1.14 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?
As you can see in the diagram above, 105 per cent is equivalent to £30.
Therefore, 1 per cent is equivalent to 
, and multiplying this by 100 creates 100 per cent, which results in £0.2857 x 100 = £28.57 (rounded down to the nearest penny).
The maximum amount that can be spent then is £28.57. You can check your answer by working out the 5 per cent postage charge on £28.57 to see that the grand total will be £30.
Activity 8 Working backwards
The price of a tennis racquet, including 20 per cent VAT, is £45.24. What was the price before VAT?
Comment
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.
Answer
120 per cent of the cost is £45.24.
1 per cent is £45.24 ÷ 120 and 100 per cent is:
Thus, the price of the tennis racquet before VAT was applied was £37.70.
In the next section we are going to look what ‘percentage points’ means. You may have heard or read about these in news reports, and it is important to understand how these relate to percentages.
Percentage points
When comparing percentages, the difference between the percentages is described in terms of ‘percentage points’. Subtracting percentages gives percentage points. Note that this is not the same as the percentage increase or decrease. This distinction is something to be aware of in media reports. For example, if 72 per cent of students starting at a college were studying maths one year, but only 63 per cent the next, the college had a decrease of nine percentage points in students studying maths.
The next activity illustrates the difference between percentage points and percentage change.
Activity 9 Mortgages
You took out mortgage when buying your new home. If you see a headline reading ‘Interest rates jump 2 per cent’, should you be worried?
Comment
Most people read this headline to mean that the interest rate on the loan increases by two percentage points. What would that mean for you? Is there another way to think about the headline that isn’t as sensational?
Answer
The headline is ambiguous. It could mean ‘Interest rates increase by 2 percentage points’ – for example, from 10 per cent to 12 per ecnt. This is a big deal – your interest rate just went up 20 per cent: 0.12 – 0.10 x 100% = 20%.
On the other hand, the headline could mean ‘Interest rates increase by 2 per cent based on your previous interest rate’. In our example, that would mean rates go up from 10% to 10% + 2% of 10% = 0.10 + 0.02 x 0.10 = 0.10 + 0.002 = 0.102 = 10.2%. Not great news, but hopefully manageable and much better than our first interpretation.
When people mean percentage points, they should say so! Unfortunately, many news articles do not. It’s up to you to find out from the article itself what it is about. In newspapers and magazines, you will often run into articles that contain percentages. The activity below illustrates the importance of interpreting this information correctly.
Activity 10 Percentage points
A newspaper headline reads ‘Exam pass rate increases from 50 per cent last year to 75 per cent this year.’ Does this mean that the pass rate has increased by 25 per cent? Explain why or why not.
Comment
Did you convert the percentages to their decimal values before using the percentage increase formula?
Answer
Writing the pass rates in decimal form as 0.5 and 0.75, the percentage increase is:
So the pass rate has increased by 50 per cent, not 25 per cent. However, it would be correct to say that there was a 25 percentage point increase in the exam pass rate.
The bottom line is to be sure that you pay attention when you run across percentages. A statistic might not mean what you initially think it does. Keep your eyes open and your brain turned on!
Now we’ve looked at percentages we are going to turn to another way to represent numbers – ratios.