Day 1 · Find evidence
Review a returned paper or take the short topic check. Sort errors into knowledge, method, accuracy or timing. Pick three specific topics.
Free GCSE maths · online and printable
GCSE maths mock revision: a practical two-week pack
Choose Foundation or Higher, identify three priorities and build a realistic two-week plan around them. Use original mixed practice, a mistake-review sheet and a calm exam-day checklist.
This is a selected practice resource, not a full specification, mock paper or grade prediction. Completing it cannot guarantee a grade. Ask your teacher which tier and topics are appropriate.
Robinson Tuition · www.robinsontuition.com/gcse-maths-mock-revision
Use the pack on paper
Choose the sections you need. Your browser can print or save them as a PDF. Questions include working space; worked answers begin separately.
Choose A4, standard margins and 100% scale. Switch off browser headers and footers if you prefer. No answers or notes are sent anywhere.
How to use this pack
- Try the questions on paper first. Follow each calculator label and keep your working.
- Open the worked answer when you are ready. Compare the method as well as the result; equivalent correct methods count.
- Record where you worked independently, needed a hint, made an error or skipped. Use the topic map to choose what to practise next.
Marks are suggested maximums for these original practice questions, not an exam-board mark scheme. The worked steps show the method; ask a teacher if you are unsure about partial credit. No score from this page changes Maths Streak or tracker progress.
Open your two-week plan
Your two-week plan
Move days to fit school, homework and rest. If a session feels too long, split it. A missed day is a reason to adjust the plan, not to double the next day’s workload.
Day 2 · Number priority
Choose one number gap. Read two worked examples, cover them, then attempt 6–10 questions. Rework two errors.
Day 3 · Algebra priority
Choose one algebra gap. Write every rearrangement or expansion and check answers by substitution where possible.
Day 4 · Ratio and percentages
Practise a ratio or percentage topic. State which amount represents 100% and keep ratio units consistent.
Day 5 · Geometry priority
Practise a geometry gap. Sketch the diagram, label the facts and write the reason or formula before calculating.
Day 6 · Mixed set: first half
Attempt questions 1–9 of your tier’s mixed set. Follow each calculator label. Record questions you skip.
Day 7 · Review and rest
Mark questions 1–9. Complete one mistake-review row for each error; redo two questions, then stop.
Day 8 · Graphs and measures
Practise reading scales, units and graphs. Write the quantity represented by a gradient where relevant.
Day 9 · Probability and data
Work on a probability or averages gap. Explain the denominator or the average you chose before calculating.
Day 10 · Return to a priority
Retry the hardest topic from days 2–5 with fresh Topic Workout questions. Compare the method with your mistake sheet.
Day 11 · Mixed set: second half
Attempt questions 10–18 of your tier’s mixed set. Keep exact values until a question requests rounding.
Day 12 · Review under time pressure
Mark questions 10–18. Redo two errors, then practise a short group of familiar questions with a sensible time limit.
Day 13 · Practise paper routines
Use your school’s paper or an official past paper. Read instructions, allocate time and practise checking. This short pack is not a full mock paper.
Day 14 · Light retrieval and prepare
Read your mistake sheet, explain three key methods from memory and use the exam-day checklist. Leave time to rest.
Foundation mixed practice
18 original questions · 49 suggested marks · 45–60 minutes, or two shorter sessions
Work on paper. Write a reason or method, even when the final answer is short. Leave a question and return later if needed.
Foundation mixed practice · Question 1
Work out 16 − 42 × 3
Working
Foundation mixed practice · Question 2
Work out 34 of 92
Working
Foundation mixed practice · Question 3
Work out 425 − 134
Give your answer as a mixed number in its simplest form.
Working
Foundation mixed practice · Question 4
In 2019 Femi bought a flat for £180 000.
In 2026 he sold the flat for £207 000.
Work out his percentage profit.
Working
Foundation mixed practice · Question 5
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.
Working
Foundation mixed practice · Question 6
- Simplify 4c × 2d
- Work out the value of 4c × 2d when c = 3 and d = 1.5
Working
Foundation mixed practice · Question 7
Solve 2x + 53 = 7
Working
Foundation mixed practice · Question 8
- Expand and simplify 3(x + 2) + 2(x − 5)
- Solve 5(x − 3) = 20
Working
Foundation mixed practice · Question 9
The nth term of a sequence is 6n − 1
Write down the first three terms of the sequence.
