Ratio and proportion
Maths · Key Stage 3 · The School
Ratio is about parts
A ratio 3:2 says: for every 3 of these, 2 of those. To share £40 in the ratio 3:2, count the PARTS first: 3 + 2 = 5 parts. One part is £40 ÷ 5 = £8. So the shares are 3 × 8 = £24 and 2 × 8 = £16 — and they add back to £40, which is your check. Always find one part first; nearly every ratio mistake comes from skipping that step.
The unitary method
If 7 pens cost £2.10, what do 12 cost? Find ONE first: £2.10 ÷ 7 = £0.30 each. Then × 12 = £3.60. This "find one, then scale" method solves recipe scaling, currency, speed, and best-buy questions. It is the single most useful arithmetic technique in adult life, and it is three lines long.
Direct or inverse?
Direct proportion: more workers, more pay — both go up together, and the ratio stays fixed. Inverse: more workers, LESS time — one goes up as the other goes down, and it is the PRODUCT that stays fixed. Check which you have before calculating: if 4 painters take 6 days (24 painter-days), 8 painters take 24 ÷ 8 = 3 days, not 12.
Six workers build a wall in 12 days. (a) How long would 6 workers take on a wall twice as long? DIRECT: double the work, double the time — 24 days. (b) How long would 12 workers take on the original wall? INVERSE: twice the workers, half the time — 6 days. Both involve doubling, and the answers move in opposite directions. Before calculating, ask: if this goes up, does that go up or down?