💯 Percentage calculator — every way to work one out
A percentage question comes in about eight shapes, and the awkward part is never the arithmetic — it is working out which sum answers the question you actually have. So every shape is on this page as a line you can type into: the answer appears at the end of the line, the workings sit under it, and the line you are typing in goes on the card.
The line you are on
The same number at the percentages people ask for most
A percentage is always a share of something, and most percentage mistakes are that something changing without saying so: two percentages only add up when they are shares of the same thing, which is why a price cut by 20% and then by 20% again is 36% off rather than 40% — the second cut is taken off the smaller number. Every line above stands on its own and is worked out from its own figures, so nothing typed in one line moves another. Answers are rounded to four decimal places, which is exact enough to act on without pretending to be exact, and a line that cannot be worked out — a percentage of nothing, or what percent one thing is of zero — says so instead of printing a figure.
Why use this calculator?
Almost nobody struggles with percentage arithmetic — that is multiplying and dividing. What people struggle with is knowing which sum their question actually is: a share of a number, one number as a share of another, a change, a discount, or the whole before any of it. Get the sum wrong and the answer comes out confidently wrong, which is what makes percentages feel slippery. So this page puts every shape of the question on one screen, as a line you type into, with the answer at the end of the line and the workings underneath it.
Where percentages come from
Percentages are Latin, and both of their famous traps are centuries old:
- Per centum is Latin for "by the hundred", and the word arrived in English in the sixteenth century for rates and interest. The mark itself has a murkier history: it is usually traced to Italian merchants shortening cento in their ledgers — often as a c with a stroke through it — in the fifteenth and sixteenth centuries, and the % is what that shorthand settled into. Nobody sat down and designed it.
- A percentage is always relative, so percentages only add up when they share a base. A price cut by 20% and then by 20% again is 36% off, not 40%: the first cut takes $20 off $100, and the second takes $16 off the $80 that is left. Turn it round and a 20% rise followed by a 20% fall leaves 96, not 100 — which is how a fund can fall a smaller percentage than it rose and still be worth less than it was.
- Percentage points exist because "percent" is used two ways at once. Going from 5% to 7% is two percentage points — the gap between the rates — and it is also a 40% increase, because 2 is 40% of 5. Both statements are true, which is why the distinction was given a name. Interest rates, inflation and election swings are all reported both ways, and which one a headline meant changes what it says.
How does this calculator work?
Every line on the page is one of these, and nothing else:
p% of n = n × p ÷ 100
- A percentage of a number: n × p ÷ 100, which is the same as n × 0.15 for fifteen percent. Dividing by 100 first is worth doing out loud, because the answer to "one percent of it" is the figure most people are really after.
- One number as a percentage of another: part ÷ whole × 100. This is the line to use for a mark, a share of a bill, or how much of a total one item is.
- The change from one number to another: (new − old) ÷ old × 100. The change is measured against where it started, which is why a rise from 80 to 100 is a 25% increase and a fall from 100 to 80 is a 20% decrease — the same gap of 20, and both figures are right.
- Adding or taking off a percentage: n × (1 + p/100) to add it, n × (1 − p/100) to take it off. A 15% discount leaves 85% of the price, so the sum is one multiplication rather than a subtraction of a subtraction.
- Working back to the whole: the figure you have ÷ (p ÷ 100) — for when you know a part of something and want the whole of it.
- The difference between two numbers: the gap over the average of the two, × 100 — because neither number is the starting point, which is what separates it from a change.
- A figure that already has the percentage in it: divide by (1 + p/100). A number that is 15% more than what you started with is 115% of it, so 92 ÷ 1.15 gives 80. Taking 15% off 92 gives 78.2 instead, which is the sum that looks right and is not.
- Percentage points: the gap between the two rates, b − a, with the relative change beside it as (b − a) ÷ a × 100. Both are printed because both get quoted.
- Rounding: answers are printed to four decimal places and trailing zeros are dropped, so 33% of 100 is 33 and 1 ÷ 3 as a percentage is 33.3333%. A line that cannot be worked out — a percentage of nothing, or one number as a percentage of zero — prints a dash and says so underneath rather than showing an infinity.
Frequently asked questions
What is the difference between a percentage change and a percentage difference?
The change is measured against one of the numbers — the one that came first — and the difference is measured against the average of both, because with a difference neither number is the starting point. Going from 80 to 100 is a 25% change, and the two numbers differ by about 22%. Use the change when one figure came before the other, and the difference when you are comparing two things side by side with no order to them.
Why is a 20% rise then a 20% fall not back where it started?
Because the second percentage is taken from the bigger number. 100 up 20% is 120, and 20% of 120 is 24, so it falls to 96. Percentages are shares, and a share of a bigger number is a bigger amount — which is also why a price cut twice by 20% is 36% off rather than 40%, and why an investment that falls 50% needs a 100% rise to get back to where it was.
How do I work back to a figure before a percentage was added?
Divide by the percentage rather than take it off. A figure that is 15% more than what you started with is 115% of it, so dividing by 1.15 gives the figure you started from — 92 ÷ 1.15 is 80, which is the line above working backwards. Taking 15% off 92 gives 78.2 instead, and that is the mistake the division exists to avoid: the percentage was a share of the smaller figure, and 15% of a bigger number is a bigger amount.
What is the difference between a percentage and a percentage point?
A percentage point is the gap between two rates; a percentage is a share of something. Going from 5% to 7% is two percentage points — that is the whole difference — and it is also a 40% increase, because 2 is 40% of 5. Both are true, and both get quoted: "the rate rose two points" and "the rate rose 40%" describe the same change from different sides.
Why is a 200% increase three times as much?
Because an increase sits on top of what was already there. A 100% increase adds the whole of the original number, so 100 becomes 200 — double. A 200% increase adds twice the original, so 100 becomes 300 — triple. Falls work the same way in reverse, and nothing can fall by more than 100%: at that point it is all gone.
How do I work out a discount in my head?
Build it from ten percent, which is a decimal point moved one place. 10% of $80 is $8; 5% is half of that, $4; 15% is both, $12; 20% is two tens, $16; 25% is a quarter. Then take it off, or take the leftover: 15% off $80 leaves 85%, which is also $68. The same trick works for a tip, and for splitting a bill: 10% of $120 is $12, so 15% is $18.
Why does a line sometimes say it cannot be worked out?
Because one of the sums has nothing to divide by. "What percent is 12 of 0?" has no answer — every percentage of nothing is still nothing — and a percentage of a number that is itself zero is zero, which is a real answer rather than a dash. Where there is no answer the page prints a dash and says underneath which figure has to change, rather than printing Infinity.