Ratios & Proportion
Ratio questions — sharing in a ratio, comparing rates, direct & inverse proportion, fractions of a quantity.
This page is a sequence, not a document: try a question cold, learn the methods, drill until they’re automatic, then prove it under a clock. Skipping straight to reading is the least effective thing you can do here — that’s not opinion, it’s the testing effect.
Try one first
Before any teaching — a real question tells you honestly where you stand.
£1140 is shared in the ratio 2:5. How much is the larger share?
The methods
Every ratios & proportion pattern the tests use — the method in a few lines, then a worked example.
Rate comparison
Convert each series to the same per-unit rate (value ÷ count) before comparing — never compare raw totals.
Worked example
Which region had the higher sales per store, and what was that figure?
Answer: Region A — £300k
Sales per store = sales ÷ stores. Region A: 2.40m/8 = £300k. Region B: 3.50m/14 = £250k. A is higher at £300k. The 'Region B' foil picks the region with higher total sales (absolute, not per store); the cross-divide foils pair the wrong sales with the wrong store count; £268k pools both regions.
Ratio split
Total the ratio parts, find the value of one part (amount ÷ parts), then scale by the share's parts.
Worked example
£840 is shared between two people in the ratio 3:4. How much does the person with the larger share receive?
Answer: 480
Total parts = 3 + 4 = 7. The larger share is 4 parts: 4/7 × 840 = £480. The £360 foil gives the smaller share; £420 splits evenly; £1,120 uses 4/3 instead of 4/7; £210 divides by a part.
Direct proportion
Find the unit value first (divide), then scale up (multiply).
Inverse proportion
Inverse proportion: the total work stays constant, so new value = (old count × old value) ÷ new count.
Fraction of a quantity
a/b of x = x ÷ b × a.
Three-part ratio
Total all the ratio parts, get the value per part, then scale by the asked share's parts.
Worked example
£960 is divided among three departments in the ratio 2:3:5. How much does the department with the largest share receive?
Answer: 480
Total parts = 2 + 3 + 5 = 10. The largest share is 5 parts: 5/10 × 960 = £480. The £288 and £192 foils give the middle/smallest shares; £320 splits evenly; £96 is a single part.
Mixtures
Weight each component by its volume: total attribute ÷ total volume. Never average the percentages alone.
Worked example
A 5-litre solution that is 20% acid is mixed with 3 litres of a 40% acid solution. What is the acid concentration of the final mixture?
Answer: 27.5%
Acid: 5×0.20 + 3×0.40 = 1.0 + 1.2 = 2.2 litres in 8 litres total → 2.2/8 = 27.5%. The 30.0% foil averages the two percentages (the classic mixture error); the others use a wrong total or add the percentages.
Cross-multiplication
Set the two ratios equal and cross-multiply (or find the unit rate, then scale).
Worked example
A car travels 240 km on 16 litres of fuel. At the same rate, how far will it travel on 28 litres?
Answer: 420 km
Set up the proportion: 240 km / 16 L = x / 28 L. Cross-multiply: x = 240 × 28 ÷ 16 = 420 km. (Equivalently the rate is 240 ÷ 16 = 15 km per litre, and 15 × 28 = 420.) The 137 foil inverts the ratio (240 × 16 ÷ 28). The 252 foil adds the extra litres to the kilometres, confusing the units. The 405 foil uses 27 litres instead of 28 (a misread). The 480 foil rounds 28 litres to roughly double 16 and doubles the distance.
Ratio change
Compute the ratio for each period, then the percentage change between the ratios: (r₂ − r₁) ÷ r₁.
Worked example
A company's cost-to-income ratio was £60m/£100m in 2022 and £66m/£120m in 2024. By what percentage did the cost-to-income ratio change?
Answer: 8.3% decrease
Ratios: 60/100 = 0.60 and 66/120 = 0.55. Change = (0.55−0.60)/0.60 = −8.3% (a decrease). The 5-point foil gives percentage points; 10% looks only at costs, 20% only at income.
Drill it
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Prove it under a clock
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