (a + b)/3; ON = b/2","reason":"Use the midpoint and the 2:1 division"},{"math":"AP = (−2a + b)/3; AN = (−2a + b)/2","reason":"Subtract OA from each position vector"},{"math":"AN = (3/2)AP; both start at A","reason":"A, P and N are collinear"},{"math":"PN = AP/2; AP:PN = 2:1","reason":"Subtract AP from AN to find the remaining segment"}],"alt":"OP = (a + b)/3; ON = b/2: Use the midpoint and the 2:1 division. AP = (−2a + b)/3; AN = (−2a + b)/2: Subtract OA from each position vector. AN = (3/2)AP; both start at A: A, P and N are collinear. PN = AP/2; AP:PN = 2:1: Subtract AP from AN to find the remaining segment"}}]},"revision":"005962736b16"},{"id":"g89-vectors-proof-questions-q10","topicId":"g89-vectors-proof-questions","topic":"Vectors Proof Questions","grade":"89","topicGrade":"89","strand":"G","difficulty":10,"marks":6,"tier":"H","calc":true,"text":"

OAB is a triangle.

Vector OA = 3a and vector OB = 3b.

P is the point on AB such that AP : PB = 1 : 2.

  1. Find vector AB in terms of a and b.
  2. Show that vector OP = 2a + b.
  3. The point Q lies on OP produced, so that BQ is parallel to OA. Work out OP : PQ.
","answerType":"multi","answer":"a) AB = 3b - 3a; b) OP = 3a + (1/3)(3b - 3a) = 2a + b; c) OP : PQ = 1 : 2","answerDisplay":"a) AB = 3b - 3a; b) OP = 3a + 13(3b - 3a) = 2a + b; c) OP : PQ = 1 : 2","accept":[],"parts":[{"label":"a","answer":"AB = 3b - 3a"},{"label":"b","answer":"OP = OA + (1/3)AB = 3a + b - a = 2a + b","answerDisplay":"OP = OA + 13AB = 3a + b - a = 2a + b"},{"label":"c","answer":"1 : 2"}],"solution":["a) Vector AB = OB − OA = 3b − 3a.","b) AP : PB = 1 : 2, so AP = 13AB = 13(3b − 3a) = b − a.","Vector OP = OA + AP = 3a + b − a = 2a + b, as required.","c) Q lies on OP produced, so OQ = t(2a + b) for some scalar t > 1.","Vector BQ = OQ − OB = 2ta + tb − 3b = 2ta + (t − 3)b.","BQ is parallel to OA = 3a, so BQ has no b component: t − 3 = 0, which gives t = 3.","So OQ = 3 × OP = 6a + 3b, and check: BQ = 6a + 3b − 3b = 6a, which is parallel to OA.","PQ = OQ − OP = 3OP − OP = 2OP.","Therefore OP : PQ = 1 : 2."],"diagram":{"type":"shape","notAccurate":true,"points":[{"at":[0,0],"label":"O","style":"none","labelPos":"below-left"},{"at":[7,0],"label":"A","style":"none","labelPos":"below-right"},{"at":[2,5],"label":"B","style":"none","labelPos":"above"},{"at":[5.333333333333334,1.6666666666666665],"label":"P","style":"dot","labelPos":"below-left"},{"at":[16,5],"label":"Q","style":"none","labelPos":"above"}],"segments":[{"from":[0,0],"to":[7,0],"label":"3a","arrow":"end","parallel":1},{"from":[0,0],"to":[2,5],"label":"3b","arrow":"end"},{"from":[7,0],"to":[2,5]},{"from":[0,0],"to":[16,5]},{"from":[2,5],"to":[16,5],"parallel":1}],"angles":[],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"proofSupport":{"plan":["Choose a route and write the required vectors in terms of the given vectors.","Compare the coefficients or find a common scalar multiplier.","Justify parallelism or collinearity; for collinearity identify a shared point."],"checks":["AB = 3b − 3a; AP = b − a — P divides AB in the ratio 1:2","OP = OA + AP = 2a + b — Follow the route O → A → P","OQ = t(2a + b); BQ = 2ta + (t − 3)b — Q lies on OP produced","t = 3, so PQ = 2OP — BQ parallel to OA removes the b component","OP:PQ = 1:2 — Compare the two lengths on the same line"],"visuals":[{"caption":"Directed routes used in the calculation. Arrow direction matters; the algebra proves the general result, not the appearance of this sketch.","diagram":{"type":"shape","notAccurate":true,"points":[{"label":"O","at":[0,0],"style":"dot","labelPos":"above-left"},{"label":"A","at":[7,0],"style":"dot","labelPos":"above-left"},{"label":"B","at":[2,5],"style":"dot","labelPos":"above-left"},{"label":"P","at":[5.333333333333333,1.6666666666666667],"style":"dot","labelPos":"above-left"},{"label":"Q","at":[16,5],"style":"dot","labelPos":"above-left"}],"segments":[{"from":[0,0],"to":[7,0],"label":"3a","answer":true,"arrow":"end"},{"from":[0,0],"to":[2,5],"label":"3b","answer":true,"arrow":"end"},{"from":[7,0],"to":[2,5],"label":"3b − 3a","answer":true,"arrow":"end"},{"from":[0,0],"to":[5.333333333333333,1.6666666666666667],"label":"2a + b","answer":true,"arrow":"end"},{"from":[5.333333333333333,1.6666666666666667],"to":[16,5],"label":"2(2a + b)","answer":true,"arrow":"end"},{"from":[2,5],"to":[16,5],"label":"6a","answer":true,"arrow":"end"}],"angles":[],"alt":"Worked construction. Green lines and labels show the reasoning."}},{"caption":"Follow the reasoning: each arrow is a justified step, not just a numerical example.","diagram":{"type":"reasoning","steps":[{"math":"AB = 3b − 3a; AP = b − a","reason":"P divides AB in the ratio 1:2"},{"math":"OP = OA + AP = 2a + b","reason":"Follow the route O → A → P"},{"math":"OQ = t(2a + b); BQ = 2ta + (t − 3)b","reason":"Q lies on OP produced"},{"math":"t = 3, so PQ = 2OP","reason":"BQ parallel to OA removes the b component"},{"math":"OP:PQ = 1:2","reason":"Compare the two lengths on the same line"}],"alt":"AB = 3b − 3a; AP = b − a: P divides AB in the ratio 1:2. OP = OA + AP = 2a + b: Follow the route O → A → P. OQ = t(2a + b); BQ = 2ta + (t − 3)b: Q lies on OP produced. t = 3, so PQ = 2OP: BQ parallel to OA removes the b component. OP:PQ = 1:2: Compare the two lengths on the same line"}}]},"revision":"683924474c58"},{"id":"g89-velocity-time-graphs-q01","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":1,"marks":3,"tier":"H","calc":false,"text":"

