Direct & inverse proportion (algebraic) – GCSE Maths

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Translate proportion statements into formulae like $y = k x^{2}$ or $y = \frac{k}{x}$ , find $k$ , then use the formula. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.

Algebraic proportion writes the sentence as a formula. Direct: $y = k x$ . Direct to a square: $y = k x^{2}$ . Inverse: $y = \frac{k}{x}$ . Inverse to a square: $y = \frac{k}{x ^{2}}$ . Joint: $E = k \frac{1}{2} m v^{2}$ style, $k$ found from a given set of values, then the formula is used for a new set.

Find $k$ first, write the formula with that $k$ , then substitute the new numbers. If $x$ is multiplied by $3$ and $y \propto x^{2}$ , $y$ is multiplied by $9$ . A table can find $k$ from one row and check it on another.

$y \propto x^{n}$ means $y = k x^{n}$ . Find $k$ from the given pair. Inverse square: $y = \frac{k}{x ^{2}}$ . Direct: doubling $x$ doubles $y$ . Inverse: doubling $x$ halves $y$ .

Objectives

Find y when y is proportional to the square of x.
Find y when y is inversely proportional to x squared.
Find the constant k.

Key points

Read the symbol $\propto$ as 'is proportional to' and write the matching formula with a constant $k$ .
Handle squares, cubes and square roots: $y \propto x^{2}$ , $y \propto x$ , $y \propto \frac{1}{x ^{2}}$ .
Find $k$ from one pair of values.
Use the formula to find a missing $y$ or a missing $x$ (which may need a root).

Worked examples

Example 1

y is proportional to the square of x. y = 18 when x = 3. Find y when x = 5.

2 marks

Write proportion model: y = kx².
Find k: 18 = k × 3² = 9k, so k = 18 ÷ 9 = 2.
Substitute x = 5: y = 2 × 5² = 2 × 25 = 50.

Answer: 50

Common slips

Slip: Write linear model: y = kx. The relationship states square of x.
Slip: Divide 18 by 3 to get k = 6. Remember to square x when finding k.
Slip: Multiply 2 × 5 = 10. Square the new x before multiplying by k.

Example 2

y is inversely proportional to x squared. y = 12 when x = 2. Find y when x = 4.

2 marks

Write model: y = k / x².
Find k: k = y × x² = 12 × 2² = 12 × 4 = 48.
Substitute x = 4: y = 48 / 4² = 48 / 16 = 3.

Answer: 3

Common slips

Slip: Write y = kx². Inverse proportion means dividing by x².
Slip: Divide 12 by 4. For inverse proportion, k is the product y × x².
Slip: Divide 48 by 4 to get 12. Square the new x before dividing.

Example 3

y is proportional to the square of x. y = 18 when x = 3. Find x when y = 50.

3 marks

Find constant k: k = 18 / 3² = 2.
Set up equation for x: 50 = 2x², so x² = 25.
Take the positive square root: x = √25 = 5.

Answer: 5

Common slips

Slip: Use k = 6. k is 18 divided by 9.
Slip: Solve 50 = 2x to get 25. Remember x remains squared.
Slip: Leave x = 25. Take square root to find x.

Questions and answers

GCSE Maths direct & inverse proportion (algebraic) questions and answers. Try each one, then open the answer. Easy questions come first.

Question 1 Easy2 marks

y is proportional to the square of x. y = 8 when x = 2. Find y when x = 4.

Show answer

Answer: 32

k = 8/4 = 2.
y = 2 × 16 = 32.

M1 — k = 8/4 = 2.
A1 — y = 2 × 16 = 32.

Question 2 Easy2 marks

y is proportional to the square of x. y = 12 when x = 2. Find y when x = 5.

Show answer

Answer: 75

k = 12/4 = 3.
y = 3 × 25 = 75.

M1 — k = 12/4 = 3.
A1 — y = 3 × 25 = 75.

Question 3 Medium2 marks

y is inversely proportional to x squared. y = 8 when x = 3. Find y when x = 6.

Show answer

Answer: 2

k = 8 × 9 = 72.
y = 72 / 36 = 2.

M1 — k = 8 × 9 = 72.
A1 — y = 72 / 36 = 2.

Question 4 Medium2 marks

y is inversely proportional to x squared. y = 18 when x = 3. Find y when x = 6.

Show answer

Answer: 4.5

k = 18 × 9 = 162.
y = 162 / 36 = 4.5.

M1 — k = 18 × 9 = 162.
A1 — y = 162 / 36 = 4.5.

Question 5 Hard2 marks

y is proportional to the square of x. y = 50 when x = 5. Find the constant k.

Show answer

Answer: 2

k = 50 / 5².
k = 50 / 25 = 2.

M1 — k = 50 / 5².
A1 — k = 50 / 25 = 2.

Question 6 Hard2 marks

y is proportional to the square of x. y = 12 when x = 2. Find x when y = 75.

Show answer

Answer: 5

k = 3, so 3x² = 75.
x² = 25, so x = 5.

M1 — k = 3, so 3x² = 75.
A1 — x² = 25, so x = 5.

Question 7 Exam-style4 marks

y is proportional to the square of x. y = 20 when x = 2. Find y when x = 5.

Show answer

Answer: 125

Write equation: y = kx².
Substitute known values: 20 = k(2²) = 4k.
Solve for k: k = 20 ÷ 4 = 5.
Find y when x = 5: y = 5 × (5²) = 5 × 25 = 125.

M1 — Form model y = kx²
M1 — Find constant k = 5
M1 — Substitute x = 5: 5 × 25
A1 — State y = 125

Common mistakes

Most lost marks on this topic are one slip.

Dividing by x when the law is a square
Divide by x² or multiply by x²
Using y = kx for a square law
The model is y = kx² or y = k/x²
Forgetting to square the new x
Square first, then multiply or divide by k

Spot the slip

y is proportional to the square of x, and y = 18 when x = 3. Kim divides 18 by 3 to get k. What is k?

Show the slip

2 k = y / x² = 18 / 9 = 2.

Exam tips

A calculator is useful here. Show the calculation you typed, not only the final line.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the formula with $k$ first; this line is almost always worth a mark on its own.
State the value of $k$ clearly and rewrite the formula — questions often ask for 'a formula for $y$ in terms of $x$ '.

Printable worksheet

The same questions are in a PDF you can print. Answers start on a later page.

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Questions students ask

Is Direct & inverse proportion (algebraic) Foundation or Higher?

Direct & inverse proportion (algebraic) is Higher only, aimed at grade 7.

What is a common slip on Direct & inverse proportion (algebraic)?

Dividing by x when the law is a square. Divide by x² or multiply by x².

Where can I get Direct & inverse proportion (algebraic) questions and answers?

The questions on this page are free. Open Show answer under each one, or download the worksheet PDF. Start free to practise this topic in the app.

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AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.

Direct & inverse proportion (algebraic) – GCSE Maths

Objectives

Key points

Worked examples

Example 1

Example 2

Example 3

Questions and answers

Question 1 Easy2 marks

Question 2 Easy2 marks

Question 3 Medium2 marks

Question 4 Medium2 marks

Question 5 Hard2 marks

Question 6 Hard2 marks

Question 7 Exam-style4 marks

Common mistakes

Spot the slip

Exam tips

Printable worksheet

Questions students ask

Is Direct & inverse proportion (algebraic) Foundation or Higher?

What is a common slip on Direct & inverse proportion (algebraic)?

Where can I get Direct & inverse proportion (algebraic) questions and answers?

Related topics

More on Direct & inverse proportion (algebraic)

Practise Direct & inverse proportion (algebraic) free