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Hard Percentage Worksheet

KS3-GCSE / Year 9-11Hard25 questions

Free to print and use in your classroom. No sign-up required.

Key formulae

Compound interest / growth: A = P(1 + r/100)^n

Depreciation / decay: A = P(1 − r/100)^n

Reverse percentage: Original = Final value ÷ multiplier

P = starting value, r = rate (%), n = number of years, A = final amount

These questions cover all percentage topics including reverse percentages, compound interest, and percentage change. Show full working for each question.

Section A: Percentage increase and decrease with multipliers

  1. 1. Increase £340 by 15%. What multiplier do you use? What is the result?

  2. 2. Decrease 860 by 35%. Write down the multiplier and calculate the result.

  3. 3. A salary of £28,000 increases by 4.5%. Find the new salary.

  4. 4. A price of £76 is reduced by 12.5%. Find the new price.

  5. 5. A value is increased by 20% and then decreased by 20%. What is the overall percentage change? Use a starting value of £100 to show your working.

  6. 6. A shop increases its prices by 8% in January, then reduces them by 5% in June. What is the overall percentage change to 2 decimal places?

  7. 7. The value of a house increases from £180,000 to £216,000. Calculate the percentage increase.

Section B: Reverse percentages

  1. 8. After a 20% increase, a price is £660. Find the original price.

  2. 9. After a 15% reduction, a price is £272. Find the original price.

  3. 10. An item costs £94 after VAT at 17.5% is added. Find the pre-VAT price.

  4. 11. After a 6% pay rise, a worker earns £37,100 per year. What was the original salary?

  5. 12. A coat is sold for £112.50 after a 25% discount. What was the original price?

  6. 13. After two successive increases of 10%, a value is £6,050. Find the original value.

Section C: Compound interest and depreciation

  1. 14. £2,500 is invested at 4% compound interest per year for 3 years. Find the total amount to the nearest penny.

  2. 15. A car worth £16,000 depreciates at 18% per year. Find its value after 2 years to the nearest penny.

  3. 16. £800 is invested at 5.5% compound interest per year for 4 years. Find the total amount to the nearest penny.

  4. 17. A machine worth £30,000 depreciates at 12% per year for 3 years. Find its value to the nearest penny.

  5. 18. Compare: £6,000 at 3% simple interest for 4 years versus £6,000 at 3% compound interest for 4 years. What is the difference to the nearest penny?

  6. 19. A house is worth £220,000 and increases in value at 2.5% per year compound. Find its value after 5 years to the nearest pound.

Section D: Mixed word problems

  1. 20. A television costs £480. It is reduced by 15% in a sale. A customer then uses a voucher for a further 10% off the sale price. How much does the customer pay?

  2. 21. A worker earns £32,000 per year. They receive a 3% pay rise, then a further 2% rise the following year. What is their salary after both rises?

  3. 22. In 2022, a town had a population of 45,000. The population fell by 4% in 2023 and then increased by 6% in 2024. What was the population at the end of 2024, to the nearest whole number?

  4. 23. A bank account has £5,200 after interest is added at 4% per year compound for 2 years. How much was originally invested? Give your answer to the nearest penny.

  5. 24. A laptop costs £960 including VAT at 20%. An electrician can reclaim the VAT. How much does the electrician effectively pay?

  6. 25. A share is worth £3.60. It rises by 25% in March and then falls by 20% in April. What is the final value of the share? Has it returned to its original value? Explain why or why not.

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Worked Answers

Section A: Percentage increase and decrease with multipliers

  1. 1. Multiplier = 1.15; Result = £391

    £340 × 1.15 = £391

  2. 2. Multiplier = 0.65; Result = 559

    860 × 0.65 = 559

  3. 3. New salary = £29,260

    £28,000 × 1.045 = £29,260

  4. 4. New price = £66.50

    £76 × 0.875 = £66.50

  5. 5. Overall change = -4% (a decrease)

    £100 × 1.20 = £120; £120 × 0.80 = £96. Overall change = -4%. The two changes do not cancel out because the 20% decrease is applied to a larger value.

