GCSE — mathsHigher2 min read

Direct and Inverse Proportion

Understand direct and inverse proportion relationships for GCSE Maths. Set up and solve proportion equations.

Tutor AI Team2026-02-25T01:47:42.511Z

Direct proportion means as one quantity increases, the other increases at the same rate. Inverse proportion means as one increases, the other decreases. Understanding these relationships is key to Higher GCSE.

Core Concepts

Direct Proportion

$y \propto x$ means $y = kx$ where $k$ is the constant of proportionality.

Also: $y \propto x^2$ means $y = kx^2$ , and $y \propto \sqrt{x}$ means $y = k\sqrt{x}$ .

Inverse Proportion

$y \propto \frac{1}{x}$ means $y = \frac{k}{x}$ .

Also: $y \propto \frac{1}{x^2}$ means $y = \frac{k}{x^2}$ .

Finding k

Substitute known values to find $k$ , then use the formula.

Worked Example: Direct

Problem

$y \propto x^2$ . When $x = 3$ , $y = 36$ . Find $y$ when $x = 5$ .

$36 = k(9)$ → $k = 4$ . So $y = 4x^2$ . When $x = 5$ : $y = 4(25) = 100$ .

Solution

Worked Example: Inverse

Problem

$y \propto \frac{1}{x}$ . When $x = 4$ , $y = 6$ . Find $y$ when $x = 8$ .

$6 = \frac{k}{4}$ → $k = 24$ . When $x = 8$ : $y = \frac{24}{8} = 3$ .

Solution

Practice Problems

1. $y \propto x^3$ . $y = 16$ when $x = 2$ . Find $y$ when $x = 3$ .
1. $y \propto \frac{1}{\sqrt{x}}$ . $y = 10$ when $x = 4$ . Find $y$ when $x = 16$ .

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Key Takeaways

✓
Direct: $y = kx^n$ — both increase together.
✓
Inverse: $y = \frac{k}{x^n}$ — one increases as the other decreases.
✓
Find $k$ first, then use the formula.

Tags:

#gcse#maths#ratio#proportion#direct#inverse

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Core Concepts

Direct Proportion

Inverse Proportion

Finding k

Worked Example: Direct

Worked Example: Inverse

Practice Problems

Key Takeaways

Related Topics

Evolution and Natural Selection

Biodiversity and Conservation

Sampling and Field Investigations

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