Percentage formulas explained
A percentage is a number expressed as a fraction of 100: 25% means 25 out of 100, or 0.25. Almost every percentage question is one of the three forms below — the calculators above just rearrange the same formula Part = Percentage ÷ 100 × Whole.
| Question | Formula | Example |
|---|---|---|
| What is P% of X? | P ÷ 100 × X | 15% of 80 = 0.15 × 80 = 12 |
| X is what % of Y? | X ÷ Y × 100 | 25 ÷ 200 × 100 = 12.5% |
| X is P% of what? | X ÷ (P ÷ 100) | 30 ÷ 0.20 = 150 |
| Percentage change | (New − Old) ÷ |Old| × 100 | 50 → 65: 15 ÷ 50 × 100 = +30% |
| Increase by P% | X × (1 + P ÷ 100) | 80 × 1.20 = 96 |
| Decrease by P% | X × (1 − P ÷ 100) | 80 × 0.80 = 64 |
| Percentage difference | |A − B| ÷ ((A + B) ÷ 2) × 100 | 10 and 15: 5 ÷ 12.5 × 100 = 40% |
| Original before a change | Final ÷ (1 ± P ÷ 100) | 120 after +20%: 120 ÷ 1.2 = 100 |
Percentage change vs. percentage difference
Use percentage change when there is a clear “before” and “after” — a price, a salary, last year’s sales. The result depends on the direction: from 50 to 65 is +30%, but from 65 back to 50 is only −23.08%. Use percentage difference when you compare two values on equal footing, such as two stores’ prices; it divides by the average, so the answer is the same either way.
Percent vs. percentage points
If a mortgage rate rises from 4% to 5%, it went up by 1 percentage point — but by 25 percent relative to where it started. News reports often mix the two. In finance a percentage point is also 100 basis points (bps).
Common percentage mistakes
- Up and down don’t cancel. +10% followed by −10% leaves you at 99% of where you started (1.1 × 0.9 = 0.99). Use the successive-changes calculator.
- Stacked discounts don’t add up. 20% off plus an extra 10% is 28% off, not 30%.
- Removing tax is not subtracting the rate. 119 € including 19% VAT is 100 € net, not 96.39 € (119 − 19%). Divide by 1.19 instead.
- Margin ≠ markup. Buying at 60 and selling at 100 is a 40% margin but a 66.7% markup.
Number formats in the US and Europe
The US and UK write 1,234.56; Germany, Spain, Italy and most of Europe write 1.234,56; France uses a space: 1 234,56; Switzerland often uses 1’234.56. German and French style guides also put a space before the percent sign (15 %). This calculator reads all of these and lets you choose how results are shown.
Quick reference: common percentages
| Percent | Decimal | Fraction |
|---|---|---|
| 1% | 0.01 | 1/100 |
| 5% | 0.05 | 1/20 |
| 10% | 0.1 | 1/10 |
| 12.5% | 0.125 | 1/8 |
| 20% | 0.2 | 1/5 |
| 25% | 0.25 | 1/4 |
| 33.33% | 0.3333 | 1/3 |
| 50% | 0.5 | 1/2 |
| 66.67% | 0.6667 | 2/3 |
| 75% | 0.75 | 3/4 |
Mental math shortcuts
- 10%: move the decimal point one place left (10% of 64 = 6.4). 5% is half of that, 20% is double.
- Swap trick: x% of y = y% of x. 8% of 50 is the same as 50% of 8 = 4.
- Tips: for 20%, take 10% and double it; for 15%, add half of 10% to 10%.