To find a percentage of a number, multiply the number by the percentage and divide by 100: 20% of 150 is 150 × 20 ÷ 100 = 30. To find what percent one number is of another, divide the part by the whole and multiply by 100: 30 is 20% of 150. Percentage change is (new − old) ÷ old × 100: a rise from 100 to 120 is 20 ÷ 100 × 100 = +20%.
Percentages turn up in almost every practical calculation: a discount in a shop, a tip on a bill, an exam score, a pay rise, an interest rate, a tax band. This calculator handles the three questions that account for nearly all everyday percentage work, and the sections below explain the formulas so you can do them on paper when you need to. For the two most common special cases there are dedicated tools: the tip calculator and the salary calculator for pay rises.
Percentage Calculator
Checked against: 20% of 150 = 30; a change from 100 to 120 = +20% Last reviewed:
Sources and method
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The three percentage questions
Almost every percentage problem is one of these three, and the hardest part is usually recognising which one you are looking at.
- What is X% of Y? You know the percentage and the whole, and you want the part. This is the discount, the tip, the commission, the tax.
- X is what percent of Y? You know the part and the whole, and you want the percentage. This is the test score, the market share, the completion rate.
- What is the percentage change from X to Y? You have a before and an after, and you want the movement between them. This is the pay rise, the price increase, the year-on-year growth.
The formulas
Finding a percentage of a number
Multiply the whole by the percentage and divide by 100:
Part = (Percentage ÷ 100) × Whole
A quicker route is to convert the percentage to a decimal by moving the decimal point two places left, then multiply once. 20% becomes 0.20, so 20% of 150 is 0.20 × 150 = 30. This is the version worth memorising, because it works cleanly on any calculator.
Finding what percentage one number is of another
Divide the part by the whole, then multiply by 100:
Percentage = (Part ÷ Whole) × 100
The wording tells you which number goes where. Whatever follows the word of is the whole, and belongs on the bottom of the fraction. In “42 out of 60”, the 60 is the whole.
Finding percentage change
Take the difference, divide by where you started, then multiply by 100:
Change = ((New − Old) ÷ Old) × 100
The critical detail is that you divide by the original value, never the new one. A positive answer is an increase and a negative answer is a decrease, so you only need one formula for both.
Worked examples
A discount in a shop
A jacket costs 89.99 and is marked 35% off. The discount is 0.35 × 89.99 = 31.50, so you pay 89.99 − 31.50 = 58.49.
There is a shortcut worth knowing. If 35% comes off, you are paying 65% of the original, so you can go straight to the answer: 0.65 × 89.99 = 58.49. One multiplication instead of two steps, and fewer chances to slip.
A test score
You scored 42 marks out of a possible 60. That is (42 ÷ 60) × 100 = 70%. If the pass mark is 65%, you needed 0.65 × 60 = 39 marks, so you passed with three marks to spare.
A pay rise
Your salary moves from 48,000 to 52,500. The difference is 4,500, and 4,500 ÷ 48,000 = 0.09375, which is a 9.38% increase. Note that dividing by the new salary instead would give 8.57% — a plausible-looking number that is simply wrong.
Fraction, decimal and percentage equivalents
Most everyday percentages are simple fractions in disguise, and recognising them makes mental arithmetic far quicker.
| Fraction | Decimal | Percentage | Of 200 |
|---|---|---|---|
| 1/2 | 0.5 | 50% | 100.00 |
| 1/3 | 0.3333 | 33.33% | 66.67 |
| 2/3 | 0.6667 | 66.67% | 133.33 |
| 1/4 | 0.25 | 25% | 50.00 |
| 3/4 | 0.75 | 75% | 150.00 |
| 1/5 | 0.2 | 20% | 40.00 |
| 2/5 | 0.4 | 40% | 80.00 |
| 1/6 | 0.1667 | 16.67% | 33.33 |
| 1/8 | 0.125 | 12.5% | 25.00 |
| 3/8 | 0.375 | 37.5% | 75.00 |
| 5/8 | 0.625 | 62.5% | 125.00 |
| 1/10 | 0.1 | 10% | 20.00 |
| 1/16 | 0.0625 | 6.25% | 12.50 |
| 1/20 | 0.05 | 5% | 10.00 |
| 1/100 | 0.01 | 1% | 2.00 |
Percentages of percentages
Percentage changes do not add up, they multiply. This is the single most common source of real-world percentage errors.
