Ratio and Proportion

GCSE Maths · Number

Ratio

A ratio compares amounts, showing how much of one thing there is compared to another, written with a colon, e.g. 2 : 3.

Simplifying ratios

Divide all parts by their HCF (like simplifying a fraction).

10 : 15 → divide by 5 → 2 : 3.
Units must match first: 50 cm : 2 m → 50 cm : 200 cm → 1 : 4.

Sharing in a ratio

1. Add the parts to find the total number of parts.

2. Divide the amount by the total parts to find the value of one part.

3. Multiply each side of the ratio by the value of one part.

Example: share £60 in the ratio 2 : 3.

Total parts = 2 + 3 = 5.
One part = 60 ÷ 5 = £12.
Shares: 2 × 12 = £24 and 3 × 12 = £36. (Check: 24 + 36 = 60 ✓)

Proportion

Two quantities are in direct proportion if they increase (or decrease) at the same rate — double one, double the other.

Unitary method: find the value of one first.

5 pens cost £2. What do 8 pens cost? One pen = 2 ÷ 5 = £0.40 → 8 pens = 8 × 0.40 = £3.20.

Best value problems

Find the cost per unit (e.g. price per 100 g) for each option and compare — the lower price per unit is better value.

Inverse proportion

As one quantity increases, the other decreases (their product stays constant). E.g. more workers → less time. If 4 workers take 6 hours, 8 workers take 6 × 4 ÷ 8 = 3 hours.

Worked example

Divide 35 sweets between Amy and Ben in the ratio 3 : 4.

Parts = 3 + 4 = 7; one part = 35 ÷ 7 = 5.
Amy = 3 × 5 = 15, Ben = 4 × 5 = 20. (15 + 20 = 35 ✓)

Common mistakes

Forgetting to make units the same before simplifying a ratio.
Dividing by the wrong number of parts — remember to add the parts first.
Confusing direct and inverse proportion.

Exam tips

For sharing, always find the value of one part first.
Use the unitary method ("find one") for proportion and best-value questions.
Check your shares add back up to the total.

Key facts to remember

Simplify ratios by dividing all parts by the HCF (units must match).
Sharing: add the parts, find one part (total ÷ parts), then multiply.
Direct proportion increases together (unitary method); inverse proportion — one up, one down (product constant).