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3 Responses
Quick test: reduce the fraction, then strip factors of 2 and 5 from the denominator-if nothing’s left it terminates; if something’s left, it repeats (e.g., 5/8 ends, 7/12 repeats thanks to that 3). I’m 99% sure that’s the whole story.
Quick test: reduce the fraction, then look at the prime factors of the denominator. If, after simplifying, the denominator is made only of 2s and/or 5s, the decimal terminates; any other prime factor (like 3, 7, 11, …) means it repeats. Example: 5/8 is already reduced and 8 = 2^3, so it terminates (indeed 0.625). But 7/12 has 12 = 2^2·3; that factor 3 forces a repeating decimal (7/12 = 0.58̅3). A handy shortcut is to keep dividing the denominator by 2 and 5 until you can’t-if you end at 1, it terminates; if not, it repeats. Also, when it does terminate, the number of decimal places equals max(exponent of 2, exponent of 5) in the simplified denominator (e.g., 3/20 = 3/(2^2·5) has max 2, so 0.15).
I picked this up when a teacher said “10’s best friends are 2 and 5.” I started jotting a tiny “2–5 check” in the margin: cancel common factors, strip 2s and 5s, and decide. It made exercises much quicker, and it’s still my go-to before I ever touch long division. If you want a clear refresher with more examples, this Khan Academy piece is solid: https://www.khanacademy.org/math/arithmetic/fraction-arithmetic/decimals-fractions/v/terminating-and-repeating-decimals.
Quick gut-check I use: reduce the fraction, then peek at the denominator-if its prime factors are only 2s and/or 5s, the decimal ends; if any other prime sneaks in, it repeats. For example, 5/8 is reduced and 8=2^3 so 5/8=0.625 (terminates), while 7/12 is reduced but 12=2^2·3 (that 3 spoils it) so 7/12=0.58333…; nice explainer: https://www.khanacademy.org/math/arithmetic/fraction-arithmetic/decimals-to-fractions/v/terminating-and-repeating-decimals