# 5.3 Reverse percentages

Say you are out shopping and see a pair of shoes advertised with a sale price of £35 and the shop is advertising 20% off everything. Whilst it is clear that £35 is the sale price after the 20% discount has been applied, how much did the shoes cost originally? In this case, you cannot simply work out 20% of the £35 and add it on – you need to use reverse percentages.

## Example: 80% = £35

We know that 80% = £35. Since we want to know what 100% is, we can start by finding what 1% is worth. We can do this by dividing both sides by 80.

Once we know what 1% is worth, we can find 100% (original price) by multiplying both sides by 100.

### Method

**80% = £35**- [
*÷ both sides of the equation by 80*] **1% = 0.4375**- [
*× both sides of the equation by 100*] (to two d.p.)**100% = £43.75**

The original price of the shoes was therefore £43.75. Whilst these calculations often require some rounding to be done with the answer, it is important that you do not round until the final stage. If you round too early, you risk altering the answer.

Take a look at one more example before trying one yourself. Let’s say that you find a kettle that costs £24.50. You can see from the ticket that it already has 15% taken off the original price.

## Example: 85% = £24.50

This time then, we know 85% of the original cost is worth £24.50 (this is calculated by doing 100% − 15%).

### Method

**85% = £24.50**- [
*÷ both sides of the equation by 85*] **1% = 0.28823…**- [
*× both sides of the equation by 100*] **100% = £28.82**

This approach can also be used if an item has increased by a given percentage. If, for example, you had a 5% pay rise and now earn £400 a week, you could calculate in the same way with the starting figure of 105% = £400.

## Activity 11: Reverse percentage problems

Work out the original values in each question. Where rounding is required, round to two decimal places.

- You buy a pair of trousers for £40. They have had a 20% discount applied. What was the original cost of the trousers?
- The local gym has a special offer on that gives 35% off when you buy an annual pass. The amount you would pay for the year for the annual pass would be £120. How much would you have paid without the discount?
- Your mobile phone provider has just put their prices up by 10%. You will now be paying £25 a month for your contract. How much were you paying before the increase?

### Answer

20% discount means that 80% of the cost is £40. Dividing each side by 80 gives 1% = 0.5.

To get to 100% we multiply each side by 100 giving 100% = £50. Original cost = £50.

35% discount means that 65% of the cost is £120. Dividing each side by 65 gives 1% = 1.84615….

To get to 100% we multiply each side by 100 giving 100% = £184.62 (to two d.p.).

A 10% increase means that 110% of the cost is £25. Dividing each side by 110 gives 1% = 0.22727….

To get to 100% we multiply each side by 100 giving 100% = £22.73 (to two d.p.).

Congratulations, you now know everything you need to know about percentages! As you have seen, percentages come up frequently in many different areas of life and having completed this section, you now have the skills and confidence to deal with all situations that involve them.

You saw at the beginning of the section that percentages are really just fractions. Decimals are also closely linked to both fractions and percentages. In the next section you will see just how closely related these three concepts are and also learn how to convert between each of them.

## Summary

In this section you have learned:

- the connection between fractions and decimals and used this to enable you to calculate a percentage of an amount
- how to calculate percentage change using the formula
- ‘reverse percentages’ and practised using them in practical situations.