Nearly every percentage problem is one of these, and the mistakes almost always come from solving the wrong one:
| Question | Formula | Example |
|---|---|---|
| What is P% of N? | N × P ÷ 100 | 20% of 50 is 10 |
| X is what percent of Y? | X ÷ Y × 100 | 37 of 240 is 15.42% |
| Change from A to B? | (B − A) ÷ A × 100 | 180 to 234 is +30% |
| N is P% of what? | N ÷ (P ÷ 100) | 96 is 120% of 80 |
The fourth is the one people get wrong most often, and it has a name: the reverse percentage. If a price of 96 includes 20% tax, the tax is not 20% of 96. Dividing gives the base: 96 ÷ 1.20 = 80, so the tax is 16. Taking 20% of the gross instead gives 19.20 and a base of 76.80, which is wrong by 3.20 on every line of an invoice.
A 50% fall followed by a 50% rise does not get you back. Starting at 100: down 50% is 50, up 50% is 75. The reason is that the second percentage applies to a smaller base. The gain needed to undo a loss is d ÷ (1 − d):
| Fall | Rise needed to recover |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 25% | 33.3% |
| 50% | 100% |
| 80% | 400% |
This is why a portfolio down 30% and then up 30% is at 91, and why a metric described as having recovered its percentage drop usually has not.
Twenty percent off, then a further ten percent off, is not thirty percent off. The multipliers combine: 0.80 × 0.90 = 0.72, so the total discount is 28%.
| Two discounts | Sum (wrong) | Actual |
|---|---|---|
| 20% then 10% | 30% | 28% |
| 30% then 20% | 50% | 44% |
| 25% then 25% | 50% | 43.75% |
| 50% then 50% | 100% | 75% |
The same arithmetic settles a question shoppers argue about: whether a discount should be applied before or after tax. Since both are multiplications, the order does not change the total. A 100 item with 10% off and 8% tax comes to 97.20 either way — 90 × 1.08 and 108 × 0.90 are the same number. Order does matter when one of them is a fixed amount rather than a percentage, and some jurisdictions specify which base tax applies to, so a receipt can differ from your arithmetic for legal rather than mathematical reasons.
This one has real consequences for anyone pricing goods. Buy for 60, sell for 100. The profit is 40. Divided by the selling price that is a 40% margin; divided by the cost it is a 66.7% markup. Same transaction, two numbers, and quoting one when you mean the other underprices the product.
| Margin | Equivalent markup |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 30% | 42.9% |
| 40% | 66.7% |
| 50% | 100% |
To convert, markup = margin ÷ (1 − margin). Applying a 30% markup when you wanted a 30% margin leaves you with a 23% margin, which on thin-margin goods is most of the profit.
If an interest rate moves from 4% to 6%, that is a rise of two percentage points and also a rise of fifty percent. Both are correct and they describe the same event with numbers that differ by a factor of twenty-five.
This is the single most exploited ambiguity in reporting about rates — conversion rates, tax rates, interest rates, unemployment, poll shares. When a claim about a rate uses the word percent without saying which, the figure is uninterpretable, and the more impressive-sounding version is usually the one chosen. If you are writing the number, say percentage points when that is what you mean.
Ten percent growth per month for a year is not 120% growth. It is 1.10^12 = 3.138, or +213.8%. Conversely, going from 100 to 180 over five years is not 16% a year; the annualised rate is 1.8^(1/5) − 1 = 12.47%, and checking it forwards, 1.1247 to the fifth power returns 1.80.
Averaging percentages is valid only when the bases are equal. Consider two shops: one converts 2 of 10 visitors, the other 30 of 1,000. The average of 20% and 3% is 11.5%. The actual combined rate is 32 ÷ 1,010 = 3.17%. Always recombine the raw counts and divide once; averaging the rates weights a tiny sample equally with a large one.
Percentages of very small samples, of changing populations, or of figures that were themselves estimates carry all the uncertainty of their inputs without showing it. Keeping the raw numbers next to the percentage is the cheapest defence.
Divide by one plus the rate. For 20%, divide the gross by 1.2 to get the net; the difference is the tax. Do not multiply the gross by the rate.
No, 43.75% off. Multiply 0.75 by 0.75 to get the remaining share of 0.5625.
Divide the cost by 0.60, not multiply by 1.40. On a cost of 60 that is 100 rather than 84.
No. Every calculation runs in your browser, and nothing is transmitted, logged or stored — which is why this is safe for pricing and salary figures you would rather not paste into an online form.
Every tool comes with a written guide, and every category is one click away.