Equations for direct and inverse proportion - Edexcel GCSE Maths (Calculator) Exam Questions

Practise 37 exam-style questions on Equations for direct and inverse proportion for Edexcel GCSE Maths (Calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

37 questions
1–5 marks each
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20 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

y is directly proportional to x.
When x = 4, y = 13.6

Find the value of y when x = 9.5

Model answer

y = kx
13.6 = k × 4, so k = 3.4
y = 3.4x
When x = 9.5, y = 3.4 × 9.5 = 32.3

Explanation

Write the proportion as an equation y = kx, use the given pair of values to find k, then substitute the new value of x. Because y is directly proportional to x, y ÷ x is always 3.4.

Mark scheme

1 mark
M1 for a correct method to find k, e.g. 13.6 = 4k or k = 3.4
oe, e.g. 13.6 ÷ 4 × 9.5
1 mark
A1 for 32.3
cao; correct answer with no working scores full marks

Question 2

3 marks

The extension, E cm, of a spring is directly proportional to the force, F newtons, applied to it.
When F = 5, E = 3.2

Find the force needed to give an extension of 7.8 cm.
Give your answer correct to 3 significant figures.

Model answer

E = kF
3.2 = k × 5, so k = 0.64
E = 0.64F
7.8 = 0.64F
F = 7.8 ÷ 0.64 = 12.1875
F = 12.2 newtons (3 s.f.)

Explanation

Write the relationship as E = kF and use the given values to find k. Then substitute E = 7.8 and rearrange to find F. A common mistake is to substitute 7.8 for F instead of for E.

Mark scheme

1 mark
M1 for E = kF
or 3.2 = 5k
1 mark
M1 for k = 0.64, or for 7.8 ÷ 0.64
oe, e.g. 7.8 ÷ 3.2 × 5
1 mark
A1 for 12.2 N
awrt 12.2; accept 12.1875; correct answer with no working scores full marks

Question 3

3 marks

y is directly proportional to the square of x.

x increases by 20%.

Work out the percentage increase in y.

Model answer

\(y = kx^2\)
New x = 1.2x
New y = \(k(1.2x)^2 = 1.44kx^2\)
So y is multiplied by 1.44, which is an increase of 44%.

Explanation

An increase of 20% means multiplying by 1.2. Because y depends on x², y is multiplied by 1.2² = 1.44, a 44% increase. A common mistake is to say 40% (doubling the 20%) or 20%.

Mark scheme

1 mark
M1 for using the multiplier 1.2 for x, e.g. \(y = k(1.2x)^2\) or a numerical example
correct answer with no working scores full marks
1 mark
M1 for 1.2² (= 1.44)
1 mark
A1 for 44%
cao

Question 4

5 marks

y is directly proportional to the square of x.
x is inversely proportional to z.

When z = 4, x = 3 and y = 36

Find the value of y when z = 5

Model answer

\(x = \frac{a}{z}\): 3 = \(\frac{a}{4}\), so a = 12 and \(x = \frac{12}{z}\)
\(y = bx^2\): 36 = b × 3², so b = 4 and \(y = 4x^2\)
When z = 5: \(x = \frac{12}{5}\) = 2.4
y = 4 × 2.4² = 4 × 5.76 = 23.04

Explanation

Use a different letter for each constant. Work through the chain: find x from z, then y from x. Combining the two gives \(y = 4\left(\frac{12}{z}\right)^2 = \frac{576}{z^2}\), so y is inversely proportional to \(z^2\).

Mark scheme

1 mark
P1 for \(x = \frac{a}{z}\) and a = 3 × 4 (= 12)
correct answer with no working scores full marks
1 mark
P1 for \(y = bx^2\) and b = 36 ÷ 3² (= 4)
1 mark
P1 for x = \(\frac{12}{5}\) (= 2.4)
1 mark
P1 for 4 × 2.4²
oe, e.g. \(y = \frac{576}{z^2}\)
1 mark
A1 for 23.04
cao

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