A car starts from rest and accelerates uniformly to a velocity of 24 m/s in 8 seconds.

It then travels at a constant velocity of 24 m/s for a further 20 seconds.

  1. Work out the acceleration of the car during the first 8 seconds.
  2. Work out the total distance travelled by the car in the 28 seconds.

","answerType":"multi","answer":"a) 3 m/s^2; b) 576 m","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3 m/s^2"},{"label":"b","answer":"576 m"}],"solution":["a) Acceleration = change in velocity divided by time = 248 = 3 m/s^2.","b) Distance = area under the velocity-time graph.","First 8 seconds (triangle): 12 x 8 x 24 = 96 m.","Next 20 seconds (rectangle): 20 x 24 = 480 m.","Total distance = 96 + 480 = 576 m."],"diagram":{"type":"axes","x":{"min":0,"max":30,"step":2,"label":"Time (s)"},"y":{"min":0,"max":30,"step":5,"label":"Velocity (m/s)"},"curves":[{"pts":[[0,0],[8,24],[28,24]]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"3faedf1d9454"},{"id":"g89-velocity-time-graphs-q02","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":2,"marks":3,"tier":"H","calc":true,"text":"

A runner's velocity-time graph is a smooth curve. The table gives her velocity v m/s at time t seconds.

t: 0, 10, 20, 30   v: 0, 4.2, 6.6, 7.8

Use the trapezium rule with 3 strips of equal width to estimate the distance the runner travels in the first 30 seconds.

","answerType":"exact","answer":"147 m","answerDisplay":null,"accept":["147","147 metres"],"parts":[],"solution":["Distance is the area under the velocity-time graph. Use 3 trapezia of width 10.","First strip: 12 x 10 x (0 + 4.2) = 21.","Second strip: 12 x 10 x (4.2 + 6.6) = 54.","Third strip: 12 x 10 x (6.6 + 7.8) = 72.","Estimate = 21 + 54 + 72 = 147 m."],"diagram":null,"draw":false,"revision":"508238cd0f66"},{"id":"g89-velocity-time-graphs-q03","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":3,"marks":4,"tier":"H","calc":true,"text":"

A cyclist accelerates away from traffic lights. Her speed-time graph is a smooth curve that gets less steep as time increases. The table gives her speed v m/s at time t seconds.

t: 0, 5, 10, 15, 20   v: 0, 3.6, 5.4, 6.4, 7.0

  1. Use the trapezium rule with 4 strips of equal width to estimate the distance travelled in the first 20 seconds.
  2. State, with a reason, whether your answer to part (a) is an underestimate or an overestimate of the actual distance.

