1.4 Working backwards
In some situations, you may be given the total after the discount or VAT has been applied and asked to find the original amount.
For example, suppose an online shop charges an extra 5 per cent of the cost of the purchases for postage and packing. You have a gift voucher for £30. What is the maximum you can spend at the shop so that the total including the postage is less than £30?
Figure 1 shows how the money you have to spend and the percentage values relate to each other. The cost is 100 per cent, as it is the whole value being dealt with. The postage is an additional 5 per cent – so as you can see in the diagram, 105 per cent is equivalent to £30.
First start by finding 1 per cent of £30 by dividing it by 105, which is then multiplied by 100 to give 100% and therefore the amount that can be spent.
1% of £30 = £30 ÷ 105 ≈ £0.2857 (to 4 decimal places)
Amount available to spend = (£30 ÷ 105) × 100 = £28.57 (to the nearest penny)
The maximum amount that can be spent then is £28.57. The answer can be checked by working out the 5 per cent postage charge on £28.57 to see that the grand total will be £30.
Many people in this situation may think that they can work out 5 per cent of £30 and subtract this from the £30 to find out how much they can spend. Try this now.
5% of £30 = 0.05 × £30 = £1.50
£30 – £1.50 = £28.50
This answer is 7p less than using our other method! So something must be wrong.
Checking this by calculating 5 per cent of £28.50 and adding this to £28.50, gives £29.93. This shows that you can’t undo the addition of 5 per cent to a value by subtracting 5 per cent from the total. That’s because the value with 5 per cent added to it is greater than the original value, so working out 5 per cent of the new value will give you a larger number than 5 per cent of the original.
This is an important idea to remember. Have a go at this next activity, and if you need to, try drawing yourself something like Figure 1 to help.
Activity 6 Working backwards
The price of a tennis racquet, including 20 per cent VAT, is £45.24. What was the price before VAT?
If you need a hint, click on the ‘Reveal comment’ button.
The £45.24 includes VAT. What percentage does this total represent? Find the monetary amount that represents 1 per cent, and use that figure to arrive at 100 per cent.
The price before VAT was added is our original 100 per cent.
So £45.22, which includes VAT, is equivalent to 120 per cent of the cost (100% + 20%)
1 per cent of £45.24 = £45.24 ÷ 120
Thus, the price of the tennis racquet before VAT was applied was £37.70.
You have very nearly finished your work on percentages for this week, but before you move onto ratios you’re going to take a quick look at the idea of percentage points. You may have heard or read about these in news reports, but not really thought about what these mean. Now it’s your chance to find out about the important difference between percentages and percentage points.