Percentage calculator

What this works out

Percentages are five questions wearing one word, and most calculators answer only the first. “What is 15% of 200” and “15 is what per cent of 200” use the same two numbers and give different answers, and the difference between “the price rose 25%” and “the price is 25% of what it was” is somebody’s rent.

So the mode selector is the tool. Pick the question, put the same two numbers in both boxes whichever question you picked, and the panel shows all four related quantities at once: the answer, the amount, that amount as a rate, and the starting value with the amount applied. Reading them together is what stops a percentage being mistaken for an amount.

The method

Underneath the five modes there is one calculation. A percentage question always has a base — the number the rate is measured against — a rate, an amount equal to base x rate / 100, and a total afterwards equal to base + amount. Choosing a mode chooses which two of those four you are supplying; the other two follow.

Two choices in that are worth stating. The first is that percentage change divides by the starting value, never the ending one. From 40 to 50 the difference is 10, and 10 over 40 is 25%; 10 over 50 would be 20%, which is the answer to “50 fell to 40”, a different question. The second is that a decrease is stored as a negative rate rather than as a subtraction. That keeps the amount signed and keeps after = base + amount true in all five modes, so the panel never has to special-case a direction — and a negative rate typed into the increase mode does the sensible thing on its own.

The arithmetic runs in decimal rather than in the browser’s binary floating point. A JavaScript number is an IEEE 754 double, which cannot hold 1.15 or 0.1 exactly, so 1.15 x 0.1 evaluates to 0.11499999999999999. On a page where somebody is checking a discount that is a wrong answer, however small, so every figure here goes through arbitrary-precision decimal arithmetic and is rounded once, at the end, to ten places.

Before you read on

From 40 to 50 is a rise of 25%. What is the fall from 50 back to 40?

• A25% — it is the same ten either way.

  The difference is the same. The base it is measured against is not.

• B20%.

  Yes: 10 over 50, because 50 is now the starting value.

• C22.5% — the average of the two.

  Percentage change has no such symmetry to average.

20%. Percentage change divides by the starting value, never the ending one — so going up it is 10 over 40, and coming back down it is 10 over 50. The two directions of the same move are not the same percentage and never can be, which is why a 50% fall needs a 100% rise to undo it. That single rule is behind most of the arguments people have about this arithmetic.

1. amount = base x rate / 100

   The only multiplication there is. The base is whatever the rate is measured against.

2. after = base + amount

   And the total afterwards is always the base plus it. A decrease is a negative rate, not a subtraction, which keeps this line true in every mode.

3. rate = amount / base x 100

   The same equation rearranged, for when you have the amount and want the rate. That is the "A is what % of B" mode.

4. rate = (b - a) / a x 100

   Percentage change is that line again, with the difference as the amount and the starting value as the base. From 40 to 50 is 10 over 40, not 10 over 50.

A worked example

The same handful of numbers through all five modes:

Question	A	B	Answer	The amount	As a percentage	Start plus the amount
A% of B	15	200	30	30	15%	230
A is what % of B	30	200	15	30	15%	230
From A to B	40	50	25	10	25%	50
A increased by B%	200	15	230	30	15%	230
A decreased by B%	80	20	64	-16	-20%	64

The arithmetic, written out:

15% of 200
  200 x 15 / 100   =  30
  200 + 30         = 230

From 40 to 50
  50 - 40          =  10
  10 / 40 x 100    =  25

80 decreased by 20%
  80 x -20 / 100   = -16
  80 + -16         =  64

And the case that decides whether the arithmetic is done properly:

10% of 1.15
  decimal          = 0.115
  binary float     = 0.11499999999999999

The table below the result runs the same base past the percentages people ask for most, with your own rate slotted into place among them. At a base of 200 that is 2, 4, 10, 20, 25, 30, 40, 50, 60, 80, 100, 120, 150, 180 and 200, and it downloads as a spreadsheet.

These are the same numbers asserted in this tool’s test file, so if the formula ever changes without this page changing with it, the build fails.

What it does not do

It does not know what your numbers are. There is no currency, no unit and no rounding to pence, because 0.435 is the right answer to 5% of 8.70 and turning it into 0.44 is a decision only you can make. It does not compound: five consecutive 10% rises are not a 50% rise, and working that out is a job for a growth calculator rather than this one. It does not handle percentage points as a separate quantity — the FAQ explains the difference, but the boxes take plain numbers. And with a negative starting value it reports the textbook sign, which is the size of the change rather than its direction on the number line.

That growth calculator is compound interest, which applies the rise again to the result instead of once to the start. It is the same percentage doing a different job.

Every percentage question is the same four quantities:

  base     the number the percentage is measured against
  percent  the rate
  amount   base x percent / 100
  after    base + amount

Each mode fixes two of them and the other two follow:

  A% of B             base = B,  percent = A
  A is what % of B    base = B,  amount  = A,      percent = amount / base x 100
  From A to B         base = A,  amount  = B - A,  percent = amount / base x 100
  A increased by B%   base = A,  percent = B
  A decreased by B%   base = A,  percent = -B

after = base + amount, in every mode, including the ones where the amount
is negative.

When base = 0 the percentage is undefined and the tool says so rather than
dividing.