Reverse percentages

The method

1. Decide what percentage your amount represents: 100% plus the increase, or 100% minus the decrease.
2. Write that percentage as a decimal multiplier, for example 1.25 for a 25% increase or 0.85 for a 15% decrease.
3. Divide the amount by the multiplier to find the original amount.
4. Check by applying the percentage change to your answer and seeing whether you get the amount in the question.

Mistakes that lose marks

• Taking 20% of the final price instead of dividing by 0.8, which does not undo the reduction.
• Using 0.25 for a 25% increase. The multiplier is 1.25, and 0.25 only finds the increase itself.
• Adding 20% VAT to a price that already includes VAT.
• Rounding the original amount before using it to find a second quantity such as the VAT or the profit.
• Treating a profit of 15% on the cost price as 15% of the selling price, which gives the wrong cost price.

7 exam-style questions

These are our own questions written in the style of the exam, not past-paper questions. The mark codes are M = method, A = accuracy and B = independent mark. Try each one, then open the solution.

1. Exam-style core In a sale a jacket is reduced by 20% and now costs £72. Find the price of the jacket before the sale. [3 marks]
   Show full solution

   1. A 20% reduction leaves 80%, so £72 is 80% of the original price M1
   2. Original price = 72 ÷ 0.8 M1
   3. = £90.00 A1

   Answer: £90.00

2. Exam-style core After a pay rise of 6%, a worker earns £477 per week. Find the weekly pay before the rise. [3 marks]
   Show full solution

   1. £477 is 106% of the old pay, so the multiplier is 1.06 M1
   2. Old pay = 477 ÷ 1.06 M1
   3. = £450.00 A1

   Answer: £450.00

3. Exam-style extended A shop reduces a bicycle by 15% in a sale and sells it for £357. Find the price of the bicycle before the sale. [3 marks]
   Show full solution

   1. A 15% reduction leaves 85%, so £357 is 85% of the original price M1
   2. Original price = 357 ÷ 0.85 M1
   3. = £420.00 A1

   Answer: £420.00

4. Exam-style extended The price of a laptop includes 20% VAT and is £486. Find the price before VAT and the amount of VAT charged. [4 marks]
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   1. £486 is 120% of the price before VAT, so the multiplier is 1.2 M1
   2. Price before VAT = 486 ÷ 1.2 M1
   3. = £405.00 A1
   4. VAT = 486 − 405 = £81.00 A1

   Answer: Price before VAT £405.00, VAT £81.00

5. Exam-style extended A shop sells a chair for £138 and makes a profit of 15% on the cost price. Find the cost price of the chair and the profit. [4 marks]
   Show full solution

   1. The selling price is 115% of the cost price, so the multiplier is 1.15 M1
   2. Cost price = 138 ÷ 1.15 M1
   3. = £120.00 A1
   4. Profit = 138 − 120 = £18.00 A1

   Answer: Cost price £120.00, profit £18.00

6. Exam-style extended In a sale a television is reduced by 20%. The sale price is then reduced by a further 10%, and the final price is £432. (a) Find the original price. (b) Find the overall percentage reduction. [5 marks]
   Show full solution

   1. The two reductions give a single multiplier: 0.8 × 0.9 = 0.72 M1
   2. Original price = 432 ÷ 0.72 M1
   3. (a) = £600.00 A1
   4. (b) The final price is 0.72 of the original, so the reduction is 1 − 0.72 = 0.28 M1
   5. (b) = 28% A1

   Answer: (a) £600.00 (b) 28%

7. Exam-style higher The value of a painting fell by 10% in the first year and then rose by 5% in the second year. After the two changes it was worth £15,120. (a) Find the value before the two changes. (b) Find the overall percentage change in the value. [5 marks]
   Show full solution

   1. The two changes give a single multiplier: 0.9 × 1.05 = 0.945 M1
   2. Value before the changes = 15,120 ÷ 0.945 M1
   3. (a) = £16,000.00 A1
   4. (b) 0.945 = 1 − 0.055, so the value is 5.5% lower than at the start M1
   5. (b) 5.5% decrease A1

   Answer: (a) £16,000.00 (b) 5.5% decrease

Where this is in the syllabus

Reverse percentages appears in Cambridge IGCSE Mathematics 0580 and Pearson Edexcel International GCSE Mathematics A 4MA1. For the official specifications and past papers, use the board's own website: Cambridge IGCSE Mathematics 0580 and Pearson Edexcel International GCSE Mathematics A 4MA1.

Questions students ask