Working
Foundation mixed practice · Question 10
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.
Working
Foundation mixed practice · Question 11
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.
Working
Foundation mixed practice · Question 12
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.
Working
Foundation mixed practice · Question 13
Here is a table of values for y = 3x − 4
x: −1, 0, 1, 2, 3
y: ?, −4, −1, 2, ?
- Work out the two missing values of y.
- Does the point (10, 26) lie on the line y = 3x − 4? You must show how you get your answer.
Working
Foundation mixed practice · Question 14
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.
Working
Foundation mixed practice · Question 15
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.
Working
Foundation mixed practice · Question 16
A fair eight-sided spinner has sections numbered 1 to 8.
The spinner is spun once.
- Write down the word that best describes the probability that the spinner lands on 9.
- Find the probability that the spinner lands on a multiple of 3.
Working
Foundation mixed practice · Question 17
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.
Working
Foundation mixed practice · Question 18
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.
Working
Higher mixed practice
18 original questions · 55 suggested marks · 55–70 minutes, or two shorter sessions
Work on paper. Write a reason or method, even when the final answer is short. Leave a question and return later if needed.
Higher mixed practice · Question 1
A trader buys a table for £68.
She sells the table for £85.
Work out her percentage profit.
Working
Higher mixed practice · Question 2
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.
Working
Higher mixed practice · Question 3
- Sam expands 3(2x + 4) and gets the answer 6x + 4
Explain the mistake Sam has made. - Factorise fully 12x2 + 18x
Working
Higher mixed practice · Question 4
Solve x2 - 11x + 28 = 0
Working
Higher mixed practice · Question 5
Solve the simultaneous equations
3x + 2y = 19
x + 4y = 13
Working
Higher mixed practice · Question 6
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.
Working
Higher mixed practice · Question 7
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
Working
Higher mixed practice · Question 8
Write these numbers in order of size, starting with the smallest.
3.2 × 104 2.9 × 105 8.7 × 103 3.05 × 104
Working
Higher mixed practice · Question 9
2n = 18
Find the value of n.
Working
Higher mixed practice · Question 10
Express √80 + √45 in the form k√5, where k is an integer.
Working
Higher mixed practice · Question 11
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
- angle ACD,
- angle ADC.
Give a reason for each stage of your working.
Working
Higher mixed practice · Question 12
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.
Working
Higher mixed practice · Question 13
The stem and leaf diagram shows the marks of 15 students in a spelling test.
| Stem | Leaves |
|---|---|
| 1 | 2 5 8 |
| 2 | 0 3 4 7 9 |
| 3 | 1 4 6 8 |
| 4 | 0 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.
Working
Higher mixed practice · Question 14
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.
Working
Higher mixed practice · Question 15
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.
- Find a formula for F in terms of d.
- Find the positive value of d when F = 15.
Working
Higher mixed practice · Question 16
Solve 3x2 - 11x + 6 = 0
Working
Higher mixed practice · Question 17
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.
Working
Higher mixed practice · Question 18
By completing the square, solve
x2 + 6x + 4 = 0
Give your answers in the form a ± √b where a and b are integers.
Working
Turn your answers into priorities
Record “independent”, “partly”, “revisit” or “skipped” beside each question. A single answer is only one piece of evidence. Start with two or three topics where the method was unclear, and retry a different question after practice. If you are unsure about partial marks, ask a teacher to look at your working.
