# Repeated Percentage Change

## Repeated Percentage Change Revision

**Repeated Percentage Change**

A **repeated percentage change** question involves calculating the **overall** percentage change that has resulted from **more than one consecutive** percentage changes.

**Calculating Percentage Change**

**Repeated percentage changes** can include percentage **increases** or percentage **decreases**:

**Percentage increase** example:

Increase 100 by 20 \%:

100\times1.2=120

**Percentage decrease** example:

Decrease 100 by 20 \%:

100\times0.8=80

Using our knowledge on how to increase and decrease by a percentage, we can work out **overall percentage changes** as follows:

**Example:**

Calculate the **overall percentage change** when an increase of 10 \% is followed by a decrease of 30 \%.

An increase of 10 \% would mean multiplying by 1.1, and a decrease of 30 \% would mean multiplying by 0.7.

To find the overall change, we need to multiply these together:

1.1\times0.7=0.77\\

As 0.77 is below 1, we know it is an overall decrease, so we can minus this from 1 to calculate the decrease:

1-0.77=0.23The overall percentage change was a decrease of 23 \%

**Example 1**

Calculate the **overall percentage increase** when 70 is **increased** by 20 \%, and then **increased** by a further 15 \%

**[2 marks]**

The quickest way to calculate this is by using the percentage changes directly:

20 \% **increase**: 1.2

15 \% **increase**: 1.15

**Overall**:

As this is higher than 1, we can confirm it is an overall increase, and we can minus 1 to calculate the overall percentage:

1.38-1=0.38Therefore, the **overall percentage increase** was 38 \%

**Example 2**

In a shop, a t-shirt is firstly **increased** in price by 25 \%, but as it did not sell well at this new price, the price was later **decreased** by 30 \%. Calculate the **overall percentage change**.

**[2 marks]**

25 \% **increase**: 1.25

30 \% **decrease**: 0.7

**Overall**:

Therefore, the t-shirt had an **overall percentage decrease** of 12.5 \%

## Repeated Percentage Change Example Questions

**Question 1:** The value of a car increases by 15 \% in one year, but then decreases by 40 \% in the next. Calculate the overall percentage change in the car’s value over the 2 years.

**[2 marks]**

15 \% increase: 1.15

40 \% decrease: 0.6

Overall:

1.15\times0.6=0.69\\ 1-0.69=0.31Therefore, the car’s value had an overall decrease of 31 \%

**Question 2: **A population of ants increases by 20 \% each year. Calculate the overall percentage increase of ants over 3 years.

**[2 marks]**

20 \% increase: 1.2

Overall:

1.2\times1.2\times1.2=1.728\\ 1.728-1=0.728Therefore, the ant population had an overall increase of 72.8 \%

**Question 3: **The number of people in a city had an increase of 5 \% in January, an increase of 8 \% in February, and then a decrease of 2 \% in March.

Calculate the overall percentage change in the 3 months, to the nearest whole number.

**[3 marks]**

5 \% increase: 1.05

8 \% increase: 1.08

2 \% decrease: 0.98

Overall:

1.05\times1.08\times0.98=1.11132\\ 1.11132-1=0.11132Therefore, the city’s population had an overall percentage increase of 11 \% (to the nearest whole number).

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