# 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.

## Multipliers

A multiplier measures how much a quantity has changed by comparing the new value to the old one. A multiplier greater than $1$ suggests an increase in value, whereas a multiplier less than $1$ suggests a decrease in value.

### Formula to calculate the multiplier

The multiplier follows the formula below:

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

##### Example 1

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

*Substitute values into the equation:*

*$\begin{aligned}\text{new\,value}&=\text{multiplier}\times \text{initial\,value}\\2500&=\text{multiplier}\times 2000\end{aligned}$***

*Rearrange and solve for the multiplier:*

$\begin{aligned}\text{multiplier}&=2500\div2000\\&=1.25\end{aligned}$

*The multiplier is $\underline{1.25.}$*

### 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:

#### Procedure

1. | Convert the percentage into a decimal by dividing it by $100$. |

2. | Establish whether you have a percentage increase or decrease. |

3. | If you have a percentage **increase**, add the value from Step One to the number $1$ If you have a percentage **decrease**, subtract the value from Step One from the number $1$. |

##### Example 2

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

*Firstly, calculate the multiplier.*

*$\begin{aligned}10\%&=0.1\\\text{multiplier}&=1+0.1\\\text{multiplier}&=1.1\end{aligned}$*

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

*$\begin{aligned} \text{new\,value}&=\text{multiplier}\times \text{initial\,value}\\£2.20&=1.1\times \text{initial value}\\\dfrac{£2.20}{1.1}&=\text{initial value}\\\text{initial value}&=£2.00\end{aligned}$*

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

## 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:

$\text{percentage \ change} = \dfrac{\text{new \ value} - \text{initial \ value}}{\text{initial \ value}} \times 100$

**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 $200$ to $130$.*

*Substitute the numbers into the formula*

$\begin{aligned}\text{percentage \ change} &= \dfrac{130-200}{200}\times100\\&= -\dfrac{70}{200}\times100\\&=-35\end{aligned}$

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

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

## 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 $10\%$ a year. If the car is initially worth $£5000$, how much is it worth after $8$ years?*

*Work out the monetary increase for one year:*

*$10\%$ of $£5000$ is $0.1\times£5000=£500$*

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

*Work out the increase for $8$ years:*

*$1$ year = $£500$* *increase*

*$8$ years = $£500 \times8 =£4000$ increase*

*The car increases in value by $£4000$ after $8$ years.*

*Work out the total value after $8$ years:*

*$£5000+£4000=£9000$*

**

*The car is worth *$\underline{£9000}$ *after $8$ years.*