Proportion: equality of ratios
A proportion is a statement that two ratios are equal. If a/b = c/d, the four numbers are said to be in proportion. This is the engine behind unit conversion, similar shapes, recipes, percentages and direct proportion.
The cross-multiplication rule
For any proportion a/b = c/d (with b, d ≠ 0): a × d = b × c.
This is the most-used identity in GCSE Maths — every "find the missing value" proportion question reduces to it.
Worked example: solve x/12 = 5/8.
- Cross-multiply: 8x = 12 × 5 = 60.
- x = 60/8 = 7.5.
Setting up proportions from words
Worked example: 7 books cost £21. How much do 11 books cost?
- 7 / 21 = 11 / x (or equivalently 21/7 = x/11).
- Cross-multiply: 7x = 11 × 21 = 231.
- x = £33.
Or recognise £3 per book and multiply: 11 × £3 = £33.
Recipe & scaling problems
Worked example: a recipe for 4 people uses 200 g of butter. How much butter for 7 people?
- 200/4 = x/7 → 200 × 7 = 4x → x = 1400/4 = 350 g.
Mixing units / unit ratios
Worked example: a car uses 12 litres of fuel for 150 km. How far for 18 litres?
- 12/150 = 18/x → 12x = 18 × 150 = 2700 → x = 225 km.
Best buys & rates
Use unit pricing — convert each option to "price per unit" and compare.
Worked example: 250 g costs £1.20; 400 g costs £1.84. Which is better value?
- £1.20 / 250 = 0.48 p/g.
- £1.84 / 400 = 0.46 p/g.
- 400 g is cheaper per gram.
Setting up the proportion correctly
The most common error is mismatched ordering. Use units to check:
If 5 apples cost 80p, then for 12 apples: price/quantity must be the same on both sides. 80/5 = x/12 ✓ (price ÷ quantity = constant rate, here 16 p per apple).
⚠Common mistakes
- Inverting the ratio — pairing wrong numerators with wrong denominators.
- Using addition instead of multiplication for scaling.
- Forgetting consistent units — pence with pounds, litres with cm³, etc.
- Confusing direct with inverse proportion — "more workers, less time" is inverse, not direct.
- Rounding mid-calculation — keep exact values and round at the end.
➜Try this— Quick check
Solve x/15 = 8/20. Cross-multiply: 20x = 120 → x = 6.
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