Percentages

What are Percentages? %

"Percent" means "per hundred" or "out of 100." A percentage is a way of expressing a fraction with a denominator of 100. The symbol for percent is %.

Percentages are commonly used in everyday life, for example, in discounts, interest rates, statistics, and test scores.

Key Idea:

  • 50% means 50 out of 100, which is equivalent to the fraction 50⁄100 or the decimal 0.50.
  • 100% represents the whole amount.
  • You can have percentages greater than 100% (e.g., 150% means 1.5 times the original amount).

Working with Percentages: Examples

Converting a Percentage to a Fraction

Problem: Convert 75% to a fraction in its simplest form.

  1. Write the percentage as a fraction over 100: 75% = 75⁄100.
  2. Simplify the fraction by dividing the numerator and denominator by their highest common factor (HCF). The HCF of 75 and 100 is 25.
  3. 75 ÷ 25 = 3
  4. 100 ÷ 25 = 4

Answer: 3⁄4

Converting a Percentage to a Decimal

Problem: Convert 45% to a decimal.

  1. To convert a percentage to a decimal, divide by 100 (or move the decimal point two places to the left).
  2. 45% = 45 ÷ 100 = 0.45

Answer: 0.45

Converting a Decimal to a Percentage

Problem: Convert 0.62 to a percentage.

  1. To convert a decimal to a percentage, multiply by 100 (or move the decimal point two places to the right) and add the % symbol.
  2. 0.62 × 100 = 62

Answer: 62%

Converting a Fraction to a Percentage

Problem: Convert 2⁄5 to a percentage.

  1. Divide the numerator by the denominator to get a decimal: 2 ÷ 5 = 0.4.
  2. Convert the decimal to a percentage by multiplying by 100: 0.4 × 100 = 40.

Answer: 40%

Alternatively, make the denominator 100: 2⁄5 = (2×20)⁄(5×20) = 40⁄100 = 40%.

Finding a Percentage of an Amount

Problem: Calculate 20% of £80.

  1. Convert the percentage to a decimal or fraction: 20% = 0.20 or 20⁄100 = 1⁄5.
  2. Multiply the decimal/fraction by the amount:
    • Using decimal: 0.20 × £80 = £16.
    • Using fraction: 1⁄5 × £80 = £80 ÷ 5 = £16.

Tip: You can find 10% first (£80 ÷ 10 = £8), then double it for 20% (£8 × 2 = £16).

Answer: £16

Percentage Increase

Problem: Increase £50 by 10%.

  1. Find 10% of £50: 0.10 × £50 = £5.
  2. Add this increase to the original amount: £50 + £5 = £55.

Alternatively, an increase of 10% means you have 100% + 10% = 110% of the original. So, multiply by 1.10: £50 × 1.10 = £55.

Answer: £55

Percentage Decrease

Problem: Decrease £200 by 25%.

  1. Find 25% of £200: 0.25 × £200 = £50. (Or 1⁄4 of £200 = £50)
  2. Subtract this decrease from the original amount: £200 - £50 = £150.

Alternatively, a decrease of 25% means you have 100% - 25% = 75% remaining. So, multiply by 0.75: £200 × 0.75 = £150.

Answer: £150

Finding the Original Amount After a Percentage Increase

Problem: A price was increased by 10% and is now £110. What was the original price?

  1. The new price (£110) represents the original price (100%) plus the 10% increase. So, £110 is 110% of the original price.
  2. Convert 110% to a decimal: 110 ÷ 100 = 1.10.
  3. To find the original price, divide the new price by this decimal multiplier: £110 ÷ 1.10 = £100.

Answer: £100

Finding the Original Amount After a Percentage Decrease

Problem: After a 20% discount, a pair of trainers costs £48. What was the original price?

  1. The sale price (£48) represents the original price (100%) minus the 20% discount. So, £48 is 100% - 20% = 80% of the original price.
  2. Convert 80% to a decimal: 80 ÷ 100 = 0.80.
  3. To find the original price, divide the sale price by this decimal multiplier: £48 ÷ 0.80 = £60.

Answer: £60

Percentage Change

Percentage change is used to express the difference between an old value and a new value as a percentage of the old value. It can be an increase or a decrease.

Formula: Percentage Change = (New Value - Old Value)⁄Old Value × 100%

Example: Percentage Increase

Problem: The price of a product increased from £50 to £60. What is the percentage increase?

  1. Calculate the difference (New Value - Old Value): £60 - £50 = £10.
  2. Divide the difference by the Old Value: £10 ÷ £50 = 0.2.
  3. Multiply by 100 to get the percentage: 0.2 × 100 = 20%.

