A reverse percentage problem gives you the result after a percentage increase or decrease, and asks you to find the original value.
Key principle: the value after the change is a percentage of the original. Divide by the multiplier, not the percentage.
Example: After a 20% increase, a price is £180. Find the original price. $$180 \div 1.20 = £150$$
Example: After a 15% decrease, a value is 68. Find the original. $$68 \div 0.85 = 80$$
Common error: students subtract 20% of the new value ($180 \times 0.20 = 36$, giving £144) — this is wrong because 20% of £180 ≠ 20% of the original. Always divide by the multiplier.
The multiplier for:
- Increase of $r%$: divide by $\left(1 + \frac{r}{100}\right)$
- Decrease of $r%$: divide by $\left(1 - \frac{r}{100}\right)$