GCSE Maths Ratio & Proportion Notes
Ratio and proportion compare quantities and scale them up or down. They appear across recipes, maps, best-buy and mixture questions in every GCSE paper.
Key concepts
- Ratio
- A comparison of amounts in the same units, e.g. 2 : 3. It has no units itself.
- Sharing in a ratio
- Add the parts to find the total number of shares, find one share, then multiply out.
- Direct proportion
- As one quantity increases, the other increases at the same rate: y = kx.
- Inverse proportion
- As one increases, the other decreases: y = k/x. More workers, less time.
- Simplify ratios by dividing every part by their highest common factor.
- A unit ratio (1 : n) makes comparing 'best value' quick.
- Proportion questions almost always start by finding the value of one unit.
Key formulas
Direct proportion
y = kx
k is the constant of proportionality.
Inverse proportion
y = k/x
Worked example
Share £60 in the ratio 2 : 3.
- 1Add the parts: 2 + 3 = 5 shares.
- 2Find one share: 60 ÷ 5 = £12.
- 3Multiply out: 2 × 12 = £24 and 3 × 12 = £36.
Answer: £24 and £36
Common mistakes
✗ Dividing the amount by one part instead of the total parts.
✓ Add all parts of the ratio first, then divide the amount by that total.
✗ Confusing direct and inverse proportion.
✓ Ask: as one goes up, does the other go up (direct) or down (inverse)?
✗ Leaving a ratio unsimplified.
✓ Divide each part by the HCF to give the simplest form.
Quick check
Try these, then reveal the answers.
Simplify the ratio 12 : 18.Reveal answer
2 : 3
Share 40 in the ratio 3 : 5.Reveal answer
15 and 25
If 4 pens cost £2, how much do 10 cost?Reveal answer
£5
Exam tips
- Underline whether the question wants one share, the difference, or the total.
- For best-buy questions, work out the price per unit (e.g. per 100 g) and compare.
- Write proportion as an equation with k, then find k from the given pair of values.
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Frequently asked questions
How do I share an amount in a ratio?+
Add the parts to get total shares, divide the amount by that total to get one share, then multiply each part by one share.
What is the difference between direct and inverse proportion?+
In direct proportion both quantities rise together (y = kx); in inverse proportion one rises as the other falls (y = k/x).
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