Percentages are a way of expressing a number as a fraction of 100. The term “percent” comes from the Latin phrase “per centum,” which means “by the hundred.” Percentages are widely used in various contexts, such as finance, statistics, and everyday calculations. Understanding percentages is essential for the 11+ exam as it forms the basis for many mathematical concepts.

Key Concepts in Percentages

1. Understanding Percentages

  • A percentage is represented by the symbol “%”.
  • For example, 50% means 50 out of 100 or half of a whole.

2. Calculating Percentages

To calculate a percentage of a number:

  • Use the formula:
    \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100

For example, to find 20% of 50:
20\% \text{ of } 50 = \frac{20}{100} \times 50 = 10

3. Finding the Whole from a Percentage

To find the whole when given a part and a percentage:

  • Use the formula:
    \text{Whole} = \frac{\text{Part}}{\text{Percentage}} \times 100

For example, if 25 is 20% of a number:
\text{Whole} = \frac{25}{20} \times 100 = 125

4. Increasing and Decreasing by a Percentage

  • To increase a number by a percentage, add the percentage to the original number.
  • To decrease a number by a percentage, subtract the percentage from the original number.

For example, increasing 200 by 10%:
200 + (10\% \text{ of } 200) = 200 + 20 = 220

5. Percentage Change

To calculate the percentage change between two numbers:

  • Use the formula:
    \text{Percentage Change} = \frac{\text{New Value} – \text{Old Value}}{\text{Old Value}} \times 100

For example, if a price increases from £50 to £60:
\text{Percentage Change} = \frac{60 – 50}{50} \times 100 = 20\%

Practice Questions on Percentages

Easy Level

  1. What is 25% of 80?
  2. If 50 is 20% of a number, what is the number?
  3. Calculate 10% of 150.
  4. What is 75% of 200?
  5. A shirt costs £40. What is 20% off the price?
  6. If a student scored 45 out of 60, what percentage did they achieve?
  7. What is 30% of 90?
  8. If 30 is 15% of a number, what is the number?
  9. Calculate 5% of 200.
  10. What is 50% of 120?
  11. A book costs £12. What is 10% off the price?
  12. If a class has 20 students and 15 are girls, what percentage are girls?
  13. What is 40% of 50?
  14. If you have £100, what is 25% of that amount?
  15. What is 60% of 80?
  16. A toy costs £15. What is 20% off the price?
  17. If a fruit basket has 8 apples out of 32 fruits, what percentage are apples?
  18. Calculate 15% of 200.
  19. What is 10% of 500?
  20. A TV costs £400. What is 5% off the price?

Medium Level

  1. What is 15% of 250?
  2. If 60 is 30% of a number, what is the number?
  3. Calculate 12% of 250.
  4. What is 80% of 150?
  5. A jacket costs £60. What is 25% off the price?
  6. If a student scored 36 out of 45, what percentage did they achieve?
  7. What is 20% of 300?
  8. If 24 is 40% of a number, what is the number?
  9. Calculate 18% of 500.
  10. What is 75% of 240?
  11. A laptop costs £800. What is 10% off the price?
  12. If a class has 40 students and 28 are boys, what percentage are boys?
  13. What is 50% of 180?
  14. If you have £250, what is 15% of that amount?
  15. What is 35% of 70?
  16. A meal costs £50. What is 20% off the price?
  17. If a car’s value decreases from £20,000 to £15,000, what is the percentage decrease?
  18. Calculate 7.5% of 400.
  19. If a product is sold for £120 after a 20% discount, what was the original price?
  20. What is 90% of 300?

Hard Level

  1. What is 28% of 250?
  2. If 75 is 60% of a number, what is the number?
  3. Calculate 25% of 1,200.
  4. A pair of shoes costs £90. What is 30% off the price?
  5. If a student scored 80 out of 100, what percentage did they achieve?
  6. What is 85% of 500?
  7. If 90 is 30% of a number, what is the number?
  8. Calculate 22% of 450.
  9. A phone costs £700. What is 15% off the price?
  10. If a class has 30 students and 18 are girls, what percentage are boys?
  11. What is 65% of 320?
  12. If a store has a sale of 20% on all items, and an item originally costs £40, what is the sale price?
  13. If a car’s value decreases from £25,000 to £20,000, what is the percentage decrease?
  14. Calculate 17.5% of 800.
  15. If a product is sold for £180 after a 10% discount, what was the original price?
  16. What is 37.5% of 160?
  17. A student scored 75 out of 90. What percentage did they achieve?
  18. If 48 is 12% of a number, what is the number?
  19. A jacket is on sale for £50 after a 20% discount. What was the original price?
  20. What is the percentage increase if a value goes from 150 to 180?

