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Reverse percentages
Sometimes a question will ask you to work backwards and find the original price of something after the price has increased. If you are given a quantity after a percentage increase or decrease, and you need to find the original amount, use this method:
Example 1
A radio sells for £63, after a 40% increase in the cost price. Find the cost price.
Solution
Start with the original amount as 100%.
Cost price = 100%
We are told the selling price is a 40% in the cost price.
So the selling price is 100% + 40% = 140% of the cost price.
We know that the selling price is £63, so 140% = £63.
Now calculate 1%:
140% = £63
1% = £63/140
1% = £0.45
The cost price is 100%, so multiply £0.45 by 100.
Cost price = 0.45 × 100 = £45.
Example 2
A new car falls in value by 30% in a year. After a year, it is worth £8,400.
Find the price of the car when it was new.
Solution
Remember that the original price of the car is 100%.
Original price = 100%.
Secondhand price = 100%  30% = 70%.
So £8,400 = 70% of the original price.
So 1% of original price = £8,400 ÷ 70
Original price = 100% = 100 x 1% = 100 x (£8,400 ÷ 70)
= £12,000.
It is easy to go wrong in this type of question. Always check that your answer is realistic.
Suggested reading…
Work out reverse percentage problems
The usual question here has to do with sales discounts. For example, a coat is on offer, 25% off. Its sale price is £200. What was the original price before the reduction?
The mistake most people make is find 25% of £200 (£50), add it on, and then say that was the original price: £250.
This is WRONG!
25% was taken off the original price, which was higher than £200. That means the reduction was more than just fifty quid. There are two ways to work out what the original price was. One is the unitary method (also used when calculating with ratios). The other is much simpler if you have a calculator to hand, but less intuitive.
Unitary method – the original price was 100%, and the sale price is 75% of the original.
 100% Original Price ? 75% Sale Price £200
Divide the sale price by 75 to work out the value of 1% = 200/75 = £2.67.
Multiply 1% by 100 to get 100%, the original price: £2.67 x 100 = £267.
2) The second method is just to convert 75% into a decimal, and then divide the sale price by the decimal number:
£200 / 0.75 = £267
This works because to reduce the original price (P) by 25%, you would multiply it by 0.75
0.75P = 200
So rearranging this equation, we can see that
P = 200/0.75
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