","answerType":"multi","answer":"a) 94.5 m; b) underestimate, the trapezia lie below the curve","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"94.5 m"},{"label":"b","answer":"underestimate, because the curve bends above the chords so each trapezium misses some area under the curve"}],"solution":["a) Area = 12 x 5 x (0 + 3.6) + 12 x 5 x (3.6 + 5.4) + 12 x 5 x (5.4 + 6.4) + 12 x 5 x (6.4 + 7.0).","= 9 + 22.5 + 29.5 + 33.5 = 94.5 m.","(Equivalently 12 x 5 x (0 + 2 x 3.6 + 2 x 5.4 + 2 x 6.4 + 7.0) = 2.5 x 37.8 = 94.5.)","b) The graph is a curve that gets less steep (concave), so each chord lies below the curve and each trapezium misses some of the area. The estimate is an underestimate."],"diagram":null,"draw":false,"revision":"28e6f0ec0e31"},{"id":"g89-velocity-time-graphs-q04","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":4,"marks":4,"tier":"H","calc":true,"text":"

The speed-time graph for a car is a smooth curve for 0 ≤ t ≤ 12, where t is the time in seconds and the speed is in m/s.

A tangent drawn by eye to the curve at t = 6 passes through the points (2, 8) and (10, 32).

  1. Use the tangent to estimate the acceleration of the car at t = 6.
  2. Explain why your answer is only an estimate.

","answerType":"multi","answer":"a) 3 m/s^2; b) the tangent is drawn by eye to a curve, so the gradient is only approximate","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"3 m/s^2"},{"label":"b","answer":"the acceleration is the gradient of the curve, and a tangent drawn by eye only approximates that gradient"}],"solution":["a) Acceleration at t = 6 is the gradient of the tangent at t = 6.","Gradient = 32 - 810 - 2 = 248 = 3.","So the acceleration is approximately 3 m/s^2.","b) The tangent is drawn by eye to a curved graph, so its gradient (and therefore the acceleration) is only an estimate."],"diagram":{"type":"axes","x":{"min":0,"max":12,"step":2,"label":"Time (s)"},"y":{"min":0,"max":45,"step":5,"label":"Speed (m/s)"},"curves":[{"fn":"0.05*(x-6)*(x-6)+3*x+2"}],"lines":[{"through":[[2,8],[10,32]],"from":2,"to":10,"label":"Tangent","dashed":true}],"points":[{"at":[2,8],"label":"(2, 8)"},{"at":[10,32],"label":"(10, 32)"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"ab24689f92e5"},{"id":"g89-velocity-time-graphs-q05","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":5,"marks":4,"tier":"H","calc":false,"text":"

A train travels at a constant velocity of 30 m/s for 40 seconds.

It then decelerates uniformly, coming to rest in a further 25 seconds.

  1. Work out the deceleration of the train.
  2. Work out the total distance the train travels in the 65 seconds.

","answerType":"multi","answer":"a) 1.2 m/s^2; b) 1575 m","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1.2 m/s^2"},{"label":"b","answer":"1575 m"}],"solution":["a) Deceleration = 3025 = 1.2 m/s^2.","b) Distance = area under the velocity-time graph.","Constant phase (rectangle): 40 x 30 = 1200 m.","Decelerating phase (triangle): 12 x 25 x 30 = 375 m.","Total = 1200 + 375 = 1575 m."],"diagram":{"type":"axes","x":{"min":0,"max":70,"step":10,"label":"Time (s)"},"y":{"min":0,"max":35,"step":5,"label":"Velocity (m/s)"},"curves":[{"pts":[[0,30],[40,30],[65,0]]}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"93614629cbdc"},{"id":"g89-velocity-time-graphs-q06","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":6,"marks":4,"tier":"H","calc":true,"text":"

A motorbike accelerates from rest. Its speed-time graph is a smooth curve that gets less steep throughout the first 12 seconds. The table gives the speed v m/s at time t seconds.

t: 0, 4, 8, 12   v: 0, 7, 11, 13

  1. Use 3 strips of equal width to estimate the distance travelled in the first 12 seconds.
  2. Amir says, \"The acceleration at t = 8 is 11 ÷ 8 = 1.375 m/s².\" Explain why Amir is wrong.

