One idea, five questions
“Percent” means “per hundred”, so every percentage problem is the same relationship — part = percent × whole ÷ 100 — with a different piece missing. The hard part is usually deciding which number is the whole (the base). In “what is 15% of 80”, the whole is 80. In “12 is what percent of 80”, the whole is still 80 and 12 is the part. For change from an old value to a new one, the base is always the old value.
The formulas
P% of Y = P / 100 × Y X is what % of Y = X / Y × 100 X is P% of what = X / (P / 100) Change A → B = (B − A) / |A| × 100 Difference A, B = |A − B| / ((A + B) / 2) × 100
Worked examples
Percent of: 15% of 80 = 0.15 × 80 = 12.
What percent: 12 out of 80 = 12 ÷ 80 = 0.15 → 15%. (A test score of 12/80.)
Find the whole: 12 is 15% of 12 ÷ 0.15 = 80. (A $12 discount that was 15% off means the original price was $80.)
Change: a price going from 80 to 100 rose (100 − 80) ÷ 80 = 25%. Going back from 100 to 80 is a 20% decrease — the base changed.
Difference: 80 and 100 differ by 20 ÷ 90 ≈ 22.22% of their average.
Things to watch
- Percent change from zero is undefined — any increase from nothing is infinitely large in relative terms.
- For a negative starting value the change is measured against its size (|A|), so a move from −50 to −25 reads as a 50% increase.
- Results are shown to up to four decimal places; the steps show the unrounded arithmetic.
Frequently asked questions
How do I calculate a percentage of a number?
Convert the percentage to a decimal by dividing by 100, then multiply. For 15% of 80: 15 ÷ 100 = 0.15, and 0.15 × 80 = 12. A handy shortcut: X% of Y equals Y% of X, so 8% of 50 is the same as 50% of 8, which is 4.
What is the difference between percent change and percent difference?
Percent change has a direction: it measures how much a value moved from an old value to a new one, relative to the old value. Percent difference has no direction: it compares two values of equal standing relative to their average. Use change for "before and after", difference for comparing two prices or measurements side by side.
Why isn't a 50% gain undone by a 50% loss?
Because each percentage is taken of a different base. $100 up 50% is $150; $150 down 50% is $75. To get back to $100 from $75 you would need a gain of 33.3%.
What is a percentage point?
A percentage point is the plain difference between two percentages. If a rate rises from 4% to 5%, it went up one percentage point, but that is a 25% increase in the rate itself. News reports often mix the two.
How do I add a percentage to a price?
Multiply by 1 plus the rate. Adding 7% sales tax to $40 is 40 × 1.07 = $42.80. To remove a percentage that was already included, divide instead: $42.80 ÷ 1.07 = $40.