Answer: 20% increase

Example: Percentage Decrease

Problem: The number of students attending a club dropped from 80 to 68. What is the percentage decrease?

  1. Calculate the difference (New Value - Old Value): 68 - 80 = -12.
  2. Divide the difference by the Old Value: -12 ÷ 80 = -0.15.
  3. Multiply by 100 to get the percentage: -0.15 × 100 = -15%.

Answer: 15% decrease (or -15% change)

Test Your Percentage Skills!

1. Convert 60% to a simplified fraction.

Answer: 3⁄5

Explanation: 60/100, HCF is 20. (60÷20=3), (100÷20=5).

2. Convert 0.07 to a percentage.

Answer: 7%

Explanation: 0.07 × 100 = 7.

3. What is 15% of £120?

Answer: £18

Explanation: 0.15 × 120 = 18. (Or 10% is £12, 5% is £6. £12+£6 = £18).

4. A price of £80 is increased by 5%. What is the new price?

Answer: £84

Explanation: 5% of £80 = £4. New price = £80 + £4 = £84. (Or £80 × 1.05 = £84).

5. Convert 125% to a decimal.

Answer: 1.25

Explanation: 125% = 125 ÷ 100 = 1.25.

6. Convert the fraction 3⁄8 to a percentage.

Answer: 37.5%

Explanation: (3 ÷ 8) × 100 = 0.375 × 100 = 37.5%.

7. A t-shirt costing £25 is reduced by 15%. What is the new price?

Answer: £21.25

Explanation: 15% of £25 = 0.15 × £25 = £3.75. New price = £25 - £3.75 = £21.25. (Or £25 × 0.85 = £21.25).

8. After a 20% pay rise, Sarah's weekly wage is £360. What was her wage before the rise?

Answer: £300

Explanation: £360 represents 120% of the original wage (100% + 20%). Original wage = £360 ÷ 1.20 = £300.

9. A shop offers a 30% discount on all items. If a bag costs £42 in the sale, what was its original price?

Answer: £60

Explanation: £42 represents 70% of the original price (100% - 30%). Original price = £42 ÷ 0.70 = £60.

10. The temperature dropped from 15°C to 9°C. What was the percentage decrease in temperature?

Answer: 40% decrease

Explanation: Change = 9°C - 15°C = -6°C. Percentage decrease = (-6⁄15) × 100% = -0.4 × 100% = -40%. So, a 40% decrease.

11. Calculate 35% of 280kg.

Answer: 98kg

Explanation: 0.35 × 280kg = 98kg. (Or 10% is 28kg, so 30% is 3 × 28kg = 84kg. 5% is 28kg ÷ 2 = 14kg. Total = 84kg + 14kg = 98kg).

12. Convert 2.5% to a simplified fraction.

Answer: 1⁄40

Explanation: 2.5% = 2.5⁄100. To remove the decimal, multiply top and bottom by 10: 25⁄1000. HCF of 25 and 1000 is 25. (25÷25=1), (1000÷25=40).

13. A population of 500 birds increases by 12% in one year. How many birds are there after the increase?

Answer: 560 birds

Explanation: 12% of 500 = 0.12 × 500 = 60. New population = 500 + 60 = 560. (Or 500 × 1.12 = 560).

14. A car valued at £12,000 depreciates by 8% in its first year. What is its value after one year?

Answer: £11,040

Explanation: 8% of £12,000 = 0.08 × £12,000 = £960. Value after 1 year = £12,000 - £960 = £11,040. (Or £12,000 × 0.92 = £11,040).

15. A runner's time improved from 60 seconds to 54 seconds. What was the percentage improvement (decrease in time)?

Answer: 10% improvement

Explanation: Change = 54 seconds - 60 seconds = -6 seconds. Percentage change = (-6⁄60) × 100% = -0.1 × 100% = -10%. So, a 10% improvement.

Exam-Style Percentage Problems

1. A coat originally priced at £150 is on sale with a 20% discount. What is the sale price of the coat?

Answer: £120

Explanation:

  1. Calculate the discount amount: 20% of £150.
    • Convert 20% to a decimal: 20 / 100 = 0.20.
    • Multiply by the original price: 0.20 × £150 = £30.
  2. Subtract the discount from the original price: £150 - £30 = £120.
  3. Alternative method: If there's a 20% discount, you pay 100% - 20% = 80% of the price.
    0.80 × £150 = £120.

2. John invests £2000 in a savings account that pays 3% simple interest per year. How much interest will he earn in 5 years?