","answerType":"multi","answer":"a) 98 m; b) 11/8 is the average acceleration over the first 8 s (chord gradient), not the gradient of the tangent at t = 8","answerDisplay":"a) 98 m; b) 118 is the average acceleration over the first 8 s (chord gradient), not the gradient of the tangent at t = 8","accept":[],"parts":[{"label":"a","answer":"98 m"},{"label":"b","answer":"Amir has found the gradient of the chord from (0, 0) to (8, 11), which is the average acceleration over the first 8 seconds. The acceleration at t = 8 is the gradient of the tangent to the curve at t = 8, which is smaller here."}],"solution":["a) Trapezium strips of width 4: 12 x 4 x (0 + 7) + 12 x 4 x (7 + 11) + 12 x 4 x (11 + 13).","= 14 + 36 + 48 = 98 m.","b) 11 divided by 8 is the gradient of the straight line from the origin to (8, 11): the average acceleration over the first 8 seconds.","The acceleration at the instant t = 8 is the gradient of the tangent to the curve at t = 8. Because the curve is getting less steep, this tangent gradient is smaller than 1.375 m/s^2, so Amir's method is wrong."],"diagram":null,"draw":false,"revision":"7a059c99a9f4"},{"id":"g89-velocity-time-graphs-q07","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":7,"marks":5,"tier":"H","calc":true,"text":"

A lorry starts from rest and accelerates uniformly for 10 seconds until it reaches a velocity of 20 m/s.

It then travels at a constant velocity of 20 m/s for T seconds.

The lorry travels a total distance of 700 m.

Work out the total time of the journey.

","answerType":"exact","answer":"40 seconds","answerDisplay":null,"accept":["40","40 s","T = 30 so total 40 s"],"parts":[],"solution":["Distance in the first 10 seconds (triangle): 12 x 10 x 20 = 100 m.","Distance remaining at constant velocity: 700 - 100 = 600 m.","Time at 20 m/s: 60020 = 30 seconds, so T = 30.","Total time = 10 + 30 = 40 seconds."],"diagram":{"type":"shape","notAccurate":true,"segments":[{"from":[0,0],"to":[9,0],"label":"Time (s)","arrow":"end"},{"from":[0,0],"to":[0,6],"label":"Velocity (m/s)","arrow":"end"},{"from":[0,0],"to":[3,4]},{"from":[3,4],"to":[8,4]},{"from":[3,0],"to":[3,4],"dashed":true},{"from":[8,0],"to":[8,4],"dashed":true},{"from":[3,-1],"to":[8,-1],"label":"T seconds","arrow":"both"}],"texts":[{"at":[-0.7,4],"text":"20"},{"at":[3,-0.4],"text":"10"},{"at":[0,-0.4],"text":"0"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"b9a853d282f7"},{"id":"g89-velocity-time-graphs-q08","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":8,"marks":5,"tier":"H","calc":true,"text":"

A ball rolls down a slope. Its velocity-time graph is a smooth curve. The table gives the velocity v m/s at time t seconds.

t: 0, 2, 4, 6   v: 0, 2.2, 5.2, 9.0

  1. A tangent drawn to the curve at t = 4 passes through the points (1, 0.1) and (6, 8.6). Use the tangent to estimate the acceleration of the ball at t = 4.
  2. Use the trapezium rule with 3 strips of equal width to estimate the distance the ball travels in the 6 seconds.

","answerType":"multi","answer":"a) 1.7 m/s^2; b) 23.8 m","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"1.7 m/s^2 (accept about 1.7 m/s^2)"},{"label":"b","answer":"23.8 m"}],"solution":["a) Acceleration at t = 4 is the gradient of the tangent: (8.6 - 0.1) ÷ (6 - 1) = 8.5 ÷ 5 = 1.7 m/s^2.","b) Trapezium strips of width 2:","12 x 2 x (0 + 2.2) = 2.2","12 x 2 x (2.2 + 5.2) = 7.4","12 x 2 x (5.2 + 9.0) = 14.2","Estimate = 2.2 + 7.4 + 14.2 = 23.8 m."],"diagram":{"type":"axes","x":{"min":0,"max":6,"step":2,"label":"Time (s)"},"y":{"min":0,"max":10,"step":2,"label":"Velocity (m/s)"},"curves":[{"fn":"0.1*x*x+0.9*x"}],"lines":[{"through":[[1,0.1],[6,8.6]],"from":1,"to":6,"dashed":true,"label":"Drawn tangent"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"2ff2e8d1709f"},{"id":"g89-velocity-time-graphs-q09","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":9,"marks":6,"tier":"H","calc":true,"text":"

A car makes a journey in three stages.

