Percentage Calculator

What is a percentage calculator?

A percentage calculator takes two numbers and works out the relationship between them as a percentage, or applies a percentage to a number to get a new one. Instead of one box that only does one kind of sum, this one has five modes, picked with buttons, so you never have to rearrange the formula in your head.

• What is P% of X? answers questions like 25% of 80 is 20.
• X is what % of Y? answers questions like 20 is 25% of 80.
• Percentage change from X to Y works out that going from 50 to 75 is a 50% increase, and shows the plain absolute difference alongside it.
• Increase or decrease X by P% has an Increase or Decrease toggle, so 80 increased by 25% gives 100.
• Reverse a percentage change takes the value after the change and the percentage, and recovers the original: something that costs 80 after 20% off started out at 100.

Every mode shows the answer in large type, a line showing how it was worked out, a badge saying whether that answer is exact or rounded, and a copy button.

How to use it

1. Pick a mode. Choose the button for the question you actually have — what a percentage of a number is, what percentage one number is of another, a change between two values, an increase or decrease, or a reverse calculation.
2. Enter your two numbers. Type them into the two fields for that mode. A comma works as the decimal separator as well as a period, so 10,5 and 10.5 are both read correctly.
3. Read the result. The answer appears instantly in large type, with the working shown underneath, a badge marking it exact or rounded, and a button to copy it.

Percent versus percentage points, and why increases and decreases do not cancel out

Two mistakes account for most of the confusion around percentages, and both are easy to check once you know to look for them.

The first is percent versus percentage points. If a rate goes from 5% to 6%, that is a rise of one percentage point — the plain difference between the two numbers. But expressed as a relative change, it is a 20% increase, because 1 is 20% of 5. Both statements are true. They just answer different questions, and mixing them up is the single most common percentage error in news and reporting. The percentage change mode shows the absolute difference next to the percentage change for exactly this reason, so both numbers are in front of you at once and neither can be mistaken for the other.

The second is that a percentage increase and the same percentage decrease do not cancel out, even though it feels like they should. 100 increased by 10% is 110. Decrease that same 110 by 10% and you get 99, not 100. The percentage is the same, but it is 10% of a different number each time, and that gap is real, not a rounding artifact — it is worth working through once by hand so it stops being surprising.

Why the answer is exact, not just close

Percentages are decimal fractions, and binary floating point — the number format almost every calculator and spreadsheet uses internally — cannot represent most decimal fractions exactly. It is the same defect behind the famous 0.1 + 0.2 printing as 0.30000000000000004 instead of 0.3.

Across 20,000 random percent-of-a-number calculations, each with inputs of at most two decimal places, doing the arithmetic the naive way in ordinary JavaScript numbers printed a different answer than the true one in 42.4% of cases — 8,475 out of 20,000. Every single one of those failures had the same cause: spurious digits trailing off past the six decimal places the exact answer can ever actually need. 1.1% of 70 comes out as 0.7700000000000001 the naive way, when the exact answer is 0.77. 35% of 5.7 gives 1.9949999999999999 instead of 1.995. 17.5% of 1.15 gives 0.20124999999999998 instead of 0.20125.

A calculator that prints 0.7700000000000001 is not broken so much as naive — it is doing the arithmetic in a format that was never built to hold decimal fractions cleanly, and just showing whatever comes out. This calculator sidesteps the problem instead of hiding it: every number you type is converted to an exact fraction, all the arithmetic is done exactly, and rounding happens exactly once, only when the answer is displayed.

That makes it possible to answer a question most percentage calculators never even ask: is the number on screen exact, or was it rounded to display? That is a decidable fact, not a guess — a fraction in lowest terms has a terminating decimal only if the sole prime factors of its denominator are 2 and 5, since 10 is 2 times 5. A third of something is flagged rounded; an eighth is exact and prints as 0.125. When rounding is needed, it rounds half away from zero, the same rule spreadsheets use for ROUND, so 0.125 to two places reads 0.13 — consistent with the spreadsheet sitting next to it.

Two situations have no answer at all, and the tool says so plainly rather than showing something misleading: asking what percentage a number is of zero, and reversing a 100% decrease, since everything collapses to zero and the original value cannot be recovered. Silently rounding is worse than saying which answers are rounded, because an exact number and a correctly-rounded approximation look identical on screen unless something tells them apart. The badge is that something.

Why is it free?

The whole calculation runs in your browser. Nothing you type is uploaded anywhere, there is no account to create, and there is no watermark on anything you copy out. It is free the same way every tool on this site is free: because it costs nothing to run.