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The pattern for determining whether or not a fraction can be converted into a finite decimal or not is quite simple: take the denominator, and if the prime factorization of said denominator contains anything other than 2s or 5s, than the decimal form of that fraction is infinite. If the denominator's factors are purely some sequence of 2s and 5s, though, then the fraction has a finite decimal form.
Here's why: we know that if the denominator is some power of 10, than the decimal form of that fraction will just be the numerator with a decimal point.
(the placement of the decimal point is determined by the difference between the number of digits in the numerator and the denominator)
However, the prime factorizations of some 60% of all possible numerators contain either a 5 or a 2, thus allowing a power-of-10 denominator to be somewhat reduced. (60% because of the 10 possible ones-place digits, 6 of them are divisible by either 2 or 5)
When this reduction takes place, the denominator will still factor out to some collection of 2s and 5s, and if this new denominator is multiplied by some number that equalizes the number of 2s and 5s in the prime factorization of the denominator, then we attain the original denominator, namely, a power of 10.
Example: 375/1000. Both 375 and 1000 are divisible by 125 (5 * 5 * 5), which means the fraction can be simplified down to 3/8. The denominator now has a 2-to-5 factor ratio of 3:0, so bringing that ratio back to 3:3 (2*2*2*5*5*5) gives us a power of 10 in the denominator (namely, 1000), and we can thus see that the decimal form is 0.375. This method can be used in every case where the denominator in simplest form of the fraction can be factored into purely 2s and 5s.