Four calculations that cover almost everything
Most percentage questions reduce to one of four forms, and this page handles each separately so you do not have to work out which formula applies. Every result shows the arithmetic, which is useful if you need to reproduce the calculation elsewhere or explain it to someone.
Finding a percentage of a number
This is the calculation behind sales tax, tips, commission and discounts. Divide the percentage by 100 to turn it into a decimal, then multiply. Fifteen percent of 200 is 0.15 multiplied by 200, which is 30.
Expressing one number as a percentage of another
Divide the part by the whole and multiply by 100. This is how you convert a test score into a percentage, work out what share of a budget a line item represents, or express a conversion rate.
Percentage change
This is where mistakes are most common. Percentage change is the difference between two values divided by the original value, not by the new one, and not by the average of the two. Going from 80 to 100 is a 25 percent increase, because the 20-point difference is measured against the starting figure of 80.
The asymmetry this creates surprises people. A price that rises by 50 percent and then falls by 50 percent does not return to where it started — it ends up 25 percent below. This is because the second calculation is applied to the larger intermediate figure. The same effect explains why an investment that loses 50 percent needs to gain 100 percent to break even.
Percentage points versus percent
These are different things and confusing them causes real misunderstandings. If an interest rate moves from 4 percent to 6 percent, that is a rise of two percentage points, but it is a 50 percent increase in the rate itself. News reports and marketing material sometimes exploit this ambiguity, so it is worth being precise about which you mean.
Compounding
Percentage changes applied in sequence do not add up. Three consecutive increases of 10 percent produce a total increase of 33.1 percent, not 30, because each increase applies to the already-increased figure. This is the mechanism behind compound interest, and it is why small recurring percentage changes matter far more over time than their headline size suggests.
Frequently Asked Questions
Why is a 50% rise followed by a 50% fall not back to the start?
Because the decrease applies to the larger intermediate value. Starting at 100, a 50 percent rise gives 150, and a 50 percent fall from 150 gives 75.
What is the difference between percent and percentage points?
A move from 4 percent to 6 percent is a rise of two percentage points, but a 50 percent increase in the rate. They describe different things.
Which number do I divide by when calculating percentage change?
Always the original starting value, never the new value or the average of the two.
How do I calculate a tip?
Use the first calculator: enter the tip percentage and the bill amount. Fifteen to twenty percent is customary in the United States, though conventions vary widely by country.
Do consecutive percentage increases add together?
No. They compound. Two successive 10 percent increases give 21 percent in total, not 20, because the second applies to the already-increased amount.