What a percentage is
The word comes from the Latin per centum, “by the hundred”. Writing 15% means 15 parts out of every 100, which is simply the number 0.15. Measurement guides such as NIST’s SI style guide treat the % sign exactly that way: as a symbol for the number 0.01. So “P%” is shorthand for P × 0.01, and taking a percentage of something is always a multiplication.
Any percentage statement links three quantities: a whole (also called the base), a part of it, and the percent that the part represents. If you know two of them, the third follows. That is why the calculator has three rows:
- What is P% of X? You know the percent and the whole, and want the part.
- A is what percent of B? You know the part and the whole, and want the percent.
- A is P% of what number? You know the part and its percent, and want the whole.
The same equation written three ways
Each line is the first one rearranged, so they always agree. The arithmetic is exact; only results such as 1 ÷ 3 that never end are rounded for display. A whole of zero in the second row, or a percent of zero in the third, has no answer because it would mean dividing by zero.
Moving between percents, decimals and fractions
To turn a percent into a decimal, divide by 100, which moves the decimal point two places left: 37.5% becomes 0.375. To go back, multiply by 100. For a fraction, write the percent over 100 and simplify: 12.5% is 12.5/100, which is 1/8. In the other direction, divide the top of the fraction by the bottom and multiply by 100, so 3/8 is 0.375, or 37.5%.
| 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 |
| 33.33…% | 0.333… | 1/3 |
| 50% | 0.5 | 1/2 |
| 75% | 0.75 | 3/4 |
| 150% | 1.5 | 3/2 |
The swap trick
P% of X always equals X% of P, because both are P × X ÷ 100. This turns awkward sums into easy ones. 8% of 50 looks fiddly, but 50% of 8 is plainly 4, and so is the original. Likewise 4% of 25 is the same as 25% of 4, which is 1. When one of the two numbers is a friendly percentage such as 10, 25 or 50, swap them and work out that version instead.
Worked examples
- Turn the percent into a decimal: 15 ÷ 100 = 0.15.
- Multiply by the whole: 0.15 × 80 = 12.
- Divide the part by the whole: 18 ÷ 24 = 0.75.
- Multiply by 100: 0.75 × 100 = 75%.
- Write 30% as a decimal: 0.30.
- Divide the part by it: 45 ÷ 0.30 = 150.
- Check with the first row: 30% of 150 is 45.
- Divide: 7 ÷ 40 = 0.175.
- Multiply by 100 to get 17.5%.
Where percentage sums usually go wrong
- Dividing the wrong way round. “18 is what percent of 24” means 18 ÷ 24 = 75%. Dividing 24 by 18 gives 133.33%, which answers a different question: what percent 24 is of 18.
- Leaving out the division by 100. 15% of 80 is 12, not 15 × 80 = 1,200. If the answer is bigger than the whole for a percent under 100, the 100 is missing.
- Using the first row when you need the third. If 45 was 30% of a bill, the bill is 45 ÷ 0.30 = 150. Taking 30% of 45 gives 13.5, which is the answer to a different question.
- Rounding a repeating percent too early. One third written as 33% understates it: 33% of 900 is 297, while a third of 900 is 300. Keep the full value until the last step.
- Rejecting results above 100%. 150% of 40 is 60, a perfectly valid answer when the part is larger than the reference. The limit applies only to shares of a single whole: no more than 100% of a class can be present.
The part, the whole and the percent must refer to the same thing. “30% of the students” and “30% of the girls” use different wholes, so their parts cannot be added or compared directly.
Questions people ask
How do I find 10% of a number in my head?
Move the decimal point one place to the left: 10% of 64 is 6.4. Halve that for 5% (3.2) and double it for 20% (12.8).
Can a percentage be more than 100 or less than zero?
A share of one whole sits between 0% and 100%, but a comparison can exceed 100%, as in 150% of 40. Negative percentages appear when describing a fall, which is what the percent change calculator handles.
Sources
- Guide for the Use of the International System of Units (SI), NIST SP 811, Chapter 7 (Sec. 7.10.2, percent), National Institute of Standards and Technology.