Percentage Calculator

Five percentage questions, kept apart

Most percentage mistakes are not arithmetic errors; they are answering a different question from the one asked. "What is 15% of 200" and "15 is what percent of 200" use the same two numbers and give 30 and 7.5. This calculator makes you pick the question first — percent of a number, what percent one number is of another, percent change between two numbers, increase by a percent, or decrease by a percent — and then states the answer as a sentence so it is obvious if you chose the wrong one.

The five formulas

• X% of Y is (X ÷ 100) × Y. 15% of 200 is 30.
• X is what percent of Y is (X ÷ Y) × 100. 37 out of 120 is 30.8333%.
• Percent change from A to B is (B − A) ÷ |A| × 100. From 80 to 100 is +25%.
• Increase A by B% is A × (1 + B ÷ 100). 120 increased by 30% is 156.
• Decrease A by B% is A × (1 − B ÷ 100). 120 decreased by 30% is 84.

Percent change divides by the absolute value of the starting number, so a move from a negative figure still reads in the direction you would expect: −20 to 10 is reported as an increase of 150%, not a decrease.

Why percentages do not reverse

A rise of 25% and a fall of 20% are the same journey in opposite directions. From 80 to 100 is +25%, because the change of 20 is measured against 80. From 100 back to 80 is −20%, because the same change of 20 is now measured against 100. Nothing is inconsistent — the base changed.

The same trap catches successive changes. Take 120, cut it by 30% to get 84, then add 30% back and you land on 109.20, not 120. Percentages of different bases cannot be added or subtracted, which is why a "30% off, then 20% off" promotion is 44% off rather than 50%.

It also explains the standard reverse-tax mistake. To remove 8.25% sales tax from a $54.11 total, divide by 1.0825 to get $49.99. Subtracting 8.25% of $54.11 gives $49.65, which is wrong by 34 cents because the tax was never a percentage of the total.

Points, percentages, and other things to be careful with

A change from 4% to 6% is a rise of two percentage points and a rise of 50%. Both are true, they are wildly different numbers, and the word "points" is what separates them. Interest rates, unemployment, market share and poll results are all quoted both ways, sometimes in the same paragraph.

Two more: percent change from zero is undefined, because there is nothing to measure against, and the calculator says so rather than returning infinity. And a percentage of a percentage is rarely what someone means — "response rates improved 10%" from a base of 20% could be 22% or 30% depending on which reading was intended, so state the base.