Reverse compound change

Objective. Recover original values from repeated growth or depreciation.

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Learn: the key idea

Reverse compound change finds an original value before a known percentage growth or decay was applied, by dividing the final amount by the overall multiplier rather than taking a percentage of the final amount. If a quantity decayed by r% to give amount A, the original value is A ÷ (1−r/100); for growth it is A ÷ (1+r/100). This works because the original value, multiplied by the multiplier, equals A, so dividing undoes the multiplication. For example, if a price after a 10% reduction is £63, the original price is 63 ÷ 0.9 = £70, not 63 × 1.1.

Worked example

After a 15% pay rise, Sam's salary is £27600. Find his salary before the rise.

1 $Growth multiplier = 1 + 0.15 = 1.15$
2 $Original salary \times 1.15 = 27600$
3 $Original salary = 27600 \div 1.15$
4 $Compute 27600 \div 1.15 = 24000$

Answer: £24000

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Common mistake. A common error is to add or subtract the percentage directly from the final value, calculating £63 + 10% = £69.30 instead of dividing by the decay multiplier 0.9 to get £70; percentage change must be undone by division, not addition.

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