Stage 1: it accelerates uniformly from rest to a velocity of V m/s in 12 seconds.

Stage 2: it travels at the constant velocity V m/s for 48 seconds.

Stage 3: it decelerates uniformly from V m/s to rest in 20 seconds.

The total distance of the journey is 960 m.

  1. Work out the value of V.
  2. Work out the acceleration of the car during stage 1.

","answerType":"multi","answer":"a) V = 15; b) 1.25 m/s^2","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"15"},{"label":"b","answer":"1.25 m/s^2"}],"solution":["a) Total distance = area under the velocity-time graph (a trapezium made of a triangle, a rectangle and a triangle).","Stage 1: 12 x 12 x V = 6V. Stage 2: 48V. Stage 3: 12 x 20 x V = 10V.","6V + 48V + 10V = 64V = 960, so V = 15.","Check: 6(15) + 48(15) + 10(15) = 90 + 720 + 150 = 960.","b) Acceleration in stage 1 = 1512 = 1.25 m/s^2."],"diagram":{"type":"shape","notAccurate":true,"segments":[{"from":[0,0],"to":[10,0],"label":"Time (s)","arrow":"end"},{"from":[0,0],"to":[0,6],"label":"Velocity (m/s)","arrow":"end"},{"from":[0,0],"to":[2,4]},{"from":[2,4],"to":[7,4]},{"from":[7,4],"to":[9,0]},{"from":[2,0],"to":[2,4],"dashed":true},{"from":[7,0],"to":[7,4],"dashed":true}],"texts":[{"at":[-0.6,4],"text":"V"},{"at":[2,-0.5],"text":"12"},{"at":[7,-0.5],"text":"60"},{"at":[9,-0.5],"text":"80"},{"at":[0,-0.5],"text":"0"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"a566853db4be"},{"id":"g89-velocity-time-graphs-q10","topicId":"g89-velocity-time-graphs","topic":"Velocity Time Graphs","grade":"89","topicGrade":"89","strand":"A","difficulty":10,"marks":7,"tier":"H","calc":true,"text":"

A drone accelerates from rest. Its velocity-time graph is a smooth curve that gets less steep throughout the first 12 seconds. The table gives the velocity v m/s at time t seconds.

t: 0, 3, 6, 9, 12   v: 0, 4.5, 7.8, 9.9, 10.8

  1. Use the trapezium rule with 4 strips of equal width to estimate the distance travelled by the drone in the 12 seconds.
  2. A tangent drawn to the curve at t = 6 passes through the points (0, 2.4) and (12, 13.2). Use the tangent to estimate the acceleration of the drone at t = 6.
  3. Is your answer to part (a) an underestimate or an overestimate of the actual distance? Give a reason.

","answerType":"multi","answer":"a) 82.8 m; b) 0.9 m/s^2; c) underestimate, the chords lie below the curve","answerDisplay":null,"accept":[],"parts":[{"label":"a","answer":"82.8 m"},{"label":"b","answer":"0.9 m/s^2"},{"label":"c","answer":"underestimate: the velocity increases at a decreasing rate, so the curve bows above each chord and the trapezia miss some area"}],"solution":["a) Trapezium strips of width 3:","12 x 3 x (0 + 4.5) = 6.75","12 x 3 x (4.5 + 7.8) = 18.45","12 x 3 x (7.8 + 9.9) = 26.55","12 x 3 x (9.9 + 10.8) = 31.05","Estimate = 6.75 + 18.45 + 26.55 + 31.05 = 82.8 m.","b) Gradient of the tangent = (13.2 - 2.4) ÷ (12 - 0) = 10.8 ÷ 12 = 0.9, so the acceleration at t = 6 is about 0.9 m/s^2.","c) The question states that the curve gets less steep throughout the interval. Each straight top edge of a trapezium lies below the curve, so the trapezia miss some area: the estimate is an underestimate. The table values alone would not prove this shape between the recorded times."],"diagram":{"type":"axes","x":{"min":0,"max":12,"step":2,"label":"Time (s)"},"y":{"min":0,"max":14,"step":2,"label":"Velocity (m/s)"},"curves":[{"fn":"1.7*x-x*x/15"}],"lines":[{"through":[[0,2.4],[12,13.2]],"from":0,"to":12,"dashed":true,"label":"Drawn tangent"}],"alt":"Question diagram. Use the labels and information in the question."},"draw":false,"revision":"71bda9ecfd9f"}]}