Reverse Percentages

Reverse percentage questions ask you to work backwards — you are given the amount after a percentage change and must find the original amount before the change happened. These questions appear at Foundation and Higher tier on AQA, Edexcel and OCR papers, and they catch out a surprising number of students who try to simply reverse the percentage. The trick is understanding that you cannot just subtract or add the same percentage — you need to use the multiplier method in reverse. This guide explains exactly how to do it, step by step.

What Are Reverse Percentages?

A reverse percentage problem gives you a value that has already been increased or decreased by a given percentage, and asks you to find the original value before the change.

Why you cannot just reverse the percentage

Suppose a price is increased by 20% to give £60. You might think: "20% of £60 is £12, so the original was £48." But this is wrong. The 20% was applied to the original price, not to £60.

The correct approach uses the multiplier:

A 20% increase means the new value is 120% of the original, i.e. the multiplier is 1.20.

So: Original = £60 ÷ 1.20 = £50

Check: £50 × 1.20 = £60 ✓

The general formula

For a percentage increase of r%:

For a percentage decrease of r%:

Step-by-Step Method

1. Identify the percentage change and whether it is an increase or a decrease.
2. Calculate the multiplier:
   • Increase of r% → multiplier = 1 + r/100
   • Decrease of r% → multiplier = 1 − r/100
3. Divide the given (new) value by the multiplier.
4. State the original value.
5. Check by applying the percentage change to your answer — you should get back to the given value.

Alternative: the "percentage bar" approach

Some students prefer to think of it as:

• After a 20% increase, the new amount represents 120% of the original.
• Find 1% by dividing the new amount by 120.
• Find 100% by multiplying by 100.

Both methods give the same answer.

Worked Example 1 — Foundation Level

Question: In a sale, a jacket is reduced by 30%. The sale price is £56. What was the original price?

Step 1: This is a 30% decrease. Multiplier = 1 − 0.30 = 0.70.

Step 2: The sale price represents 70% of the original.

Step 3: Original price = £56 ÷ 0.70 = £80

Check: £80 × 0.70 = £56 ✓

Common wrong answer: £56 + 30% of £56 = £56 + £16.80 = £72.80 — this is incorrect because 30% should be of the original price, not the sale price.

Worked Example 2 — Higher Level

Question: A company's profits increased by 12% this year to £89,600. What were the profits last year?

Step 1: This is a 12% increase. Multiplier = 1 + 0.12 = 1.12.

Step 2: This year's profit represents 112% of last year's.

Step 3: Last year's profit = £89,600 ÷ 1.12 = £80,000

Check: £80,000 × 1.12 = £89,600 ✓

Common Mistakes

• Adding or subtracting the percentage of the new value. This is the single most common error. Remember: the percentage was applied to the original, not the new value. You must divide by the multiplier, not add/subtract a percentage of what you have been given.
• Using the wrong multiplier. For a decrease, the multiplier is less than 1 (e.g., 0.70 for a 30% decrease). For an increase, it is greater than 1 (e.g., 1.12 for a 12% increase). Mixing these up reverses the direction.
• Confusing the question type. Not every percentage question is a reverse percentage. If the question gives you the original and asks for the new value, that is a standard percentage increase/decrease. Reverse percentage starts from the result and asks for the original.
• Not checking the answer. Always multiply your original by the multiplier to verify you get back to the number in the question.

Exam Tips

1. Identify reverse percentage questions by looking for phrases like "after an increase of ...", "the sale price is ...", "after depreciation the value is ...". These all signal that you have the new value and need the original.
2. Write the multiplier explicitly. On AQA and Edexcel mark schemes, identifying the correct multiplier earns a method mark even if your division is wrong.
3. For VAT questions, if a price includes 20% VAT, the multiplier is 1.20. To find the price before VAT, divide by 1.20. This is a very common exam context.
4. Show your check. While not always required, writing "Check: £80 × 0.70 = £56 ✓" demonstrates confidence and can earn a quality-of-written-communication mark.

Practice Questions

Question 1 (Foundation): A TV costs £540 after a 10% increase. What was the original price?

Question 2 (Foundation/Higher): A shop reduces all prices by 25%. A pair of trainers now costs £63. Find the original price.

Question 3 (Higher): After two years of depreciation at 15% per year, a car is worth £12,138.75. What was the car's original value?

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Related Topics

• Percentages: Increase and Decrease
• Ratio Basics and Sharing
• Direct and Inverse Proportion
• Fractions, Decimals and Percentages

Summary

• Reverse percentages find the original value before a percentage increase or decrease.
• You cannot simply add or subtract the percentage of the new value — the percentage was applied to the original.
• Use the multiplier method: divide the new value by the multiplier.
• For an increase of r%: multiplier = 1 + r/100. For a decrease: multiplier = 1 − r/100.
• Original = New value ÷ Multiplier.
• Always check by multiplying your answer by the multiplier to get back to the given value.
• Common contexts include sales, VAT, profit/loss and depreciation.