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.)​

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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.​

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