Five percentage questions, five formulas
A percentage is a number out of 100, so every percentage problem is a small piece of arithmetic with a hundred in it. The trouble is that the same word covers several different questions, and the right formula depends on which one you are asking. This calculator separates them: a percentage of a number, what percent one number is of another, percentage change, increasing or decreasing by a percentage, and working backwards to the original. Pick the question, enter the two numbers and the steps are written out so you can check the logic.
What is a percentage of a number?
Divide the percentage by 100 and multiply by the number. For 20% of 150 that is 20 ÷ 100 × 150 = 30. Moving the decimal point is a shortcut: 10% of any number is the number divided by 10, and 1% is the number divided by 100. Percentages above 100 work the same way, so 150% of 40 is 60, and a negative percentage gives a negative answer.
This number is what percent of another?
Divide the part by the whole and multiply by 100. If 45 of 180 questions were answered correctly, 45 ÷ 180 × 100 = 25%. The whole is the number after the word of, and it cannot be zero, because nothing is a percentage of zero. The calculator tells you when the whole is zero instead of showing an infinite or empty result.
Percentage change between two values
Percentage change measures how far a value moved compared with where it started: subtract the old value from the new one, divide by the old value and multiply by 100. From 80 to 100 the change is (100 − 80) ÷ 80 × 100 = 25%. Going back from 100 to 80 is not the mirror image: (80 − 100) ÷ 100 × 100 = -20%. The two answers differ because each change is measured against a different starting point. That asymmetry is why a price that rises 25% needs to fall only 20% to return to where it began.
The starting value is the base, so it cannot be zero, and the formula is easiest to read when it is positive. If the starting value is negative the sign can mislead. Going from −50 to −25 is an improvement of 25, yet the signed formula gives -50%, because it divides by a negative number. When that happens the calculator shows the signed result and also the result that divides by the size of the starting value, which is 50%, with a note explaining the difference.
Increase or decrease a number by a percentage
To raise a number by a percentage, multiply it by 1 plus the percentage as a decimal. To lower it, multiply by 1 minus the percentage. Taking 15% off 120 means 120 × (1 − 0.15) = 102. Successive changes do not cancel. Increase 100 by 20% and you reach 120; decrease that by 20% and you arrive at 96, not 100, because the second 20% is taken from a larger number. A decrease cannot be larger than 100%, since that would make the value negative, so the calculator refuses it.
Reverse percentage: find the original value
If a price is 92 after a 20% decrease, the original was not 20% above it. The discount was taken from the original, so divide the final value by the share that remains: 92 ÷ 0.80 = 115. Adding 20% to 92 would give 110.4, which is wrong. For an increase, divide by 1 plus the percentage instead. A 100% decrease leaves nothing, so the original cannot be recovered, and the calculator explains that case.
Percent and percentage points
Percentage points describe the arithmetic gap between two percentages, and percent describes the relative change. A rate that moves from 5% to 7% has risen by 2 percentage points, which is a 40% relative increase. News stories and reports often use the two interchangeably, so check which one is meant. To work out a relative change between two rates, enter the rates as the starting and ending values in the change mode.
Worked examples
| Question | Answer |
|---|---|
| 20% of 150 | 30 |
| 45 is what percent of 180 | 25% |
| Change from 80 to 100 | 25% |
| 120 decreased by 15% | 102 |
| Original of 92 after a 20% decrease | 115 |
Where percentages show up
Shopping uses percentages for sales, so see the discount calculator for coupons that stack and the sales tax calculator or VAT calculator for tax on a price. Businesses use them for pricing: the margin and markup calculator shows why a 25% markup is a 20% margin. Students can turn category scores into a grade with the weighted grade calculator, and a pay change is a percentage question too, which the salary calculator handles for annual, monthly and hourly pay.
Percentage calculator questions
How do I calculate a percentage of a number?
Multiply the number by the percentage and divide by 100. For 20% of 150, 20 × 150 ÷ 100 = 30.
How do I calculate percentage increase?
Subtract the old value from the new one, divide by the old value and multiply by 100. From 80 to 100 that is 25%.
Why is going up 20% and down 20% not back to the start?
Each change is measured against the value before it. Up 20% takes 100 to 120, and 20% of 120 is larger than 20% of 100, so the way down ends at 96.
How do I find the original price before a discount?
Divide the final price by the share that remains after the discount. A final price of 92 after 20% off is 92 ÷ 0.80 = 115.
What is the difference between percent and percentage points?
Percentage points are the plain gap between two percentages. A move from 5% to 7% is 2 points, but a 40% relative increase.
Can a percentage be negative or above 100?
Yes. A negative percentage means a decrease or a value in the opposite direction, and a percentage above 100 means more than the whole, such as a price that has more than doubled. The one thing you cannot do is divide by zero.