Percentages in simple words
“Percent” means “out of 100”. 25% means 25 out of every 100, which is the same as a quarter. 100% is the whole thing, and 50% is half.
To find a percent of a number, divide the percent by 100 and multiply. 15% of 200 is (15 ÷ 100) × 200 = 0.15 × 200 = 30. So if 200 children are in a school and 15% walk to school, 30 children walk.
To find what percent one number is of another, divide and multiply by 100. 30 out of 200 is (30 ÷ 200) × 100 = 15%.
To find a percent change, compare the change with where you started. A price that goes from 120 to 150 rises by 30, and 30 ÷ 120 × 100 = 25%, so it went up by 25%.
Choose the correct reference value
Percentage change compares with the original value; percentage difference compares with the average of two nonnegative values. This page keeps increases and decreases in the same change tool rather than creating duplicate calculators. For negative original values, change uses the original magnitude as its denominator.
Going from 120 to 150 is a 25% increase. Going back from 150 to 120 is a 20% decrease. Their symmetric percentage difference is 30/135 × 100 ≈ 22.2222%. A change from zero has no finite percentage; two zero values have no defined percentage difference in this convention.
The three percentage questions
Almost every percentage problem is one of three forms, and knowing which one you have tells you the operation to use.
- Finding a part: what is 15% of 200? Multiply — (15 ÷ 100) × 200 = 30.
- Finding a rate: 30 is what percent of 200? Divide, then scale — (30 ÷ 200) × 100 = 15%.
- Finding a whole: 30 is 15% of what? Divide by the rate — (30 ÷ 15) × 100 = 200.
| Percent | of 40 | of 80 | of 120 | of 250 | of 1,000 |
|---|---|---|---|---|---|
| 5% | 2 | 4 | 6 | 12.5 | 50 |
| 10% | 4 | 8 | 12 | 25 | 100 |
| 12.5% | 5 | 10 | 15 | 31.25 | 125 |
| 15% | 6 | 12 | 18 | 37.5 | 150 |
| 20% | 8 | 16 | 24 | 50 | 200 |
| 25% | 10 | 20 | 30 | 62.5 | 250 |
| 33⅓% | 13.3333 | 26.6667 | 40 | 83.3333 | 333.3333 |
| 50% | 20 | 40 | 60 | 125 | 500 |
| 75% | 30 | 60 | 90 | 187.5 | 750 |
Percent, decimal and fraction
A percentage is a fraction with 100 as the denominator, so every percentage has a decimal and a fraction form, and switching between them is often the fastest way to compute. Multiplying by 25% is dividing by 4; 12.5% is dividing by 8.
| Percent | Decimal | Fraction |
|---|---|---|
| 1% | 0.01 | 1/100 |
| 5% | 0.05 | 1/20 |
| 10% | 0.1 | 1/10 |
| 12.5% | 0.125 | 1/8 |
| 20% | 0.2 | 1/5 |
| 25% | 0.25 | 1/4 |
| 33⅓% | 0.333… | 1/3 |
| 50% | 0.5 | 1/2 |
| 66⅔% | 0.667… | 2/3 |
| 75% | 0.75 | 3/4 |
| 100% | 1 | 1/1 |
| 150% | 1.5 | 3/2 |
Percentage change, points, and the direction trap
Percentage change is (new − old) ÷ |old| × 100. The absolute value in the denominator keeps the sign meaningful when the starting value is negative.
Percentage change is not symmetric, which catches people out constantly. Going from 100 to 150 is a 50% increase, but going from 150 back to 100 is a 33.3% decrease. A 50% loss requires a 100% gain to recover. Similarly, a 20% increase followed by a 20% decrease does not return you to the start: 100 becomes 120, then 96.
Percentage points are a separate unit. If an interest rate moves from 3% to 4%, that is a rise of one percentage point but a 33% relative increase. News reports conflate the two often enough that it is worth checking which is meant.
| From | To | Change | Change needed to return |
|---|---|---|---|
| 100 | 150 | +50.0% | -33.3% |
| 150 | 100 | -33.3% | +50.0% |
| 100 | 50 | -50.0% | +100.0% |
| 50 | 100 | +100.0% | -50.0% |
| 80 | 100 | +25.0% | -20.0% |
| 100 | 80 | -20.0% | +25.0% |
Percentage points versus percent
When a rate changes, two different numbers describe the move. The difference in points is the arithmetic gap between the two rates; the percent change is that gap relative to where it started. Both are correct; they answer different questions, and the smaller-sounding one is usually the one quoted by whoever wants the change to sound small.
| Rate moves | In percentage points | As a percent change |
|---|---|---|
| 4% → 5% | +1 point | +25% |
| 10% → 12% | +2 points | +20% |
| 50% → 40% | −10 points | −20% |
| 2% → 4% | +2 points | +100% |
Percentage difference
When neither value is the starting point — two shops' prices, two students' scores — percentage change is the wrong tool, because it gives a different answer depending on which you call the base. Percentage difference divides the gap by the average of the two values instead, so it is symmetric: 50 and 70 are 33.3% apart whichever you name first. Use change when there is a before and an after, and difference when there is not.
Reverse percentages
When you know a total that already includes a percentage, you must divide rather than subtract. If a price with 20% tax is $120, the pre-tax price is 120 ÷ 1.20 = $100, not 120 − 20% = $96. The percentage was applied to the smaller original figure, so subtracting it from the larger total overshoots.
The same logic applies to discounts. An item marked 30% off at $70 had an original price of 70 ÷ 0.70 = $100. This is the calculation to use whenever you need to recover an original value from a discounted or tax-inclusive one.
Mental arithmetic shortcuts
Percentages are commutative, which is the most useful trick available: 4% of 75 is the same as 75% of 4, and the second is far easier. Beyond that, 10% is a decimal shift, 1% is two shifts, and 5% is half of 10%. Combining these covers most everyday cases — 15% is 10% plus half of it, and 20% is double the 10% figure.
Decimals, percents and shares of a total
A percent is a decimal multiplied by 100: 0.25 × 100 = 25%, and 12.5% ÷ 100 = 0.125. Moving the decimal point two places is the same operation, which is why the converter never introduces rounding.
Each option's share is its count divided by the total, times 100. With 5,200, 4,100 and 700 votes the total is 10,000, so the shares are 52%, 41% and 7%. The lead is 1,100 votes, or 11 percentage points, and 52% is a majority. If 16,000 people were eligible, turnout is 10,000 ÷ 16,000 = 62.5%.