Answers and Explanations

Easy Level

  1. 25\% \text{ of } 80 = \frac{25}{100} \times 80 = 20
  2. 50 = 20\% \text{ of } x \Rightarrow x = \frac{50 \times 100}{20} = 250
  3. 10\% \text{ of } 150 = \frac{10}{100} \times 150 = 15
  4. 75\% \text{ of } 200 = \frac{75}{100} \times 200 = 150
  5. 20\% \text{ off } 40 = \frac{20}{100} \times 40 = 8 \Rightarrow 40 – 8 = 32
  6. \frac{45}{60} \times 100 = 75\%
  7. 30\% \text{ of } 90 = \frac{30}{100} \times 90 = 27
  8. 30 = 15\% \text{ of } x \Rightarrow x = \frac{30 \times 100}{15} = 200
  9. 5\% \text{ of } 200 = \frac{5}{100} \times 200 = 10
  10. 50\% \text{ of } 120 = \frac{50}{100} \times 120 = 60
  11. 10\% \text{ off } 12 = \frac{10}{100} \times 12 = 1.2 \Rightarrow 12 – 1.2 = 10.8
  12. \frac{15}{20} \times 100 = 75\%
  13. 40\% \text{ of } 50 = \frac{40}{100} \times 50 = 20
  14. 25\% \text{ of } 100 = \frac{25}{100} \times 100 = 25
  15. 60\% \text{ of } 80 = \frac{60}{100} \times 80 = 48
  16. \frac{10}{20} = 50\%
  17. 15\% \text{ of } 200 = \frac{15}{100} \times 200 = 30
  18. 30\% \text{ of } 100 = \frac{30}{100} \times 100 = 30
  19. $$ \frac{15}{20} \times 100 =

75\% $$

  1. 60\% \text{ of } 100 = \frac{60}{100} \times 100 = 60

Medium Level

  1. 15\% \text{ of } 250 = \frac{15}{100} \times 250 = 37.5
  2. 60 = 30\% \text{ of } x \Rightarrow x = \frac{60 \times 100}{30} = 200
  3. 12\% \text{ of } 250 = \frac{12}{100} \times 250 = 30
  4. 80\% \text{ of } 150 = \frac{80}{100} \times 150 = 120
  5. 25\% \text{ off } 60 = \frac{25}{100} \times 60 = 15 \Rightarrow 60 – 15 = 45
  6. \frac{36}{45} \times 100 = 80\%
  7. 20\% \text{ of } 300 = \frac{20}{100} \times 300 = 60
  8. 24 = 40\% \text{ of } x \Rightarrow x = \frac{24 \times 100}{40} = 60
  9. 18\% \text{ of } 500 = \frac{18}{100} \times 500 = 90
  10. 75\% \text{ of } 240 = \frac{75}{100} \times 240 = 180
  11. 10\% \text{ off } 800 = \frac{10}{100} \times 800 = 80 \Rightarrow 800 – 80 = 720
  12. \frac{28}{40} \times 100 = 70\%
  13. 50\% \text{ of } 180 = \frac{50}{100} \times 180 = 90
  14. 15\% \text{ of } 250 = \frac{15}{100} \times 250 = 37.5
  15. 35\% \text{ of } 70 = \frac{35}{100} \times 70 = 24.5
  16. 20\% \text{ off } 50 = \frac{20}{100} \times 50 = 10 \Rightarrow 50 – 10 = 40
  17. \frac{10}{20} \times 100 = 50\%
  18. 4:3 \text{ ratio} = 7\%
  19. 18\% \text{ of } 400 = \frac{18}{100} \times 400 = 72
  20. 12:8 = 5:3

Hard Level

  1. 75 = 60\% \text{ of } x \Rightarrow x = \frac{75 \times 100}{60} = 125
  2. \frac{12}{100} \times 3 = 3.6 \text{ cups of flour}
  3. 10\% \text{ off } 70 = \frac{10}{100} \times 70 = 7 \Rightarrow 70 – 7 = 63
  4. \frac{90 – 70}{70} \times 100 = 28.57\%
  5. 80 + 4 = 12 \text{ litres}
  6. 20\% \text{ of } 50 = \frac{20}{100} \times 50 = 10
  7. \frac{30 – 60}{60} \times 100 = 50\%
  8. 12\% \text{ of } 50 = \frac{12}{100} \times 50 = 6
  9. 20 = 100 \Rightarrow 20\%
  10. \frac{2}{100} = 20\%
  11. \frac{9}{45} = 5\%
  12. 25\% \text{ of } 50 = \frac{25}{100} \times 50 = 12.5
  13. 20\% \text{ off } 30 = 24 – 6 = 18
  14. 75\% \text{ of } 300 = 75 + 45 = 120
  15. 25\% \text{ off } 80 = \frac{25}{100} \times 80 = 20
  16. 12 – 3 = 15 \text{ students}
  17. \frac{4 \times 60}{100} = 240
  18. 25 = \frac{90 + 50}{25}
  19. \frac{8}{12} = 75\%
  20. 35 = 28\%