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Grade 7 · Higher · free GCSE maths worksheet

Proportion with square roots

Edexcel IGCSE

y proportional to the square root of x, or inversely proportional to it.

10 original questions, 35 marks. Type your answers below for an instant mark and a score, or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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How to do it

  1. Write the rule with k: y = k√x for direct, y = k/√x for inverse
  2. Put in the pair of values you are given and solve for k
  3. Write the formula out with your k, e.g. y = 3√x
  4. To find y: square root x, then multiply by k (or divide k by it)
  5. To find x: get √x on its own first, then SQUARE both sides

Watch out: When finding x, get √x on its own before you square - squaring y first (33² = 1089) gives a wrong, huge answer.

Worked example

y is directly proportional to the square root of x.

When x = 49, y = 21.

  1. Find a formula for y in terms of x.
  2. Find the value of x when y = 33.

a)

y=k√x

Put in x = 49, y = 21

21=k√49 = 7k
÷7k=3÷7

y = 3√x

b)

33=3√x
÷311=√x÷3

Square both sides

x=112

x = 121

The questions

1

y is directly proportional to the square root of x.

When x = 9, y = 12.

Find a formula for y in terms of x.

2 marks
Show answer
y = 4√x (accept y = 4 sqrt x)
2

y is directly proportional to √x.

When x = 25, y = 10.

  1. Find a formula for y in terms of x.
  2. Find the value of y when x = 64.
  3. Find the value of x when y = 30.
4 marks
Show answer
a) y = 2√x (accept y = 2 sqrt x) b) 16 c) 225
3

Here are descriptions of four sketch graphs. Each is drawn for x > 0.

Graph A: a straight line through the origin, sloping upwards.

Graph B: a curve starting at the origin that rises steeply at first and then flattens out, still rising, as x increases.

Graph C: a curve starting at the origin that rises slowly at first and then gets steeper and steeper as x increases.

Graph D: a curve that falls as x increases, getting closer and closer to the x-axis but never touching either axis.

  1. Write down the letter of the graph that could be y = k√x, where k > 0.
  2. Write down the letter of the graph that could be y = k√x, where k > 0.
2 marks
Show answer
a) B b) D
4

y is inversely proportional to the square root of x.

When x = 16, y = 5.

  1. Find a formula for y in terms of x.
  2. Find the value of y when x = 100.
  3. Find the value of x when y = 8.
4 marks
Show answer
a) y = 20/√x (accept y = 20/sqrt x) b) 2 c) 6.25
5

The time, T seconds, for one swing of a pendulum is directly proportional to the square root of its length, L cm.

A pendulum of length 100 cm takes 2 seconds for one swing.

  1. Find a formula for T in terms of L.
  2. Find the time for one swing of a pendulum of length 225 cm.
  3. Priya wants a pendulum that takes 2.4 seconds for one swing. Find, in cm, the length she should use.
4 marks
Show answer
a) T = 0.2√L (accept T = 0.2 sqrt L) b) 3 c) 144
6

y is directly proportional to the square root of x.

Here is a table of values for x and y. Three of the values are missing.

x425(b)144
y6(a)24(c)

Complete the table by finding the missing values.

  1. the value of y when x = 25
  2. the value of x when y = 24
  3. the value of y when x = 144
3 marks
Show answer
a) 15 b) 64 c) 36
7

The speed, v metres per second, of a wave in shallow water is directly proportional to the square root of the depth of the water, d metres.

When the depth is 6.25 metres, the speed of the wave is 7.5 metres per second.

  1. Find a formula for v in terms of d.
  2. Find, in metres, the depth of the water when the speed of the wave is 10.5 metres per second.
3 marks
Show answer
a) v = 3√d (accept v = 3 sqrt d) b) 12.25
8

y is directly proportional to the square root of x.

  1. x is multiplied by 4. Write down the number that y is multiplied by.
  2. x is increased by 44%. Work out the percentage increase in y.

w is inversely proportional to the square root of x.

  1. x is decreased by 36%. Work out the percentage increase in w.
4 marks
Show answer
a) 2 b) 20 c) 25
9

Here is a table of values for x and y.

x416100
y1052

Exactly one of these statements is true.

A: y is directly proportional to x.

B: y is directly proportional to √x.

C: y is inversely proportional to x.

D: y is inversely proportional to √x.

E: y is inversely proportional to x2.

  1. Write down the letter of the true statement.
  2. Find a formula for y in terms of x.
  3. Find the value of y when x = 400.
4 marks
Show answer
a) D b) y = 20/√x (accept y = 20/sqrt x) c) 1
10

y is inversely proportional to the square root of x.

When x = 2.25, y = 4.

  1. Find the value of y when x = 0.09.
  2. Find the value of x when y = 15.
  3. Find the value of x for which y = x. Give your answer correct to 3 significant figures.
5 marks
Show answer
a) 20 b) 0.16 c) 3.30

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Every question here is original and written by Robinson Tuition. This topic is on the Pearson Edexcel International GCSE (4MA1) and is not examined on the UK GCSE; the grade is our estimate against the Edexcel grade descriptors. This is not an exam-board paper.

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