Ratio and Proportion

Simplifying ratios

Divide all parts by the HCF: 12 : 18 = 2 : 3.

To the form 1 : n, divide both by the first number: 5 : 8 = 1 : 1.6.

Sharing in a ratio

Share £40 in the ratio 3 : 5:

Total parts = 3 + 5 = 8
One part    = 40 ÷ 8 = £5
3 parts = £15,  5 parts = £25

"Difference" problems

Amy and Ben share money 5 : 2 and Amy gets £18 more.

The difference is 5 − 2 = 3 parts = £18 → 1 part = £6 → Amy £30, Ben £12.

Direct proportion (unitary method)

5 pens cost £2 → 1 pen = £0.40 → 8 pens = £3.20.

• Direct proportion graph is a straight line through the origin: y = kx.

Inverse proportion

As one rises, the other falls: y = k/x.

4 workers take 6 hours → 1 worker takes 24 hours → 3 workers take 8 hours.

Best buy

Compare cost per unit (price ÷ quantity); cheapest per unit wins.

Exam tip

Underline whether the ratio gives you the total or a difference — they need different first steps.