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Mathematics · pattern A, Numeric fields

Percentage of, percentage change and percentage difference.

Enter two values to get one as a percentage of the other, the change between them and the percentage difference — three questions that are constantly confused for one another.

Inputs

Value AValue BA percentage%For the of-value rows below.

The formula used

A of B = A ÷ B × 100; change = (new − old) ÷ old × 100

Percentage change divides by the original value. Percentage difference divides by the average of the two, which is what makes it symmetric.

A as a percentage of B

25.0000 %

Percentage change from B to A: -75.0000 %
Percentage difference: 120.0000 %
That percentage of B: 27.0000
B increased by that percentage: 207.0000
B decreased by that percentage: 153.0000
Percentage that reverses the change: 300.0000 %

A rise and a fall of the same percentage do not cancel.

A 20% rise followed by a 20% fall leaves you 4% down, because the fall is calculated on the larger amount. The last row above gives the percentage that actually reverses a change, and it is never the same figure — which is behind a great deal of misleading statistical writing.

Questions about percentage

Change or difference?: Change is directional and divides by the original value; difference is symmetric and divides by the average. Use change whenever one value came first.
Why does a 50% loss need a 100% gain to recover?: Because the gain is measured against the reduced amount. Halving then doubling returns to the start, and doubling is a 100% rise.
Can a percentage exceed 100?: Yes for increases. It cannot for a share of a whole whose parts are exhaustive.

A calculator handles the arithmetic. It cannot teach you the method.

If the number is not the part you are stuck on, that is what the service is for — a specialist who explains the working, not just the answer.

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Arithmetic runs in double-precision floating point, so results beyond about fifteen significant figures are not exact. Where a question wants an exact fraction or surd, keep the exact form rather than a decimal.