Extended. When a quantity changes by the same percentage every year, it is multiplied by the same number every year. Populations grow and cars lose value this way, and one formula handles them all.
Value after \(n\) years \(= \text{start} \times \left(1 \pm \dfrac{r}{100}\right)^{n}\). Growth uses a multiplier bigger than 1, decay a multiplier between 0 and 1. The power is the number of times the change happens.
Fig. 1: a car bought for $18 000 loses 15% of its value each year, so its value is multiplied by 0.85 every year. The curve falls by a smaller amount each year but never reaches zero
Section 01
One multiplier, many years.
Open the lab on Population growth, then Depreciation. Move the years slider and watch the curve: each step multiplies by the same number, so a growth curve gets steeper and a decay curve flattens out without reaching zero.
Change each year
Multiplier
Type
increase of 3.5%
1.035
growth
increase of 7%
1.07
growth
decrease of 15%
0.85
decay
decrease of 8%
0.92
decay
Worked example Depreciation
A car costs \(\$18\,000\). Its value falls by 15% each year. Find its value after 4 years.
A 15% decrease leaves 85%, so the multiplier is 0.85.
After 4 years: \(18\,000 \times 0.85^{4}\).
\(= 18\,000 \times 0.522\,006... = 9396.1125\)
Answer: \(\$9396.11\) (Fig. 1)
Worked example Population growth
A town has 24 000 people. The population grows by 3.5% each year. Estimate the population after 10 years.
Multiplier: \(1.035\).
\(24\,000 \times 1.035^{10} = 33\,854.37...\)
A population is a whole number of people.
Answer: About 33 900 (3 s.f.)
Examiner trap
Exponential decay is NOT a fixed amount off each year. \(18\,000 - 4 \times 0.15 \times 18\,000 = \$7200\) is wrong, because each year's 15% is taken from a smaller value.
Section 02
How many years?
To find when a value first passes a target, try whole numbers of years in the formula. Write down the values either side of the target to show your method.
Worked example First year below a target
A population of 5000 birds falls by 8% each year. After how many complete years is it first below 3000?
Multiplier: \(0.92\). Try values of \(n\) in \(5000 \times 0.92^{n}\).
The power is the number of times the change happens, not the number of hours. In 24 hours a 6-hour halving happens 4 times, so the power is 4, not 24.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which multiplier gives a decrease of 4% each year?
Quick check+10 XP
\(\$5000\) grows by 2% each year. What is it worth after 3 years?
Quick check+10 XP
Which statement describes exponential decay?
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800 mg of a substance halves every 6 hours. How much is left after 24 hours?
Section 03
Working backwards.
If you know the value after the change, divide by the multiplier to the power \(n\) to get the start value. If you know the start and end values, find the multiplier with a root.
Worked example The original value
A machine loses 10% of its value each year. After 3 years it is worth \(\$14\,580\). What was it worth when new?