Question

Without actual division, classify the decimal expansion of the following numbers as terminating or non-terminating and recurring.
(i) $\frac{7}{16}$
(ii) $\frac{13}{150}$
(iii) $\frac{-11}{75}$
(iv) $\frac{17}{200}$

Answer 1

(i) $16=2^4$
$\frac{7}{16}=\frac{7}{2^4}=\frac{7}{2^4\times 5^0}$. So, $\frac{7}{16}$ has a terminating decimal expansion.
(ii) $150=2\times 3\times 5^2$
$\frac{13}{150}=\frac{13}{2\times 3\times 5^2}$
Since it is not in the form $\frac{p}{2^m\times 5^n},\frac{13}{150}$ has a non-terminating and recurring decimal expansion.
(iii) $\frac{-11}{75}=\frac{-11}{3\times 5^2}$
Since it is not in the form $\frac{p}{2^m\times 5^n}.\frac{-11}{75}$ has a non-terminating and recurring decimal expansion.
(iv) $\frac{17}{200}=\frac{17}{8\times 25}=\frac{17}{2^3\times 5^2}$. So $\frac{17}{200}$ has a terminating decimal expansion.