Question

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

Answer 1

Let

$\frac{77}{210}=\frac{11}{30}$

$\frac{p}{q}$ is terminating if ,

$11$ and $30$ have no common factor,

Therefore, $11$ and $30$ are co prime.

For denominator $=30=2\times 3\times 5$

Denominator is not of the form $2^n{5}^m$

Thus $\frac{11}{30},\frac{77}{210}$ is not terminating repeating decimal.