Suppose a stock falls 20% and then rises 20%. Starting at 100, the fall takes it to 80. The 20% rise is then calculated on 80, not on 100, so it adds 16 and you finish at 96 — down 4%, not back to even. To combine successive changes, multiply the factors: 0.80 × 1.20 = 0.96.
The same logic applies to stacked discounts. “30% off, then a further 20% off at the till” is not 50% off. It is 0.70 × 0.80 = 0.56, which is 44% off.
Percent versus percentage points
These are different units, and confusing them can distort a number badly.
If an interest rate rises from 4% to 6%, the rate has risen by 2 percentage points. But as a percentage change, it has risen by (6 − 4) ÷ 4 = 50%. Both statements are correct and they describe the same event, yet “rates rose 50%” and “rates rose 2 points” land very differently on a reader.
The rule: use percentage points when subtracting two percentages, and percent when expressing the change relative to the starting figure.
Common mistakes
- Dividing by the wrong number in a percentage change. The denominator is always the original value. This is the error that produces most wrong answers.
- Adding successive percentages. Two 10% rises are a 21% rise, not 20%, because the second applies to an already larger base.
- Reversing a percentage by subtracting it. If a price includes 20% tax, the pre-tax price is not the price minus 20%. Divide by 1.20 instead.
- Averaging percentages that have different bases. A 50% pass rate in a class of 10 and a 90% rate in a class of 90 do not average to 70%; weight each by its class size.
- Losing the decimal point. 5% is 0.05, not 0.5. Writing the decimal out before multiplying prevents a factor-of-ten error.
Frequently asked questions
How do I calculate a percentage of a number?
Multiply the number by the percentage, then divide by 100. For example, 20% of 150 is (150 x 20) / 100 = 30. You can also convert the percentage to a decimal first and multiply once: 0.20 x 150 = 30.
How do I work out what percentage one number is of another?
Divide the part by the whole, then multiply by 100. If you scored 42 out of 60, that is (42 / 60) x 100 = 70%. The number after the word “of” always goes on the bottom of the fraction.
What is the formula for percentage increase?
Subtract the original value from the new value, divide by the original value, then multiply by 100. Going from 80 to 100 is ((100 – 80) / 80) x 100 = 25% increase. Use the same formula for a decrease; the answer simply comes out negative.
Why is a 50% rise followed by a 50% fall not back where it started?
Because each percentage is taken from a different base. 100 rises by 50% to 150, but the 50% fall is then taken from 150, not from 100, so you lose 75 and end at 75. Percentage changes multiply rather than add, which is why they do not simply cancel out.
What is the difference between percent and percentage points?
A percentage point is the arithmetic gap between two percentages. If interest rises from 4% to 6%, that is a rise of 2 percentage points, but a 50% increase in the rate itself. News reports frequently mix these up, and the difference can be very large.
How do I reverse a percentage to find the original price?
Divide by 1 plus the percentage as a decimal. If a price is 120 after a 20% markup, the original was 120 / 1.20 = 100. To reverse a discount, divide by 1 minus the decimal: a sale price of 80 after 20% off was 80 / 0.80 = 100.
Can a percentage be more than 100?
Yes. Any time the part is larger than the whole you get more than 100%. If revenue grows from 50,000 to 150,000 that is a 200% increase. Percentages above 100 are only impossible when the quantity is bounded, such as the share of a single fixed total.
Is percentage change the same as percentage difference?
No. Percentage change has a clear before and after, and divides by the starting value. Percentage difference compares two values with no natural order, and divides by their average. Use change for growth over time, and difference when comparing two independent measurements.