Ratio & proportion

Splitting in a ratio, scaling recipes, direct and inverse proportion.

A ratio question is really a sharing question. Once you can split something into equal parts, every ratio question uses the same three moves.

The method — three steps, every time

Step 1 · total parts

4 + 5 = 9

Step 2 · one part

£45 ÷ 9 = £5

Step 3 · answer

5 × £5 = £25

That was: Abir and Guneet share £45 in the ratio 4:5. How much does Guneet get? Add the parts, find what one part is worth, then count up Guneet's parts. £25.

Remember it like this

“Parts… one part… answer.”

Say it, then write all three steps down — even when the question feels easy. That is where the marks hide.

⚠ Watch out

Divide by the right side of the ratio. If red : blue = 3 : 7 and there are 42 blue, the 42 belongs to the 7 — so one part is 42 ÷ 7 = 6, not 42 ÷ 3. Label each side before you divide.

One side given, find the other

The ratio of red to blue counters is 3:7. There are 42 blue counters. How many red counters are there?

  1. Label the ratio: red : blue = 3 : 7, and 42 is the blue number
  2. 7 parts = 42, so one part = 42 ÷ 7 = 6
  3. Red has 3 parts: 3 × 6

18 red counters

Scaling a recipe

A recipe for 4 people uses 300g of rice. How much rice for 10 people?

  1. Find the amount for ONE person: 300 ÷ 4 = 75g
  2. Multiply up for 10 people: 75 × 10

750g

⚠ Watch out

Workers-and-time questions run BACKWARDS: more workers means LESS time. If 5 workers take 6 hours, first find the total work (5 × 6 = 30 worker-hours), then divide: 3 workers take 30 ÷ 3 = 10 hours. Ask yourself: with more workers, should my answer get bigger or smaller?

✏ Your turn

Mia and Jo share 63 stickers in the ratio 2:7. How many stickers does Jo get?

Show the answer
  1. Total parts: 2 + 7 = 9
  2. One part: 63 ÷ 9 = 7
  3. Jo has 7 parts: 7 × 7

49 stickers

For parents — the full topic guide

Ratio & Proportion — 11+ Topic Guide

Part of the GrammarMock topic guides.

Why examiners love ratio

Ratio questions separate the well-drilled from the truly fluent, because they can be dressed up in endless contexts — recipes, paint mixing, sharing money, map scales, class compositions — while always being the same underlying machine. GL papers typically carry 3-5 ratio/proportion questions, and they cluster in the harder half of the paper.

The one method that solves almost everything: find one part

"Abir and Guneet share £45 in the ratio 4:5. How much does Guneet get?"

  1. Total parts: 4 + 5 = 9
  2. One part: £45 ÷ 9 = £5
  3. Answer: Guneet has 5 parts = £25

Teach this as a ritual: parts, one part, answer. Written down every time, even when it feels easy — the errors come from doing step 1 in the head and mis-adding.

The four variations GL uses

1. Share in a ratio (above) — the base case

2. Given one side, find the other

"The ratio of red to blue is 3:7. There are 42 blue. How many red?"

Here the total isn't given — one side is. 7 parts = 42, so one part = 6, so red = 3 × 6 = 18. The trap: dividing 42 by 3 instead of 7 (dividing by the wrong side of the ratio). Fix: write the ratio with labels — red:blue = 3:7 — and point at which number 42 belongs to.

3. Three-way ratios

"Flour, sugar and butter in the ratio 6:3:2. 480g of flour — how much butter?"

Same machine: 6 parts = 480g, one part = 80g, butter = 2 × 80 = 160g. Nothing new, just more labels to keep straight.

4. Recipe scaling (proportion)

"A recipe for 4 people uses 300g of rice. How much for 10 people?"

Method: find the amount for ONE, then multiply. 300 ÷ 4 = 75g per person; 75 × 10 = 750g. This "unitary method" also solves best-buy questions ("3 pens for 60p or 5 pens for 90p — which is better value?" → 20p vs 18p per pen).

5. Inverse proportion (the hard variant)

"5 workers take 6 hours to paint a fence. How long would 3 workers take?"

More workers = LESS time — the relationship runs backwards. Method: find the total work first. 5 × 6 = 30 worker-hours. Then 30 ÷ 3 = 10 hours. Children who apply the normal unitary method get 3.6 hours and pick the trap option. The tell: ask "if I had MORE workers, would the answer get bigger or smaller?" If smaller, it's inverse — compute total work first.

Simplifying ratios (quick marks)

45:6 → divide both by 3 → 15:2. Same skill as simplifying fractions. Watch for units: "20p to £1" must become 20:100 = 1:5 first — mixed units are the planted trap.

Common errors checklist

  1. Dividing by the wrong side of the ratio (variation 2 above)
  2. Forgetting to add ALL parts in three-way ratios
  3. Answering the number of PARTS instead of the actual amount
  4. Applying direct proportion to inverse problems (workers/time, speed/time)
  5. Not simplifying to match the answer options

Practice plan

Four days: one variation per day (share / one-side-given / three-way / scaling), 6 questions each. Day five: inverse proportion only. Then the mixed questions in GrammarMock Maths Papers 3 and 5 (the hard papers both carry ratio in the last third).

Ready to practise this topic?

Try a full paper — questions on ratio & proportion show up in most 11+ maths and VR papers.

Browse papersMore topics