Foundation mixed practice: question-to-topic map
Q1: BIDMAS (Order of Operations)
Evidence / next step: ____________________
Practise this skill (Foundation)
Q2: Fractions of an Amount
Evidence / next step: ____________________
Practise this skill (Foundation)
Q3: Fractions
Evidence / next step: ____________________
Practise this skill (Foundation)
Q4: Percentage Change
Evidence / next step: ____________________
Practise this skill (Foundation)
Q5: Ratio
Evidence / next step: ____________________
Practise this skill (Foundation)
Q6: Substitution
Evidence / next step: ____________________
Practise this skill (Foundation)
Q7: Solving Equations
Evidence / next step: ____________________
Practise this skill (Foundation)
Q8: Expanding and Factorising
Evidence / next step: ____________________
Practise this skill (Foundation)
Q9: Sequences (Nth Term)
Evidence / next step: ____________________
Practise this skill (Foundation)
Q10: Angles
Evidence / next step: ____________________
Practise this skill (Foundation)
Q11: Pythagoras
Evidence / next step: ____________________
Practise this skill (Foundation)
Q12: Volume of a Prism
Evidence / next step: ____________________
Practise this skill (Foundation)
Q13: Drawing Linear Graphs
Evidence / next step: ____________________
Practise this skill (Foundation)
Q14: Two Way Tables
Evidence / next step: ____________________
Practise this skill (Foundation)
Q15: Averages
Evidence / next step: ____________________
Practise this skill (Foundation)
Q16: Probability
Evidence / next step: ____________________
Practise this skill (Foundation)
Q17: Proportion
Evidence / next step: ____________________
Practise this skill (Foundation)
Q18: SOHCAHTOA (Trigonometry)
Evidence / next step: ____________________
Practise this skill (Foundation)
Higher mixed practice: question-to-topic map
Q1: Percentage Change
Evidence / next step: ____________________
Practise this skill (Higher)
Q2: Ratio
Evidence / next step: ____________________
Practise this skill (Higher)
Q3: Expanding and Factorising
Evidence / next step: ____________________
Practise this skill (Higher)
Q4: Solving Quadratics
Evidence / next step: ____________________
Practise this skill (Higher)
Q5: Simultaneous Equations
Evidence / next step: ____________________
Practise this skill (Higher)
Q6: SOHCAHTOA (Trigonometry)
Evidence / next step: ____________________
Practise this skill (Higher)
Q7: Equation of a Line
Evidence / next step: ____________________
Practise this skill (Higher)
Q8: Standard Form
Evidence / next step: ____________________
Practise this skill (Higher)
Q9: Fractional and Negative Indices
Evidence / next step: ____________________
Practise this skill (Higher)
Q10: Surds
Evidence / next step: ____________________
Practise this skill (Higher)
Q11: Circle Theorems
Evidence / next step: ____________________
Practise this skill (Higher)
Q12: Similar Shapes (Area and Volume)
Evidence / next step: ____________________
Practise this skill (Higher)
Q13: Box Plots
Evidence / next step: ____________________
Practise this skill (Higher)
Q14: Probability Trees
Evidence / next step: ____________________
Practise this skill (Higher)
Q15: Direct and Inverse Proportion (Algebraic)
Evidence / next step: ____________________
Practise this skill (Higher)
Q16: Factorising Harder Quadratics
Evidence / next step: ____________________
Practise this skill (Higher)
Q17: Conditional Probability
Evidence / next step: ____________________
Practise this skill (Higher)
Q18: Completing the Square
Evidence / next step: ____________________
Practise this skill (Higher)
For a guided summary, try the short topic check. Neither this pack nor that check predicts a GCSE grade or assesses the whole course.
Mistake-review sheet
Choose useful errors, not just the hardest questions. Cover the answer and redo the question after writing the corrected method.
Review 1
Question / topic: __________________________________________
Error: knowledge / method / accuracy / timing / skipped
What happened? __________________________________________
Corrected method: ________________________________________
________________________________________________________
Next practice and date: ____________________________________
Review 2
Question / topic: __________________________________________
Error: knowledge / method / accuracy / timing / skipped
What happened? __________________________________________
Corrected method: ________________________________________
________________________________________________________
Next practice and date: ____________________________________
Review 3
Question / topic: __________________________________________
Error: knowledge / method / accuracy / timing / skipped
What happened? __________________________________________
Corrected method: ________________________________________
________________________________________________________
Next practice and date: ____________________________________
Review 4
Question / topic: __________________________________________
Error: knowledge / method / accuracy / timing / skipped
What happened? __________________________________________
Corrected method: ________________________________________
________________________________________________________
Next practice and date: ____________________________________
Your blank two-week timetable
Print this sheet and pencil in a manageable time, a specific skill and a review activity. Leave room for schoolwork and rest.
Week 1
| Day | Time / minutes | Topic and activity | Check / next step |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 | |||
| 7 |
Week 2
| Day | Time / minutes | Topic and activity | Check / next step |
|---|---|---|---|
| 8 | |||
| 9 | |||
| 10 | |||
| 11 | |||
| 12 | |||
| 13 | |||
| 14 |
Exam-day checklist
- Confirm the date, time, room and tier with school.
- Pack black pens, pencil, ruler, rubber, protractor, compasses and any other equipment your school requests.
- For calculator papers, check your calculator works and know how to set degrees and enter fractions.
- Follow the paper’s instructions. Check units, rounding and whether working is required.
- Leave an attempt at questions you can start; move on and return if stuck.
