Reverse Percentages

A reverse percentage question tells you an amount after a percentage change and asks for the amount before it, the original amount. For example, a jumper costs £36 in a sale with 20\% off. What was its normal price?

The £36 is the price after the reduction, so it is not 100\%. It is only 80\% of the normal price. So 10\% is \pounds 36\div8=\pounds 4.50, and the normal price is 10\times\pounds 4.50=\pounds 45

The trap is to take 20\% of £36 and add it on. That gives £43.20, which is wrong: taking 20\% off £43.20 gives £34.56, not £36.

In the exam the question says something like "Work out the normal price", "the original price" or "the value before the increase", usually for 2 or 3 marks. It is often one of the last questions on a paper, and examiners say the most common mistake is treating the amount you are given as 100\%.

The original amount is always 100\%. After a decrease, take the percentage away from 100: after 30\% off, the new amount is 70\% of the original. After an increase, add it on: after a 20\% rise, the new amount is 120\% of the original.

Fractions work in the same way. If a price is reduced by \frac{1}{3}, the new price is \frac{2}{3} of the original. Divide by 2 to find \frac{1}{3}, then multiply by 3.

Use this to check your answer. After a decrease, the original amount must be more than the amount you are given. After an increase, it must be less.

Once you know which percentage you have, divide by it to find 1\%. Then multiply by 100 to get back to the original. For example, if 85\% is 3400, then 1\% is 3400\div85=40 and 100\% is 40\times100=4000.

\text{original}=\text{amount you have}\div\text{its percentage}\times100

This works for every percentage, including ones over 100, such as 105\%.

When the percentage you have is a multiple of 10, you can find 10\% instead and multiply by 10, as with the jumper. Whichever way you choose, write each step down: examiners give marks for a correct method.

The multiplier is the percentage you have, written as a decimal. So 70\% is 0.7, 82\% is 0.82 and 115\% is 1.15.

Going forwards, you multiply the original amount by the multiplier to get the new amount. Going backwards, you do the opposite, so you divide:

\text{original}=\text{new amount}\div\text{multiplier}

For example, after a 15\% rise a ticket costs £46. The multiplier is 1.15, and \pounds 46\div1.15=\pounds 40. Check by going forwards: \pounds 40\times1.15=\pounds 46. Multiplying £46 by 1.15 would give £52.90, which is more than £46, so it can't be the price before a rise.

VAT is a tax added to many prices. The question tells you the rate, usually 20\%. With VAT at 20\%, a price including VAT is 120\% of the price before VAT, so divide it by 1.2

Some questions tell you how much was added or taken off, not the new amount. For example, prices are cut by 15\%, and the price of a lamp is cut by £6. Here the £6 is the 15\%.

So work back from it in the same way: 1\% is \pounds 6\div15=\pounds 0.40, and 100\% is 100\times\pounds 0.40=\pounds 40. Don't work out 15\% of £6.

Then read the question again. If it asks for the amount after the change, there is one more step: add the change on for an increase, or take it off for a decrease. The lamp's sale price is \pounds 40-\pounds 6=\pounds 34

In a longer question the percentage change may come with something else, such as a delivery charge, a voucher or some people leaving. Work backwards through the story: undo the last thing that happened first, then undo the percentage change.

Write down what each number stands for as you go, such as "sale price = £390". Then check your answer by going forwards through the story to see if you get back to the number in the question.

• Treating the amount you are given as 100\%. It is the amount after the change, so work out which percentage it is first: after 30\% off it is 70\%, and adding 30\% of it back on does not undo the change.
• Taking the percentage off again. The change has already happened, so don't take 30\% off the sale price; work back to the normal price instead.
• Using the percentage taken off as the percentage you have. After 30\% off, the sale price is 70\% of the normal price, not 30\%.
• Multiplying by the multiplier instead of dividing. To go back to the original, divide: \pounds 46\div1.15, not \pounds 46\times1.15.
• Taking a percentage of the change. If a 15\% cut is £6, then £6 is the 15\%: divide by 15, don't find 15\% of £6.
• Not checking the answer makes sense. After a decrease the original must be bigger than the amount given, and after an increase smaller.