Reverse percentages Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Reverse percentages questions. See exactly how to solve problems on reverse percentage, decrease multiplier, sale price, quarter off.

reverse percentagedecrease multipliersale pricequarter offincrease multiplierVAT
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
A coat is reduced by 20% in a sale. Its sale price is £48. Work out the price of the coat before the sale.

Worked solution

  1. Decide what the given amount stands for

    £48=80% of the original\pounds 48 = 80\% \text{ of the original}

    The 20% was taken off the original, not taken off £48. So £48 is 80% of the original — this is a reverse percentage problem, and the original is the number we have to find.

  2. Divide by the multiplier

    original=£48÷0.8=£60\text{original} = \pounds 48 \div 0.8 = \pounds 60

    80% as a decimal multiplier is 0.8. The original was multiplied by 0.8 to get £48, so dividing by 0.8 undoes it and gives £60.

  3. Check by working forwards

    £60×0.8=£48\pounds 60 \times 0.8 = \pounds 48

    The round-trip check: £60 with the change applied gives £48, exactly as the question says. (Doing £48 ×\times 1.2 = £57.60 instead would be wrong — the 20% was a percentage of the original, not of £48.)

Answer
£60\pounds 60
Question 2
1 markeasy
A book is reduced by 10% in a sale. The sale price is £45. Work out the original price of the book.

Worked solution

  1. Decide what the given amount stands for

    £45=90% of the original\pounds 45 = 90\% \text{ of the original}

    The 10% was taken off the original, not taken off £45. So £45 is 90% of the original — this is a reverse percentage problem, and the original is the number we have to find.

  2. Divide by the multiplier

    original=£45÷0.9=£50\text{original} = \pounds 45 \div 0.9 = \pounds 50

    90% as a decimal multiplier is 0.9. The original was multiplied by 0.9 to get £45, so dividing by 0.9 undoes it and gives £50.

  3. Check by working forwards

    £50×0.9=£45\pounds 50 \times 0.9 = \pounds 45

    The round-trip check: £50 with the change applied gives £45, exactly as the question says. (Doing £45 ×\times 1.1 = £49.50 instead would be wrong — the 10% was a percentage of the original, not of £45.)

Answer
£50\pounds 50
Question 3
2 markseasy
In a sale, all prices are reduced by 25%. A kettle costs £36 in the sale. Work out the price of the kettle before the sale.

Worked solution

  1. Decide what the given amount stands for

    £36=75% of the original\pounds 36 = 75\% \text{ of the original}

    The 25% was taken off the original, not taken off £36. So £36 is 75% of the original — this is a reverse percentage problem, and the original is the number we have to find.

  2. Divide by the multiplier

    original=£36÷0.75=£48\text{original} = \pounds 36 \div 0.75 = \pounds 48

    75% as a decimal multiplier is 0.75. The original was multiplied by 0.75 to get £36, so dividing by 0.75 undoes it and gives £48.

  3. Check by working forwards

    £48×0.75=£36\pounds 48 \times 0.75 = \pounds 36

    The round-trip check: £48 with the change applied gives £36, exactly as the question says. (Doing £36 ×\times 1.25 = £45 instead would be wrong — the 25% was a percentage of the original, not of £36.)

Answer
£48\pounds 48
Question 4
2 markseasy
The price of a train ticket is increased by 10%. The new price is £55. Work out the price of the ticket before the increase.

Worked solution

  1. Decide what the given amount stands for

    £55=110% of the original\pounds 55 = 110\% \text{ of the original}

    The 10% was added on to the original, not added on to £55. So £55 is 110% of the original — this is a reverse percentage problem, and the original is the number we have to find.

  2. Divide by the multiplier

    original=£55÷1.1=£50\text{original} = \pounds 55 \div 1.1 = \pounds 50

    110% as a decimal multiplier is 1.1. The original was multiplied by 1.1 to get £55, so dividing by 1.1 undoes it and gives £50.

  3. Check by working forwards

    £50×1.1=£55\pounds 50 \times 1.1 = \pounds 55

    The round-trip check: £50 with the change applied gives £55, exactly as the question says. (Doing £55 ×\times 0.9 = £49.50 instead would be wrong — the 10% was a percentage of the original, not of £55.)

Answer
£50\pounds 50
Question 5
2 markseasy
A printer costs £96 including VAT at 20%. Work out the price of the printer before VAT was added.

Worked solution

  1. Decide what the given amount stands for

    £96=120% of the original\pounds 96 = 120\% \text{ of the original}

    The 20% was added on to the original, not added on to £96. So £96 is 120% of the original — this is a reverse percentage problem, and the original is the number we have to find.

  2. Divide by the multiplier

    original=£96÷1.2=£80\text{original} = \pounds 96 \div 1.2 = \pounds 80

    120% as a decimal multiplier is 1.2. The original was multiplied by 1.2 to get £96, so dividing by 1.2 undoes it and gives £80.

  3. Check by working forwards

    £80×1.2=£96\pounds 80 \times 1.2 = \pounds 96

    The round-trip check: £80 with the change applied gives £96, exactly as the question says. (Doing £96 ×\times 0.8 = £76.80 instead would be wrong — the 20% was a percentage of the original, not of £96.)

Answer
£80\pounds 80

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