  6. 6. Overall increase = 2.60%

    Combined multiplier: 1.08 × 0.95 = 1.026. This represents a 2.60% increase.

  7. 7. Percentage increase = 20%

    Change = £216,000 − £180,000 = £36,000. Percentage = (36,000 ÷ 180,000) × 100 = 20%

Section B: Reverse percentages

  1. 8. Original price = £550

    After a 20% increase, the price is 120% of original. Original = £660 ÷ 1.20 = £550

  2. 9. Original price = £320

    After a 15% reduction, the price is 85% of original. Original = £272 ÷ 0.85 = £320

  3. 10. Pre-VAT price = £80

    With VAT at 17.5%, the price is 117.5% of original. Original = £94 ÷ 1.175 = £80

  4. 11. Original salary = £35,000

    After a 6% rise, the salary is 106% of original. Original = £37,100 ÷ 1.06 = £35,000

  5. 12. Original price = £150

    After a 25% discount, the price is 75% of original. Original = £112.50 ÷ 0.75 = £150

  6. 13. Original value = £5,000

    Two increases of 10%: combined multiplier = 1.10 × 1.10 = 1.21. Original = £6,050 ÷ 1.21 = £5,000

Section C: Compound interest and depreciation

  1. 14. Total amount = £2,812.16

    A = 2500 × 1.04³ = 2500 × 1.124864 = £2,812.16

  2. 15. Value after 2 years = £10,763.20

    A = 16000 × 0.82² = 16000 × 0.6724 = £10,758.40. More precisely: 16000 × 0.82 = 13120; 13120 × 0.82 = £10,758.40

  3. 16. Total amount = £990.47

    1.055&sup4; = 1.055 × 1.055 = 1.113025; × 1.055 = 1.174241; × 1.055 = 1.238825; A = 800 × 1.238825 = £991.06. (More precisely: 800 × 1.055^4 = 800 × 1.238825 = £991.06)

  4. 17. Value after 3 years = £20,347.78

    A = 30000 × 0.88³. 0.88² = 0.7744; 0.88³ = 0.681472; A = 30000 × 0.681472 = £20,444.16

  5. 18. Compound interest gives £54.81 more

    Simple: 6000 + (6000 × 0.03 × 4) = 6000 + 720 = £6,720. Compound: 6000 × 1.03&sup4; = 6000 × 1.125509 = £6,753.19. Difference = £6,753.19 − £6,720 = £33.19

  6. 19. Value after 5 years = £248,914

    1.025&sup5; = 1.025 × 1.025 = 1.050625; × 1.025 = 1.076891; × 1.025 = 1.103813; × 1.025 = 1.131408; A = 220000 × 1.131408 = £248,910 (to nearest pound)

Section D: Mixed word problems

  1. 20. Customer pays £367.20

    Sale price: £480 × 0.85 = £408. After voucher: £408 × 0.90 = £367.20

  2. 21. Salary after both rises = £33,971.20

    After first rise: £32,000 × 1.03 = £32,960. After second rise: £32,960 × 1.02 = £33,619.20

  3. 22. Population at end of 2024 = 45,842 (to nearest whole number)

    End of 2023: 45,000 × 0.96 = 43,200. End of 2024: 43,200 × 1.06 = 45,792

  4. 23. Original investment = £4,807.69 (to nearest penny)

    £5,200 = P × 1.04² = P × 1.0816. P = £5,200 ÷ 1.0816 = £4,807.69

  5. 24. Electrician pays £800

    Pre-VAT price: £960 ÷ 1.20 = £800

  6. 25. Final value = £3.60; it has returned to its original value but this is a coincidence of these specific numbers.

    £3.60 × 1.25 = £4.50; £4.50 × 0.80 = £3.60. The value is the same as the start. This happens because 1.25 × 0.80 = 1.00 exactly. In general, a 25% rise followed by a 20% fall gives an overall change of 0% (the multipliers cancel). This is different from a 20% rise then 20% fall, which gives −4%.

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