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Fractions, decimals and percentages

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Maths

# Percentage change  0%

Summary

# 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.

FAQs

• Question: What is simple interest?

Answer: Simple interest is increasing the value of an object by a constant amount in regular intervals.

• Question: How do you calculate a multiplier using percentage change values?

Answer: Firstly, convert the percentage to a decimal by dividing it by 100. Then, if you have a percentage increase, add this value to the number 1. If you have a percentage decrease, subtract this value from the number 1.

• Question: How do you calculate the percentage change?

Answer: Percentage change = (new value - initial value)/initial value * 100

Theory

Exercises