- Use spare time to check signs, scales, substituted answers and whether results are sensible.
Calculator and non-calculator practice
Each question is labelled. Put the calculator aside for non-calculator items. For calculator questions, estimate first and keep unrounded values until the final line. Your school or exam board sets the instructions for the actual paper.
Use a full paper when you are ready
These 18-question sets are mixed practice, not complete mock papers. For full coverage and timing, use a paper provided by school or an official AQA, Pearson Edexcel or OCR past paper.
Worked answers
Keep this section covered until you have tried the questions. A matching final answer does not establish that all the reasoning is correct.
Credit an equivalent correct method. A correct method can earn its checkpoint even if later arithmetic is wrong; only award an accuracy checkpoint when its stated result is correct. Do not award the same checkpoint twice. For a proof, the reasons are essential. This original guidance is not an exam-board scheme.
Foundation mixed practice · Answer 1: BIDMAS (Order of Operations)
Foundation mixed practice · Answer 1
1 suggested marks · compare each step, not just the final answer.
Work out the top of the fraction: 16 - 4 = 12
Work out the bottom of the fraction: 2 × 3 = 6
12 ÷ 6 = 2
2
Suggested marking checkpoints
- 1 mark: 2.
Next practice: BIDMAS (Order of Operations) workout · Learn this method
Foundation mixed practice · Answer 2: Fractions of an Amount
Foundation mixed practice · Answer 2
1 suggested marks · compare each step, not just the final answer.
92 ÷ 4 = 23.
23 × 3 = 69.
69
Suggested marking checkpoints
- 1 mark: 69.
Next practice: Fractions of an Amount workout · Learn this method
Foundation mixed practice · Answer 3: Fractions
Foundation mixed practice · Answer 3
3 suggested marks · compare each step, not just the final answer.
Write as improper fractions: 425 = 225 and 134 = 74.
Use denominator 20: 225 = 8820 and 74 = 3520.
8820 - 3520 = 5320 = 21320.
21320
Suggested marking checkpoints
- 1 mark: Correct improper fractions 22/5 and 7/4.
- 1 mark: Correct common-denominator subtraction 88/20 − 35/20 = 53/20 (or equivalent whole/fraction method).
- 1 mark: Simplest mixed number 2 13/20.
Next practice: Fractions workout · Learn this method
Foundation mixed practice · Answer 4: Percentage Change
Foundation mixed practice · Answer 4
3 suggested marks · compare each step, not just the final answer.
Profit = 207 000 − 180 000 = £27 000.
Percentage profit = 27 000 ÷ 180 000 = 0.15.
0.15 × 100 = 15%.
15%
Suggested marking checkpoints
- 1 mark: Profit £27,000.
- 1 mark: Divide by original £180,000.
- 1 mark: 15%.
Next practice: Percentage Change workout · Learn this method
Foundation mixed practice · Answer 5: Ratio
Foundation mixed practice · Answer 5
4 suggested marks · compare each step, not just the final answer.
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.
£753
Suggested marking checkpoints
- 1 mark: Use eight ratio parts: 30 candles per part.
- 1 mark: 90 large and 150 small.
- 1 mark: Correct costs £378 and £375, with a method to add them.
- 1 mark: Total £753.
Next practice: Ratio workout · Learn this method
Foundation mixed practice · Answer 6: Substitution
Foundation mixed practice · Answer 6
3 suggested marks · compare each step, not just the final answer.
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.
(a) 8cd; (b) 36
Suggested marking checkpoints
- 1 mark: (a) 8cd.
- 1 mark: (b) Substitute 8 × 3 × 1.5.
- 1 mark: 36.
Next practice: Substitution workout · Learn this method
Foundation mixed practice · Answer 7: Solving Equations
Foundation mixed practice · Answer 7
3 suggested marks · compare each step, not just the final answer.
Multiply both sides by 3: 2x + 5 = 21
Subtract 5 from both sides: 2x = 16
Divide both sides by 2: x = 8
x = 8
Suggested marking checkpoints
- 1 mark: Multiply by 3 to get 2x + 5 = 21.
- 1 mark: Subtract 5 to get 2x = 16.
- 1 mark: x = 8.
Next practice: Solving Equations workout · Learn this method
Foundation mixed practice · Answer 8: Expanding and Factorising
Foundation mixed practice · Answer 8
4 suggested marks · compare each step, not just the final answer.
