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

- 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

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

** **

- 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

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

** **

- 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

- 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*n*th root) - To find
*n*, use trial and improvement (test different whole-number values for*n*until both sides of the equation balance)

- Remember,
*r*is a percentage not a decimal- For example, an increase of 25% means
*r*= 25, not 0.25

- For example, an increase of 25% means
- 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

(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

Substitute *P* = 25 000, *r* = 4 and *n* = 13 into the formula

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

Substitute *P* = 60 and *n* = 3 into the formula

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

This is now an equation in terms of *r *

*
*

Menu