Express the following in the form $\frac{p}{q}$, where p and q are integers and q ≠ 0.

(i) $0 . \overline{6}$

(ii) $0.4 \overline{7}$

(iii) $0 . \overline{001}$
Solution:

(i) $0 . \overline{6}=0.666 \ldots$

Let $x=0.666 \ldots$

$10 x=6.666 \ldots$

$10 x=6+x$

$9 x=6$

$x=\frac{2}{3}$

(ii) $0 . \overline{47}=0.4777 \ldots . .$

$=\frac{4}{10}+\frac{0.777}{10}$

$10 x=7.777 \ldots$

$10 x=7+x$

$x=\frac{7}{9}$

$\frac{4}{10}+\frac{0.777 \ldots}{10}=\frac{4}{10}+\frac{7}{90}$

$=\frac{36+7}{90}=\frac{43}{90}$

(iii) $0 . \overline{001}=0.001001 \ldots$

Let $x=0.001001 \ldots$

$1000 x=1.001001 \ldots$

$1000 x=1+x$

$999 x=1$

$x=\frac{1}{999}$
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