Percentage Calculator
Three related questions get confused with each other constantly: what is X percent of Y, X is what percent of Y, and what is the percent change from A to B. Each has a different formula, and picking the wrong one produces an answer that looks plausible.
How to use it
- Choose which of the three calculations you want.
- Enter the two values.
- Read the result and the formula used, so you can check the interpretation.
The three calculations
Percent of multiplies: 25 percent of 200 is 0.25 times 200, which is 50. This is the tip, tax, and discount case.
What percent divides: 50 is what percent of 200 is 50 divided by 200, which is 25 percent. This is the share-of-total case — conversion rates, market share, pass rates.
Percent change divides the difference by the original: from 200 to 250 is 50 divided by 200, a 25 percent increase. The critical detail is which value goes in the denominator. Change is always relative to the starting point, and swapping them gives a different answer: from 250 to 200 is a 20 percent decrease, not 25.
Percentage changes do not cancel out
This is the single most consequential property of percentages, and it catches experienced people regularly.
A 50 percent gain followed by a 50 percent loss does not return you to where you started. $100 rises to $150, then loses 50 percent of $150 — which is $75, not $50 — leaving $75. You are down 25 percent overall.
The reason is that each percentage applies to a different base. The gain applies to $100 and the loss applies to $150. Reversing a 50 percent loss requires a 100 percent gain, and reversing an 80 percent loss requires a 400 percent gain. This asymmetry is why volatility erodes compounded returns, and why the average of a series of percentage changes overstates the actual outcome.
The correct way to combine successive changes is to multiply the factors: a 20 percent rise then a 10 percent fall is 1.20 times 0.90, which is 1.08, an 8 percent net rise. Adding and subtracting the percentages gives 10 percent, which is wrong.
Percentage points are not percentages
A rate moving from 2 percent to 3 percent has risen by one percentage point and by 50 percent. Both statements are true and they describe the same event, which is why the distinction gets exploited.
The relative figure sounds dramatic when the base is small. An interest rate going from 0.5 percent to 1 percent is a 100 percent increase, which is technically accurate and rhetorically misleading. Conversely, a conversion rate improving from 2 percent to 2.4 percent is only 0.4 percentage points but a 20 percent lift in conversions, which is the figure that matters commercially.
The convention that avoids ambiguity is to use percentage points for absolute differences between rates and percent for relative changes, and to state which you mean whenever the base is small enough for the two to diverge sharply.
Discounts, taxes, and reversing them
Successive discounts compound rather than add. Thirty percent off followed by another 20 percent off is 0.70 times 0.80, which is 0.56 — a 44 percent total discount, not 50 percent. Stacked-coupon promotions rely on this reading larger than it is.
Order does not matter for pure multiplication, so applying a discount before or after another discount gives the same result. It does matter when tax is involved and the jurisdiction specifies whether tax applies to the pre-discount or post-discount amount.
Reversing a percentage requires division, not subtraction. To find the pre-discount price from a sale price after 30 percent off, divide by 0.70. To find the pre-tax amount from a total including 8 percent tax, divide by 1.08. Subtracting the percentage from the final figure is the most common percentage error in everyday use and always produces a number that is close enough to look right.
At a glance
| Percent of | (percent / 100) times value |
|---|---|
| What percent | part divided by whole, times 100 |
| Percent change | (new minus old) divided by old, times 100 |
| Transmitted | Nothing |
Frequently asked questions
Why does a 50 percent gain then a 50 percent loss lose money?
Each percentage applies to a different base. $100 rises to $150, then loses 50 percent of $150, leaving $75. Reversing a 50 percent loss needs a 100 percent gain.
How do I combine two percentage changes?
Multiply the factors. A 20 percent rise then a 10 percent fall is 1.20 times 0.90 equals 1.08, an 8 percent net rise. Adding the percentages gives the wrong answer.
What is the difference between percent and percentage points?
2 percent to 3 percent is one percentage point and a 50 percent relative increase. Use percentage points for differences between rates and percent for relative change.
How do I reverse a discount?
Divide, do not subtract. After 30 percent off, divide the sale price by 0.70 to recover the original.
Read more
Everyday math — A 50 percent gain and a 50 percent loss leave you down 25 percent, and that is the least surprising thing here.