Percentage Calculator

How to calculate a percentage

To find P% of a number X, multiply the number by the percentage and divide by 100: X × P ÷ 100. So 15% of 200 is 200 × 15 ÷ 100 = 30. Equivalently, convert the percentage to a decimal first (15% = 0.15) and multiply: 200 × 0.15 = 30. “Percent” literally means “per hundred”, which is why both routes are the same calculation.

How to find what percentage one number is of another

Divide the part by the whole and multiply by 100: X ÷ Y × 100. 30 as a share of 200 is 30 ÷ 200 × 100 = 15%. If the whole is zero, no percentage exists — the calculator reports that state explicitly instead of showing a number.

The part does not have to be smaller than the whole. Percentages can exceed 100% and can be negative when the context permits: 300 is 150% of 200, and a −20 result against a 200 budget is −10% of it.

Percentage increase and decrease

Two different questions hide behind these words, and the calculator keeps them apart:

• Measure a change that already happened (old → new): the percentage change from 80 to 100 is (100 − 80) ÷ 80 × 100 = +25%. Use the “% change” mode.
• Apply a change to a value (change a value by P%): increasing 200 by 15% gives 200 × 1.15 = 230, and decreasing 200 by 15% gives 200 × 0.85 = 170. Use the “Increase / decrease” mode.

The first divides by the starting value; the second multiplies by a factor. Mixing them up is how “a 25% increase” gets misapplied as “multiply by 25”.

Percentage change vs percentage difference

Change requires a baseline. Difference does not. Percentage change measures a move from a starting value, so direction matters: 80 → 100 is +25%, while 100 → 80 is -20%. Percentage difference compares two values symmetrically, dividing the gap by their mean: |80 − 100| ÷ 90 × 100 ≈ 22.222222% — the same answer whichever value you name first.

Use percentage change when one value is genuinely the “before” (a price rise, a year-over-year move). Use percentage difference when neither value is privileged — comparing two labs’ measurements, two prices from different shops, two candidates’ scores. Saying “80 and 100 differ by 25%” silently picks 80 as the baseline; the symmetric difference is 22.222222%, and the Compare mode above shows every reading side by side.

Percentage points vs percent

A rate moving from 5% to 5.5% rises 0.5 percentage points (50 basis points) — and 10% in relative terms. Both numbers are correct; they answer different questions, and citing one as the other is the most common percentage error in reporting.

When a central bank raises interest rates from 5% to 5.5%, that is a 0.5 percentage point increase — but it is a 10% increase in the rate (0.5 ÷ 5 = 10%). Conflating the two is a pervasive percentage-reporting error. The convention: 'percentage points' (or basis points = 1/100 of a percentage point) refers to the absolute change in a percentage value; 'percent' refers to the relative change.

Example: a tax rate going from 20% to 25% is a 5 percentage point increase OR a 25% relative increase. A loan default rate moving from 2% to 3% is a 1 percentage point increase but a 50% relative increase. Interest-rate reporting prefers basis points (a 25 basis point cut = a 0.25 percentage point cut = colloquially 'a quarter point').

The reverse problem: small percentage-point moves often map to very large relative changes. A company moving from 2% net margin to 4% doubled its margin (a 100% relative increase) with only a 2 percentage point improvement. Both numbers are correct and answer different questions, which is why financial disclosures often state the absolute move in points or basis points alongside the relative change in percent.

Reverse percentages

A reverse percentage recovers the original whole from a known part. If a price tag reads $80 after a 20% discount, the original price was 80 ÷ (1 − 0.20) = $100.

Adding 20% back to 80 does not work: 80 × 1.20 = 96, not 100. The discount removed 20% of the original 100 (that is, 20), but adding 20% back adds 20% of the smaller 80 (only 16). Reversing a percentage always divides by the retained fraction instead of adding the percentage back. The same logic answers “gross from net” questions: a receipt of 108 including 8% tax is 108 ÷ 1.08 = 100 before tax.

Why percentage increases and decreases do not cancel

Sequential percentage changes multiply, they never add. +50% followed by −50% of 100 is 100 × 1.5 × 0.5 = 75 — a net -25%, even though the changes “sum” to zero. Recovering from a loss therefore always takes a larger percentage gain than the loss itself:

Percentages do not compose linearly. A 20% gain followed by a 20% loss does NOT return you to the original — it leaves you at 96% (1.2 × 0.8 = 0.96). A 50% gain followed by a 50% loss leaves you at 75%. This asymmetry compounds: to recover from a 50% loss requires a 100% gain, not 50%. The general rule: recovering from an L% loss requires a gain of L ÷ (100 − L) × 100 percent, which exceeds L for any L > 0.

