How To Calculate Reverse Percentages

To reverse a percentage, divide the final value by the percentage it represents, then multiply by 100 to get the original value. Equivalently, divide by the decimal multiplier: after a 20% discount you are paying 80% of the original price, so divide the sale price by 0.8. The same method works backwards from an increase, such as removing VAT from a price.

What are reverse percentages?

A reverse percentage (sometimes called back-calculating a percentage) works out the original price or value of something after a percentage discount or increase has been applied. You know the final figure and the percentage change, and the original value is the unknown.

You’re most likely to come across reverse percentages in numerical reasoning tests, as they examine your general aptitude for numbers, data and mathematical principles, often through questions about discounted prices, tax or changing values.

The reverse percentage formula

Original value = final value ÷ percentage the final value represents x 100

Or, using the decimal multiplier:

Original value = final value ÷ multiplier

• After a decrease of x%, the final value represents (100 - x)% of the original. A 20% discount means you divide by 80 and multiply by 100, or simply divide by 0.8.
• After an increase of x%, the final value represents (100 + x)% of the original. A price including 20% VAT is 120% of the net price, so divide by 120 and multiply by 100, or divide by 1.2.

Key points to remember

1. Always start by working out what percentage of the original the final value represents: 100 minus the decrease, or 100 plus the increase.

2. Don’t just take the percentage of the final value. Adding 20% back onto a price that was reduced by 20% does not return the original, because the 20% was taken from a larger number.

3. If you’re working without a calculator, find the value of 1% first (divide the final value by the percentage it represents), then multiply by 100. Breaking values into their 1%, 10% or 25% form makes the mental arithmetic easier.

4. Sense-check your answer by working forwards: apply the original percentage change to your answer and you should land back on the figure given in the question.

Step 1: Start at 100% and minus the figure you've been given

The first thing you need to do is start at 100% and minus the figure you’ve been given.

So, if you’ve been asked to find out what the original price of an item is in a sale where everything is 20% off, you’d do: 100 - 20 = 80.

This means that you’re only paying 80% of the original price in a 20% off sale.

Step 2: Find the value of 1%

If you know that the sale item you’re purchasing is £160, you know that £160 is 80% of the original cost.

Next, you’ll need to divide the sale price by the percentage of the original price to find the value of 1%: 160 ÷ 80 = 2.

Step 3: Multiply 1% by 100 to get the value of 100%

Then, multiply 2 (or 1%) by 100 to get the value of 100% (or the original price). So, 2 x 100 = 200.

This means that the original price of the sale item was £200, and in a 20% off sale it’s now £160.

Example questions

Question 1: original price after a discount

Sarah buys a scarf in the sale. It’s marked as 75% off and now costs £20. How does Sarah work out the original cost of the scarf?

1. The scarf has been reduced by 75%, so the sale price of £20 represents 100% - 75% = 25% of the original price.

2. Divide the sale price by the percentage it represents to find the value of 1%: 20 ÷ 25 = 0.80

3. Multiply the value of 1% by 100 to find the original price: 0.80 x 100 = £80. The original price of the scarf was £80.

Question 2: original price after an increase

There’s been a 10% rise in prices in Mark’s local coffee shop. If an Americano now costs Mark £2.75, how much did it cost before the increase?

1. Prices have risen by 10%, so the new price represents 100% + 10% = 110% of the old price, and 110% is £2.75.

2. Work out the value of 1%: 2.75 ÷ 110 = 0.025

3. Multiply by 100: 0.025 x 100 = 2.50. Mark’s Americano cost £2.50 before the 10% price increase.

Question 3: taking VAT off a price

A laptop costs £660 including VAT at 20%. What is the price before VAT?

1. Including 20% VAT, the price represents 100% + 20% = 120% of the net price.

2. Using the multiplier method, divide by 1.2: 660 ÷ 1.2 = £550. The price before VAT is £550.

3. Check by working forwards: 550 x 1.2 = 660, which matches the price in the question.

Question 4: a test-style decrease

A company’s headcount fell by 15% over the year to 2,040 employees. How many employees did it have at the start of the year?

1. After a 15% fall, 2,040 represents 100% - 15% = 85% of the original headcount.

2. Find the value of 1%: 2,040 ÷ 85 = 24

3. Multiply by 100: 24 x 100 = 2,400. The company started the year with 2,400 employees.

Practice reverse percentage questions

Reverse percentages appear regularly in employer numerical tests, usually wrapped in financial data such as discounted revenue or prices including tax. To build speed and accuracy, work through our numerical reasoning tests and basic numeracy tests, or start with a selection of free aptitude tests.