GCSE & IGCSE maths practice

Direct and inverse proportion questions and answers

Two quantities are in direct proportion if one doubles when the other doubles ( $y = kx$ ). They're in inverse proportion if one halves when the other doubles ( $y = \dfrac{k}{x}$ ). For Higher, the relationship can involve squares, cubes or roots: find $k$ from the values you're given, then use it.

15 questions · Foundation and Higher tier · Every answer is checked as you type it, with a full worked solution. Updated 10 Nov 2026.

Start the questions Direct and inverse proportion guide

Where direct and inverse proportion comes up

GCSE (AQA, Edexcel, OCR)	Foundation and Higher: solve problems involving direct and inverse proportion. Higher only: construct and interpret equations that describe direct and inverse proportion.
Cambridge IGCSE (0580)	Core: ratio and proportion problems. Extended only: express direct and inverse proportion in algebraic terms (including squares, cubes and roots) and find the constant of proportionality.

The method

Direct (unitary method): find the value for 1, then multiply.
Inverse: the product stays the same (e.g. workers × days).
Algebra: write y = kx (or kx², k/x...), substitute to find k, then use the equation.

Direct and inverse proportion questions

Try each one before looking at the hint. If you're stuck, the AI tutor can talk you through it step by step.

Warm-up

Question 1Warm-upFoundation

5 pens cost £3.50. How much do 8 pens cost?

Question 2Warm-upFoundation

$y$ is directly proportional to $x$ . When $x = 4$ , $y = 12$ . Find $y$ when $x = 10$ .

Question 9Warm-upFoundation

4 cinema tickets cost £26. How much do 7 tickets cost?

Question 10Warm-upFoundation

A recipe uses 3 eggs to serve 4 people. How many eggs are needed for 12 people?

Exam-style

Question 3Exam-styleFoundation

6 workers take 10 days to build a wall. How long would 4 workers take, working at the same rate?

days

Question 4Exam-styleFoundation

3 kg of apples cost £7.20. How much do 5 kg cost?

Question 5Exam-styleHigher

$y$ is directly proportional to $x^2$ . When $x = 3$ , $y = 18$ . Find $y$ when $x = 5$ .

Question 6Exam-styleHigher

$y$ is inversely proportional to $x$ . When $x = 3$ , $y = 8$ . Find $y$ when $x = 6$ .

Question 11Exam-styleFoundation

8 taps fill a tank in 9 hours. How long would 12 taps take?

hours

Question 12Exam-styleFoundation

$y$ is directly proportional to $x$ . When $x = 2$ , $y = 7$ . Find $y$ when $x = 9$ .

Question 13Exam-styleHigher

$y$ is directly proportional to $x^2$ . When $x = 4$ , $y = 48$ . Find the positive value of $x$ when $y = 75$ .

Question 14Exam-styleHigher

$y$ is inversely proportional to $x$ . When $x = 4$ , $y = 10$ . Find $x$ when $y = 8$ .

Challenge

Question 7ChallengeHigher

$y$ is inversely proportional to $x^2$ . When $x = 2$ , $y = 20$ . Find $y$ when $x = 4$ .

Question 8ChallengeHigher

$y$ is directly proportional to $\sqrt{x}$ . When $x = 9$ , $y = 12$ . Find $y$ when $x = 25$ .

Question 15ChallengeHigher

The distance $d$ an object falls is directly proportional to the square of the time $t$ . If $t$ is tripled, by what factor does $d$ increase?

Common mistakes

Treating inverse proportion as direct (more workers means fewer days).
Forgetting to square or root x when finding k.
Not writing the equation with the found k.

Stuck on one of these?

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Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

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