Reverse Percentages

I`m fascinated by students misconceptions because the most common ones are repeated over a long period of time and across a wide geographical area. With reverse percentages questions, a very common misconception is to think that the figure that they are given is 100%. Here is an example of a past exam question:

In a sale, normal prices are reduced by 20%. Andrew bought a saddle for his horse in the sale. The sale price of the saddle was £220. Calculate the normal price of the saddle.

Error: 20% of £220 = £44. £220 + £42 = £262.

Correct way: We always take the original amount to be 100%. This means that the percentage that we have after the reduction is 80%, meaning that £220 = 80%. As the original price is always 100%, 20% is required. If £220 = 80%, then 20% is 80% / 4. Thus £55 = 20% So if 80% = £220 and 20% = £55, then 100% = £275.

Similarly, if you are given the amount after an increase, then the increased amount will be greater than 100% and then you will need to work your way back to 100%.

e.g. The price of all rail season tickets to London increased by 20% to £6. What was the original fare?

A: £6 = 120% so £1 = 20%. If £1 is 20%, then 100% is found by multiplying by 5. The original price was £5.