Number – Ratio, Proportion & Percentages

Ratio, direct & inverse proportion, reverse percentages and repeated percentage change.

Ratio and proportion appear in almost every paper. Master sharing in a ratio and reverse percentages – these are high-frequency exam skills.

Sharing in a given ratio

1. Add the parts of the ratio 2. Divide the total by the sum of the parts → one part 3. Multiply by each part of the ratio

Worked example

Share £84 in the ratio 3 : 5

Solution

Total parts = 8 One part = 84 ÷ 8 = £10.50 3 parts = **£31.50** 5 parts = **£52.50**

Reverse percentages

When given the amount after a percentage change: Original = New amount ÷ multiplier Example: after 20% increase the price is £72. Original = 72 ÷ 1.20 = £60

Worked example

A laptop is reduced by 15% and now costs £425. What was the original price?

Solution

Multiplier = 0.85 Original = 425 ÷ 0.85 = **£500**

Direct proportion

If y is directly proportional to x: y ∝ x or y = kx. Find k from a known pair, then use the equation.

Worked example

y is directly proportional to x. When x = 6, y = 15. Find y when x = 10.

Solution

15 = k × 6 → k = 2.5 y = 2.5 × 10 = **25**

Practice questions

Try these without looking at the answers first.

Q1.Share £120 in the ratio 2 : 3 : 7

Show answer

£20 : £30 : £70

Q2.A coat is reduced by 30% and now costs £56. Original price?

Hint: 56 ÷ 0.7

Show answer

£80

Q3.y ∝ x. When x = 4, y = 18. Find y when x = 7.

Show answer

31.5

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