How do you calculate a percentage?

Percent means per hundred. 25% is 25 per hundred, which is 25/100, which is 0.25. Every percentage question is the same relationship between three numbers:

part  =  percent ÷ 100  ×  whole

You always know two of them and want the third. That gives exactly three question shapes, and each one is a single multiplication or division. The hard part is never the arithmetic. It is working out which number is the whole.

Shape 1: What is X% of Y?

You know the percent and the whole. Turn the percent into a decimal by dividing by 100, then multiply.

15% of 80      →  0.15 × 80     =  12
17.5% of 240   →  0.175 × 240   =  42
8% of 1,250    →  0.08 × 1,250  =  100

This is the one behind tips, tax, and commission. The word "of" is doing the work: it means multiply, and the number after it is the whole.

Shape 2: X is what percent of Y?

You know the part and the whole. Divide the part by the whole, multiply by 100. This is the shape most people mean when they ask how to calculate a percentage.

30 out of 120   →  30 ÷ 120 × 100  =  25%
43 out of 50    →  43 ÷ 50 × 100   =  86%
18 out of 24    →  18 ÷ 24 × 100   =  75%
1 out of 3      →  1 ÷ 3 × 100     =  33.33…%   (33 1/3 exactly)

The whole goes on the bottom. Swap them and 30 out of 120 becomes 400%, which is a good sign something went wrong, but not every mistake is that loud.

Shape 3: X is Y% of what?

You know the part and the percent, and want the whole. Divide the part by the percent as a decimal.

12 is 15% of what?    →  12 ÷ 0.15  =  80
42 is 17.5% of what?  →  42 ÷ 0.175 =  240

Check it by running shape 1 on the answer: 15% of 80 is 12. If the check does not land back on your part, the division was set up wrong.

The 25% off trap

Shape 3 is where the real money mistakes live. A jacket costs 60 after 25% off. What was the original price? The tempting move is to add 25% back:

60 × 1.25   =  75         wrong
60 ÷ 0.75   =  80         right

The discount was 25% of the original, not of the sale price. After 25% off you are paying 75% of the original, so 60 is 75% of something, and that something is 60 ÷ 0.75 = 80. Check: 25% of 80 is 20, and 80 − 20 = 60. The same logic runs through working out a discount and through percentage change, where down 50% then up 50% leaves you at 75% of where you started.

Doing it in your head

A few anchors cover most everyday percentages. Build the rest out of them.

10%   move the decimal point one place left     10% of 340  =  34
 1%   move it two places                        1% of 340   =  3.4
 5%   half of 10%                               5% of 340   =  17
20%   double 10%                                20% of 340  =  68
25%   divide by 4                               25% of 340  =  85
15%   10% plus 5%                               34 + 17     =  51

One more trick that feels like cheating: X% of Y is always the same as Y% of X, because both are X × Y ÷ 100. So 8% of 50 is awkward, but 50% of 8 is 4, and they are the same number. 4% of 75 becomes 75% of 4, which is 3.

Fractions you already know as percentages

A percentage is just a fraction with 100 on the bottom, so any fraction converts the same way: divide top by bottom, then multiply by 100.

Over 100%, and percentages of percentages

Nothing stops the part from being bigger than the whole. 150 compared with 60 is 150 ÷ 60 × 100 = 250%. That is normal for growth figures and price comparisons. It only signals an error when the part is supposed to sit inside the whole, like the share of a class that passed.

What you cannot do is add or average percentages that come from different wholes. 86% on a 50-mark paper and 70% on a 100-mark paper is not 78% overall. Add the marks instead: 43 + 70 = 113 out of 150, which is 75.33%. The long version is in how to average percentages.

Why 7 out of 25 comes out as 28.000000000000004%

Shape 2 is a division followed by a multiplication, and that pair is exactly where binary floating point shows its seams. Ask JavaScript or Python for 7 out of 25 as a percentage, 7 / 25 * 100, and both print 28.000000000000004. Ask for 57% as a whole number, 0.57 * 100, and you get 56.99999999999999. The cause is that 0.28 and 0.57 have no exact binary form, the same fault behind 0.1 + 0.2 = 0.30000000000000004. A spreadsheet will hide it behind formatting until you round, sort, or compare against 57. Prism's keypad works in exact decimal arithmetic rather than binary floating point, so 7 ÷ 25 × 100 is 28 and nothing else.

Or let the percentages tool pick the whole for you

Prism's percentages tool has a row for each common shape: type 15 and 80 to read 12 off X% of Y, or 30 and 120 into Part and Whole to read 25% off Share. Shape 3 is one division on the keypad. Percentages are free forever, with no ads and no subscription. See the app.