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Reverse Percentages

FoundationHigherAQAEdexcelOCR

Get to grips with reverse percentages using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on solving reverse percentage problems, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. Treat what you have as the percentage left after the change, then scale back to 100%.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Reverse percentage questions give you the amount after a percentage change and ask for the amount before it. The wording is often a sale price, a price including VAT, or a salary after a rise.

The key insight is that the number you are given does not represent 100 percent. If a price has been reduced by 20 percent, the price you can see is 80 percent of the original, so dividing by 100 would be meaningless.

The reliable method is to divide by the multiplier that produced the change. Because the change multiplied the original, dividing undoes it. Students who instead find a percentage of the new amount get a subtly wrong answer, which is exactly what the question is testing.

Revision notes

Identifying the multiplier

First decide what percentage the amount you are given represents. After a 20 percent decrease it is 80 percent, so the multiplier was \(0.8\). After a 15 percent increase it is 115 percent, so the multiplier was \(1.15\).

Write this down before calculating anything. Choosing the multiplier correctly is the whole question.

Dividing to reverse the change

The original amount was multiplied by the multiplier to give the new amount, so divide the new amount by the multiplier to get back.

If a coat costs \(£48\) after a 20 percent reduction, then \(48 \div 0.8 = 60\), so the original price was \(£60\). Checking is easy: 20 percent of \(£60\) is \(£12\), and \(60 - 12 = 48\).

Spotting a reverse percentage question

Look for words such as after, now, including or has been reduced to. If the percentage change has already happened to the number you are given, the question is a reverse percentage.

The giveaway is that the answer must be found before the change, not after it.

Key points

  • The amount given is not 100 percent.
  • Work out what percentage the given amount represents.
  • Divide by the multiplier to reverse the change.
  • After a \(p\) percent decrease, divide by \(1 - \frac{p}{100}\).
  • After a \(p\) percent increase, divide by \(1 + \frac{p}{100}\).
  • Always check by applying the change forwards to your answer.

Worked examples

Example 1

A jacket costs \(£63\) after a 10 percent reduction. Find the original price.

Working

\[100\% - 10\% = 90\%\]the sale price represents 90 percent of the original
\[63 \div 0.9\]divide by the multiplier 0.9 to reverse the decrease
\[= £70\]the original price was £70

Example 2

A salary is \(£26\,450\) after a 15 percent rise. Find the salary before the rise.

Working

\[100\% + 15\% = 115\%\]the new salary represents 115 percent of the old one
\[26\,450 \div 1.15\]divide by the multiplier 1.15
\[= £23\,000\]the salary before the rise was £23 000

Example 3

A bill is \(£90\) including VAT at 20 percent. Find the bill before VAT.

Working

\[100\% + 20\% = 120\%\]the total includes the VAT, so it is 120 percent
\[90 \div 1.2\]divide by 1.2 to remove the VAT
\[= £75\]the bill before VAT was £75

Common mistakes

  • Finding the percentage of the new amount and adding it back on.

    Adding 20 percent of £48 gives £57.60, not £60. The percentage was taken from the original, which is larger, so this always undershoots.

  • Dividing by the percentage itself rather than the multiplier.

    Dividing £48 by 0.2 gives £240. The multiplier is 0.8, not 0.2, because 80 percent remains.

  • Treating the given amount as 100 percent.

    The whole point of the question is that the change has already happened, so the given figure is 80 or 115 percent, not 100.

  • Using 1.2 to reverse a 20 percent decrease.

    A decrease uses 0.8. Mixing up the increase and decrease multipliers reverses the wrong change.

Exam tips

  • Write down what percentage the given amount represents before doing any arithmetic.
  • Divide, never multiply, when reversing a percentage change.
  • Check your answer by applying the original change forwards.
  • Watch for the words after, now and including, which signal a reverse percentage.

Key terms

Reverse percentage
Finding the original amount when only the amount after a percentage change is known.
Multiplier
The decimal that produced the change, such as 0.8 for a 20 percent decrease.
VAT
Value Added Tax, a percentage added to the price of many goods and services.
Original amount
The value before the percentage change took place.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.