Answer: £300

Explanation:

  1. Calculate the interest earned in one year: 3% of £2000.
    • Convert 3% to a decimal: 3 / 100 = 0.03.
    • Multiply by the principal amount: 0.03 × £2000 = £60 per year.
  2. Calculate the total interest earned in 5 years: £60/year × 5 years = £300.

3. A mobile phone company increased its monthly plan from £25 to £28. What was the percentage increase in the price of the monthly plan?

Answer: 12%

Explanation:

  1. Calculate the actual increase in price: £28 - £25 = £3.
  2. Calculate the percentage increase: (Actual Increase / Original Price) × 100%.
    • (£3 / £25) × 100%
    • 0.12 × 100% = 12%.

4. In a survey of 240 people, 45% said their favourite colour was blue. 25% said red, and the rest said green. How many people said green was their favourite colour?

Answer: 72 people

Explanation:

  1. Calculate the total percentage for blue and red: 45% + 25% = 70%.
  2. Calculate the percentage for green: 100% - 70% = 30%.
  3. Calculate the number of people who said green: 30% of 240.
    • Convert 30% to a decimal: 30 / 100 = 0.30.
    • Multiply by the total number of people: 0.30 × 240 = 72 people.

5. A recipe requires 250g of flour. If you want to make a batch that is 40% larger, how much flour will you need?

Answer: 350g

Explanation:

  1. Calculate the increase in flour: 40% of 250g.
    • 0.40 × 250g = 100g.
  2. Add the increase to the original amount: 250g + 100g = 350g.
  3. Alternative method: A 40% larger batch means 100% + 40% = 140% of the original.
    1.40 × 250g = 350g.

6. A shop buys a watch for £50 and sells it for £80. What is the percentage profit made on the watch?

Answer: 60%

Explanation:

  1. Calculate the profit amount: £80 (selling price) - £50 (cost price) = £30.
  2. Calculate the percentage profit based on the cost price: (Profit / Cost Price) × 100%.
    • (£30 / £50) × 100%
    • 0.60 × 100% = 60%.

7. The price of a train ticket is £45 after a 10% increase. What was the price before the increase?

Answer: £40.91 (approx)

Explanation:

  1. The new price (£45) represents 100% + 10% = 110% of the original price.
  2. Convert 110% to a decimal: 1.10.
  3. Divide the new price by the decimal multiplier: £45 ÷ 1.10 = £40.9090...
  4. Round to two decimal places for currency: £40.91.

8. A library had 1500 books. After a clear-out, they had 1275 books left. What was the percentage decrease in the number of books?

Answer: 15%

Explanation:

  1. Calculate the decrease in books: 1500 - 1275 = 225 books.
  2. Calculate the percentage decrease: (Decrease / Original Number) × 100%.
    • (225 / 1500) × 100%
    • 0.15 × 100% = 15%.

9. A phone bill is £36, which includes VAT at 20%. What was the price of the phone bill before VAT was added?

Answer: £30

Explanation:

  1. The price including VAT (£36) represents 100% (original price) + 20% (VAT) = 120% of the original price.
  2. Convert 120% to a decimal: 1.20.
  3. Divide the total price by the decimal multiplier: £36 ÷ 1.20 = £30.

10. A company's profits increased from £80,000 to £104,000 in one year. Calculate the percentage increase in profits.

Answer: 30%

Explanation:

  1. Calculate the actual increase in profit: £104,000 - £80,000 = £24,000.
  2. Calculate the percentage increase: (Actual Increase / Original Profit) × 100%.
    • (£24,000 / £80,000) × 100%
    • 0.30 × 100% = 30%.

Interactive Percentage Tools

Fraction to Decimal & Percentage

Enter a fraction to convert it to a decimal and a percentage.

/

Decimal: ?

Percentage: ?%

Percentage to Decimal & Fraction

Enter a percentage to convert it to a decimal and a simplified fraction.

%

Decimal: ?

Fraction: ?

Percentage of an Amount

% of
Result: ?

Percentage Increase

by %
New Amount: ?

Percentage Decrease

by %
New Amount: ?

Number as a Percentage of Another

Calculate what percentage one number (the 'part') is of another number (the 'whole').

is what % of
Result: ?%

Original Amount After Percentage Increase

Find the original amount before a percentage increase was applied.

after a % increase
Original Amount: ?

Original Amount After Percentage Decrease

Find the original amount before a percentage decrease was applied.

after a % decrease
Original Amount: ?

Percentage Change

Calculate the percentage change (increase or decrease) between two values.

to
Result: ?

Key Percentage Facts