Hard GCSE Percentage change Questions

Challenging, exam-style GCSE Percentage change questions with worked solutions. Stretch yourself on the hardest successive percentage change, combined multiplier, percentage profit, mark-up problems.

successive percentage changecombined multiplierpercentage profitmark-upbuilding up cost and revenuecomparing percentage changes
GCSE Foundation34 questionsStep-by-step solutions
Question 1
6 markschallenging
A hotel room costs £80 per night. In August the price rises by 40%. In October the August price falls by 30%. In January the October price falls by a further 25%. Work out the overall percentage change from the original £80.
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Worked solution

  1. Write down the original price

    original=£80\text{original} = \pounds 80

    The overall percentage change is measured against this £80.

  2. August: find the multiplier

    100%+40%=140%=1.4100\% + 40\% = 140\% = 1.4

    A 40% rise means multiplying by 1.4.

  3. August: work out the price

    £80×1.4=£112\pounds 80 \times 1.4 = \pounds 112

    The August price is £112.

  4. October: find the multiplier

    100%−30%=70%=0.7100\% - 30\% = 70\% = 0.7

    A 30% fall means multiplying by 0.7.

  5. October: apply it to the AUGUST price

    £112×0.7=£78.40\pounds 112 \times 0.7 = \pounds 78.40

    The 30% comes off £112 (that is £33.60), not off the original £80. The October price is £78.40.

  6. January: find the multiplier

    100%−25%=75%=0.75100\% - 25\% = 75\% = 0.75

    A further 25% fall means multiplying by 0.75.

  7. January: apply it to the OCTOBER price

    £78.40×0.75=£58.80\pounds 78.40 \times 0.75 = \pounds 58.80

    The January price is £58.80.

  8. Multiply the three multipliers together

    1.4×0.7×0.751.4 \times 0.7 \times 0.75

    Successive percentage changes combine by multiplying their multipliers, never by adding the percentages.

  9. Work out the product in stages

    1.4×0.7=0.98,0.98×0.75=0.7351.4 \times 0.7 = 0.98, \quad 0.98 \times 0.75 = 0.735

    The single equivalent multiplier is 0.735.

  10. Check the single multiplier against the price

    £80×0.735=£58.80\pounds 80 \times 0.735 = \pounds 58.80

    The single multiplier gives the same January price. ✓

  11. Compare the multiplier with 1

    0.735<10.735 < 1

    The multiplier is below 1, so the room is cheaper in January than it was originally.

  12. Turn the multiplier into a percentage change

    (0.735−1)×100=−26.5(0.735 - 1) \times 100 = -26.5

    The overall change is a 26.5% decrease.

  13. Confirm from the prices

    80−58.8080×100=21.2080×100=26.5%\frac{80 - 58.80}{80} \times 100 = \frac{21.20}{80} \times 100 = 26.5\%

    The price has fallen by £21.20, and £21.20 out of the original £80 is 26.5%. ✓

  14. Watch the common error

    +40%−30%−25%=−15%≠−26.5%+40\% - 30\% - 25\% = -15\% \ne -26.5\%

    Adding the percentages gives a 15%15\% fall, which is badly wrong. Notice too that +40%+40\% then −30%-30\% is not +10%+10\%: 1.4×0.7=0.981.4 \times 0.7 = 0.98, already a 2%2\% fall.

  15. State the overall percentage change

    26.5% decrease26.5\% \text{ decrease}

    The January price is 26.5% below the original £80.

Answer
26.5% decrease26.5\% \text{ decrease}
Question 2
5 markschallenging
A tank holds 250 litres of water. On Monday 18% of the water is used. On Tuesday 25% of the water that is left is used. Work out what percentage of the original 250 litres is still in the tank.
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Worked solution

  1. Write down the original amount of water

    original=250 litres\text{original} = 250 \text{ litres}

    Everything at the end must be compared back to this 250 litres.

