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Compound percentage problems involve successive percentage changes — one after another. These are trickier than single percentage changes because of a counter-intuitive result: successive percentage changes do not simply add up. This lesson explains why, shows you the correct method, and covers the common traps.
Example: A house worth £200,000 increases by 10% one year and then by 10% the next year.
Intuitive (but wrong): 10% + 10% = 20% increase → £200,000 × 1.20 = £240,000
Correct:
The correct answer (£242,000) is £2,000 more than the intuitive answer (£240,000). This extra £2,000 is the "interest on interest" — the second 10% is applied to £220,000 (which includes Year 1's increase), not the original £200,000.
The Rule: When percentages are applied successively, multiply the multipliers rather than adding the percentages.
For each percentage change, find the multiplier:
| Change | Multiplier |
|---|---|
| +10% | ×1.10 |
| +25% | ×1.25 |
| −10% | ×0.90 |
| −20% | ×0.80 |
| +5% | ×1.05 |
| −15% | ×0.85 |
For successive changes, multiply the multipliers:
Example: +10% then +10%
Example: +20% then −20%
Key Insight: A 20% increase followed by a 20% decrease does NOT get you back to the start. You end up 4% lower.
Example: A price increases by 15% and then by 10%.
If the original price is £400:
Example: An investment grows by 5%, then 8%, then 3%.
Example: A car depreciates by 15% in Year 1 and 10% in Year 2.
If the car was worth £24,000:
Note: 15% + 10% = 25%, but the actual decrease is only 23.5%. With successive decreases, the overall decrease is always less than the sum, because the second decrease is applied to an already-reduced value.
Example: A salary increases by 8% then decreases by 5%.
Example: Membership drops by 12% then increases by 15%.
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