Interest, Growth and Decay

Number

An item losing a fixed percentage of its value each year behaves exactly like compound interest with the sign reversed
Use a multiplier below 1 and raise it to the power of the number of years
- A yearly loss of 12% has multiplier 0.88
A value P falling by r% for n years becomes P × (1 − r/100)ⁿ
Where the question asks how much value was lost, subtract the final value from the starting value

Worked example

Value falling each year

A machine bought for £7500 loses 12% of its value each year. Find its value after 4 years, and the value lost.

Solution:

A repeated percentage decrease works the same way as compound interest, with a multiplier below 1
Losing 12% leaves 100 − 12 = 88% of the value, so the multiplier is 0.88
Using 0.12 instead would give the value lost in one year, not the value remaining, which is the usual slip
Applying the multiplier once per year for 4 years: 7500 × 0.88⁴
0.88⁴ = 0.59969536
Value after 4 years = £4497.72 to the nearest penny
The value lost is the fall from the starting price, so subtract
Value lost = 7500 − 4497.72 = £3002.28
A check on the scale: after 4 years a little under 60% remains, and £4497.72 is a little under 60% of £7500