Percentage Calculator

Background.

Most percentage questions come in one of three shapes. What is 18% of a 64.50 restaurant bill? By what percentage did rent rise when it went from 80 to 100? What share of a 60-question quiz is 45 correct answers? The first scales a number, the second compares a new value with an old one, and the third measures a part against its whole. Each has its own arithmetic, and most percentage errors come from running the right arithmetic on the wrong question, so you choose the question first and the calculator applies that one formula.

The multiplication is rarely what goes wrong. The reference value is. A percentage is a ratio to something, and when that something changes, the same number of dollars becomes a different percentage. A price that falls from 100 to 80 has dropped 20%, yet climbing back from 80 to 100 takes a 25% rise, because the second change is measured against the smaller number. A 20% discount followed by a 20% markup leaves the price 4% below where it began. A tax rate that moves from 20% to 24% has risen four percentage points but 20% in relative terms. None of this is a trick; in each case the base has changed under your feet.

Those distinctions run through ordinary money decisions: discounts and sales tax at the till, tips, pay rises set against inflation, test scores, investment returns, interest rates reported in percentage points or basis points. The explanation and questions below take each trap in turn, including the two cases where the arithmetic has no clean answer: a starting value of zero, where percentage change has no meaning at all, and a negative starting value, where the sign of the result stops telling you which way the number moved.

What is percentage calculator?

A percentage is a ratio expressed per hundred; the word comes from the Latin per centum, "per hundred." The % sign stands for the number 0.01, so 25% is the fraction 25/100, the decimal 0.25 and the ratio 1:4 written another way. Percentages do three distinct jobs: they scale a quantity (35% of a paycheck withheld for tax), they describe relative change (sales up 12% on last year), and they describe composition (oxygen is about 21% of dry air by volume). Each job has its own arithmetic, which is why there are three modes rather than one.

A percentage is dimensionless, and it means nothing until you know which whole it refers to. "50% more" of a $10 item is $5; "50% more" of a $1,000 item is $500. Losing track of the reference value is the most frequent source of percentage mistakes, and keeping it in view resolves most of them.

Two related units describe changes in rates that are themselves percentages. A percentage point is the plain difference between two percentages: 20% to 24% is a rise of 4 percentage points. A basis point is one hundredth of a percentage point, the unit in which moves in bond yields and central-bank rates are usually described, so a policy rate moving from 5.25% to 5.50% has risen 25 basis points.

How to use this calculator.

1. 01Choose the mode that matches your question. "What is X% of a number" covers tips, taxes, discounts, and any time you're scaling a value down or up by a percentage.
2. 02If you picked "percent of a number", enter the percent (e.g. 18 for an 18% tip) and the base value (e.g. 64.50 for a restaurant bill). The result updates instantly.
3. 03If you picked "percent change", enter the starting value in the From field and the ending value in the To field. A positive result is growth and a negative one a decline, as long as the starting value is positive.
4. 04If you picked "what percent", enter the Part (the subset) and the Whole (the total). The result is the percentage the part represents of the whole — for example, 18 correct answers out of 20 questions returns 90.
5. 05Read the result as a percentage in the change and what-percent modes (25 means 25%) and as a plain number in the percent-of mode, in whatever units the value was in.
6. 06For a sale price or a tax-inclusive total, enter the combined percentage in percent-of mode and get the final figure in one step: 75 for 25% off, or 107.25 for a 7.25% tax added on.

The formula.

Each mode runs one formula. Percent of a number is result = value × (percent ÷ 100), so 20% of 250 is 250 × 0.20 = 50. Percent change is result = ((to − from) ÷ |from|) × 100, the difference measured against the size of the starting value. What percent is result = (part ÷ whole) × 100: the same division as percent change, but comparing a piece with its total rather than a later value with an earlier one.

Because the % sign is just 0.01, every percentage converts freely. Divide by 100 for a decimal; write it over 100 and reduce for a fraction. 75% is 0.75 is 3/4. That conversion is what makes successive changes easy to combine: turn each change into a multiplier (a 10% rise is × 1.10, a 20% fall is × 0.80), multiply the multipliers, and subtract 1. Two 10% rises give 1.10 × 1.10 = 1.21, a 21% rise rather than 20%; a 20% fall followed by a 20% rise gives 0.80 × 1.20 = 0.96, a 4% loss. Working backwards uses the same multiplier in reverse: if a price after a 25% markup is 60, the original was 60 ÷ 1.25 = 48, not 60 less 25%.

Percent change is directional; it always divides by the starting value. When neither number is the natural baseline, as when comparing two shops' prices or two lab measurements, people use the percentage difference instead, which divides the gap by the average of the two: |a − b| ÷ ((a + b) ÷ 2) × 100. For 80 and 100 that is 20 ÷ 90 × 100 = 22.2222%, between the 25% and 20% you get by taking either number as the base. That symmetric version is not one of the three modes, so work it by hand when you need it.

Two inputs have no meaningful answer. A change from a starting value of zero, and a part of a whole of zero, both divide by zero. Instead of showing infinity or a misleading 0, the calculator refuses those two inputs and asks for a non-zero starting value or whole.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST Guide to the SI, §7.10.2 — guidance on expressing relative differences as dimensionless percentages.
2. [2]Khan Academy — Percent word problems and the three canonical percentage operations.
3. [3]Investopedia — Percentage Change: definition, formula, and uses in finance and economics.
4. [4]BBC Bitesize (KS3 Maths) — Percentages, percentage change, and reverse percentages explained for a pedagogical audience.
5. [5]International Bureau of Weights and Measures (BIPM) — SI Brochure, 9th edition, §5.4.7 on the use of the percent symbol with dimensionless quantities.
6. [6]Federal Reserve Board — explanatory note on the distinction between basis points, percentage points, and percent change when describing interest rate moves.