Understanding percentage results
Percentage calculations contain several common pitfalls. The table below describes the most frequently encountered errors and how to avoid them.
| Common pitfall | Explanation | Correct approach |
|---|---|---|
| Percentage points vs. percent | A rate rising from 10% to 15% increases by 5 percentage points, but by 50 percent. These are different measures. | Use 'percentage points' for absolute differences between percentage values; use 'percent change' only for relative change. |
| Reversing a percent change | A +50% increase followed by a −50% decrease does not return to the original value. 100 × 1.5 = 150; 150 × 0.5 = 75. Net change: −25%. | To reverse a percent change, use the reciprocal: to undo a +50% increase, apply −33.3% (not −50%). |
| Base value matters | '10% off then 10% off' is not the same as '20% off'. Two successive 10% discounts give a net reduction of 19%, not 20%. | Apply each percentage to the value that exists at that step, not to the original value. |
| Percent of a percent | '10% of 20%' equals 2 percentage points, not 2%. This phrasing is often ambiguous. | Convert percentages to decimals before multiplying: 0.10 × 0.20 = 0.02 = 2 percentage points. |
- Percent change is symmetric in neither direction nor value. A 100% increase doubles a quantity, but reversing that requires only a 50% decrease. This asymmetry is mathematically correct but counterintuitive.
- When A is negative, Mode 3 (percent change) uses the absolute value of A in the denominator, following the standard mathematical convention for relative change: ((B − A) ÷ |A|) × 100. The sign of the result still reflects whether the value increased (positive) or decreased (negative).
- Rounding: this calculator rounds results to four decimal places of precision. When chaining percentage calculations, use unrounded intermediate values to minimise accumulated rounding error.
What is a percentage?
A percentage is a dimensionless ratio expressed as a fraction of 100. The word derives from the Latin 'per centum', meaning 'by the hundred'. Writing a value as a percentage multiplies it by 100 and appends the percent sign (%). For example, the ratio 0.25 equals 25%, and the ratio 1.0 equals 100%. Percentages provide a convenient, scale-independent way to compare proportions across different totals.
Three fundamental question types cover the vast majority of percentage calculations encountered in everyday life, finance, science and statistics. The first asks for a part of a whole: 'What is X% of Y?' The second asks for a ratio: 'X is what percent of Y?' The third measures relative change: 'By what percentage did a value change from X to Y?' This calculator addresses all three.
Percentage calculations underpin a wide range of applied fields. In finance, interest rates, returns and discounts are expressed as percentages. In statistics, frequencies, proportions and confidence intervals often use percentage notation. In everyday commerce, sales tax, tips and price reductions are routinely quoted as percentages of a base value.
How to use this percentage calculator
- Select the calculation type from the dropdown. Choose 'What is A% of B?' to find a fraction of a number, 'A is what percent of B?' to find the ratio, or 'Percent change from A to B' to measure a relative change.
- Enter the first value (A) in the first field. Depending on the mode, this is the percentage, the numerator, or the starting value.
- Enter the second value (B) in the second field. Depending on the mode, this is the base number, the denominator, or the ending value.
- The result appears immediately. For percent change, a positive result indicates an increase and a negative result indicates a decrease.
Percentage formulas with worked examples
The three modes use distinct but related formulas, all rooted in the definition of percentage as a fraction of 100.
Mode 1 — 'What is A% of B?': multiply B by A and divide by 100. This finds the portion of B that corresponds to A percent. Example: 25% of 200 = (25 ÷ 100) × 200 = 50.
Mode 2 — 'A is what percent of B?': divide A by B and multiply by 100. This expresses A as a proportion of B in percentage form. Example: 50 is what percent of 200? = (50 ÷ 200) × 100 = 25%.
Mode 3 — 'Percent change from A to B': subtract A from B, divide by the absolute value of A, and multiply by 100. This measures the relative change from starting value A to ending value B. Example: from 200 to 250 = ((250 − 200) ÷ |200|) × 100 = +25%. A result of −25% means B is 25% less than A. Note: percent change is undefined when A = 0.
Frequently asked questions
How do I calculate a percentage of a number?
To find A percent of B, divide A by 100 and multiply by B. For example, 25% of 200 = (25 ÷ 100) × 200 = 50. Equivalently, move the decimal point two places to the left in the percentage and multiply: 0.25 × 200 = 50. This operation finds the portion of a whole that corresponds to a given fraction expressed as a percent.
What is the formula for percent change?
Percent change = ((new value − original value) ÷ |original value|) × 100. A positive result means the value increased; a negative result means it decreased. For example, a price rising from 200 to 250 yields ((250 − 200) ÷ 200) × 100 = +25%. A price falling from 200 to 150 yields ((150 − 200) ÷ 200) × 100 = −25%. Percent change is undefined when the original value is zero.
What is the difference between percentage points and percent change?
Percentage points measure an absolute arithmetic difference between two percentages. Percent change measures a relative difference. If an interest rate rises from 2% to 5%, it rises by 3 percentage points. It also rises by 150 percent ((5 − 2) ÷ 2 × 100 = 150%). These are numerically distinct and not interchangeable. The distinction matters in finance, public policy and statistics, where conflating them leads to misleading comparisons.
If I increase a value by 50% and then decrease it by 50%, do I return to the original?
No. A 50% increase followed by a 50% decrease returns only 75% of the original value, a net loss of 25%. For example: 100 increased by 50% becomes 150; 150 decreased by 50% becomes 75. This happens because the second percentage is applied to the new, larger value. To fully reverse a 50% increase, you need to apply a 33.3% decrease (since 150 × (1 − 1/3) = 100).
How do I find what percentage one number is of another?
Divide the first number by the second, then multiply by 100. The result is the first number expressed as a percentage of the second. For example, to find what percentage 50 is of 200: (50 ÷ 200) × 100 = 25%. This means 50 is 25% of 200. Use the 'A is what percent of B?' mode of this calculator for this calculation.
Can percentages exceed 100%?
Yes. A percentage above 100% simply means the value in question is greater than the reference base. For example, if sales increase from 100 units to 250 units, the percent change is +150%, meaning sales are now 250% of the original level (an increase of 150 percentage points beyond 100%). Percentages below 0% are also valid and indicate a decrease.
How do I calculate a reverse percentage (working backwards)?
To find the original value before a percentage was added, divide the final value by (1 + the percentage as a decimal). For example, if a price including 20% tax is £120, the pre-tax price is 120 ÷ 1.20 = £100. To find the original value before a percentage was subtracted, divide by (1 − the percentage as a decimal). For example, a price of £80 after a 20% discount implies an original price of 80 ÷ 0.80 = £100.
References
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods. Section 1.3.6.7: Percentage. nist.gov/sematech-e-handbook.
- Corder GW, Foreman DI. Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach. Wiley, 2009. (Standard coverage of percentage and proportional reasoning.)
- Office for National Statistics (ONS). Style Guide: Percentages and percentage points. ons.gov.uk.