Reverse percentages: finding the original value

A reverse percentage question gives you the result and asks what you started with: what is the whole when 20% of it is 35, or what did the scooter cost before VAT was added. The usual hurdle is the temptation to take the percentage off again, which does not work, so we go through why.

When do I use this?

There are two versions, and both are in the list of percentage problem types in what is a percentage?

• You know what a certain percentage of the whole is, and you want the whole (100%).
• You know a value after a percentage has been added, such as a price with VAT, and you want the value before.

Finding the whole from a percentage

Let's say we know that 20% is 35, and we want the whole. We start by finding 1%, and then we can multiply by 100 to get the full number, that is 100%, the whole. We find 1% by dividing by 20, because we have 20%:

$$\frac{35}{20} \times 100 = 175$$

So the whole is 175. More generally we can say: if $x\%$ corresponds to $y$, how big is the whole?

$$\frac{y}{x} \times 100$$

It is the same 1% stepping stone as in finding a percentage of an amount, just used in the other direction. There we divided the whole by 100 to get 1%. Here we divide the part by its number of per cent to get 1%.

Finding the price before VAT

Let's say a scooter costs £240 including VAT, and we would like to find out what the price was before VAT. In the UK, VAT is 20%. (my Danish notes use 25% here, because Danish VAT is 25%. The reasoning is the same.)

You might be tempted to just take 20% off the £240, but unfortunately that is not right. What we know is that 20% was added to some other amount, and that gave £240. If we call that amount $x$, we can write it as an equation:

$$x + 20\% = 240$$

What it says is: "some amount, plus 20% of it, gives £240". Because adding 20% is the same as multiplying by 1.2, we can write it as

$$x \times 1.2 = 240$$

When we solve this equation, we find the price before VAT. We divide by 1.2 to get $x$ on its own:

$$x = \frac{240}{1.2} = 200$$

So the scooter cost £200 before VAT. We can quickly check by adding 20% to this amount and seeing whether it gives 240:

$$200 \times 1.2 = 240$$

It does.

Setting the equation up is a slightly long-winded way of doing it, and the point of writing it out is to give an understanding of what is going on. Once you see it, the shortcut is simply: the price with VAT is the price without VAT times 1.2, so to go back you divide by 1.2.

Why taking 20% off does not work

Look at what happens if we take 20% off the £240 anyway. 20% of 240 is 48, and $240 - 48 = 192$, not 200. The 20% that was added was 20% of £200, which is £40. The 20% we took off was 20% of £240, which is £48. Same percentage, different whole, different amount.

So when we find the price with VAT, we add 20%, but when we find the price without VAT, we do not take 20% off. In my Danish notes, where VAT is 25%, the shortcut going back is to take 20% off, and that gap between 25 and 20 shows how lopsided the two directions are. With UK VAT the clean move is to divide by 1.2.

Common mistakes

• Taking the percentage off the final value. The classic one. 20% was added to the original price, so you must undo a multiplication by 1.2, which means dividing by 1.2. Taking 20% of the bigger number off removes too much.
• Forgetting to check. A reverse percentage is easy to check: run it forwards. If your original price times 1.2 lands on the price you were given, you are right. If not, something slipped.
• Mixing up the two versions. "20% of the whole is 35" is solved by finding 1% and multiplying by 100. "£240 after 20% was added" is solved by dividing by 1.2. In the first, 35 is a part of the whole. In the second, 240 is the whole plus a bit more.

Related

• What is a percentage?
• Percentage increase and decrease with a multiplier
• How to find a percentage of an amount
• What is an equation?