Without actual division, classify the decimal expansion of the following numbers as terminating or non-terminating and recurring.(i) (ii) (iii) (iv)

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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) $\frac{17}{200}$

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(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.