Percentage Changes

Level 7

 

The objectives of this section are to

*  find percentage increases or decreases using multipliers;

*  express an increase or decrease as a percentage.

 

 

Percentage increases and decreases

 

Example:

A television costs £240.  In a sale, its price is reduced by 15%.  Find the new price of the television.

 

Solution:

We first find 15% of £240 = 0.15 × 240 = £36.

So, the new price of the television is £240 - £36 = £206.

 

* * *

 

Example 2:

In 2004, 180 parents applied to a school for a place for their child.  The following year saw an increase of 35% in the number of applications.  Find the number of applications in 2005.

 

Solution:

We first find 35% of 180 = 0.35 × 180 = 63.

So in 2005, the number of applications is 180 + 63 = 243.

 

Although we can solve percentage increase and decrease problems in the above way, it is usually more convenient to solve them using multipliers.

 

 

Multipliers

 

If a price increases by 50%, the new price will be 150% of the old price, or 1.5 times as much.  So to increase by 50%, you multiply by 1.5.  This number is referred to as the multiplier.

 

If a price increases by 20%, the new price will be 120% of the old price, or 1.2 times as much. So the multiplier for an increase of 20% is 1.2.

 

The table below shows other examples of percentage increases and the corresponding multipliers:

 

Increase

Multiplier

25% increase

1 + 0.25 = 1.25

34% increase

1 + 0.34 = 1.34

7% increase

1 + 0.07 = 1.07

17.5% increase

1 + 0.175 = 1.175

4.8% increase

1 + 0.048 = 1.048

 

If a price is decreased by 25%, then the new price will be 75% of the old price.  So the multiplier is 0.75.

 

If a price is decreased by 10%, then the new price will be 90% of the old price.  The multiplier therefore is 0.9.

 

The table below shows other examples of percentage decreases and the corresponding multipliers:

 

Decrease

Multiplier

35% decrease

1 - 0.35 = 0.65

18% decrease

1 – 0.18 = 0.82

4% decrease

1 - 0.04 = 0.96

2.8 decrease

1 – 0.028 = 0.972

42.5% decrease

1 – 0.425 = 0.575

 

 

Example:  Gas prices increases by 14%.  A customer used to pay £84 per month.  Find her new monthly gas bill.

 

Solution:  The multiplier for an increase of 14% is 1 + 0.14 = 1.14.

So, the new monthly gas bill is £84 × 1.14 = £95.76.

 

* * *

 

Example 2: Following the opening of a new supermarket nearby, the number of customers using a small store decreased by 21%.  If 2400 customers used to use the store each week, find the number of customers after the store opened.

 

Solution:  The multiplier for a decrease of 21% is 1 – 0.21 = 0.79.

So the number of customers after the supermarket opened is 2400 × 0.79 = 1896.

 

 

Successive increases and decreases

 

Introductory example

A couple buy a house for £185000.  Its value increases by 12% in the first year and by a further 7% in the following year.  Find the value of the house after two years.

 

Solution:

The diagram below summarises the two increases.  The multiplier for a 12% increase is 1.12 and the multiplier for a 7% increase is 1.07.

Text Box: After 2 years
£221704
Text Box: Initial price
£185000
Text Box: After 1 year
£207200
Text Box: × 1.12
-------->
---
Text Box: × 1.07
--------->

 

 

 

 

 

 


So the value of the house after 2 years is £221704.

 

* * *

 

Example:

In August, a shop charges £350 for a washing machine.  The shop puts the washing machine up by 15% in September.  In October the shop has a sale and it then reduces the price of the washing machine by 20%.  What is the sale price of the washing machine?

 

Solution:

The multiplier for an increase of 15% is 1.15.

The multiplier for a decrease of 20% is 0.8.

 

 

Text Box: Oct price
£322
Text Box: Sept price
£402.50
Text Box: × 1.15
-------->
Text Box: × 0.8
--------->
Text Box: August price
£350

 

 

 

 

 


So the sale price of the washing machine is £322.

 

* * *

 

Note:

To get the overall effect of several percentage changes, you multiply together the corresponding multipliers.

 

Example:

A train company increases is rail fares by 4% one year and by 6.5% the following year.  Find the percentage increase in cost over the two years.

 

Solution:

The multiplier for a 4% increase is 1 + 0.04 = 1.04.

The multiplier for a 6.5% increase is 1 + 0.065 = 1.065.

So, the overall multiplier is 1.04 × 1.065 = 1.1076.

This corresponds to a 10.76% increase in prices in two years.

 

* * *

 

Example 2:

A petrol station increases the price of petrol by 5%.  The following week, it reduces the cost of petrol by 4%.  Find the overall change in the cost of the petrol.

 

Solution:

The multiplier for a 5% increase is 1.05.

The multiplier for a reduction of 4% is 0.96

The overall multiplier corresponding to these two changes is 1.05 × 0.96 = 1.008.

This corresponds to an increase of 0.8% (as the decimal 0.008 is equivalent to 0.8%).

 

 

Calculating a percentage increase or decrease

 

 

Introductory example

A house was purchased for £200000 in 2001.  The owner sold it five years later for £250000.  Find the percentage by which the house increased in price.

 

Solution

The increase in the price of the house is £50000.

We can express this increase as a fraction of the original cost of the house:

           

Therefore the increase, expressed as a percentage of the original cost, is 25%.

 

* * *

 

We usually use the following formula to find a percentage increase or decrease:

 

 

 

Example

The population of a town decreased from 36000 to 35400 over five years.  Find the percentage decrease in population.

 

Solution:

The population decreased by 600.  Therefore:-

              (to 2 decimal places)

 

 

* * *

 

Example 2:

The price of a loaf of bread increases from 96p to £1.05.  Find the percentage increase in price.

 

Solution:

The loaf of bread increases by 9p.

So the percentage increase is: