Percentage Calculator
Three different questions get called "a percentage calculation" and they need different arithmetic. Pick which one you are asking, put in two numbers, and this shows the answer and the working behind it.
15% of 250
37.50
- Answer
- 37.50
- Working
- 15 ÷ 100 × 250
About the percentage calculator
Per cent means per hundred, and that is the whole idea — a way of expressing a proportion against a fixed base so that different proportions can be compared. The arithmetic is trivial; what causes trouble is that three quite different questions all get described the same way.
The first is straightforward: what is 15% of 250? Divide by a hundred and multiply. The second reverses it: 37.5 is what per cent of 250? Divide and multiply by a hundred. The third asks about change between two values, and it is the one that catches people.
Percentage change is not symmetrical, and that surprises almost everyone the first time. Going from 80 to 100 is a 25% rise. Going from 100 back to 80 is a 20% fall. The same twenty units, two different percentages, because the base you divide by changed.
What it works out
- What X per cent of a number is
- What percentage one number is of another
- The percentage change between two values
- The working, written out
The formula
A% of B = A ÷ 100 × B A as % of B = A ÷ B × 100 Change = (B − A) ÷ A × 100
All three are the same relationship rearranged. A part, a whole, and a proportion between them — give any two and the third follows. Which formula you need depends only on which of the three you are missing.
For percentage change, the number you divide by is the starting value, and that is the entire source of the asymmetry. From 80 to 100 the change is 20 over a base of 80, which is 25%. From 100 to 80 it is 20 over a base of 100, which is 20%.
The same trap appears in discounts, and it costs real money. A price reduced by 20% is not restored by adding 20% back: a hundred reduced by a fifth is eighty, and eighty increased by a fifth is ninety-six. To reverse a percentage you divide rather than add, which is why undoing a discount needs dividing by 0.8 rather than multiplying by 1.2.
Percentage points are worth keeping separate too. If a rate moves from 4% to 6%, that is a rise of two percentage points but a fifty per cent increase. Both statements are true and they sound very different, which is why the distinction gets exploited.
- A
- The first number — the percentage, the part, or the starting value depending on the question.
- B
- The second number — the whole, or the ending value.
A worked example
Fifteen per cent of two hundred and fifty.
That works out to 37.50 .
- Answer
- 37.50
- Working
- 15 ÷ 100 × 250
Questions
How do I work out a percentage of a number?
Divide the percentage by a hundred and multiply. 15% of 250 is 0.15 × 250, which is 37.5. As a shortcut, 10% is the number with the decimal point moved one place left, and you can build most percentages from there.
How do I work out percentage change?
Subtract the old value from the new, divide by the old value, then multiply by a hundred. From 80 to 100 that is 20 ÷ 80 × 100, which is a 25% rise.
Why is a 25% rise cancelled by a 20% fall?
Because the base changes. The rise is measured against 80 and the fall against 100. Twenty units is a quarter of eighty but only a fifth of a hundred, so the same movement gives different percentages depending on direction.
How do I reverse a discount?
Divide rather than add. If 20% off gave a price of 80, the original was 80 ÷ 0.8, which is 100. Adding 20% to 80 gives 96 and is wrong, because the 20% was taken from the larger number.
What is the difference between per cent and percentage points?
A move from 4% to 6% is two percentage points, but a fifty per cent increase. Both describe the same change and sound very different, which is why the distinction matters when reading anything about rates.
How do I add tax or a tip?
Multiply by one plus the rate. Adding 20% is multiplying by 1.2; adding 8.5% is multiplying by 1.085. To find the pre-tax figure from a total, divide by the same number.