Number
Grade 5-6Reverse Percentages
Reverse percentage questions give the amount after a change and ask for the amount before it, which trips students up because the instinct is to multiply or find a percentage of the wrong number. This lesson covers reversing a percentage increase, reversing a percentage decrease, and VAT problems, which are the most common real-world use of reverse percentages.
What you need to know
- If a price already includes a percentage change, that price is not 100%.
- After a 20% rise the amount is 120%, so divide by 1.2 to get the original.
- After a 15% sale the price is 85%, so divide by 0.85 to find the original.
- Never take the percentage straight off the new amount; that gives the wrong original.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Reversing a percentage increase
If an amount has been increased by a percentage, the given value represents more than 100% of the original. Add the percentage to 100%, convert to a decimal, and divide the given amount by that decimal to find the original.
A number is increased by 20% to give 96. The decimal multiplier for a 20% increase is 1.2, so the original number is 96 ÷ 1.2 = 80.
Reversing a percentage decrease
If an amount has been decreased by a percentage, the given value represents less than 100% of the original. Subtract the percentage from 100%, convert to a decimal, and divide the given amount by that decimal to find the original.
A number is decreased by 15% to give 102. The decimal multiplier for a 15% decrease is 0.85, so the original number is 102 ÷ 0.85 = 120.
VAT and reverse percentages
VAT questions almost always need a reverse percentage, since the price given includes VAT already added, and the question asks for the price before VAT. UK VAT is usually 20%, so the price including VAT represents 120% of the original.
A washing machine costs £258 including 20% VAT. The price before VAT is 258 ÷ 1.2 = £215.
Spotting a reverse-percentage question
The key phrase to look for is a description of a change followed by the result, with the original amount as the unknown: "a number is increased/decreased by X% to give Y" or "a price includes X% VAT and is Y". These always need dividing by a decimal multiplier, not multiplying.
It is easy to mistake this for a normal percentage-of-an-amount question and take X% of the given value instead, which gives the wrong answer because the given value is not the original amount.
Worked examples
Three exam-style questions, fully solved
A number is increased by 20% to give 96. Find the original number.
Easy- 1.Add the percentage to 100%: 100% + 20% = 120%
- 2.Convert to a decimal: 1.2
- 3.Divide the given amount by the decimal: 96 ÷ 1.2
Answer: 80
A washing machine costs £258 including 20% VAT. Find the price before VAT was added.
Medium- 1.Recognise that the price given already includes VAT, so it represents 120% of the original price
- 2.Convert 120% to a decimal: 1.2
- 3.Divide the given price by the decimal: 258 ÷ 1.2
Answer: £215
A water tank's volume decreased by 12% due to evaporation, leaving 440 litres. What was the original volume?
Hard- 1.Subtract the percentage from 100%: 100% - 12% = 88%
- 2.Convert to a decimal: 0.88
- 3.Divide the given amount by the decimal: 440 ÷ 0.88
Answer: 500 litres
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
In a sale, all prices are reduced by 20%. A coat now costs £48. What was the original price?
A bill is £120 including 20% VAT. What was the amount before VAT was added?
After a 15% pay rise, someone earns £23 000. What did they earn before the rise?
A number is decreased by 40% to give 90. What was the original number?
A shop reduces a TV by 30% to £280. What was the original price?
60% of a class are girls. There are 18 girls. How many students are in the class altogether?
Avoid these
The mistakes students make most often
Finding a percentage of the given (final) amount and adding or subtracting it, instead of dividing by the correct decimal multiplier.
Multiplying by the decimal multiplier instead of dividing, which finds a different, incorrect amount rather than reversing the change.
Using a decimal multiplier for the wrong direction, for example dividing by 1.15 for a 15% decrease instead of 0.85, or dividing by 0.85 for a 15% increase instead of 1.15.
Treating a reverse-percentage question as an ordinary percentage-of-an-amount question, since the given number is not the original amount.
With VAT questions, forgetting that the price given already includes VAT, and mistakenly adding VAT again to find the original price.
FAQ
Questions students and parents ask
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