## What are reverse percentages?

In a reverse percentage question, we are given an amount that has recently **increased** or **decreased** by a certain **percent**.

We must then use this information to find out what the **original amount** was.

### Examples

- The price of a t-shirt has
**decreased**by 10% in a sale. It is now selling for £18.

What was the original price of the t-shirt? - The price of a TV has
**increased**by 20% and is now £240. How much was the TV originally?

The first example here involves a decreased amount, whereas the second example asks about an increased amount.

How can we work out the answers to these questions?

## How to find the original amount after a percentage decrease

### Example question

The cost of a t-shirt has been **decreased** by 10% in a sale.

Its new price is £18.

What was its **original price**?

### Step 1

We know that the t-shirt is now at 90% of its original value, because it started at 100% of its value and decreased by 10%.

So, 90% of the original price = £18.

### Step 2

If we can work out 10% of the amount, we can then multiply this by 10 to find 100%.

If 90% = £18, then £18 divided by 9 will give us 10%.

£18/9 = £2

### Step 3

So, 10% = £2. We can multiply this by 10 to find 100%.

£2 x 10 = £20.

In summary, we have found that:

90% of the original amount = £18

10% of the original amount = £2

100% of the original amount = £20.

The answer is £20!

### Top Tip

We could easily find 10% of the amount in this question. For other questions, it may well be necessary to find 1% of the original amount instead, and then multiply this by 100 to find 100%.

In the above example, we could have done this by dividing 18 by 90, to find 1%, and then multiplying 0.2 by 100 to reach 20.

## How to find the original amount after a percentage increase

### Example question

The price of a TV has **increased** by 20% and is now £240.

How much was the TV originally?

### Step 1

We know that the TV is now at 120% of its original value, because it started at 100% of its value and the question tells us that it **increased** by 20%.

So, the 120% of the original price = £240.

### Step 2

If we work out 10% of the original price, then we can find 100% of the original price.

If 120% = £240, we can divide £240 by 12 to find 10%.

220/12 = 20.

### Step 3

So now we know that 10% = £20.

If multiply £20 by 10, we will find the original price of the TV, as this will be 100% of the original value.

20 x 10 = 200.

Final answer: the original price of the TV, before it was increased in price, was **£200**!

### Top Tip

We could also have worked this out by first finding 1% of the original price, and then multiplying this by 100 to find 100%.

In the above example, we could have done this by dividing 240 by 120, to find 1%, and then multiplying 2 by 100 to reach 200.

## Further Practice

For questions on **reverse percentages** in expert 11+ Maths practice exam papers, take a look at the papers in our shop: