Turn 1/2 into a decimal and it comes out clean: 0.5. But turn 1/3 into a decimal and you get 0.333333..., with the 3 repeating forever. Same kind of division — so why does one end and the other never does? It turns out this isn't luck at all; there's a rule you can spot just by looking at the denominator.
Let's start with the terms. A decimal like 0.5, where the digits after the decimal point stop after a fixed count, is called a terminating decimal. A decimal like 0.333..., where some digit (or block of digits) after the decimal point repeats forever, is called a repeating decimal. The part that repeats is called the "repetend" — for 1/3, the repetend is "3," and for 1/7, the entire six-digit block "142857" is the repetend.
Turning a fraction into a decimal is, in the end, just division. But as you keep dividing, the number of possible remainders is fixed — a natural number smaller than the denominator (from 0 up to denominator−1). So if you keep dividing for as many steps as the denominator, eventually a remainder is bound to repeat one that already showed up before. The moment the remainder repeats, the digit that comes next has no choice but to repeat too. That's exactly why repeating decimals happen.
So when does the remainder hit 0 (meaning the division finishes) and give you a terminating decimal? Reduce the fraction to lowest terms, factor the denominator, and if the only prime factors are 2 and 5, you get a terminating decimal. The reason is that in our base-10 number system, 10 is 2×5. If the denominator is made up only of powers of 2 and 5, you can multiply the numerator and denominator by the right number to turn the denominator into a power of 10 — 10, 100, 1000, and so on — and then the division comes out exact. On the other hand, if the denominator has even a single prime factor other than 2 or 5 — like 3, 7, or 11 — there's no way to multiply your way to a power of 10, so you get a repeating decimal.
Let's take 5/6 as an example. Factor 6 and you get 2×3. Since there's a 3 in there in addition to the 2, you can predict ahead of time that it'll be a repeating decimal. Work it out and, sure enough, 5/6 = 0.8333..., a repeating decimal where "3" repeats. What about 7/8, on the other hand? 8 = 2×2×2, so its only prime factor is 2, meaning it should be a terminating decimal. Work it out and 7/8 = 0.875 exactly, right on the nose.
Knowing this rule lets you tell, just from the denominator, whether a fraction "will be terminating or repeating" without actually carrying the division all the way out. On a test, you don't have to sit there dividing for a long time — factoring the denominator alone predicts the answer. And whether a number is a terminating or a repeating decimal, every number that can be written as a fraction belongs to one big category called "rational numbers" — the name itself means "a number that can be expressed as a ratio."
On our activity page, you can guess ahead of time whether a fraction will be terminating or repeating, then watch the actual long division play out one digit at a time and see with your own eyes exactly where the repetend begins. It also shows the denominator's prime factorization alongside, so you can check for yourself that the rule really does hold.