  2. Find the multiplier for Monday

    100%−18%=82%=0.82100\% - 18\% = 82\% = 0.82

    If 18% is used, 82% is left, so multiply by 0.82.

  3. Work out the water used on Monday

    18% of 250=45 litres18\% \text{ of } 250 = 45 \text{ litres}

    45 litres are used on Monday.

  4. Work out the water left after Monday

    250×0.82=205 litres250 \times 0.82 = 205 \text{ litres}

    250−45=205250 - 45 = 205 litres are left. ✓

  5. Read the Tuesday condition carefully

    25% of the water that is LEFT25\% \text{ of the water that is LEFT}

    Tuesday's 25% is 25% of the 205 litres left, NOT 25% of the original 250 litres.

  6. Find the multiplier for Tuesday

    100%−25%=75%=0.75100\% - 25\% = 75\% = 0.75

    If 25% of what is left is used, 75% of it remains, so multiply by 0.75.

  7. Work out the water used on Tuesday

    25% of 205=51.25 litres25\% \text{ of } 205 = 51.25 \text{ litres}

    51.25 litres are used on Tuesday — not the 62.5 litres that 25% of 250 would be.

  8. Work out the water left after Tuesday

    205×0.75=153.75 litres205 \times 0.75 = 153.75 \text{ litres}

    205−51.25=153.75205 - 51.25 = 153.75 litres remain in the tank.

  9. Combine the multipliers

    0.82×0.75=0.6150.82 \times 0.75 = 0.615

    The two stages together are a single multiplier of 0.615.

  10. Check the combined multiplier

    250×0.615=153.75250 \times 0.615 = 153.75

    The single multiplier reproduces the 153.75 litres left. ✓

  11. Read the percentage straight off the multiplier

    0.615=61.5%0.615 = 61.5\%

    A multiplier of 0.615 means 61.5% of the original water is still there.

  12. Confirm from the litres

    153.75250×100=61.5%\frac{153.75}{250} \times 100 = 61.5\%

    153.75 out of the original 250 litres is 61.5%. ✓

  13. Work out how much has gone

    100%−61.5%=38.5%100\% - 61.5\% = 38.5\%

    So 38.5% of the original water has been used altogether.

  14. Watch the common error

    18%+25%=43%≠38.5%18\% + 25\% = 43\% \ne 38.5\%

    Adding the percentages would suggest 43% used and 57% left. That is wrong, because the 25% is a percentage of the reduced 205 litres.

  15. State the percentage of the original that is left

    61.5%61.5\%

    61.5% of the original 250 litres is still in the tank.

Answer
61.5%61.5\%
Question 3
5 markschallenging
The number of trees in a wood rises by 12% one year and falls by 12% the next year. Work out the overall percentage change in the number of trees.
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Worked solution

  1. Choose a number of trees to work with

    start=10000 trees\text{start} = 10000 \text{ trees}

    The overall percentage change does not depend on the starting number, so pick 10000 to keep the arithmetic exact.

  2. Find the multiplier for the 12% rise

    100%+12%=112%=1.12100\% + 12\% = 112\% = 1.12

    A 12% increase means multiplying by 1.12.

  3. Apply the rise

    10000×1.12=1120010000 \times 1.12 = 11200

    After the first year there are 11200 trees. 1200 trees were added.

  4. Find the multiplier for the 12% fall

    100%−12%=88%=0.88100\% - 12\% = 88\% = 0.88

    A 12% decrease means multiplying by 0.88.

  5. Notice what the fall acts on

    12% of 11200=134412\% \text{ of } 11200 = 1344

    The 12% fall is 12% of the LARGER 11200, so 1344 trees are lost — more than the 1200 that were gained.

  6. Apply the fall

    11200×0.88=985611200 \times 0.88 = 9856

    After the second year there are 9856 trees, fewer than the 10000 at the start.

  7. Combine the multipliers

    1.12×0.88=0.98561.12 \times 0.88 = 0.9856

    The single equivalent multiplier is 0.9856.

