Exponential Growth & Decay

The ideas of compound interest and depreciation can be applied to other (non-money) situations, such as increasing or decreasing populations.

What is exponential growth?

When a quantity grows exponentially it is increasing from an original amount, P, by r % each year for n years

Some questions use a different timescale, such as each day, or each minute

Real-life examples of exponential growth include population increases, bacterial growth and the number of people infected by a virus
The same formula from compound interest is used

Final amount of the quantity is

$P {(1 + \frac{r}{100})}^{n}$

Substitute values of P, r and n from the question into the formula to find the final amount

What is exponential decay?

When a quantity exponentially decays it is decreasing from an original amount, P, by r % each year for n years

Some questions use a different timescale, such as each day, or each minute

Real-life examples of exponential decay include the temperature of hot water cooling down, the value of a car decreasing over time and radioactive decay (how radioactive a substance is over time)
The same formula from compound interest is used, but with +r replaced by -r

Final amount of the quantity is

$P {(1 - \frac{r}{100})}^{n}$

Substitute values of P, r and n from the question into the formula to find the final amount

How do I use the exponential growth & decay formula?

To find a final amount, substitute the values of P, r and n (from the question) into the formula
If the final amount is given in the question, F, set the whole formula equal to this final amount

$P {(1 + \frac{r}{100})}^{n} = F$

Some questions then ask to find P, r or n

To find P or r, rearrange the formula to make P or r the subject (for r, one of the steps involves taking an nth root)
To find n, use trial and improvement (test different whole-number values for n until both sides of the equation balance)

Examiner Tip

Remember, r is a percentage not a decimal
- For example, an increase of 25% means r = 25, not 0.25
Look out for how the question wants you to give your final answer
- It may want the final amount to the nearest thousand
- If finding n, your answer should be a whole number

Worked example

(a) An island has a population of 25 000 people. The population increases exponentially by 4% every year. Find the population after 13 years, giving your answer to the nearest hundred.

The question says “increases exponentially” so use $P {(1 + \frac{r}{100})}^{n}$
Substitute P = 25 000, r = 4 and n = 13 into the formula

$25 000 {(1 + \frac{4}{100})}^{13}$

Work out this value on your calculator

41626.83…

Round this value to the nearest hundred

41 600 people

(b) The temperature of a cup of coffee exponentially decays from 60°C by r % each hour. After 3 hours, the temperature is 18°C.

Write down an equation in terms of r.

The question says “exponentially decays” so use $P {(1 - \frac{r}{100})}^{n}$
Substitute P = 60 and n = 3 into the formula

$60 {(1 - \frac{r}{100})}^{3}$

The final value is 18, so set the whole formula equal to 18

$60 {(1 - \frac{r}{100})}^{3} = 18$

This is now an equation in terms of r

$60 {(1 - \frac{r}{100})}^{3} = 18$

Exponential Growth & Decay (Cambridge O Level Maths)

Revision Note