What Is a Percentage?
A percentage is a fraction expressed out of one hundred. The word comes from the Latin per centum, meaning "by the hundred", and the symbol % evolved from a handwritten abbreviation of the Italian per cento. Saying that 45% of a class passed an exam is the same as saying 45 out of every 100 students passed, or that the fraction 45/100 — equivalently the decimal 0.45 — passed.
Percentages exist because comparing raw counts is misleading when the totals differ. A team that wins 8 of 10 games and a team that wins 40 of 50 have different records in absolute terms, but both won 80% of their games. Converting to a common base of 100 makes them directly comparable.
The Percentage Formula
Every percentage problem is the same equation rearranged. Three quantities are involved — the percentage, the whole, and the part — and knowing any two gives the third.
Part = Percentage ÷ 100 × Whole
Percentage = Part ÷ Whole × 100
Whole = Part ÷ Percentage × 100
The three calculators at the top of this page are these three forms. For example, with a percentage of 15 and a whole of 200, the part is 15 ÷ 100 × 200 = 30. Reversing it: 30 is 30 ÷ 200 × 100 = 15% of 200. And reversing again: if 30 is 15% of some number, that number is 30 ÷ 15 × 100 = 200.
Converting Between Percentages, Decimals and Fractions
These three notations describe the same quantity, and converting between them is mechanical.
| Conversion | Method | Example |
|---|---|---|
| Percentage → decimal | Divide by 100 (move the point two places left) | 62% → 0.62 |
| Decimal → percentage | Multiply by 100 (move the point two places right) | 0.075 → 7.5% |
| Percentage → fraction | Put over 100 and reduce | 40% → 40/100 → 2/5 |
| Fraction → percentage | Divide, then multiply by 100 | 3/8 → 0.375 → 37.5% |
Common Percentage Equivalents
Memorising a handful of these makes mental arithmetic much faster.
| Percentage | Fraction | Decimal |
|---|---|---|
| 1% | 1/100 | 0.01 |
| 5% | 1/20 | 0.05 |
| 10% | 1/10 | 0.1 |
| 12.5% | 1/8 | 0.125 |
| 20% | 1/5 | 0.2 |
| 25% | 1/4 | 0.25 |
| 33.33% | 1/3 | 0.3333… |
| 50% | 1/2 | 0.5 |
| 66.67% | 2/3 | 0.6667… |
| 75% | 3/4 | 0.75 |
| 100% | 1/1 | 1.0 |
| 150% | 3/2 | 1.5 |
| 200% | 2/1 | 2.0 |
Percentage Change vs. Percentage Difference
These two are frequently confused, and they answer different questions.
Percentage change has a direction. One value is the starting point and the other is where you ended up, so the starting value is the denominator:
Percentage change = (New − Old) ÷ |Old| × 100
Percentage difference has no direction. Neither value is privileged, so the denominator is the average of the two:
Percentage difference = |V1 − V2| ÷ ((V1 + V2) ÷ 2) × 100
Going from 10 to 12 is a percentage change of (12 − 10) ÷ 10 × 100 = 20% increase. But the percentage difference between 10 and 12 is |10 − 12| ÷ 11 × 100 = 18.18%. Use change when there is a before and an after; use difference when comparing two measurements of equal standing, such as results from two laboratories.
Why Percentage Increases and Decreases Do Not Cancel
This is the single most common percentage mistake. A 50% increase followed by a 50% decrease does not return you to where you started, because the two percentages are taken from different bases.
| Start | Operation | Result |
|---|---|---|
| 100 | +50% | 150 |
| 150 | −50% | 75 |
The increase was 50% of 100, but the decrease was 50% of 150 — a larger amount. The net effect is a 25% loss. In general, an increase of p% followed by a decrease of p% always leaves you below the starting value, by p2/100 percent. A stock that falls 50% must then rise 100% just to break even.
Percentage Points Are Not Percentages
If an interest rate rises from 4% to 6%, it has risen by 2 percentage points — but by 50% in relative terms. Both statements are correct and they describe the same event. News reports frequently blur the two, which can make a change sound far larger or smaller than it is. When the quantity being measured is itself a percentage, always state which one you mean.
Worked Examples
Sales tax
An item costs $48 and the sales tax rate is 8.25%. The tax is 8.25 ÷ 100 × 48 = $3.96, so the total is $51.96.
A discount
A $180 jacket is marked 35% off. The discount is 0.35 × 180 = $63, so you pay $117. A shortcut: paying "35% off" means paying 65% of the price, so 0.65 × 180 = $117 directly.
Stacked discounts
An item is 20% off, and a coupon takes a further 10% off at the register. This is not 30% off. The price becomes 0.80 × 0.90 = 0.72 of the original, which is 28% off. Successive percentages multiply; they do not add.
A tip
A restaurant bill is $86.40 and you want to leave 18%. The tip is 0.18 × 86.40 = $15.55, giving a total of $101.95.
Reversing a percentage
You paid $63.60 including 6% sales tax and want the pre-tax price. The $63.60 represents 106% of the original, so the original is 63.60 ÷ 1.06 = $60.00. Note that subtracting 6% from $63.60 would give $59.78, which is wrong — a common error.
Mental Shortcuts
- 10% of anything is the number with the decimal point moved one place left. 10% of 847 is 84.7.
- 1% moves it two places: 1% of 847 is 8.47.
- Build from 10%. 30% is three lots of 10%; 5% is half of 10%; 15% is 10% plus half of it.
- Percentages commute. x% of y always equals y% of x. Working out 4% of 75 is awkward, but 75% of 4 is obviously 3.
- 50%, 25%, 20% are just halving, quartering and dividing by five.
Frequently Asked Questions
How do I calculate a percentage of a number?
Divide the percentage by 100 and multiply by the number. 22% of 350 is 0.22 × 350 = 77.
How do I work out what percentage one number is of another?
Divide the part by the whole and multiply by 100. If 34 of 40 questions were correct, that is 34 ÷ 40 × 100 = 85%.
Can a percentage exceed 100?
Yes, whenever the part is larger than the whole. If revenue grows from $2m to $5m, the growth is 150%. Percentages are bounded by 100 only when they describe a share of a fixed total.
How do I remove a percentage that has already been added?
Divide rather than subtract. To strip 20% VAT from a gross price, divide by 1.20 — do not subtract 20%, which gives a different and incorrect answer.