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
(a) 5x - 4; (b) x = 7
Suggested marking checkpoints
- 1 mark: (a) Expand to 3x + 6 + 2x − 10.
- 1 mark: Simplify to 5x − 4.
- 1 mark: (b) Valid transformation of 5(x − 3) = 20.
- 1 mark: x = 7.
Next practice: Expanding and Factorising workout · Learn this method
Foundation mixed practice · Answer 9: Sequences (Nth Term)
Foundation mixed practice · Answer 9
2 suggested marks · compare each step, not just the final answer.
When n = 1: 6(1) - 1 = 5
When n = 2: 6(2) - 1 = 11
When n = 3: 6(3) - 1 = 17
5, 11, 17
Suggested marking checkpoints
- 1 mark: Correct substitution of n = 1, 2, 3 into 6n − 1.
- 1 mark: All three terms in order: 5, 11, 17.
Next practice: Sequences (Nth Term) workout · Learn this method
Foundation mixed practice · Answer 10: Angles
Foundation mixed practice · Answer 10
3 suggested marks · compare each step, not just the final answer.
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.
x = 21
Suggested marking checkpoints
- 1 mark: Use straight-line angles to form 5x + 4x − 9 = 180.
- 1 mark: Rearrange to 9x = 189.
- 1 mark: x = 21.
Next practice: Angles workout · Learn this method
Foundation mixed practice · Answer 11: Pythagoras
Foundation mixed practice · Answer 11
3 suggested marks · compare each step, not just the final answer.
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
6 m
Suggested marking checkpoints
- 1 mark: Identify the ladder as hypotenuse and use height² = 6.5² − 2.5².
- 1 mark: height² = 36.
- 1 mark: Take positive root: 6 m.
Next practice: Pythagoras workout · Learn this method
Foundation mixed practice · Answer 12: Volume of a Prism
Foundation mixed practice · Answer 12
3 suggested marks · compare each step, not just the final answer.
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
480 cm³
Suggested marking checkpoints
- 1 mark: Trapezium cross-section area 32 cm².
- 1 mark: Multiply by prism length 15.
- 1 mark: 480 cm³.
Next practice: Volume of a Prism workout · Learn this method
Foundation mixed practice · Answer 13: Drawing Linear Graphs
Foundation mixed practice · Answer 13
3 suggested marks · compare each step, not just the final answer.
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
(a) -7 and 5; (b) Yes, 3(10) - 4 = 26
Suggested marking checkpoints
- 1 mark: (a) Missing y = −7 when x = −1.
- 1 mark: Missing y = 5 when x = 3.
- 1 mark: (b) Yes, with substitution 3(10) − 4 = 26.
Next practice: Drawing Linear Graphs workout · Learn this method
Foundation mixed practice · Answer 14: Two Way Tables
Foundation mixed practice · Answer 14
3 suggested marks · compare each step, not just the final answer.
Men = 25 of 90 = 36, so women = 90 - 36 = 54.
Men at yoga = 36 - 21 = 15.
Yoga total = 15 + 25 = 40.
40
Suggested marking checkpoints
- 1 mark: 36 men.
- 1 mark: 15 men at yoga.
- 1 mark: 40 yoga members in total.
Next practice: Two Way Tables workout · Learn this method
Foundation mixed practice · Answer 15: Averages
Foundation mixed practice · Answer 15
3 suggested marks · compare each step, not just the final answer.
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.
3 minutes 32 seconds
Suggested marking checkpoints
- 1 mark: Sum 1060 seconds.
- 1 mark: Mean 212 seconds.
- 1 mark: 3 minutes 32 seconds.
Next practice: Averages workout · Learn this method
Foundation mixed practice · Answer 16: Probability
Foundation mixed practice · Answer 16
2 suggested marks · compare each step, not just the final answer.
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.
(a) impossible; (b) 14
Suggested marking checkpoints
- 1 mark: (a) Impossible: 9 is not on the spinner.
- 1 mark: (b) Multiples of 3 are 3 and 6, so probability 2/8 = 1/4.
Next practice: Probability workout · Learn this method
Foundation mixed practice · Answer 17: Proportion
Foundation mixed practice · Answer 17
3 suggested marks · compare each step, not just the final answer.
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.
No - 30 biscuits need 450 g of butter and she only has 425 g, so she is 25 g short
Suggested marking checkpoints
- 1 mark: Scale factor 30/12 = 2.5 (or unit amount 15 g per biscuit).
