Percentages Explained: Cheat Sheet with Worked Examples
A one-page percentages cheat sheet: convert fractions, decimals and percentages, find a percentage of an amount, increase and decrease with a multiplier, work out a percentage change and solve reverse percentage problems, each with a worked example.
Ages 11–14 · Grades 6–8 (US) · Years 7–9 (UK) · Classes 6–8 (India)
- Common Core Math 6.RP.A.3c
- Common Core Math 7.RP.A.3
- England KS3 Years 7–9
- AQA GCSE R9
- Cambridge IGCSE 0580 C1.13 / E1.13
- Australian Curriculum v9 AC9M7N06
- Australian Curriculum v9 AC9M8N05
- CBSE Class 8
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What a percentage means
A percentage is a number of parts out of 100. The word comes from the Latin per centum, "by the hundred", so 37% means 37 out of every 100, the fraction 37/100 or the decimal 0.37. Percentages make different totals easy to compare: 18 out of 24 and 30 out of 40 are both 75%.
100% is the whole amount. Percentages can be bigger than 100%: if a plant grows from 40 cm to 60 cm, its new height is 150% of the old one.
Converting fractions, decimals and percentages
- Percentage → decimal: divide by 100, so 45% = 0.45 and 7% = 0.07.
- Decimal → percentage: multiply by 100, so 0.375 = 37.5%.
- Fraction → percentage: divide the top by the bottom, then multiply by 100. 3/8 = 0.375 = 37.5%.
- Percentage → fraction: write it over 100 and simplify. 45% = 45/100 = 9/20.
Useful equivalents to remember: 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/10 = 10% and 1/3 ≈ 33.3%.
Finding a percentage of an amount
Without a calculator, build the percentage from 10%, 5% and 1%. To find 35% of 240: 10% is 24, 30% is 72, 5% is 12, so 35% is 84. To find 12% of 240: 10% is 24 and 1% is 2.4, so 12% is 24 + 2.4 + 2.4 = 28.8.
With a calculator, change the percentage to a decimal and multiply: 0.35 × 240 = 84. To write one amount as a percentage of another, divide and multiply by 100, after putting both in the same unit: 45 cm out of 2 m is 45 ÷ 200 × 100 = 22.5%.
Increase, decrease and percentage change with multipliers
A multiplier changes an amount by a percentage in one step. It is 100% plus or minus the change, written as a decimal.
| Change | Multiplier | Example |
|---|---|---|
| Increase by 15% | 1.15 | GBP 60 × 1.15 = GBP 69 |
| Decrease by 25% | 0.75 | EUR 80 × 0.75 = EUR 60 |
| Increase by 3% | 1.03 | 200 kg × 1.03 = 206 kg |
| Decrease by 8% | 0.92 | USD 50 × 0.92 = USD 46 |
To find a percentage change, divide the change by the original value and multiply by 100. A price that rises from USD 50 to USD 62 changes by 12, and 12 ÷ 50 × 100 = 24%. Dividing by the new value, 62, would give the wrong answer.
Reverse percentages
In a reverse percentage problem you know the value after the change and need the original. Write the new value as a percentage of the original, turn it into a multiplier and divide.
After a 20% discount a phone costs GBP 360. The sale price is 80% of the original, so the original is 360 ÷ 0.8 = GBP 450. A price that includes a 20% sales tax of EUR 96 was 96 ÷ 1.2 = EUR 80 before tax. Always check by multiplying back: 450 × 0.8 = 360.
Mistakes to watch for
- Finding 10% by dividing by 100 instead of 10.
- Writing 0.06 as 60% instead of 6%.
- Dividing by the new value when finding a percentage change.
- Undoing a 20% decrease by adding 20%: 360 × 1.2 = 432, not 450.
- Expecting a 10% rise and a 10% fall to cancel out: 100 × 1.1 × 0.9 = 99.
- Mixing up percentages and percentage points: a rise from 40% to 50% is 10 percentage points, but a 25% increase.
Frequently asked questions
How do you find a percentage of an amount?
Change the percentage to a decimal by dividing by 100, then multiply by the amount. For example, 35% of 240 is 0.35 × 240 = 84. Without a calculator, build it from 10% (24), 30% (72) and 5% (12).
What is a multiplier in percentages?
It is the decimal you multiply by to increase or decrease an amount in one step: 100% plus or minus the change. A 15% increase uses 1.15 and a 25% decrease uses 0.75.
How do you calculate percentage change?
Subtract to find the change, divide by the original value and multiply by 100. From USD 50 to USD 62 the change is 12, so the increase is 12 ÷ 50 × 100 = 24%.
How do you find the original price after a discount?
Work out what percentage of the original is left, turn it into a multiplier and divide by it. After 20% off, GBP 360 is 80% of the original, so the original is 360 ÷ 0.8 = GBP 450.
Sources & methodology
Every fact is checked against the sources below. We write original explanations and draw original graphics; no figures are copied from textbooks. Spotted an error? See our corrections policy.
- Prealgebra 2e, 6.1 Understand Percent (OpenStax (Rice University), accessed 1 Oct 2026)
- Prealgebra 2e, 6.2 Solve General Applications of Percent (OpenStax (Rice University), accessed 1 Oct 2026)
- Grade 7, Unit 4: Proportional Relationships and Percentages (Illustrative Mathematics (Kendall Hunt), accessed 1 Oct 2026)
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