If a coat is reduced by 10% in a sale and then three weeks later it is reduced by a further 20%, its total percentage discount is NOT 30%.
The general rule of compound percentages is NOT to add the percentages together, rather is it to find the total percentage change. Note, the second percentage change is found on a new amount and not on the original amount.
- A 15% decrease is calculated by multiplying 85% or 0.85.
- A 10% decrease is calculated by multiplying 90% or 0.9.
Therefore a 15% discount followed by a 10% discount is equivalent to multiplying the original price by 0.85 and then by 0.9.
original price × 0.85 × 0.9 = original price × 0.765
The sale price is 76.5% of the original price.
This is equivalent to a 23.5% discount.
John invests in some shares.
After one week the value goes up by 12% (to find a 12% increase we multiply by 112% or 1.12)
The following week they go down by 12% (to find a 12% decrease we multiply by 88% or 0.88)
original amount × 1.12 × 0.88 = original amount × 0.9856
John has 98.56% of his original investment and has therefore made a 1.44% loss.
Jon puts £600 into a savings account with an annual compound interest rate of 5%. How much will he have in the account at the end of 3 years if he doesn’t add or withdraw any money?
Note: at the end of each year interest is added to the total amount this means that each year 5% of an ever larger amount is added to the account. This is an example of exponential growth.
To increase the amount in the account by 5% we need to multiply it by 105% or 1.05. This is carried out for each year that the money is in the account.
At the end of year 1 Jon has £600 × 1.05 = £630
At the end of year 2 Jon has £630 × 1.05 = £661.50
At the end of year 3 Jon has £ 661.50 × 1.05 = £694.58
Note: we can also write this in a single calculation as
£600 × 1.05 × 1.05 × 1.05 = £729.31
Or using index notation as
£600 × 1.05³= £607.75
Repeated percentage change
We can use the powers in the above example to help solve many problems involving repeated percentage increase and decrease.
The population of a town increases by 2% each year, if the current population is 2345, what will it be in 3 years?
To increase the population by 2% we multiply it by 1.02.
After 5 years the population will be
2345 × 1.02³ = 2489 (to the nearest whole number)