Percentage Calculator

The three questions percentages answer

A percentage is a fraction with a denominator of 100, so every percentage question is a ratio question in disguise. Fifteen per cent of 250 is 250 × 15 ÷ 100, which is 37.5. That is the "share of a number" case, and it is the one people mean most often.

• Share: what is 15% of 250? Multiply, then divide by 100.
• Ratio: 45 is what percentage of 250? Divide, then multiply by 100 — 18%.
• Change: from 250 to 300 is an increase of 20%, because the change is measured against the starting figure.

The third case is where most errors happen. The denominator is always the original value, not the new one, so a move from 100 to 120 is a 20% rise, while a move back from 120 to 100 is a 16.7% fall. The same numbers, two different percentages.

Why a rise and a fall do not cancel out

Percentages apply to the value in front of them, so a 20% rise followed by a 20% fall does not return to the start: 100 becomes 120, and 120 less 20% is 96. The second percentage was taken of a bigger base, so it removed more than the first added.

This is the same reason discounts and markups are not symmetric. A 25% discount needs a 33.3% markup to get back to the original price. Any time two percentages are applied in sequence, multiply the factors rather than adding the percentages: 1.20 × 0.80 = 0.96.

Two traps worth knowing

Percentage points are not percentages. Moving a rate from 4% to 5% is a rise of one percentage point, but it is a rise of 25% in the rate itself. Both statements are true and they mean very different things, which is why news about interest rates or tax rates is so easy to misread.

Rounding should happen last. Rounding a percentage to a whole number before multiplying can shift a result by several units on large amounts, so keep full precision through the calculation and round only the figure you are going to quote.

What each input means

The calculator asks for 2 pieces of information. Every field is validated before the maths runs, so an impossible value is refused rather than quietly producing a wrong answer.

A worked example using the default inputs

With the number of 250, the percentage of 15 %, the calculator returns 37.50.

The same arithmetic runs whatever you type into the form above, so you can reproduce any figure by matching the inputs listed here.

How this is calculated

Part = whole × percentage ÷ 100. Increase = whole × (1 + percentage ÷ 100).

Working values are kept at full precision and rounded only for display. Impossible dates, zero denominators, negative amounts where they make no sense, and out-of-range values are rejected before any result is shown.

The methods these maths tools apply

Each tool in this section implements one standard definition. The value of knowing which is that you can verify a result with rough mental arithmetic and can spot an input mistake before it costs you anything.

Order of operations is the rule most often broken by hand and by spreadsheet. A negative number squared is positive only when the minus sign is inside the brackets, and a trigonometric function applied in degree mode returns a different number from the same input in radian mode, so the mode matters as much as the digits.

Rounding is a display decision, not a mathematical one. Keeping full precision until the final line is why a result printed as 0.33 can still be exactly one third underneath.

Three worked examples from this section

15% of 240, both directions

Fifteen per cent of 240 is 240 × 0.15 = 36. Adding it gives 276; taking it away gives 204. Going back from 276 by subtracting 15% returns 234.60, which is why "add 15% then subtract 15%" never lands where you started.

15% of 240

36

240 increased by 15%

276

240 decreased by 15%

204

276 reduced by 15%

234.60

Solving x² − 5x + 6 = 0

Here a = 1, b = −5 and c = 6, so the discriminant is 25 − 24 = 1. A positive discriminant means two real roots: (5 ± 1) ÷ 2 gives 2 and 3. The vertex sits at x = 2.5, where the curve dips to −0.25.

The spread of 4, 8, 15, 16, 23, 42

Six values summing to 108 give a mean of 18. Sorted, the middle pair is 15 and 16, so the median is 15.5. Dividing the sum of squared deviations (910) by n − 1 and taking the square root gives a sample standard deviation of about 13.5 — most of the list sits within one of those of the mean, with 42 the clear outlier.

Precision, rounding and exactness

Arithmetic here follows the standard mathematical definition of each operation. Working values are kept at full precision and rounded only when they are printed, so intermediate rounding never accumulates into the final answer.

• Where a result is inexact, it is shown to a sensible number of significant figures rather than padded with meaningless digits.
• Very large or very small inputs may be displayed in scientific notation; the underlying value is unchanged.
• Where a problem has more than one valid answer, as with quadratic equations or signed square roots, every branch is reported rather than only the positive one.

Sources and further reading

• NIST Digital Library of Mathematical Functions