# Compound percentages

**Compound percentages**

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.

For example:

- 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.

Example 2

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.

Example 3

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.

For example,

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)