Change, difference and the base underneath

Published 2026-11-01 · Last reviewed 2026-11-06

Almost every percentage argument is really a disagreement about two things that were never said out loud: which number was the base, and whether the question was a change or a difference. Get those two on the table and the arithmetic is a multiplication. Leave them off and the same pair of figures produces a rise in one report and a fall in another, both correct. This page is about the two missing words, and about the third argument - the one about digits - that follows them.

A percentage is a fraction of a hundred

A percentage is shorthand for a fraction whose denominator is a hundred, and nothing more. So 15% of 300 is a multiplication: 300 times 15 over 100, which is 45. Written as a fraction in lowest terms the same number is 3/20, which is why the calculator prints the fraction next to the answer - it is the same statement with smaller numbers, and it makes the mental arithmetic possible. The fraction is also where the exactness question starts: 3/20 is exact, while any percentage printed as a decimal is a rounded version of something.

The reverse question - an amount as a percentage of another - is a division, and it is where the base becomes unavoidable. 15 out of 300 is 5%, but 300 as a percentage of 15 is 2000%. Same two numbers, opposite reading, wildly different answers. A page that prints one without saying which number it divided by has not answered the question; it has only answered a question.

Change is measured from where you started

Percentage change divides the difference by the value you started from, so it is asymmetric on purpose. 100 rising to 125 is 25%; the same pair the other way round, 125 falling to 100, is -20%. Neither number is wrong, and that is exactly the trap: a report that says "up 25%" in one quarter and "down only 20%" after a return to the same figure is using two different denominators on the same pair of numbers.

Doubling and halving are the cleanest case. A 100% increase takes 100 to 200, and a 50% decrease takes it back. The percentages differ because a percentage is always a percentage of something, and the something changed between the two directions.

Anything that is already a percentage - a rate, a share, a mark, a tax band - moves in two kinds of number at once. From 10% to 8% is a fall of 2 points, and it is also a 20% fall relative to where it started. Both are true, both are printed, and a headline that picks one without labeling it is choosing the version that sounds bigger.

Difference has no base, on purpose

When neither number is a starting point - two prices, two measurements of the same quantity, two candidates in a poll - the honest measure is percentage difference: the gap divided by the average of the two. For 100 and 125 that is 22.22% in either direction, which is the property change does not have. It is a bigger number than 25% and a smaller one than 20%, and it says only one thing: these two figures are 22.22% apart.

The rule that follows is dull and effective: before comparing anything, name the base. If there is a base, use change. If there is not, use difference. If the quantity was itself a percentage, say whether you mean points or relative change, or print both as this site does.

Why the digits are computed with BigInt

Binary floating point cannot represent most decimal fractions exactly, and the result is not a rounding difference at the seventh decimal place - it is visibly wrong text. In JavaScript, 0.1 plus 0.2 evaluates to 0.30000000000000004. A page that prints that has told the visitor something untrue about their own input. The same addition done on the digits - 0.1 and 0.2 read as integers over a power of ten, added as integers, written back with the decimal point where it belongs - is 0.3, and that is what this site does with BigInt.

The same choice decides what a fraction looks like. 0.125 read as digits is 125 over 1000, which reduces to 1/8 - exactly, with no rounding to explain. A third cannot be written that way at all, and the honest way to say so is the repeating-cycle notation: 1/3 is 0.(3), meaning the 3 repeats for ever. Any finite string of threes is slightly less than a third, so printing six of them and calling it the answer is a small lie; printing 0.(3) is the truth, in one character more.

Floating point is still used for the part of the answer that is a rounded number anyway - a change rate, a share of a total, a percentage printed to four decimals. The rule is not "never use floats"; it is "do not use floats where the digits are the answer".

Where these numbers come from

• Every figure above is calculated, not copied in. One shared percentage calculation answers the three ordinary questions, works out the change and difference measures and the percentage-point pair, reduces the exact fraction and writes out its decimal expansion, and does the digit-by-digit addition. The calculator page and this guide use the same calculation, so a change in one shows up in both or in neither.
• Every release automatically checks the arithmetic: the repeating cycles (a third, a sixth, the ninety-six digits of one ninety-seventh), the exact reductions, and the identity that the answer, the fraction and the amount cannot disagree.
• The two standard definitions are cited, not invented. A percentage is a fraction with denominator 100; percentage change is measured against the initial value, percentage difference against the mean of the two values. The termination rule - a reduced fraction terminates exactly when its denominator has no prime factors other than two and five - is elementary number theory, and the page states it rather than assuming it.

What this guide does not cover

It is not a statistics lesson: a percentage change of a small sample is still a small sample, and nothing here tells you whether a difference is meaningful. It also does not cover compounding, where several changes in a row multiply rather than add: a 25% rise followed by a 20% fall does not return to the starting figure, which is the same base problem one step further along. And it says nothing about rounding policy for money: a discount, a tip split or a tax line needs a decision about which cent moves, and that decision belongs to the page that prints the currency, not to this one.

Work the numbers yourself in the Percentage Calculator, or take the same arithmetic to money in the Tip Calculator and the VAT Calculator.

Explanations and estimates only - not statistical or financial advice. Read the full disclaimer.