← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 1 Number · Chapter 5 · pp. 43–49

Further percentages

Sales, pay rises, exam marks and population growth are all percentages. The one idea that ties this chapter together is the multiplier: turn the percentage into a decimal and multiply. It handles “25% of”, “increase by 12%” and “what was it before?” with the same move.

Learning objectives

By the end of this chapter you can:

  1. Convert between percentages, fractions and decimals, including awkward ones like .
  2. Find a percentage of a quantity.
  3. Write one quantity as a percentage of another, including in different units.
  4. Increase or decrease by a percentage, and find a percentage increase or decrease.
  5. Find the original amount before a percentage change (reverse percentages). (Extended)

Book exercises: 5.1 to 5.4 and Student assessments 1 to 3 (pp. 43–49).

Fraction
Decimal0.1250.250.3750.50.6250.750.875
Percent12.5%25%37.5%50%62.5%75%87.5%

💡 Eighths go up in steps of 12.5%. Tenths go up in steps of 10%: = 30%, = 40%, and so on.

Next: percentage of a quantity →

Percentage of a quantity

🔑 Key ideas

  • Percentage → decimal: divide by 100. 35% = 0.35, 7% = 0.07, 250% = 2.5.
  • To find a percentage of something, multiply by that decimal.
  • Fraction → percentage when the bottom doesn't go into 100: divide, then × 100.
35% = 98 m 65% = 182 m 0%35%100% = 280 m
The whole amount is 100%. 35% of 280 m is the shaded part: 0.35 × 280 = 98 m.

📘 Example 1 · Find 35% of 280 m

  1. Change the percentage to a decimal: 35% = 35 ÷ 100 = 0.35.
  2. Multiply: 0.35 × 280 = 98 m.

📘 Example 2 · Write as a percentage

  1. 16 doesn't divide into 100, so change to a decimal first: 3 ÷ 16 = 0.1875.
  2. Multiply by 100: 0.1875 × 100 = 18.75%.

📘 Example 3 · A survey of 120 children: 55% prefer lamb

  1. 55% = 0.55.
  2. 0.55 × 120 = 66 children.
  3. Check the whole survey adds up: the four groups (55%, 20%, 15%, 10%) total 100%, so their numbers must total 120.

✏️ Try it

  1. 62.5% of 80
  2. 250% of 7
  3. as a percentage (1 d.p.)
Answers

1. 0.625 × 80 = 50 · 2. 2.5 × 7 = 17.5 · 3. 0.7333… × 100 = 73.3%

Next: one quantity as a percentage of another →

One quantity as a percentage of another

🔑 Key ideas

  • Write it as a fraction part ÷ whole, then multiply by 100.
  • Both quantities must be in the same units first.

percentage = × 100

📘 Example 4 · 69 marks out of 75

  1. Write part over whole: .
  2. Multiply by 100: × 100 = 0.92 × 100 = 92%.

📘 Example 5 · 40 cm as a percentage of 2 m

  1. Same units first: 2 m = 200 cm.
  2. Part over whole: .
  3. × 100: 0.2 × 100 = 20%.

✏️ Try it

  1. 13 out of 39
  2. 25 minutes as a percentage of 1 hour
  3. A hockey team plays 42 matches and wins 21. What percentage does it win?
Answers

1. 33.3% · 2. × 100 = 41.7% · 3. 50%

Next: increase and decrease →

Percentage increase and decrease

🔑 Key ideas

  • The original amount is always 100%.
  • Increase by 12% → new amount is 112% → multiply by 1.12.
  • Decrease by 15% → new amount is 85% → multiply by 0.85.
  • Percentage change = × 100.
Original100% · × 1 +12% → 112%× 1.12 −15% → 85%× 0.85 Reverse÷ the multiplier
Every change is one multiplication. Going back to the original is one division by the same multiplier.

📘 Example 6 · A $14 500 truck rises in price by 12%

  1. The original price is 100%. After a 12% rise it is 100% + 12% = 112%.
  2. As a multiplier: 112% = 1.12.
  3. New price = 1.12 × $14 500 = $16 240.

