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.

(x).$\frac{77}{210}$.

Answer 1

Solution.

If the denominator has factors of the form of $2^m\times 5^n$ then it has a terminating decimal expansion.

If the denominator has factors other than $2$ and $5$ then it has a non-terminating decimal expansion.

Now,

$$\begin{gathered} \frac{77}{210}=\frac{7\times 11}{30\times 7} \\ =\frac{11}{30} \end{gathered}$$

Factorize the denominator.

$30=2\times 3\times 5$

As the denominator is not in the form of $2^m\times 5^n$ .

Hence, $\frac{77}{210}$ has a non-terminating decimal expansion.