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.
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.
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: