Direct and inverse proportion equations · Higher notes

Proportion describes a relationship that stays consistent as quantities change. Direct proportion to x² means y/x² remains constant; it does not mean y/x stays constant. Inverse proportion means one quantity decreases as the linked quantity increases, according to the stated power. Replace the proportion sign with an equation containing a constant k, then use one known pair to find k. Keep that same constant for new values. Read the power or root carefully before comparing scale changes.

Learning objectives

Write direct or inverse proportion as an equation.
Find the constant from a known pair and use it for a new input.

Key ideas

Direct proportion to a positive integer power includes that power in the model.

y\propto x^n\quad\Rightarrow\quad y=kx^n,\quad n\in\mathbb{Z}_{>0}

Inverse proportion to a positive integer power places that power in the denominator.

y=\frac{k}{x^n},\quad x\ne0,\quad n\in\mathbb{Z}_{>0}

Use a known pair to find the constant before substituting a new input.

For positive inputs in this inverse-root model, quadrupling the input halves the output.

y=\frac{k}{\sqrt x},\qquad x>0

Worked examples

Example 1

The output is directly proportional to the square of the input. The known pair is three and eighteen. Find the output at five.

y=kx^2,\quad x=3,\ y=18\quad\Rightarrow\quad18=9k

k=2\quad\Rightarrow\quad y=2x^2

x=5\quad\Rightarrow\quad y=2(5^2)

Answer

y=50

Example 2

For inverse proportion, use the known pair two and twelve to find the output at six.

y=\frac{k}{x},\quad x=2,\ y=12\quad\Rightarrow\quad k=24

Answer

y=\frac{24}{6}=4

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Form and use an equation for direct or inverse proportion, including powers.

AO2 · Reasoning

Explain why an additive or simple inverse model is not appropriate.

AO3 · Problem solving

Use one observation to find a constant and predict a new value within the model.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Do not omit the specified power or root. Doubling an input does not always double its output.

Try it yourself

Attempt each question before opening its answer.

Check 1: y = k/x. When x = 2, y = 12. Find y when x = 6.

Answer: 4

k = 12 × 2 = 24, so y = 24/6.

Check 2: y is inversely proportional to x². When x = 3, y = 8. Find y when x = 6.

Answer: 2

Doubling x multiplies x² by 4, so y is divided by 4.

Explain it in your own words

Write equations for direct proportion to x² and inverse proportion to x².
Explain how one known pair determines k.
Explain what happens to y = k/√x when x is multiplied by 9.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Ratio specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.