Question

What is the shortcut to identify the terminating and non-terminating numbers by seeing the question.

Answer 1

Any terminating decimal (like $0.0376$) is, essentially, a fraction with a power of ten in the dominator. For example, $0.0376=\frac{376}{10000}=\frac{47}{1250}$.

Notice we simplified this fraction, by cancelling a factor of 8 in the numerator.

Ten has factors of 2 and 5, so any power of ten will have powers of 2 and powers of 5, and some might be canceled by factors in the numerator , but no other factors will be introduced into the denominator.

Thus, if the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, then it can be written as something over a power of ten, which means its decimal expression will terminate.