- 1 mark: 450 g butter needed.
- 1 mark: No: 425 g is less than 450 g, so she is 25 g short.
Next practice: Proportion workout · Learn this method
Foundation mixed practice · Answer 18: SOHCAHTOA (Trigonometry)
Foundation mixed practice · Answer 18
2 suggested marks · compare each step, not just the final answer.
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).
5.56 m
Suggested marking checkpoints
- 1 mark: Use sine with the ladder as hypotenuse: height = 6 sin 68°.
- 1 mark: 5.56 m (3 significant figures).
Next practice: SOHCAHTOA (Trigonometry) workout · Learn this method
Higher mixed practice · Answer 1: Percentage Change
Higher mixed practice · Answer 1
3 suggested marks · compare each step, not just the final answer.
Profit = 85 − 68 = £17.
Percentage profit = 17 ÷ 68 = 0.25.
0.25 × 100 = 25%.
25%
Suggested marking checkpoints
- 1 mark: Profit £17.
- 1 mark: Divide by the original cost £68.
- 1 mark: 25%.
Next practice: Percentage Change workout · Learn this method
Higher mixed practice · Answer 2: Ratio
Higher mixed practice · Answer 2
3 suggested marks · compare each step, not just the final answer.
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.
12
Suggested marking checkpoints
- 1 mark: Match banana parts to obtain apples:bananas:pears = 6:4:3.
- 1 mark: Use 24 apples for six ratio parts, so one part represents 4 pieces of fruit.
- 1 mark: 12 pears.
Next practice: Ratio workout · Learn this method
Higher mixed practice · Answer 3: Expanding and Factorising
Higher mixed practice · Answer 3
3 suggested marks · compare each step, not just the final answer.
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)
(a) He only multiplied the first term by 3; 3 times 4 = 12, so the answer should be 6x + 12; (b) 6x(2x + 3)
Suggested marking checkpoints
- 1 mark: (a) Explain that both terms must be multiplied by 3, giving 6x + 12.
- 1 mark: (b) Identify common factor 6x.
- 1 mark: 6x(2x + 3).
Next practice: Expanding and Factorising workout · Learn this method
Higher mixed practice · Answer 4: Solving Quadratics
Higher mixed practice · Answer 4
3 suggested marks · compare each step, not just the final answer.
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
x = 4 or x = 7
Suggested marking checkpoints
- 1 mark: Find factors with sum −11 and product 28, or correctly substitute into the quadratic formula.
- 1 mark: Obtain one correct root.
- 1 mark: Both roots 4 and 7, with a valid route from the equation.
Next practice: Solving Quadratics workout · Learn this method
Higher mixed practice · Answer 5: Simultaneous Equations
Higher mixed practice · Answer 5
3 suggested marks · compare each step, not just the final answer.
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 ✓
x = 5, y = 2
Suggested marking checkpoints
- 1 mark: Eliminate a variable using equivalent equations.
- 1 mark: x = 5.
- 1 mark: y = 2.
Next practice: Simultaneous Equations workout · Learn this method
Higher mixed practice · Answer 6: SOHCAHTOA (Trigonometry)
Higher mixed practice · Answer 6
2 suggested marks · compare each step, not just the final answer.
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).
67.4°
Suggested marking checkpoints
- 1 mark: cos KJL = 5/13.
- 1 mark: 67.4° (1 decimal place).
Next practice: SOHCAHTOA (Trigonometry) workout · Learn this method
Higher mixed practice · Answer 7: Equation of a Line
Higher mixed practice · Answer 7
3 suggested marks · compare each step, not just the final answer.
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.
y = 2x + 5
Suggested marking checkpoints
- 1 mark: Gradient (9 − 1)/(2 − (−2)) = 2.
- 1 mark: Intercept c = 5.
- 1 mark: y = 2x + 5.
Next practice: Equation of a Line workout · Learn this method
Higher mixed practice · Answer 8: Standard Form
Higher mixed practice · Answer 8
2 suggested marks · compare each step, not just the final answer.
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.
8.7 x 10^3, 3.05 x 10^4, 3.2 x 10^4, 2.9 x 10^5
Suggested marking checkpoints
- 1 mark: Correct comparison using common powers or ordinary-number conversions.
- 1 mark: All four numbers in the correct ascending order.
Next practice: Standard Form workout · Learn this method
Higher mixed practice · Answer 9: Fractional and Negative Indices
Higher mixed practice · Answer 9
2 suggested marks · compare each step, not just the final answer.
