Percentage Calculator

Pick the question you are asking, type the two numbers, and the answer appears with a one-line explanation underneath. The three modes cover what people actually mean when they reach for a percentage calculator: finding a percentage of a number, expressing one number as a share of another, and measuring how far something went up or down. The headline output is the figure you asked for; the plain-English line restates it so you can confirm you posed the question you intended rather than its mirror image.

Three questions, three formulas

The word percentage hides three different calculations, and mixing them up is the most common percent mistake there is.

x% of y = x × y ÷ 100

x as a percent of y = x ÷ y × 100

change from x to y = (y − x) ÷ x × 100

Run the defaults, x = 18 and y = 240, through all three. What is 18% of 240: 18 × 240 ÷ 100 = 43.2. What percent is 18 of 240: 18 ÷ 240 = 0.075, which is 7.5%. Percent change from 18 to 240: (240 − 18) ÷ 18 = 12.3333, an increase of 1233.33%. The same two numbers give three wildly different answers, which is why the calculator asks for the question before it shows a result.

Where per hundred comes from

Percent is the Latin per centum, "by the hundred," and the habit of reckoning in hundredths is older than the word. Roman public finance leaned on fractions of 100: Augustus levied the centesima rerum venalium, a one-in-a-hundred tax on goods sold at auction, and other hundredth-based duties recur across Roman records. That levy mattered enough that when Romans petitioned to have it lifted, Tacitus reports Tiberius refusing on the ground that the military treasury depended on it. Stating a rate as so many parts per hundred made it easy to quote and easy to check, centuries before anyone drew a percent sign.

A base of 100 stuck because it is convenient rather than mathematically privileged. One hundred divides evenly by 2, 4, 5, 10, 20, 25 and 50, so most everyday shares land on whole numbers or short decimals. Medieval Italian merchants took the practice up in earnest, quoting profit, loss and interest per cento — for each hundred — and the reckoning textbooks that trained their clerks folded these into the Rule of Three, the standard medieval method for solving proportions. Printing spread that training quickly: the anonymous Treviso Arithmetic of 1478, the earliest printed mathematics book in the West, was a commercial handbook written in the Venetian vernacular for traders. From the late fifteenth into the early sixteenth century, arithmetic texts commonly carried computations on a denominator of 100, applied to exactly those merchant questions of profit and loss, interest and proportion.

Two later developments turned that merchant shorthand into ordinary arithmetic. Hindu-Arabic numerals and place value made multiplying or dividing by 100 a matter of shifting digits rather than wrestling with Roman numerals. Then came the decimal fraction. Simon Stevin's 1585 booklet De Thiende established the everyday use of decimals in Europe, and once a share could be written as 0.075 as readily as seven and a half in every hundred, moving between a decimal and a percent became trivial. By the seventeenth century, quoting interest in hundredths was normal in financial markets.

How the percent sign took shape

The symbol is much younger than the idea. Before roughly 1425 there is no known evidence of a dedicated mark; scribes wrote per cento in full, or abbreviations such as "per 100", "p 100" and "p cento", the p often carrying a stroke through its descender, the standard scribal shorthand for per. An anonymous arithmetic text of 1339 shows those abbreviations in use. Italian pages written around 1435, added to a text of 1425, show "pc" trailed by a tiny loop, the -o ending of Italian ordinals such as primo and secondo. By 1650 that looped abbreviation had flattened into a horizontal fraction shape, a stage still visible in a 1684 arithmetic, and the "per" fell away afterwards. The modern diagonal form — two small circles split by a slanted stroke — is a late tidying. David Eugene Smith, who catalogued the arithmetics written before 1601 in his Rara Arithmetica of 1908, wrote in the second volume of his History of Mathematics in 1925 that the solidus form is modern. The sign we ended up with still reads as something set over one hundred, which is exactly what it means.

The percent sign moves for free

A trick worth memorizing: x% of y always equals y% of x, because both are x × y ÷ 100. An awkward problem can usually be flipped into an easy one. 4% of 75 feels like work; 75% of 4 is obviously 3. 16% of 25 becomes 25% of 16, which is 4. When a percent-of problem looks ugly, swap the two numbers before doing anything else.

The reverse percentage trap

Suppose a price rose by 20% and now stands at 240. The original was not 240 minus 20% — that gives 192 and is wrong. The increase was charged on the old price, not the new one, so you undo it by dividing: 240 ÷ 1.20 = 200. Check it forward: 200 plus 20% of 200 is 240. This matters most where displayed prices already include tax. UK and EU shelf prices include VAT, so getting back to the net amount is always a division. In the US and most of Canada, sales tax is added at the till, so the shelf price is the base and a straight percent-of does the job. Australian prices include the 10% GST, so the pre-tax amount is the price divided by 1.1.

The same reverse move recovers a starting figure behind any finished percentage: a discounted total, a tip-inclusive bill, a headcount after a stated cut. Whenever the percentage was applied to the number you are looking for, division undoes it and subtraction does not.

Change is measured from where you started

Percent change divides by the starting value, which makes it asymmetric. From 18 to 240 is an increase of 1233.33%, but the return trip from 240 to 18 is a decrease of only 92.5%. The same asymmetry explains why a 50% loss needs a 100% gain to break even: drop 100 by half and you hold 50, and climbing from 50 back to 100 is a 100% rise. A change from zero has no defined percentage at all — there is nothing to measure against — so the calculator reports an error there rather than inventing a number.

One further distinction saves a lot of confusion. A move from 40% to 45% can be told two ways: it is a five percentage-point rise, but a 12.5% increase relative to 40. Points compare the two figures directly; percent compares the change to the figure you began from. Financial and political reports blur the two constantly, and the gap between them widens as the base shrinks.

Frequently asked questions

How do I work out 18% of 240 in my head?

Break it into pieces of 10%, 5% and 1%. For 240, 10% is 24, 5% is 12, and 1% is 2.4, so 18% is 24 + 12 + 7.2 = 43.2. Most everyday percentages fall apart into those three building blocks within a few seconds.

What percent is 18 out of 240?

Divide 18 by 240 to get 0.075, then multiply by 100 to get 7.5%. The general rule is part divided by whole, times 100. A common slip is dividing the wrong way around — 240 divided by 18 gives 1333.33%, which answers a different question entirely.

How do I calculate percentage increase between two numbers?

Subtract the old value from the new one, divide by the old value, and multiply by 100. Going from 80 to 92 is (92 − 80) ÷ 80 = 0.15, a 15% increase. Always divide by the value you started from, never the one you ended at.

Why does a 50% loss need more than a 50% gain to recover?

Because each change is measured against a different base. Drop 100 by 50% and you hold 50; gain 50% of that and you are at 75, still a quarter short of where you began. Fully recovering from a 50% loss takes a 100% gain.

How do I find the original price before VAT was added?

Divide by one plus the rate rather than subtracting the percentage. A UK price of 240 including 20% VAT started at 240 ÷ 1.20 = 200. Subtracting 20% from 240 gives 192, which is wrong because the 20% was charged on 200, not on 240.