To calculate a percentage, divide the part by the whole and multiply by 100. If 9 of 36 questions are answered correctly, 9 ÷ 36 = 0.25, and 0.25 × 100 = 25%. The word percent means "per one hundred", so every percentage is a ratio with a denominator of 100.
What are the three basic percentage questions?
Nearly every percentage problem is one equation with three parts: the percent, the base (the whole), and the amount (the part). OpenStax writes it as percent × base = amount. Knowing two of the three gives the third.
| You want | Formula | Example |
|---|---|---|
| The amount (a percent of a number) | amount = percent (as a decimal) × base | 35% of 90 = 0.35 × 90 = 31.5 |
| The percent (part as a share of a whole) | percent = amount ÷ base × 100 | 9 of 36 = 9 ÷ 36 × 100 = 25% |
| The base (the whole) | base = amount ÷ percent (as a decimal) | 12 is 15% of what? 12 ÷ 0.15 = 80 |
How do you convert a percent to a decimal?
Write the percent over 100 (a fraction you can also turn into a decimal), or move the decimal point two places to the left and drop the % sign. 35% becomes 0.35, 7% becomes 0.07, and 125% becomes 1.25. Convert back by moving the point two places to the right.
How do you find a percent of a number?
Convert the percent to a decimal and multiply by the number.
- 35% of 90: 0.35 × 90 = 31.5.
- 125% of 28: 1.25 × 28 = 35. A percent above 100 gives an amount larger than the base.
- 8% of 45 (for example, a sales tax at an assumed 8% rate): 0.08 × 45 = 3.60.
Is there a mental shortcut?
Ten percent is the base with the decimal point moved one place left. Build other percentages from it. For 15% of 80: 10% is 8, 5% is half of that (4), and 8 + 4 = 12. For 20% of 80, double 10%: 16.
How do you calculate what percent one number is of another?
Divide the part by the whole and convert the decimal to a percent.
- Put the part over the whole: 42 out of 50 becomes 42 ÷ 50.
- Divide: 42 ÷ 50 = 0.84.
- Multiply by 100: 84%.
The same step works when the part is larger than the whole. 144 ÷ 96 = 1.5, so 144 is 150% of 96. This is the situation behind weighted grade scores: each score is first turned into a percentage of its own maximum.
How do you calculate percent increase and decrease?
Subtract to find the change, then divide by the original amount.
percent change = (new − original) ÷ original × 100
- Increase from 26 to 36: change = 10, and 10 ÷ 26 = 0.385, so a 38.5% increase.
- Decrease from $3.71 to $3.64: change = 0.07, and 0.07 ÷ 3.71 = 0.019, so a 1.9% decrease.
A negative result means a decrease. The original value is always the denominator, which is the most common place to go wrong. A fuller definition is in the glossary entry on percent change.
Why do an increase and a decrease of the same percent not cancel?
The second percentage is applied to a different base. Start at 100, increase by 20% to reach 120, then decrease by 20%: 120 × 0.80 = 96, not 100. Recovering from a 20% drop needs a 25% rise, because 80 × 1.25 = 100.
How do you work backwards to the original price?
If a price already includes a discount or markup, divide by the remaining fraction rather than adding the percentage back.
An item costs $64 after a 20% discount. It is 80% of the original, so original = 64 ÷ 0.80 = $80. Adding 20% to $64 would give $76.80, which is wrong. The discount calculator does this in both directions.
What is the difference between percent and percentage points?
Percentage points compare two percentages by simple subtraction. Percent change compares them as a ratio. If an interest rate moves from 4% to 5%, it rose by 1 percentage point, and by 1 ÷ 4 = 25% in relative terms. Reports that mix up the two can make a change look much bigger or smaller than it is.
Common mistakes
- Dividing by the new value instead of the original when computing percent change.
- Forgetting to convert the percent to a decimal before multiplying (using 35 instead of 0.35).
- Adding and subtracting percentages from different bases as if they could be combined.
- Treating "percent off" and "percentage points off" as the same thing.
What is the proportion method?
Percent problems can also be written as a proportion: amount ÷ base = percent ÷ 100. For "what is 20% of 70?", write n ÷ 70 = 20 ÷ 100, so n = 70 × 20 ÷ 100 = 14. The proportion form gives the same result as the decimal method and is useful when the numbers are easier to cross-multiply than to convert.