Sharing in a ratio, best value, direct and inverse proportion, and the graph skills that appear at Higher tier.
Ratio and proportion questions appear throughout every GCSE Maths paper — in standalone questions, in worded problems, and embedded in geometry, science-style calculations, and even algebra. The topic is broader than most students realise, and it extends significantly at Higher tier with formal proportion notation and graphs.
A ratio compares two or more quantities. To simplify a ratio, divide all parts by their highest common factor. 15:25 simplifies to 3:5 (dividing by 5). 12:18:24 simplifies to 2:3:4 (dividing by 6).
To convert a ratio to the form 1:n (useful for map scales and other applications), divide both parts by the left-hand value. 4:14 becomes 1:3.5.
The most fundamental ratio question type: divide an amount in a given ratio.
Example: Share £240 in the ratio 3:5.
Total parts: 3 + 5 = 8 parts.
Value of one part: £240 ÷ 8 = £30.
First share: 3 × £30 = £90. Second share: 5 × £30 = £150.
Check: £90 + £150 = £240 ✓
Always check your answers sum to the original total. If they don't, something has gone wrong — usually with the total number of parts.
A harder version: "Ahmed and Bella share money in the ratio 2:7. Ahmed receives £50. How much does Bella receive?"
If 2 parts = £50, then 1 part = £25. Bella has 7 parts = 7 × £25 = £175.
The "one part" method works for all ratio sharing problems. Find the value of one part, then scale up for each person's share. It avoids errors much better than trying to calculate proportions directly.
Recipe questions appear regularly. The skill is scaling ingredients up or down proportionally. Always find what one unit (one person, one item) requires, then scale to the required amount.
Example: A recipe for 6 people needs 450g of flour. How much for 10 people?
1 person needs: 450 ÷ 6 = 75g.
10 people need: 75 × 10 = 750g.
Best value questions ask which of two or more options gives the most quantity per pound spent. The method: calculate a common unit — either cost per gram, or grams per penny.
Cereal: 500g for £2.20, or 750g for £3.15. Which is better value?
Option 1: £2.20 ÷ 500 = 0.44p per gram.
Option 2: £3.15 ÷ 750 = 0.42p per gram.
Option 2 is better value (lower cost per gram).
Always state which is better value and why — examiners want a conclusion with a justification, not just the calculations.
Two quantities are in direct proportion if when one doubles, the other doubles — they always scale by the same factor. The relationship is y = kx where k is the constant of proportionality.
The graph of direct proportion is a straight line through the origin. If a question gives you one pair of values, use them to find k, then use k to answer further questions.
y is directly proportional to x. When x = 5, y = 35. Find y when x = 8.
y = kx → 35 = k × 5 → k = 7.
When x = 8: y = 7 × 8 = 56.
Two quantities are in inverse proportion if when one doubles, the other halves. The relationship is y = k/x. The graph is a reciprocal curve (a hyperbola) — not a straight line.
y is inversely proportional to x. When x = 4, y = 9. Find y when x = 12.
y = k/x → 9 = k/4 → k = 36.
When x = 12: y = 36/12 = 3.
At Higher tier, proportion questions go beyond simple linear and inverse relationships. You may be told that y is proportional to x² (y = kx²), or that y is inversely proportional to √x (y = k/√x). The method is always the same: use the given pair of values to find k, then substitute.
The symbol ∝ means "is proportional to". When you see "y ∝ x²", this is telling you the relationship is y = kx² for some constant k. Your job is to find k using the given values, then use the equation to answer the question. Never skip the step of writing out the equation with ∝ replaced by = k.
Map scale questions use ratio directly. A scale of 1:50,000 means 1 cm on the map represents 50,000 cm in real life. To convert: multiply the map measurement by the scale factor for real distance. Divide the real distance by the scale factor for map distance.
Always check units carefully. A scale of 1:25,000 with a map distance of 4 cm gives a real distance of 100,000 cm = 1,000 m = 1 km. Unit conversion errors are extremely common in map scale questions.
The full ratio and proportion content assessed at GCSE is detailed in the AQA specification and the Edexcel GCSE Maths specification.
PaperPlus generates GCSE ratio and proportion questions — including Higher tier proportion with powers — with full mark schemes for AQA, Edexcel and OCR.
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