How do you know if 7/8 a terminating or repeating decimal

Answer 1

If the prime factors of the denominator (when reduced to lowest terms) are all either 2's or 5's, then the decimal equivalent will be terminating; otherwise it will be repeating.

So in this question the fraction is in lowest terms, and the prime factorization of the denominator is 2³. So the prime factorization consists of all 2's. Therefore the decimal equivalent is terminating.

Another example:

3/20

The prime factorization of the denominator is 2²⋅5. Since there are no prime factors other than 2 or 5, this too will terminate.

Finally here is one that will not terminate:

7/30. The prime factorization of 30 = 2⋅3⋅5. The existence of a prime factor (3) other than 2 or 5 means that this fraction will be a repeating decimal.