Percentage change, reverse percentages and problem-solving
Percentages turn ratios into a "per hundred" scale. This topic combines three skill blocks: percentage change (going up/down), reverse percentages (finding the original) and multi-step percentage problems.
Percentage change formula
% change = (new − old) / old × 100.
A positive answer = increase; negative = decrease.
Worked example: a price rises from £40 to £46.
- Change = 46 − 40 = 6.
- 6/40 × 100 = 15% increase.
Multiplier method
To increase by p%, multiply by (1 + p/100). To decrease by p%, multiply by (1 − p/100).
Worked example: increase £80 by 12%.
- Multiplier = 1.12.
- 80 × 1.12 = £89.60.
Worked example: decrease 250 by 8%.
- Multiplier = 0.92.
- 250 × 0.92 = 230.
Reverse percentages
If a value AFTER a percentage change is given, divide by the multiplier to recover the original.
Worked example: a coat costs £63 after a 10% reduction. What was the original price?
- Sale multiplier = 0.9.
- Original = 63 ÷ 0.9 = £70.
Worked example: a salary, after 4% rise, is £31 200. What was it before?
- Multiplier = 1.04.
- Before = 31 200 ÷ 1.04 = £30 000.
Compound percentage change
For two changes in succession, multiply the multipliers.
Worked example: a price rises 20% then falls 15%. Net change?
- 1.20 × 0.85 = 1.02 → 2% increase overall.
⚠Common mistakes
- Reverse percentages by subtracting back — wrong. £63 + 10% = £69.30, not £70.
- Adding percentages directly — 20% rise then 20% fall is NOT no change (it's a 4% loss).
- Treating "% change" as new ÷ old × 100 — that gives the multiplier, not the change.
- Confusing original vs new in the denominator — always divide the change by the original.
- Forgetting to subtract from the multiplier for a decrease — 1 − p/100, not p/100 alone.
➜Try this— Quick check
A car loses 18% of its value. After loss it's worth £8 200. Original value?
- Multiplier = 0.82.
- 8200 / 0.82 = £10 000.
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