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Percentage Problem Solving Worksheet

KS3-Higher GCSE | Years 9-11Challenging20 questions

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These problems require more than one step. Choose your own method and show all working clearly.

  1. 1. A number is increased by 20%, then the result is increased by 25%. By what single percentage has the original number increased overall?

  2. 2. An original price is reduced by 30%, then reduced by a further 20%. What is the equivalent single percentage reduction?

  3. 3. A car depreciates by 25% in year 1 and 15% in year 2. Starting at £20,000, find its value after 2 years and calculate the total percentage decrease.

  4. 4. A value is increased by 40%. By what percentage must it then be decreased to return to the original value? Give your answer to 1 decimal place.

  5. 5. Two shops sell the same television. Shop A sells it for £800 after a 20% reduction. Shop B sells it for £860 after a 25% reduction. Which shop had the higher original price?

  6. 6. Alex invests £10,000. For the first 3 years the account pays 4% compound interest per year. For the next 2 years it pays 3% compound interest per year. Find the total amount at the end of 5 years.

  7. 7. A population grows at 3% per year. How many complete years does it take for the population to grow by more than 50%?

  8. 8. A shop reduces prices by 15%, then increases them by 15%. A customer says prices are back to the original. Show this is incorrect and find the actual percentage difference from the original.

  9. 9. Three friends each invest £5,000. Anna receives 5% compound interest per year. Beth receives 4.5% compound interest per year. Carl receives 7% simple interest per year. After 3 years, who has the most money? Show all working.

  10. 10. A price is increased by x%, and then the result is decreased by x%. Using algebra, show that the final price is always less than the original, unless x = 0.

  11. 11. A house is worth £320,000. It increases by 5% in year 1, decreases by 3% in year 2, and increases by 7% in year 3. Find the value to the nearest pound at the end of year 3.

  12. 12. An estate agent charges 1.5% commission on property sales. In one month they sell properties worth £4.8 million in total. How much commission do they earn?

  13. 13. A machine produces 2,400 items per day. 2.5% are faulty. After the machine is serviced, the fault rate falls by 40%. How many faulty items are produced per day after the service?

  14. 14. An investment grew to £12,597.12 after 3 years at 8% compound interest per year. Find the original amount invested.

  15. 15. A school's budget is cut by 12.5%. The head teacher negotiates a restoration of 8% of the new (reduced) budget. What is the overall percentage change from the original budget?

  16. 16. A price rises by 20% every year for 3 years. What is the total percentage increase over the 3 years? Explain why this is not the same as three separate 20% increases added together.

  17. 17. Maria earns £35,000 per year. She receives a 4% rise in year 1, a 2.5% rise in year 2, and a 3% rise in year 3. Find her salary at the end of year 3.

  18. 18. A number is 20% more than a second number. The second number is 25% less than a third number. What percentage of the third number is the first number?

  19. 19. A shop owner marks up goods by 40% above cost price, then offers a 10% sale discount on the marked-up price. What is the overall percentage profit on the cost price?

  20. 20. Which gives the greater return on a £10,000 investment: 5% compound interest for 10 years, or 6% compound interest for 8 years? Show all working and state the difference.

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Worked Answers

  1. 1. 1.20 × 1.25 = 1.50. Overall increase: 50%.
  2. 2. 0.70 × 0.80 = 0.56. Overall reduction: 44%.
  3. 3. 20,000 × 0.75 × 0.85 = 20,000 × 0.6375 = £12,750. Total percentage decrease: (1 − 0.6375) × 100 = 36.25%.
  4. 4. If original = 100, after 40% increase = 140. Decrease needed: 40 ÷ 140 × 100 = 28.6% (to 1 d.p.).
  5. 5. Shop A original: 800 ÷ 0.80 = £1,000. Shop B original: 860 ÷ 0.75 = £1,146.67. Shop B had the higher original price.
  6. 6. After 3 years at 4%: 10,000 × 1.04³ = 10,000 × 1.124864 = £11,248.64. After 2 more years at 3%: 11,248.64 × 1.03² = 11,248.64 × 1.0609 = £11,933.91.
  7. 7. 1.03¹³ ≈ 1.4685 < 1.50; 1.03¹&sup4; ≈ 1.5126 > 1.50. Answer: 14 years.
  8. 8. 0.85 × 1.15 = 0.9775. The price is 2.25% below the original. The customer is incorrect.
  9. 9. Anna: 5,000 × 1.05³ = 5,000 × 1.157625 = £5,788.13. Beth: 5,000 × 1.045³ = 5,000 × 1.141166 = £5,705.83. Carl: 5,000 + (5,000 × 0.07 × 3) = 5,000 + 1,050 = £6,050. Carl has the most with £6,050.
  10. 10. Let original price = P. After increase: P(1 + x/100). After decrease: P(1 + x/100)(1 − x/100) = P(1 − x²/10,000). Since x²/10,000 > 0 for any non-zero x, the final price is always less than P.
  11. 11. 320,000 × 1.05 = 336,000; × 0.97 = 325,920; × 1.07 = £348,734 (to the nearest pound).
  12. 12. 4,800,000 × 0.015 = £72,000.
  13. 13. Original faulty items: 2,400 × 0.025 = 60. After 40% reduction in fault rate: 60 × 0.60 = 36 faulty items per day.
  14. 14. 1.08³ = 1.259712. Original = 12,597.12 ÷ 1.259712 = £10,000.
  15. 15. 0.875 × 1.08 = 0.945. Overall change: 5.5% reduction. (If original = £100: after 12.5% cut = £87.50; 8% restoration = £87.50 × 1.08 = £94.50.)
  16. 16. 1.20³ = 1.728. Total increase = 72.8%. This is more than 3 × 20% = 60% because each year's increase is applied to a larger amount, so the growth compounds.
  17. 17. 35,000 × 1.04 = 36,400; × 1.025 = 37,310; × 1.03 = £38,429.30.
  18. 18. Let third number = 100. Second = 100 × 0.75 = 75. First = 75 × 1.20 = 90. The first number is 90% of the third.
  19. 19. Cost = 100; marked up to 140; 10% sale discount: 140 × 0.90 = 126. Overall profit on cost price: 26%.
  20. 20. 5% for 10 years: 10,000 × 1.05¹&sup0; = 10,000 × 1.628895 = £16,288.95. 6% for 8 years: 10,000 × 1.06&sup8; = 10,000 × 1.593848 = £15,938.48. 5% for 10 years gives more, by £350.47.

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