What is a percentage?
A percentage is a way of expressing a ratio as parts per hundred. The symbol % literally tells you that the comparison has been scaled to 100. Thus, 25% means 25 out of 100, or 0.25 as a decimal and 1/4 as a fraction.
Percentages make quantities easier to compare because the denominator is standardized. A test score of 82 out of 100 can be read directly as 82%, while 123 out of 150 becomes easier to interpret after converting it to 82%.
Four percentage questions you should recognize
Find the part
What is 15% of 200? Multiply the whole by 15/100.
Find the percent
25 is what percent of 200? Divide the part by the whole and scale by 100.
Find the whole
10 is 5% of what? Rearrange the same equation to solve for the whole.
Measure change
Compare a starting value and an ending value using percentage change.
Percentage formulas
- part
- the amount represented by the percentage
- whole
- the reference total
- P
- the percentage number, such as 15 for 15%
This form answers “what percent of the whole is the part?”
This reverse form is useful when you know the part and its percentage.
Use + for an increase and − for a decrease.
The starting value is the reference point, so the direction matters.
Percentage difference treats A and B symmetrically by using their average as the denominator.
Worked examples
15% of 200
Answer: 30
25 is what percent of 200?
Answer: 12.5%
10 is 5% of what?
Answer: 200
80 increased by 25%
Answer: 100
500 decreased by 10%
Answer: 450
Difference between 10 and 6
Answer: 50%
Percentage change vs percentage difference
From 80 to 100, the reference is the starting value 80. The increase is 20, so 20/80 × 100 = 25%.
For 10 and 6, there is no designated start. The denominator is their average, 8, so 4/8 × 100 = 50%.
Percentages in everyday life
Percentages appear in discounts, taxes, test scores, survey results, financial rates, statistics, scientific concentrations and measurement error. The calculation is the same; what changes is the meaning of the reference value.
Quick reference
| Percentage | Decimal | Fraction | Of 200 |
|---|---|---|---|
| 1% | 0.01 | 1/100 | 2 |
| 5% | 0.05 | 1/20 | 10 |
| 10% | 0.10 | 1/10 | 20 |
| 12.5% | 0.125 | 1/8 | 25 |
| 25% | 0.25 | 1/4 | 50 |
| 50% | 0.50 | 1/2 | 100 |
| 75% | 0.75 | 3/4 | 150 |
| 100% | 1.00 | 1 | 200 |
Percentage points are not the same as percent change
If a rate moves from 20% to 25%, the change is 5 percentage points. The relative percentage change is 25% because 5 ÷ 20 × 100 = 25%. Keeping these two ideas separate prevents a common interpretation error.
Reverse percentages: finding the original value
Suppose a product is priced at $90 after a 10% discount. The $90 is 90% of the original price, so the original is 90 ÷ 0.90 = $100. Dividing by the remaining percentage is the key step; simply adding 10% to $90 would not reverse the discount correctly.
FAQ
A percentage is a ratio expressed per 100. For example, 35% is 35/100 = 0.35.
Divide the part by the whole and multiply by 100. If 20 of 50 objects are selected, 20/50 × 100 = 40%.
Percentage change uses a designated starting value. Percentage difference uses the average of two values, so neither value is treated as the starting point.
Treat the discounted price as 90% of the original. Divide the discounted price by 0.90 to recover the original price.
Yes. A percentage greater than 100% means the numerator is larger than the reference whole. For example, 150% means 1.5 times the reference value.