18 = 12³ = 2−3.
So 2n = 2−3, giving n = −3.
-3
Suggested marking checkpoints
- 1 mark: Write 1/8 as 2⁻³.
- 1 mark: n = −3.
Next practice: Fractional and Negative Indices workout · Learn this method
Higher mixed practice · Answer 10: Surds
Higher mixed practice · Answer 10
3 suggested marks · compare each step, not just the final answer.
√80 = √(16 × 5) = 4√5.
√45 = √(9 × 5) = 3√5.
4√5 + 3√5 = 7√5, so k = 7.
7√5
Suggested marking checkpoints
- 1 mark: √80 = 4√5.
- 1 mark: √45 = 3√5.
- 1 mark: 7√5, hence k = 7.
Next practice: Surds workout · Learn this method
Higher mixed practice · Answer 11: Circle Theorems
Higher mixed practice · Answer 11
4 suggested marks · compare each step, not just the final answer.
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°.
(a) 29°; (b) 105°
Suggested marking checkpoints
- 1 mark: (a) 29°.
- 1 mark: Reason: angles in the same segment standing on AD.
- 1 mark: (b) Angle ABC = 75°.
- 1 mark: Angle ADC = 105°, justified by opposite angles of a cyclic quadrilateral totalling 180°.
Next practice: Circle Theorems workout · Learn this method
Higher mixed practice · Answer 12: Similar Shapes (Area and Volume)
Higher mixed practice · Answer 12
3 suggested marks · compare each step, not just the final answer.
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.
81 cm³
Suggested marking checkpoints
- 1 mark: Length factor from C to D is 3/2 (square root of area factor).
- 1 mark: Volume factor (3/2)³.
- 1 mark: Volume 81 cm³.
Next practice: Similar Shapes (Area and Volume) workout · Learn this method
Higher mixed practice · Answer 13: Box Plots
Higher mixed practice · Answer 13
3 suggested marks · compare each step, not just the final answer.
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.
(lowest) 12; (lower quartile) 20; (median) 29; (upper quartile) 38; (highest) 45
Suggested marking checkpoints
- 1 mark: Median 29.
- 1 mark: Quartiles 20 and 38.
- 1 mark: Minimum 12 and maximum 45; this question requests the five values, not a drawn plot.
Next practice: Box Plots workout · Learn this method
Higher mixed practice · Answer 14: Probability Trees
Higher mixed practice · Answer 14
4 suggested marks · compare each step, not just the final answer.
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
0.45
Suggested marking checkpoints
- 1 mark: Identify the two mutually exclusive orders for exactly one pass.
- 1 mark: Cara passes and Dan fails: 0.15.
- 1 mark: Cara fails and Dan passes: 0.30.
- 1 mark: Add to obtain 0.45.
Next practice: Probability Trees workout · Learn this method
Higher mixed practice · Answer 15: Direct and Inverse Proportion (Algebraic)
Higher mixed practice · Answer 15
4 suggested marks · compare each step, not just the final answer.
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.
(a) F = 240d2; (b) 4
Suggested marking checkpoints
- 1 mark: F = k/d².
- 1 mark: k = 240.
- 1 mark: For F = 15 obtain d² = 16.
- 1 mark: Positive d = 4.
Next practice: Direct and Inverse Proportion (Algebraic) workout · Learn this method
Higher mixed practice · Answer 16: Factorising Harder Quadratics
Higher mixed practice · Answer 16
3 suggested marks · compare each step, not just the final answer.
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.
x = 23, x = 3
Suggested marking checkpoints
- 1 mark: Correct factorisation (3x − 2)(x − 3), or correct quadratic-formula substitution.
- 1 mark: One correct root.
- 1 mark: Both roots 2/3 and 3 from valid working.
Next practice: Factorising Harder Quadratics workout · Learn this method
Higher mixed practice · Answer 17: Conditional Probability
Higher mixed practice · Answer 17
4 suggested marks · compare each step, not just the final answer.
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
0.75
Suggested marking checkpoints
- 1 mark: Identify both ways to cycle on Tuesday.
- 1 mark: 0.7 × 0.9 = 0.63.
- 1 mark: 0.3 × 0.4 = 0.12.
- 1 mark: Add to obtain 0.75.
Next practice: Conditional Probability workout · Learn this method
Higher mixed practice · Answer 18: Completing the Square
Higher mixed practice · Answer 18
3 suggested marks · compare each step, not just the final answer.
