Compound growth and decay

In a nutshell

Compound growth and decay (commonly used to work out compound interest) involves percentage changes over a period of time - usually on a yearly basis.

Compound growth and decay formula

In order to calculate compound growth and decay, a formula must be used:

$\text{Amount after n time} = \text{initial amount} \times \text{(percentage change multiplier)}^n$

This is sometimes written as:

$N = N_0 \times (\text{multiplier})^n$

Note: n could be time in years, months, days or any time period you can think of!

Example 1

Mitchell bought a car for $£10,500$ . It depreciates in value by $12\%$ each year. How much will the car be worth in six years time?

Work out the multiplier.

$1-0.12=0.88$

Use the formula.

Amount after six years $= 10500\times(0.88)^6 = \underline{£4876.24}$ (rounded to two decimal places.)

Example 2

The panda population is decreasing by two percent every year. The current population is $5000$ . What was the population three years ago? (Round your answer to the nearest whole number).

Work out the multiplier.

$1-0.02=0.98$

Fill in the formula with what you know where $x$ is the population three years ago.

$7500=x\times(0.98)^3$

Rearrange and solve for $x$ .

$\begin {aligned}&&7500&=x\times(0.98)^3 \\\div(0.98)^3&&&&\div (0.98)^3 \\&&7968.62&=x\end {aligned}$

Round to the nearest whole number.

$7968.62 \dashrightarrow \underline{7969}$

The population three years ago was $\underline{7969}$ pandas.