# Percentage calculator

> Find X% of Y, what percent one number is of another, percentage increase or decrease, percent difference, and the original before a percentage was added.

Interactive version: https://www.calcopenly.com/math/percentage-calculator
Subject: Math calculators

A percentage is a number out of 100: 15% means 15/100 = 0.15, so 15% of 240 is 0.15 × 240 = 36. The calculator answers seven questions built on that rule: X% of Y, what percent X is of Y (X ÷ Y × 100), percent change ((Y − X) ÷ |X| × 100), percent difference (the gap divided by the average of the two numbers), the whole when a part and its percentage are known, and increasing or decreasing a number by a percentage.

Typical uses are discounts, sales tax, pay rises, exam marks and price changes. A price rising from 80 to 100 has gone up 25%, but falling back from 100 to 80 is a 20% drop, because each change is measured against its own starting value.

A negative percent change is a decrease. Percent change needs a non-zero starting value and percent difference a non-zero average; the calculator reports an error rather than dividing by zero.

## Percentage formulas at a glance

Every percentage problem links three numbers: a part, a whole and a rate, with part = rate ÷ 100 × whole. Knowing any two gives the third. The % sign stands for the number 0.01 ([NIST Guide to the SI, §7.10.2](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values)), so dividing by 100 is all it takes to turn a percentage into a decimal: 7.5% is 0.075.

| Question | Formula | Example | Answer |
|---|---|---|---|
| What is X% of Y? | X ÷ 100 × Y | 7.5% of 64 | 4.8 |
| X is what percent of Y? | X ÷ Y × 100 | 27 out of 36 | 75% |
| Y is X% of what? | Y ÷ (X ÷ 100) | 54 is 45% of what? | 120 |
| Increase Y by X% | Y × (1 + X ÷ 100) | 250 plus 12% | 280 |
| Decrease Y by X% | Y × (1 − X ÷ 100) | 250 minus 12% | 220 |
| Percent change from X to Y | (Y − X) ÷ \|X\| × 100 | 250 to 280 | +12% |
| Percent difference of X and Y | \|Y − X\| ÷ ((X + Y) ÷ 2) × 100 | 250 and 280 | 11.32% |

The last two rows use the same pair of numbers and disagree because they divide by different bases: 250 for the change, and the average, 265, for the difference.

## How to work out a percentage by hand

A student scores 43 out of 52 on a test. To express that as a percentage:

1. Divide the part by the whole: 43 ÷ 52 = 0.826923…
2. Multiply by 100: 82.6923…%.
3. Round once, at the end: 82.7%.

The calculator's "X is what % of Y" mode, with X = 43 and Y = 52, returns 82.6923%. Rounding in step 1 instead, to 0.83, reports 83%, which is 0.3 points too high and can be enough to cross a grade boundary.

For mental arithmetic, build the rate out of easy pieces. Moving the decimal point one place left gives 10%, two places gives 1%, half of 10% is 5%, and dividing by 4 or 8 gives 25% or 12.5%. A 15% tip on a bill of 84 is 8.40 (10%) plus 4.20 (5%), or 12.60. The [tip and bill split calculator](/everyday/tip-and-bill-split-calculator) does the same sum and divides it between people.

Converting between fractions, decimals and percentages is the same operation in different notation. Multiply a decimal by 100 to get the percentage; divide the numerator by the denominator to get the decimal.

| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.33% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.1666… | 16.67% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 2/3 | 0.666… | 66.67% |
| 3/4 | 0.75 | 75% |
| 1/20 | 0.05 | 5% |

## Percent change and percentage points

When the two values being compared are themselves percentages, there are two ways to state the change. A mortgage rate moving from 2.5% to 3.0% rose by 0.5 percentage points, found by plain subtraction, and by 20% in relative terms, found with the percent change formula (0.5 ÷ 2.5 × 100). In basis points, where one basis point is 0.01 percentage point, the move is 50 basis points.

A fall and a rise of the same percentage do not cancel, because the rise is measured from the lower value. The rise needed to recover from a fall of p% is p ÷ (100 − p) × 100:

| Fall | Rise needed to get back to the start |
|---|---|
| 5% | 5.26% |
| 10% | 11.11% |
| 20% | 25% |
| 25% | 33.33% |
| 40% | 66.67% |
| 50% | 100% |
| 75% | 300% |
| 90% | 900% |

The gap widens quickly: an investment that loses half its value has to double to break even.

The calculator divides a change by the absolute value of the starting number. Moving from −50 to −25 therefore reads as a 50% increase, because the value rose towards zero. A spreadsheet formula written as (new − old) ÷ old returns −50% for the same move, so check which convention a report uses before comparing figures. When the starting value is 0 no percent change exists; report the absolute change instead.

## Reverse percentages: finding the original value

To undo a percentage change, divide by the multiplier that produced it. After a 15% discount a jacket costs 68. The sale price is 85% of the original, so the original was 68 ÷ 0.85 = 80. In the calculator, choose "Y is X% of what" with Y = 68 and X = 85.

The tempting shortcut, adding 15% of 68 back on, gives 68 × 1.15 = 78.20, which is 1.80 short, because the 15% was taken from 80, not from 68. The same error appears when removing sales tax or VAT from a gross price; the [GST, VAT and sales tax calculator](/finance/gst-vat-sales-tax-calculator) handles that case directly.

Markup and margin are a reverse-percentage pair. An item that costs 80 and sells for 100 has a markup of 25% (20 ÷ 80) and a margin of 20% (20 ÷ 100). The profit is the same 20 in both; only the base changes. The [discount and markup calculator](/everyday/discount-and-markup-calculator) converts between the two.

