percentages.co.uk

Percentage Reasoning Worksheet

Year 6-KS3 | Ages 10-14Reasoning and Problem Solving20 questions

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

Show your working and write your reasoning clearly. These questions require you to explain, justify and prove.

  1. 1. Is 30% of 80 the same as 80% of 30? Show your working and explain why.

  2. 2. Sophie says “50% of 50 is 25.” Ben says “50% of 50 is 50.” Who is correct? Explain your answer.

  3. 3. A price rises by 50%, then falls by 50%. Tom says the price is back to the original. Is Tom correct? Prove it using an example.

  4. 4. Explain why you cannot find the original price by subtracting 20% from a sale price when the item was discounted by 20%. Use an example to demonstrate.

  5. 5. Jack says “20% of 60 is 12, and 60% of 20 is also 12.” Is he correct? Explain why or why not.

  6. 6. A school reports that attendance improved from 90% to 95%. Is this a 5% increase? Explain the difference between a 5 percentage point increase and a 5% increase.

  7. 7. Show that increasing by 10% twice is not the same as increasing by 20%. Use £500 as your starting amount.

  8. 8. A value is reduced by 40%. What percentage increase is needed to return to the original value? Show your working.

  9. 9. Two students calculate 15% of 240 differently. Alice finds 10%, then adds half of 10% to get 15%. Ben multiplies 240 by 0.15. Show that both methods give the same answer.

  10. 10. A shop offers “25% off, then a further 10% off the sale price.” A customer claims this is the same as 35% off. Is the customer correct? Explain and show your working.

  11. 11. Is 1% of £100 the same as £1? Is 1% of £1,000 also £1? Explain your answer.

  12. 12. Sam says “If you increase a number by 25% and then decrease it by 25%, you always end up with the original number.” Show he is wrong using an example.

  13. 13. Arrange these values in order from smallest to largest without a calculator: 3/7,  42%,  0.43. Explain your method.

  14. 14. A number increases by 50%. What single multiplier represents this increase? What multiplier would reverse the change and return to the original value?

  15. 15. Prove that percentage change depends only on the ratio of the new value to the original, not on the original value itself. Use two examples with the same ratio to illustrate.

  16. 16. A teacher says “If you know 10% of a number, you can find any percentage.” Explain how, and give an example finding 17% of 350.

  17. 17. Is it possible for a percentage increase to be greater than 100%? Give an example where a value more than doubles.

  18. 18. A class has 24 girls and 16 boys. What percentage are girls? A new boy joins. Do both percentages change? Recalculate and explain.

  19. 19. Two towns each grow by 10% per year. Town A has 50,000 people. Town B has 80,000 people. After 2 years, what is the difference in population? Explain which town gains more people in absolute terms and why.

  20. 20. A student writes: “25% of 80 = 80 ÷ 4 = 20.” Write three different correct methods for finding 25% of a number and explain why they all work.

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

Worked Answers

  1. 1. Yes. 30% of 80 = 0.30 × 80 = 24. 80% of 30 = 0.80 × 30 = 24. Both give 24. Percentages are commutative: (a/100) × b = (b/100) × a.
  2. 2. Sophie is correct. 50% means half. Half of 50 is 25. Ben has confused 50% of 50 with 50 itself.
  3. 3. Tom is incorrect. Using £100: rises by 50% to £150, then falls 50%: £150 × 0.50 = £75. The final price is £75, not £100.
  4. 4. The 20% discount was applied to the original price, not the sale price. Example: original £100, 20% off = £80. Subtracting 20% from £80 gives £64, not £100. To find the original, divide by 0.80: £80 ÷ 0.80 = £100.
  5. 5. Yes, he is correct. a% of b always equals b% of a. 20% of 60 = 0.20 × 60 = 12; 60% of 20 = 0.60 × 20 = 12.
  6. 6. It is a 5 percentage point increase. The actual percentage increase is: (95 − 90) ÷ 90 × 100 = 5.56%. These are different things. A percentage point increase describes the absolute difference between two percentages.
  7. 7. Two 10% increases: 500 × 1.10 × 1.10 = 500 × 1.21 = £605. One 20% increase: 500 × 1.20 = £600. £605 ≠ £600. Two successive 10% increases give a 21% overall increase, not 20%.
  8. 8. If original = 100, after 40% decrease = 60. Increase needed: 40 ÷ 60 × 100 = 66.7% (to 1 d.p.).
  9. 9. Alice: 10% of 240 = 24; 5% = 12; 15% = 36. Ben: 240 × 0.15 = 36. Both methods give 36.
  10. 10. The customer is incorrect. Using £100: 25% off = £75; further 10% off £75 = £7.50; final price = £67.50. Total discount = 32.5%, not 35%.
  11. 11. 1% of £100 = £1. 1% of £1,000 = £10, not £1. The value of 1% depends on the amount it is a percentage of.
  12. 12. Using 100: 100 × 1.25 = 125; 125 × 0.75 = 93.75. The result is 93.75, not 100. Sam is wrong because the 25% decrease is applied to the larger number.
  13. 13. Convert to percentages: 3/7 ≈ 42.86%; 42% = 42%; 0.43 = 43%. Order: 42% < 3/7 < 0.43.
  14. 14. Increase by 50%: multiplier = 1.50. To reverse: multiply by 1 ÷ 1.50 = 2/3 ≈ 0.667. This is a decrease of 33.3%.
  15. 15. Percentage change = (B − A) ÷ A × 100 = (B/A − 1) × 100. This depends only on B/A, not on A. Example 1: 60 to 72, ratio = 1.20, 20% increase. Example 2: 100 to 120, ratio = 1.20, 20% increase. Same ratio, same percentage change.
  16. 16. Once you know 10%, divide by 10 to get 1%, then multiply to find any percentage. Example: 10% of 350 = 35; 1% = 3.5; 17% = 17 × 3.5 = 59.5.
  17. 17. Yes. If a value increases from 50 to 130, percentage increase = 80 ÷ 50 × 100 = 160%, which is greater than 100%.
  18. 18. Original: girls = 24/40 = 60%; boys = 40%. After new boy joins (total 41): girls = 24/41 ≈ 58.5%; boys = 17/41 ≈ 41.5%. Both percentages change because the total has changed.
  19. 19. Town A after 2 years: 50,000 × 1.21 = 60,500. Town B: 80,000 × 1.21 = 96,800. Difference = 36,300. Town B gains more people in absolute terms because 10% of a larger population is a larger number, even though the percentage growth rate is identical.
  20. 20. Method 1: 80 ÷ 4 = 20 (25% is one quarter; dividing by 4). Method 2: 80 × 0.25 = 20 (multiplier method). Method 3: find 50% of 80 = 40, then halve again = 20 (25% is half of 50%). All three work because they each calculate the same fraction, 1/4, of the amount.

Related resources

Percentage of a Number Calculator

Work out any percentage of any amount instantly.

GCSE Percentages Revision

Full GCSE revision guide covering all percentage topics with exam-style examples.

Problem Solving Worksheet

Percentage problem-solving questions in context.

Higher Percentage Worksheet

GCSE Higher tier percentage worksheet including reverse percentages.