Question

Without actually performing the long division,state whether the following rational numbers will have a terminating decimal expansion or a non terminating repeating decimal expansion:
$\frac{77}{210}$

Answer 1

A simplified fraction of the form $\frac{p}{q}$ has a terminating decimal expansion if and only if all the prime factors of $q$ are either $2$ or $5$.
$\frac{77}{210}=\frac{11}{30}$
Given: $q=30=2\cdot 3\cdot 5$
As the given denominator has $3$ as a prime factor, the decimal expansion will be non-terminating.