This is why multi-year investment returns are summarized with the geometric (compound) mean rather than the arithmetic mean. An investment that returns +50%, then −50% in two consecutive years has an arithmetic mean return of 0% but a geometric mean of about −13.4% — the measure that matches what actually happened to the money. Investment-education resources such as those from the CFA Institute teach compound (geometric) averaging for this reason; an arithmetic mean across volatile years overstates realized compound returns.

Practical implications: for multi-year financial data, look for CAGR (Compound Annual Growth Rate) rather than a simple average annual return, and remember that drawdowns are expensive to reverse — a portfolio that drops 50% needs a subsequent 100% gain just to break even.

Percentage of a percentage

Taking a percentage of a percentage multiplies the two rates: 50% of 20% is 20 × 0.50 = 10%. If 20% of visitors sign up and 50% of those pay, then 10% of all visitors pay.

Keep this distinct from two look-alikes: a percentage-point change (20% → 10% is a fall of 10 points) and a relative change of a rate (20% → 10% is a −50% relative change). “Half of 20%” is 10%; “20% cut by half” is also 10%; but “20% minus 10 percentage points” describes a different operation that happens to land on the same number — the points mode above keeps the three vocabularies separate.

How to average percentages correctly

The arithmetic mean of two percentages is only valid when both rest on the same-sized base. Score 5/10 (50%) on a quiz and 90/100 (90%) on an exam: the simple average is 70%, but the combined performance is 95 correct out of 110 questions = 86.363636%. The small quiz dragged the naive average down far below the real rate.

The general rule is the weighted average: combined % = Σ(rateᵢ × baseᵢ) ÷ Σ(baseᵢ) — equivalently, total parts divided by total whole. Unequal bases need weighting; equal bases make the simple mean and the weighted mean agree.

Common percentage mistakes

The arithmetic of percentages is easy; the mistakes are conceptual, and the same few appear everywhere:

• Assuming percentages cap at 100%. 150% of 80 is 120 — more than the whole is perfectly meaningful for growth, targets and comparisons.
• Assuming the part must be smaller than the whole. “300 is what % of 200” is a valid question; the answer is 150%.
• Confusing percentage points with percent. 5% → 5.5% is +0.5 pp but +10% relative. Headlines mix these constantly; the points mode reports both, labelled.
• Confusing change with difference. 80 → 100 is +25%; 100 → 80 is −20%; the symmetric difference is ≈22.222222%. “They differ by 25%” silently picks a baseline.
• Adding sequential changes. +50% then −50% is −25%, not 0% — changes multiply. Stacked discounts compound the same way: 20% off then 10% off is 28% off, not 30%.
• Reversing a discount with the same percentage. After 20% off, adding 20% back gives 96% of the original — reverse a discount by dividing by the retained fraction.
• Averaging percentages with unequal bases. 50% of 10 and 90% of 100 do not average to 70% overall; weight by the base sizes.
• Taking a change from a zero baseline. Percent change from 0 is undefined — report the absolute change instead. This calculator refuses to print a number there.
• Trusting huge percentages from tiny denominators. Going from 1 case to 3 cases is “+200%” — technically true, practically noise. Check the absolute numbers behind dramatic percentages.
• Quoting percent change over a negative baseline. From −10 to 10 there is no universally agreed percentage change; conventions disagree, so report the absolute change.

Mental percentage shortcuts

A few anchors cover most quick checks: 10% moves the decimal one place left; 1% moves it two; 5% is half of 10%; 20% is double 10%; 25% is a quarter (divide by 4); 50% is half; 75% is three quarters (half plus a quarter).

Quick reference for common percentages applied to common base amounts — useful for tips, quick checks and verifying the calculator's working in your head.

Mental shortcuts: 10% = move the decimal one place left; 1% = two places; 5% = half of 10%; 20% = double 10%; 15% = 10% + half of 10%; 25% = divide by 4; 75% = half plus a quarter.

How this calculator is verified

Every answer on this page is produced by one deterministic calculation engine, written once and reused everywhere: the same functions run the live calculator in your browser, render the worked-example and formula tables at build time, and back the claim checker. No result is generated by a language model, fetched from an API, or hand-typed into the article — a calculation never leaves your device, and a published example cannot drift from what the tool computes.

Edge cases fail closed. Division by zero, zero baselines, indeterminate forms (0 ÷ 0), impossible reverse percentages and negative-baseline changes are reported as explicit states — undefined, indeterminate, no solution, or ambiguous convention — with an explanation, never as a fabricated number. Automated tests exercise every mode, every parser phrasing shown on this page and every edge case listed above, and the site build fails if any of them regresses. The formulas used are documented in the tables and methodology on this page.