Rates of change

Percentage change

Percentage change

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Tutor: Labib


Percentage change

In a nutshell

Percentage change compares how much things have either grown or deteriorated. It can be used to calculate values of objects before and after they undergo a percentage change. However, it is best to first learn about multipliers.


A multiplier measures how much a quantity has changed by comparing the new value to the old one. A multiplier greater than 11​ suggests an increase in value, whereas a multiplier less than 11​ suggests a decrease in value.

Formula to calculate the multiplier

The multiplier follows the formula below:

new value=multiplier×initial value\text{new\,value}=\text{multiplier}\times \text{initial\,value} 

Example 1

The value of a car has increased from £2000£2000​ to £2500£2500​. What is the multiplier of this change?

Substitute values into the equation:

new value=multiplier×initial value2500=multiplier×2000\begin{aligned}\text{new\,value}&=\text{multiplier}\times \text{initial\,value}\\2500&=\text{multiplier}\times 2000\end{aligned}​​​​

Rearrange and solve for the multiplier:


The multiplier is 1.25.\underline{1.25.}

Calculating the multiplier using percentage change

You may be given the percentage increase or decrease in a question and will need to calculate the multiplier.  In this instance, use the following steps:


Convert the percentage into a decimal by dividing it by 100100​.
Establish whether you have a percentage increase or decrease.
If you have a percentage increase, add the value from Step One to the number 11​​
If you have a percentage decrease, subtract the value from Step One from the number 11​.

Example 2

The cost of a certain brand of chocolate has increased by 10%10\%. Its new price is £2.20£2.20. What was its original price?

Firstly, calculate the multiplier.


Substitute the multiplier and the value in the question into the equation below:

new value=multiplier×initial value£2.20=1.1×initial value£2.201.1=initial valueinitial value=£2.00\begin{aligned} \text{new\,value}&=\text{multiplier}\times \text{initial\,value}\\£2.20&=1.1\times \text{initial value}\\\dfrac{£2.20}{1.1}&=\text{initial value}\\\text{initial value}&=£2.00\end{aligned}​​

The original price was £2.00\underline{£2.00}.

Percentage change

To find the percentage change, you are calculating by what percentage of the original value something has increased or decreased.

Percentage change formula

To calculate percentage change, use the following formula:

percentage  change=new  valueinitial  valueinitial  value×100\text{percentage \ change} = \dfrac{\text{new \ value} - \text{initial \ value}}{\text{initial \ value}} \times 100​​

Note: A positive value for percentage change indicates a percentage increase, while a negative value indicates a percentage decrease.

Percentage change problems

You may be asked to find the value of a quantity after it changed, before it changed, or be asked to find the percentage change itself. To do this, use the percentage change formula.

Example 3

Identify and calculate the percentage change from 200200​ to 130130​.

Substitute the numbers into the formula

percentage  change=130200200×100=70200×100=35\begin{aligned}\text{percentage \ change} &= \dfrac{130-200}{200}\times100\\&= -\dfrac{70}{200}\times100\\&=-35\end{aligned}

Note the sign in front of the value to establish whether the change is an increase or decrease.

It is a percentage decrease of 35%.\underline{35\%.}

Simple interest

Simple interest is an application of percentage increase. It is about increasing the value of something by a fixed amount at regular intervals (usually once a year). It is best understood with an example.

Example 4

The value of a car increases with simple interest of 10%10\%​ a year. If the car is initially worth £5000£5000​, how much is it worth after 88​ years?

Work out the monetary increase for one year:

10%10\%​ of £5000£5000 is 0.1×£5000=£5000.1\times£5000=£500

The car increases in value by £500£500 per year.

Work out the increase for 88​ years:

11​ year = £500£500 increase

88​ years = £500×8=£4000£500 \times8 =£4000 increase

The car increases in value by £4000£4000 after 88​ years.

Work out the total value after 88​ years:


The car is worth £9000\underline{£9000} after 88​ years.

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FAQs - Frequently Asked Questions

What is simple interest?

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