## Common percentage mistakes

Successive discounts multiply rather than add. A 20% discount followed by a further 10% off leaves 0.80 × 0.90 = 0.72 of the price, a total discount of 28%, not 30%. The order of the two discounts makes no difference.

| First discount | Second discount | Combined discount |
|---|---|---|
| 10% | 10% | 19% |
| 20% | 10% | 28% |
| 20% | 20% | 36% |
| 25% | 25% | 43.75% |
| 30% | 20% | 44% |
| 50% | 20% | 60% |
| 50% | 50% | 75% |

"50% off, then an extra 50% off" takes three quarters off the price, not all of it. Other errors that recur:

- **Adding successive increases.** Two 10% rises multiply to 1.1 × 1.1 = 1.21, a 21% increase. Interest that is added to the balance each period grows the same way; the [compound interest calculator](/finance/compound-interest-calculator) runs that over many periods.
- **Dividing a total change by the number of years.** A price that rises from 100 to 150 over five years has gone up 50%, but not 10% a year: five 10% rises compound to 1.1⁵ = 1.61, a 61% increase. The steady yearly rate that produces 50% is 1.5^(1/5) − 1 = 8.45%. The [scientific calculator](/math/scientific-calculator) evaluates `1.5^(1/5) - 1` in one line.
- **Confusing "percent of" with "percent more than".** 150% of 40 is 60, while 150% more than 40 is 40 + 60 = 100. In the calculator the first is "X% of Y" and the second is "Increase Y by X%".
- **Averaging percentages with different bases.** If 30% of 200 people in one town and 60% of 50 people in another answer yes, the combined share is (60 + 30) ÷ 250 = 36%. The simple average of 30% and 60%, 45%, gives the town of 50 the same weight as the town of 200.
- **Decreasing by more than 100%.** A positive quantity cannot fall by more than 100% and stay positive. "Decrease Y by X%" with X above 100 returns a negative number: 50 decreased by 120% is −10. If that is not what the situation allows, the rate or the base is wrong.
- **Rounding the rate before using it.** A rate rounded to 8% instead of 8.45% understates five years of growth by more than 3 percentage points, since 1.08⁵ = 1.469. Keep full precision through the calculation and round only the figure you report.

## Inputs

- **I want to find** (options: X% of Y, X is what % of Y, % change from X to Y, % difference between X and Y, Y is X% of what, Increase Y by X%, Decrease Y by X%)
- **X**
- **Y**
- **X**
- **Y**
- **From (X)**
- **To (Y)**

## Results

- Result — main result
- In words

## Formula

$$
\begin{gathered} \text{X\% of Y} = \frac{X}{100} \times Y \\[10pt] \%\,\text{change} = \frac{Y - X}{|X|} \times 100 \end{gathered}
$$

## Worked examples

### 15% of 240

- I want to find: X% of Y
- X: 15%
- Y: 240
- **Result: 36**
- Checked against: 0.15 × 240

### 36 is what % of 240

- I want to find: X is what % of Y
- X: 36
- Y: 240
- **Result: 15 %**
- Checked against: 36 / 240 × 100

### Change from 80 to 100

- I want to find: % change from X to Y
- From (X): 80
- To (Y): 100
- **Result: 25 %**
- Checked against: (100 − 80) / 80 × 100

### Difference between 80 and 100

- I want to find: % difference between X and Y
- From (X): 80
- To (Y): 100
- **Result: 22.222222 %**
- Checked against: 20 / 90 × 100 = 22.2…

### 45 is 18% of what

- I want to find: Y is X% of what
- X: 18%
- Y: 45
- **Result: 250**
- Checked against: 45 / 0.18

## Questions

### How do you calculate a percentage of a number?

Divide the percentage by 100 and multiply: 15% of 240 is 0.15 × 240 = 36. The order does not matter, so X% of Y equals Y% of X; 8% of 50 is the same as 50% of 8, which is 4. Percentages above 100 follow the same rule: 150% of 240 is 1.5 × 240 = 360.

### How do you calculate percentage increase or decrease?

Subtract the old value from the new one, divide by the old value and multiply by 100. From 80 to 100 is (100 − 80) ÷ 80 × 100 = 25% up; from 100 back to 80 is (80 − 100) ÷ 100 × 100 = 20% down. The two differ because each is measured against its own starting value, so a 25% rise is undone by a 20% fall.

### What is the difference between percent change and percent difference?

Percent change has a direction and divides by the starting value; percent difference has none and divides by the average of the two numbers. For 80 and 100 the change is 25% (or −20% going the other way), while the difference is 20 ÷ 90 × 100 ≈ 22.22% whichever number comes first. Use percent difference when neither value is a baseline, such as two lab measurements.

### How do you find the original price before a percentage was added?

Divide by 1 plus the rate as a decimal. A price of 120 that includes 20% tax was 120 ÷ 1.2 = 100 before tax. Taking 20% off 120 gives 96, which is wrong because the 20% was charged on 100, not on 120. In this calculator choose “Y is X% of what” and enter Y = 120 and X = 120%.

### What is a percentage point?

A percentage point is the plain difference between two percentages. An interest rate moving from 4% to 5% rises by 1 percentage point, which is a 25% relative increase because 1 ÷ 4 = 0.25. Calling that move “a 1% rise” would instead mean 4% × 1.01 = 4.04%, which is why statistics and news reports use percentage points for changes in rates.

### How accurate is the percentage calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 5 worked examples whose answers come from independent sources; for example, “15% of 240” is checked against 0.15 × 240.

### Where does the method come from?

NIST Guide to the SI, §7.10.2 — percent.

## Sources

- [NIST Guide to the SI, §7.10.2 — percent](https://www.nist.gov/pml/special-publication-811)