Ratio & ProportionFoundation + Higher★ Core topic
Ratio.
Sharing, simplifying, scaling.
A ratio compares two (or more) quantities. Simplify by dividing by the HCF. Share by finding one part, then multiplying. Ratios can be written as fractions of the total.
I · Key definitions
Ratio
A comparison of quantities, separated by a colon.
3 : 5 or 2 : 3 : 5
Part
One 'share' in the ratio.
in 2:3, total is 5 parts
Simplified
Smallest whole-number form. Divide all by HCF.
6:10 = 3:5
1 : n form
Divide both sides by the first number.
3:12 = 1:4
Unit ratio
Useful for best value comparisons.
£5 for 250g → £0.02/g
Part-to-whole
Ratio rewritten as a fraction.
2:3 → A is 2/5, B is 3/5
II · The sharing method — visualised
Share £360 in ratio 3 : 5 : 4
A = £90
3 × £30
B = £150
5 × £30
C = £120
4 × £30
Method: Total parts = 3 + 5 + 4 = 12. One part = 360 ÷ 12 = £30. Multiply each share by that.
III · The four types of ratio problem
Simplifying
1. Find HCF of all numbers
2. Divide each by HCF
18:24:30 → HCF=6 → 3:4:5
Sharing
1. Total parts = sum of ratio
2. One part = amount ÷ total parts
3. Multiply each share
£60 in 2:3 → 1 part=£12 → £24, £36
One known → find others
If A : B = 3 : 5 and A = 12,
one part = 4 → B = 20
also: total = 4 × 8 = 32
Ratio as fraction
A : B = a : b
→ A is a/(a+b) of total
→ B is b/(a+b)
2:3 → A is 2/5, B is 3/5
IV · Worked examples
Share in a three-way ratio
3 marks
Share £360 in the ratio 3 : 5 : 4.
i.Total parts = 3 + 5 + 4 = 12
ii.One part = 360 ÷ 12 = £30
iii.Multiply: 3×£30, 5×£30, 4×£30
£90 : £150 : £120
Find a missing amount
2 marks
Ali and Ben share money in the ratio 2 : 5. Ben gets £40. How much does Ali get?
i.Ben has 5 parts worth £40
ii.One part = 40 ÷ 5 = £8
iii.Ali has 2 parts = 2 × £8
Ali gets £16
Simplify and convert
2 marks
Simplify 48 : 36 and write in the form 1 : n.
i.HCF(48, 36) = 12 → 48÷12 : 36÷12 = 4 : 3
ii.To make first = 1, divide both by 4: 1 : 0.75
Simplified: 4 : 3 · in 1 : n form: 1 : 0.75
V · Common mistakes & examiner tips
Common mistakes
Adding amounts without the ratio. If ratio is 2:3, you can't split 50/50.
Forgetting total parts. A:B = 3:5 means 8 parts total, not 3.
Treating ratio like fraction incorrectly. 3:5 does NOT mean A = 3/5 of total.
Dividing wrong amount. If A gets £12 out of A:B=3:5, one part is £4, not 12/5.
Order matters. 2:3 is not the same as 3:2.
Examiner tips
Always write out total parts first. It's the key unlock.
Draw a bar model. Especially for 3-way and tricky problems.
Check by adding back. Your shares should sum to the original amount.
For 'best value' questions, find cost per unit for each option.
In proportion questions disguised as ratio: set up equivalent fractions.