The four percentage formulas
Almost every percentage question is one of four calculations. The calculator above has a mode for each of them; here is how each one works so you can check the answer or do it by hand.
1. What is X% of Y?
Divide the percentage by 100, then multiply by the number.
Example: a jacket costs $150 and is 20% off. The discount is 20 ÷ 100 × 150 = $30, so you pay $120.
2. X is what percent of Y?
Divide the part by the whole, then multiply by 100.
Example: you scored 42 out of 60 on a test. 42 ÷ 60 × 100 = 70%.
3. Percentage change from X to Y
Find the difference, divide by the starting value, then multiply by 100. A positive answer is an increase; a negative answer is a decrease.
Example: your rent rises from 25,000 to 27,500 a month. (27,500 − 25,000) ÷ 25,000 × 100 = 10% increase.
4. Increase or decrease a number by X%
Multiply by 1 plus the percentage as a decimal. For a decrease, use a negative percentage.
Example: a salary of 80,000 with a 6% raise becomes 80,000 × 1.06 = 84,800.
How to calculate percent off a price
A discount is a percentage decrease. You can find the saving with mode 1, or the price you pay with mode 4 and a negative percentage.
Example: a phone case is $80 with 25% off. You pay 80 × 0.75 = $60, a saving of $20.
Two discounts in a row do not add up. “20% off, plus an extra 10% at the till” means 0.80 × 0.90 = 0.72 of the price, so the total saving is 28%, not 30%. The second discount applies to a price that has already been cut. For sale prices, stacked discounts and sales tax with the working shown, use the discount calculator.
How to calculate marks percentage
To turn exam marks into a percentage, divide the marks you obtained by the total marks, then multiply by 100. This is mode 2 above.
Example: in Pakistan, India and Bangladesh, results are often shown as total marks across all subjects. If your board exam totals 1,100 marks and you scored 935, your percentage is 935 ÷ 1,100 × 100 = 85%. With five subjects of 100 marks each and 437 marks in total, it is 437 ÷ 500 × 100 = 87.4%.
Always use the total of all subjects, not the average of each subject’s percentage, unless every subject has the same maximum. If one paper is out of 75 and another out of 100, averaging the two percentages gives the wrong answer. Some boards and universities also have their own rules, such as counting only your best five subjects, so check what yours uses. To convert a CGPA into a percentage, see the CGPA calculator.
Percent difference vs percent change
These two sound alike but answer different questions.
- Percent change compares a new value with an old one. The old value is the base, so direction matters.
- Percent difference compares two values when neither is “the original”, such as two shop prices or two lab results. It divides by their average.
Percent difference = |A − B| ÷ ((A + B) ÷ 2) × 100
Worked example with 40 and 50:
| Calculation | Working | Answer |
|---|---|---|
| Percent change, 40 → 50 | (50 − 40) ÷ 40 × 100 | 25% increase |
| Percent change, 50 → 40 | (40 − 50) ÷ 50 × 100 | 20% decrease |
| Percent difference, 40 and 50 | 10 ÷ 45 × 100 | 22.22% |
Percent change gives a different answer depending on which number you start from. Percent difference gives the same answer either way, which is why it suits comparisons between equals. This calculator’s % change mode does percent change; for percent difference, use the formula above.
How to calculate percent error
Science classes use percent error to compare a measured value with the accepted (true) value.
Example: you measure gravity as 9.6 m/s², and the accepted value is 9.81 m/s². The error is |9.6 − 9.81| ÷ 9.81 × 100 = 0.21 ÷ 9.81 × 100 = 2.14%. The bars around the top mean you ignore the minus sign, so percent error is never negative. You can get the same number from the % change mode by entering the true value first and the measured value second, then dropping the minus sign.
Reverse percentage: finding the original number
If you know the value after a percentage change, divide by the multiplier to get back to the start.
Examples:
- A price is 150 after a 20% increase. The original was 150 ÷ 1.20 = 125.
- You paid 68 after a 15% discount. The original price was 68 ÷ 0.85 = 80.
- A bill of £96 includes 20% VAT. The price before VAT was 96 ÷ 1.20 = £80.
The common mistake is to take 20% off the final number: 150 − 20% = 120, not 125. That fails because the 20% was worked out on the original, not on 150. The percentage increase calculator has a reverse mode that does this step for you, along with average growth over several periods.
Common mistakes to avoid
- Increases and decreases are not symmetric. A 50% fall needs a 100% rise to recover. If a share price drops from 100 to 50, it must double to get back to 100.
- Use the original value as the base. Percentage change always divides by the starting value, not the ending value.
- Percent vs percentage points. An interest rate moving from 4% to 5% is a rise of 1 percentage point, but a 25% increase in the rate itself.
- Successive percentages multiply. Two 10% increases add up to 21%, not 20%, because 1.10 × 1.10 = 1.21.
Quick reference table
| Question | Formula | Example |
|---|---|---|
| 15% of 200 | 15 ÷ 100 × 200 | 30 |
| 30 is what % of 120 | 30 ÷ 120 × 100 | 25% |
| Change from 50 to 65 | (65 − 50) ÷ 50 × 100 | 30% increase |
| 400 decreased by 12.5% | 400 × (1 − 0.125) | 350 |
| 25% off 80 | 80 × 0.75 | 60 |
| 935 marks out of 1,100 | 935 ÷ 1,100 × 100 | 85% |
| Original before a 20% rise to 150 | 150 ÷ 1.20 | 125 |
Where percentages show up
Percentages are everywhere: discounts and sales tax, exam marks, salary raises, inflation, loan interest and investment returns. For money questions with interest over time, use the compound interest calculator. For marks and grades, try the GPA calculator or the grade calculator. Our guide on how to calculate percentage walks through every formula again, with a cheat sheet of common fractions as percentages.
The calculator accepts commas as thousands separators and decimals, so you can paste figures like 1,250.75 straight from a bank statement or spreadsheet. Nothing you type is sent anywhere; the maths happens in your browser.