Three different questions that share one symbol
Most percentage mistakes come from answering the wrong one of three questions. Each has its own formula, and mixing them up gives numbers that look reasonable but are wrong.
| Question | Formula | Example | Answer |
|---|---|---|---|
| What is X% of a number? | X ÷ 100 × base | 18% of 2,450 | 441 |
| By what percent did a value change? | (new − old) ÷ old × 100 | 80 rises to 100 | +25% |
| How far did a rate move? | new rate − old rate | 10% rises to 12% | +2 percentage points |
The first question scales one number. The second compares two values of the same thing. The third only comes up when the thing being measured is already a percentage, such as an interest rate, a tax rate, a pass rate or a margin.
Percentage CalculatorFind X% of any number, including negative or above-100% percentages.Percent change always starts from the original value
Going from 80 to 100 is a rise of 20 units, and 20 ÷ 80 = +25%. Going from 100 back to 80 is also 20 units, but 20 ÷ 100 = −20%. The size of the change is the same; the starting value it is measured against is not.
That is why a 20% price rise followed by a 20% discount does not take you back to the start. A price of 100 rises to 120, and 20% off 120 brings it to 96, which is 4% below where it began. To undo a 20% fall you need a 25% rise, because the rise is measured from the lower value.
Percentage Increase CalculatorEnter the original and new values to get the signed change and the percentage of the original.Percentage points: when the value is already a percent
If a lending rate moves from 10% to 12%, it rose by 2 percentage points. Measured against the old rate, the same move is a 20% increase, since 2 ÷ 10 = 0.20. Both statements are true. "Rates up 20%" sounds dramatic, while "rates up 2 points" tells a borrower exactly what changed. If an exam pass rate moves from 40% to 50%, that is 10 percentage points, or a 25% increase in the share of students passing.
Statistics agencies run into the same issue with index numbers. The US Bureau of Labor Statistics notes that changes in Consumer Price Index points depend on how high the index already is, so it reports percent changes instead. The same 18-point move can be a large or small percentage depending on where the index started.
The same applies to business ratios. An operating margin that falls from 12% to 9% is 3 percentage points lower, while the ratio itself fell by 25% relative to its starting value. To see how a changed selling price affects gross margin specifically, use the worked lamp prices in the margin vs markup guide.
Why +50% followed by −50% leaves you 25% down
Start with 200. A 50% rise takes it to 300. A 50% fall from 300 takes it to 150. Overall, 200 to 150 is a 25% loss. The fall is taken from a larger value than the rise was, so it removes more units than the rise added.
Here a is the size of the equal rise and fall as a decimal. With a = 0.5, the net change is 1.5 × 0.5 − 1 = −0.25. The order does not matter: falling 50% first and then rising 50% also ends at 150.
| Rise then fall of | Value after both | Net change |
|---|---|---|
| 10% | 99 | −1% |
| 20% | 96 | −4% |
| 25% | 93.75 | −6.25% |
| 50% | 75 | −25% |
| 80% | 36 | −64% |
Small swings nearly cancel; large ones do not. The same arithmetic tells you how much you need to gain to recover from a loss.
For investments, the simple average of +50% and −50% is 0%, yet the money is down a quarter. The gap between an average of yearly returns and the compound growth rate is covered in CAGR vs ROI vs average return.
A checklist for percentages in news, bills and payslips
- Find the starting value. "Up 30%" means nothing until you know up from what, and since when.
- If the thing measured is a rate, look for points. Tax rates, interest rates and margins usually move in percentage points.
- Multiply successive changes; don't add them. Two 10% raises give 1.10 × 1.10 = 1.21, a 21% total rise.
- "300% more" means four times as much. Going from 25 to 100 is a 300% increase, not a tripling.
- Watch out for small starting numbers. Complaints rising from 2 to 5 is a 150% increase in just 3 cases.
- Gross percentages don't carry through to take-home pay. A 10% gross raise can change take-home pay by a different percentage when tax is charged in slabs; see how a bonus or raise affects Pakistan salary tax.
Where these formulas stop working
- A zero starting value. A change from zero has no defined percent change, and the calculator rejects it. Report the change in units instead.
- Negative starting values. A profit that moves from −50 to +25 has no single agreed percent change. The calculator only accepts a positive starting value; describe such moves in units.
- Change over time. A total percent change says nothing about the yearly rate behind it. Use a compound annual growth rate for that.
- Averaging percentages across groups. If the groups differ in size, the overall percentage needs weights. Weighted grades is a worked case.