Direct & Inverse Proportion (Years 8–11) — Maths

Two quantities are in proportion when they change together in a fixed way. Shopping, driving, cooking and building all lean on it. There are two kinds, and they behave oppositely — knowing which you're facing is half the battle.

Why this matters: proportion answers "how much for a different amount?" — the everyday question behind unit pricing, scaling recipes, fuel for a longer trip, or how many people you need to finish a job on time. Spot direct from inverse and you'll never double a time when you should be halving it.

Unitary method: find the value of one, then scale up

Direct proportion — they rise together

Direct: both rise together — a straight line through (0, 0).

Double one and the other doubles. More items cost more money; more petrol takes you further. The trick is the unitary method: find the value of one, then scale up.

5 pens cost £2. How much for 8 pens?
• One pen = £2 ÷ 5 = £0.40.
• 8 pens = 8 × £0.40 = £3.20.

Double number line: find 1 pen (40p), then scale to any amount.

Inverse proportion — one up, the other down

Inverse: fewer workers → more time; the product stays 24.

Here, as one quantity grows the other shrinks. More workers means the job takes less time. What stays the same is the total amount of work (the two quantities multiplied together).

Analogy: a fixed pile of bricks has to be laid. Whether one person lays them all or ten share the load, the pile is the same size — so more hands means each does fewer, and the job finishes sooner.

4 workers build a wall in 6 hours. How long for 3 workers?
• Total work = 4 × 6 = 24 "worker-hours".
• With 3 workers: 24 ÷ 3 = 8 hours.

Telling them apart

Ask: "if there's more of one, should there be more or less of the other?" That single question decides which method to use.

	Direct proportion	Inverse proportion
As one goes up…	the other goes up	the other goes down
What stays fixed	the rate (value of one)	the product (total work)
Method	find one, then ×	find the total, then ÷
Example	more pens cost more	more workers → less time

Common mistake: don't treat every question as direct. "Fewer workers, so it'll be quicker" is the classic slip — fewer workers means more time, not less. If more of one gives less of the other, it's inverse: multiply to find the total, then divide.

In a nutshell: direct → find the value of one and multiply; inverse → the product stays constant, so find the total and divide. Always sanity-check the direction of your answer.