# 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.*