R2Ratio & ProportionFoundation & Higher

Direct & Inverse Proportion

Two quantities are in direct proportion if they increase together at the same rate, and in inverse proportion if one goes up as the other goes down. Spotting which is which is the key skill.

40 min Video by Maths Genie AQA GCSE Maths

Direct and Inverse ProportionWatch the walkthrough, then read the notes below.

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What you'll learn

Recognise direct and inverse proportion
Solve direct proportion (unitary method)
Solve inverse proportion problems
Use proportion in recipe and best-buy questions

Direct proportion

In direct proportion, doubling one quantity doubles the other. The $unitary method$ finds the value of one item first, then scales up.

1One pen:

2 \div 5 = £0.40

2Eight pens:

0.40 \times 8 = £3.20

Answer£3.20

Inverse proportion

In inverse proportion, the product stays the same: if it takes 4 people 6 hours, then $4 \times 6 = 24$ “person-hours”. More people means fewer hours.

y \propto x (direct), y \propto \frac{1}{x} (inverse)

The two kinds of proportion.

Tip — Ask yourself: as one goes up, does the other go up (direct) or down (inverse)?

Remember these

direct: y = k x

Constant rate.

inverse: x y = k

Constant product.

Watch out for these

Treating an inverse problem (like workers and time) as direct.

Check the direction: more workers → less time, so it is inverse.

Scaling without finding the value of one item first.

Use the unitary method: find one, then multiply.

Key takeaways

Direct: both increase together (y = kx).
Inverse: one up, the other down (xy = k).
Unitary method: find the value of one, then scale.

Test yourself

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