📘 Example 7 · $130 tools with a 15% discount

  1. After a 15% cut the price is 100% − 15% = 85%, so the multiplier is 0.85.
  2. Sale price = 0.85 × $130 = $110.50.

📘 Example 8 · Percentage profit: bought for $75 000, sold for $87 000

  1. Change = 87 000 − 75 000 = $12 000.
  2. Divide by the original amount: = 0.16.
  3. × 100: 16% profit.

⚠️ A common trap

A $100 coat is cut by 25% to $75. Putting the price back up by 25% gives 1.25 × 75 = $93.75, not $100. The second 25% is of a smaller amount. To get back to $100 you need a rise of 25 ÷ 75 = 33⅓%.

🧪 Try it: percentage calculator

    ✏️ Try it

    1. Increase 80 by 12.5%.
    2. Decrease 550 to 352. What is the percentage decrease?
    3. A computer priced at $7800 is reduced by 12.5%. Find the sale price.
    Answers

    1. 1.125 × 80 = 90 · 2. 198 ÷ 550 × 100 = 36% · 3. 0.875 × 7800 = $6825

    Next: reverse percentages →

    Reverse percentages Extended

    🔑 Key ideas

    • You know the amount after a change and want the amount before.
    • Work out what percentage the new amount is (e.g. 128%), then divide by that multiplier.
    • Never take the percentage off the new amount: that uses the wrong 100%.

    📘 Example 9 · Ahmed got 92% by answering 23 questions correctly. How many did he get wrong?

    1. 92% of the questions = 23, so 1% = 23 ÷ 92 = 0.25.
    2. 100% = 0.25 × 100 = 25 questions in the test.
    3. Wrong = 25 − 23 = 2 questions.

    📘 Example 10 · A car sells for $15 360 at a 28% profit. What did it cost the seller?

    1. The selling price is 100% + 28% = 128% of the cost, a multiplier of 1.28.
    2. Divide to undo it: cost = $15 360 ÷ 1.28 = $12 000.
    3. Check: 1.28 × 12 000 = 15 360. ✔

    📘 Example 11 · Output fell by 4% to 24 000 units. What was it last year?

    1. This year is 100% − 4% = 96% of last year: multiplier 0.96.
    2. Last year = 24 000 ÷ 0.96 = 25 000 units.

    ✏️ Try it

    1. 85% of X is 765. Find X.
    2. 3500 families own a car. This is 28% of the town's families. How many families are there?
    3. Sarah got 87.5% by gaining 105 marks. How many marks did she lose?
    Answers

    1. 765 ÷ 0.85 = 900 · 2. 3500 ÷ 0.28 = 12 500 · 3. 105 ÷ 0.875 = 120 marks in total, so she lost 15

    Next: summary and quiz →

    Chapter summary

    • Percent ↔ decimal: ÷ 100 or × 100. A fraction whose bottom doesn't divide 100: divide first, then × 100.
    • p% of an amount = (p ÷ 100) × amount.
    • One as a percentage of another = part ÷ whole × 100, with the same units.
    • Increase by p%: × (1 + p/100). Decrease by p%: × (1 − p/100). Percentage change = change ÷ original × 100.
    • Reverse percentage: divide the new amount by the multiplier.

    📄 Practice questions

    The book's Exercises 5.1 to 5.4 and Student assessments 1 to 3 are in the printable assignment for this chapter.

    Mini-quiz

    1. What is 35% of 280 m?
    2. What is 3/16 as a percentage?
    3. A girl scores 69 marks out of 75. What percentage is that?
    4. Which multiplier increases an amount by 12%?
    5. Tools cost $130. In a sale they are 15% off. What is the sale price?
    6. A house bought for $75 000 sells for $87 000. What is the percentage profit?
    7. A $100 coat is cut by 25%, then raised by 25%. What is the price now?
    8. After a 28% profit a car sells for $15 360. What did it cost?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 5 (Further percentages). Explanations, the Try it answers, the tools and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.