  8. Check the combined multiplier

    10000×0.9856=985610000 \times 0.9856 = 9856

    The single multiplier reproduces the final number of trees. ✓

  9. Compare the multiplier with 1

    0.9856<10.9856 < 1

    The multiplier is below 1, so the wood ends up with fewer trees than it started with.

  10. Turn the multiplier into a percentage change

    (0.9856−1)×100=−1.44(0.9856 - 1) \times 100 = -1.44

    The overall change is a 1.44% decrease.

  11. Confirm from the tree counts

    10000−985610000×100=14410000×100=1.44%\frac{10000 - 9856}{10000} \times 100 = \frac{144}{10000} \times 100 = 1.44\%

    144 trees have been lost out of the original 10000, which is 1.44%. ✓

  12. Watch the common error

    +12%−12%≠0%+12\% - 12\% \ne 0\%

    The rise and fall do not cancel, because the fall is a percentage of a bigger number than the rise was.

  13. See where the 1.44% comes from

    0.12×0.12=0.0144=1.44%0.12 \times 0.12 = 0.0144 = 1.44\%

    For an x% rise followed by an x% fall the multiplier is (1 + x)(1 − x) = 1 − x², so the loss is exactly 0.12² = 1.44%.

  14. Check with a different starting number

    2500×1.12×0.88=24642500 \times 1.12 \times 0.88 = 2464

    25002500 trees would become 24642464, a fall of 3636, and 36÷2500×100=1.44%36 \div 2500 \times 100 = 1.44\%. The same answer appears. ✓

  15. State the overall percentage change

    1.44% decrease1.44\% \text{ decrease}

    The wood ends up with 1.44% fewer trees than it had at the start.

Answer
1.44% decrease1.44\% \text{ decrease}
Question 4
6 markschallenging
A shopkeeper buys 60 T-shirts for a total of £480. He sells 45 of them at £12 each and the other 15 at £7 each. Work out the percentage profit.
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Worked solution

  1. Write down the total cost

    cost=£480\text{cost} = \pounds 480

    The shopkeeper pays £480 for all 60 T-shirts. This is the original amount.

  2. Work out the cost of one T-shirt

    £480÷60=£8\pounds 480 \div 60 = \pounds 8

    Each T-shirt costs £8. (Useful for checking, but the percentage profit can be worked out from the totals.)

  3. Work out the money from the full-price T-shirts

    45×£12=£54045 \times \pounds 12 = \pounds 540

    The 45 T-shirts sold at £12 bring in £540.

  4. Check how many T-shirts are left

    60−45=1560 - 45 = 15

    That leaves 15 T-shirts, which matches the question.

  5. Work out the money from the cheaper T-shirts

    15×£7=£10515 \times \pounds 7 = \pounds 105

    The 15 T-shirts sold at £7 bring in £105. Note £7 is BELOW the £8 cost, so these are sold at a loss.

  6. Work out the total money taken

    £540+£105=£645\pounds 540 + \pounds 105 = \pounds 645

    Altogether the shopkeeper takes £645.

  7. Compare the takings with the cost

    £645>£480\pounds 645 > \pounds 480

    He takes more than he paid, so overall there is a profit despite the 15 sold at a loss.

  8. Work out the profit

    £645−£480=£165\pounds 645 - \pounds 480 = \pounds 165

    The overall profit is £165.

  9. Check the profit shirt by shirt

    45×£4−15×£1=£180−£15=£16545 \times \pounds 4 - 15 \times \pounds 1 = \pounds 180 - \pounds 15 = \pounds 165

    Each full-price shirt makes £12 − £8 = £4, and each cheap shirt loses £8 − £7 = £1. The totals agree. ✓

  10. Recall the percentage profit formula

    percentage profit=profitcost price×100\text{percentage profit} = \frac{\text{profit}}{\text{cost price}} \times 100

    The £480 cost is the original amount, so it is the base.

  11. Substitute the numbers

    165480×100\frac{165}{480} \times 100

    Divide the £165 profit by the £480 cost — not by the £645 taken.