Complete the square: x^2 + 6x + 4 = (x + 3)^2 - 9 + 4 = (x + 3)^2 - 5.
So (x + 3)^2 = 5.
Take square roots: x + 3 = ±√5.
x = -3 ± √5.
Check by expanding: (x + 3)^2 - 5 = x^2 + 6x + 9 - 5 = x^2 + 6x + 4.
x = -3 + sqrt(5) or x = -3 - sqrt(5)
Suggested marking checkpoints
- 1 mark: (x + 3)² − 5 = 0.
- 1 mark: x + 3 = ±√5.
- 1 mark: Both roots −3 ± √5.
Next practice: Completing the Square workout · Learn this method
Learn the methods
BIDMAS (Order of Operations)
Brackets identify a calculation to do as a unit. Next evaluate powers. Multiplication and division have equal priority: work left to right. Then add and subtract left to right. A fraction bar groups the whole numerator and denominator, so calculate each before dividing.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Sequences (Nth Term)
The letter n is the position, starting at 1, not the previous term. Substitute n = 1, 2, 3 into the formula to get the first three terms. Multiply before adding or subtracting. The common difference of a linear sequence is the coefficient of n.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Two Way Tables
A two-way table counts each person in one row and one column. Row and column totals must agree. Find missing categories by subtracting known counts from the relevant total; then add only the categories asked for. Sketching a table helps avoid counting the same person twice.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Proportion
When the unit price is constant, cost and quantity increase in the same ratio. Divide the known cost by the quantity to find the cost of one unit, then multiply by the required quantity. Keep the unit beside each value so you do not multiply by the original quantity again.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Standard Form
Standard form is a × 10ⁿ with 1 ≤ a < 10 and integer n. The power records place value, not a number to add. To compare positive numbers, compare powers first and coefficients second when powers match. Converting to ordinary numbers is an equally valid check.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Fractional and Negative Indices
A negative power means a reciprocal: a⁻ⁿ = 1/aⁿ for a ≠ 0. A fractional power a^(m/n) means take the nth root then raise to power m (use positive bases here). Thus 64^(−2/3) = 1/(cube root of 64)² = 1/16. The negative sign does not make the answer negative.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Surds
A surd is an exact root left unevaluated. To simplify a square root, take out square factors: √80 = √(16 × 5) = 4√5. Only like surds combine, just as like algebraic terms do. When squaring a bracket, multiply every term by every term; the middle terms do not disappear.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Circle Theorems
First identify the chord or arc used by both angles. Radii of the same circle are equal, giving isosceles triangles. Angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180°; the angle at the centre is twice the angle at the circumference on the same arc. State the relevant fact at each step.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Similar Shapes (Area and Volume)
Similar shapes have the same shape and one constant length scale factor k. Area scales by k² because two lengths are multiplied; volume scales by k³ because three are multiplied. Take a square root of an area ratio to recover k. Mass scales like volume only when the material has the same density.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Box Plots
Order the data first. The median splits the ordered values in half. For an odd number of values, leave out the middle value when finding the median of each half: these are the lower and upper quartiles. Plot the minimum, two quartiles, median and maximum on one consistent scale.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Direct and Inverse Proportion (Algebraic)
Replace a proportionality statement with an equation containing a constant. Direct proportion to b² gives a = kb²; inverse proportion to d² gives F = k/d². Use a known pair to find the constant, then substitute the new value. When combining rules, substitute the whole expression in brackets before taking a power.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Factorising Harder Quadratics
For ax² + bx + c, find two numbers whose product is ac and sum is b. Split bx using those numbers, then factorise in pairs. If a product equals zero, at least one factor is zero, so solve both linear equations. Check by expanding; the quadratic formula is also a valid solution method unless factorising is specifically requested.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Conditional Probability
A conditional probability uses the group that is still possible after the first event. Without replacement, both the total and the relevant colour count may change. Multiply along one path; add different mutually exclusive paths. “Exactly one” often needs two paths, such as red then blue and blue then red.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.
Completing the Square
Half the coefficient of x to form a bracket. Expanding (x + p)² gives x² + 2px + p², so subtract the extra p² to keep equality. In y = (x − a)² + b, the square is smallest at x = a, giving turning point (a,b). Solving a positive square requires both the positive and negative square roots.
Use the worked answer for your question to follow these steps with numbers, then cover it and try the linked workout.