X is what percent of Y, and why the order matters

45 is what percent of 180? The answer is 25%. Put the same two numbers in the other order and you get 400%. Both are correct arithmetic. Only one answers the question that was asked.

That is the entire difficulty with this calculation. There is one division in it, and the only way to get it wrong is to divide the wrong way round — which is easy, because nothing about two bare numbers says which is which.

Which number goes on the bottom

percent = x ÷ y × 100

The whole goes on the bottom. In plain English, the whole is the number that follows the word "of". "45 is what percent of 180" puts 180 after "of", so:

45 ÷ 180 × 100 = 25%

Reverse it and you are asking a different question: 180 ÷ 45 × 100 = 400%. That says 180 is four times 45, which is true and was not what anyone wanted.

The tell, and when the tell fails

If the part is smaller than the whole, one order gives an answer below 100% and the other gives one above it. Anything over 100% is a claim that the part is bigger than the whole. Sometimes that is real — sales can beat target — but it is the first thing to check.

The tell fails when the two numbers are close. Take 47 and 52. One way: 47 ÷ 52 × 100 = 90.38%. The other: 52 ÷ 47 × 100 = 110.64%. Both look like reasonable figures on a slide, and nothing in either flags an error.

A worked example: a budget line

A department has a budget of $2,400 and has spent $1,880.

• Spent, as a share of budget: 1,880 ÷ 2,400 × 100 = 78.33%
• Remaining: 2,400 − 1,880 = $520, and 520 ÷ 2,400 × 100 = 21.67%

Those two add to 78.33 + 21.67 = 100.00%, which is the check worth doing: the parts of a whole must come to 100. The exact values are 78.3333... and 21.6666..., so they sum to exactly 100 before rounding.

Flip the division and you get 2,400 ÷ 1,880 × 100 = 127.66%. The sentence that goes with it is "the budget is 127.66% of what has been spent" — true, useless, and easy to paste into a report as though it were the spend figure.

Percent of, versus percent more than

These are two different quantities that differ by exactly 100, and they get swapped freely.

• Percent of: x ÷ y × 100
• Percent more than: (x − y) ÷ y × 100

Take a $75 item against a $60 one.

• 75 ÷ 60 × 100 = 125% — the dearer item costs 125% of the cheaper one.
• (75 − 60) ÷ 60 × 100 = 25% — it costs 25% more than the cheaper one.

Writing "125% more" instead of "125% of" claims a price of 60 × 2.25 = $135. That is not a rounding error, it is a different number entirely.

And "less than" is not the mirror image

Go the other direction with the same two prices: 60 ÷ 75 × 100 = 80%, so the cheaper item costs 20% less than the dearer one.

So 75 is 25% more than 60, while 60 is 20% less than 75. Same pair of prices, two different percentages, and neither is wrong. The direction you compare in decides the denominator, and the denominator is the whole calculation.

A test score, which is the easy case

34 marks out of 40: 34 ÷ 40 × 100 = 85%. Marks out of a maximum are the one situation where the whole is never ambiguous, which is why this is the version everybody finds easy.

The cheapest check there is

Say the answer back as a full sentence before you use it. "45 is 25 percent of 180." If the sentence is false, the numbers went in the wrong way round. It costs a second and catches the only mistake this calculation can make.

The free Is What Percent tool takes a value X and a total Y and returns X as a percentage of Y, in your browser, with nothing sent anywhere. It cannot tell which of your two numbers is the whole — that judgement is yours, and it is the part actually worth learning.