  12. Evaluate the division

    165480=0.34375\frac{165}{480} = 0.34375

    The profit is 0.34375 of the cost.

  13. Turn it into a percentage

    0.34375×100=34.375%0.34375 \times 100 = 34.375\%

    The percentage profit is 34.375%, which is exact.

  14. Check with a multiplier

    £480×1.34375=£645\pounds 480 \times 1.34375 = \pounds 645

    A 34.375% profit has multiplier 1.34375, and £480 × 1.34375 = £645. ✓

  15. State the percentage profit

    34.375% profit34.375\% \text{ profit}

    Dividing by the £645 takings instead would give 25.58%, which is not the percentage profit.

Answer
34.375% profit34.375\% \text{ profit}
Question 5
6 markschallenging
In a sale, every price is reduced by 20%. On the final day, a further 30% is taken off the sale price. A customer says: "That is 50% off altogether." Work out the actual overall percentage reduction.
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Worked solution

  1. Write down what the customer claims

    20%+30%=50%  (claim)20\% + 30\% = 50\% \; \text{(claim)}

    The customer has added the two percentages. We will test this.

  2. Choose a price to work with

    original=£100\text{original} = \pounds 100

    The overall percentage reduction is the same for every price, so use £100 to make it clear.

  3. Find the multiplier for the first reduction

    100%−20%=80%=0.8100\% - 20\% = 80\% = 0.8

    A 20% reduction means multiplying by 0.8.

  4. Apply the first reduction

    £100×0.8=£80\pounds 100 \times 0.8 = \pounds 80

    The sale price is £80. So far £20 has come off.

  5. Find the multiplier for the second reduction

    100%−30%=70%=0.7100\% - 30\% = 70\% = 0.7

    A further 30% reduction means multiplying by 0.7.

  6. Notice what the 30% is taken from

    30% of £80=£2430\% \text{ of } \pounds 80 = \pounds 24

    The extra 30% comes off the £80 SALE price, not off the original £100. Only £24 comes off, not £30.

  7. Apply the second reduction

    £80×0.7=£56\pounds 80 \times 0.7 = \pounds 56

    The final-day price is £56.

  8. Combine the multipliers into one

    0.8×0.7=0.560.8 \times 0.7 = 0.56

    The single equivalent multiplier is 0.56.

  9. Check the combined multiplier

    £100×0.56=£56\pounds 100 \times 0.56 = \pounds 56

    The single multiplier gives the same final price. ✓

  10. Read the reduction from the multiplier

    100%−56%=44%100\% - 56\% = 44\%

    A multiplier of 0.56 means the customer pays 56% of the original price, so 44% has come off.

  11. Confirm with the money

    £100−£56=£44\pounds 100 - \pounds 56 = \pounds 44

    £44£44 has come off the original £100£100, and 44÷100×100=44%44 \div 100 \times 100 = 44\%. ✓

  12. Test the claim of 50% off

    £100×0.5=£50≠£56\pounds 100 \times 0.5 = \pounds 50 \ne \pounds 56

    A genuine 50% off would give £50, but the shopper actually pays £56. The claim is wrong.

  13. Explain the error

    20% of £100=£20,30% of £80=£2420\% \text{ of } \pounds 100 = \pounds 20, \quad 30\% \text{ of } \pounds 80 = \pounds 24

    The two percentages are taken from different amounts — £100 and then £80 — so they cannot be added.

  14. Check with a different price

    £250×0.8×0.7=£140\pounds 250 \times 0.8 \times 0.7 = \pounds 140

    £250£250 falls to £140£140, a fall of £110£110, and 110÷250×100=44%110 \div 250 \times 100 = 44\%. The same 44%44\% appears whatever the price. ✓

  15. State the actual overall reduction

    44% reduction44\% \text{ reduction}

    The true overall reduction is 44%, not the 50% the customer claimed.

Answer
44% decrease44\% \text{ decrease}

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