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

$P open parentheses 1 plus r over 100 close parentheses to the power of n$

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

$P open parentheses 1 minus r over 100 close parentheses to the power of n$

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

$P open parentheses 1 plus r over 100 close parentheses to the power of n equals F$

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)

Exam Tip

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 $P open parentheses 1 plus r over 100 close parentheses to the power of n$
Substitute P = 25 000, r = 4 and n = 13 into the formula

$25 space 000 open parentheses 1 plus 4 over 100 close parentheses to the power of 13$

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 $P open parentheses 1 minus r over 100 close parentheses to the power of n$
Substitute P = 60 and n = 3 into the formula

$60 open parentheses 1 minus r over 100 close parentheses cubed$

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

This is now an equation in terms of r

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