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Percentages

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Cambridge (CIE) IGCSE — International Maths: Extended

Percentages

🎯 The Big Idea: A percentage is just a special fraction "out of 100" — and once you learn to turn any percentage into a multiplier (a single decimal number), you can increase, decrease, reverse, or compare anything in one calculator step.

Summary — What This Chapter Covers

  • Percentages, fractions, and decimals are three ways of writing the same value — you must be able to convert freely between them.
  • You can find a percentage of an amount either by building it from 10%, 5%, 1% "blocks" (no calculator) or by using a multiplier (with a calculator).
  • To increase by a percentage, multiply by (1 + the decimal). To decrease, multiply by (1 − the decimal).
  • Several percentage changes in a row can be combined into a single "combined multiplier" by multiplying the individual multipliers together.
  • You can work out what percentage change actually happened using m = after ÷ before or the percentage-change formula.
  • The exact same logic gives you percentage profit or loss — just swap "before/after" for "cost price/selling price".
  • Reverse percentage questions give you the amount after a change and ask you to find the amount before — you divide by the multiplier, you do NOT just apply the opposite percentage change.

1. Basic Percentages

What is a percentage, really?

"Per-cent" literally means "out of 100". So 50% just means "50 out of every 100", which is exactly the same amount as the fraction 50/100 or the decimal 0.5.

Here's the key skill: any fraction can be turned into a percentage by rewriting it with a denominator of 100 (or by dividing top by bottom on your calculator and multiplying by 100). This matters because percentages let you compare things fairly — "3 out of 4" and "75 out of 100" don't look like the same thing at first glance, but once you see both as 75%, the comparison becomes instant.

Fraction → Percentage: multiply the decimal value by 100 Percentage → Decimal: divide by 100  |  Decimal → Percentage: multiply by 100

Example: 234/650 = 234 ÷ 650 = 0.36 = 36%

💡 Think of it like this A percentage is a "universal ruler" — no matter what the original total was (100 people, 650 grams, £8000), converting to percentage puts everything on the same 0–100 scale so you can compare fairly.
Q1. Write 18 out of 24 as a percentage.
Q2. Convert 0.6% into a decimal.

Finding a percentage WITHOUT a calculator

The trick here is to build every percentage out of a few easy "building blocks":

To find...Do this
50%Halve the amount
25%Halve it twice (a quarter)
10%Divide by 10
1%Divide by 100

Then combine blocks to reach any percentage you need:

  • 20% → find 10%, then double it
  • 5% → find 10%, then halve it
  • 12% → find 10% + 1% + 1% (add them together)
  • Over 100% → remember 100% IS the original amount, so 150% = original + 50%
Worked Example Find 35% of 40. Start with 10% of 40 = 4. So 30% = 4 × 3 = 12, and 5% = 4 ÷ 2 = 2. Add them: 35% of 40 = 12 + 2 = 14.
Q. Find 12% of 650 without a calculator.

Finding a percentage WITH a calculator (multipliers)

multiplier

3. Reverse Percentages

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Also in the full note
  • 2. Percentage Increases & Decreases
  • What to Memorise
  • Concepts Checklist
  • Exam Tips
  • Expressing one number as a % of another
  • Increasing by a percentage
  • Decreasing by a percentage
  • Repeated percentage changes
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More International Mathematics topics

Number Operations
Basic Percentages
Powers & Roots
Simple Interest
Prime Factor Decomposition
Converting Fractions, Decimals & Percentages
Set Notation & Venn Diagrams
Basic Fractions
Simplifying Surds
Types of Numbers