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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) \( 0 . \overline{001} \)
To do:
We have to express the given decimals in $\frac{p}{q}$ form.
Solution:
(a) $0. \overline{6}$
Let $x = 0.6666....$
Multiply both sides by 10.
$10x = 10(0.6666....)$
$10x = 6.6666.....$
Therefore,
$10x-x = 6.6666.... - 0.6666.....$
$9x = 6$
$x = \frac{6}{9}$
Therefore,
$0. \overline{6}$ in $\frac{p}{q}$ form is $\frac{6}{9}$.
(b) $0. 4\overline{7}$
Let $x = 0.47777....$
Multiply both sides by 10.
$10x = 10(0.47777....)$
$10x = 4.7777.....$
Multiply both sides by 100.
$100x = 100(0.47777....)$
$100x = 47.7777.....$
Therefore,
$100x-10x = 47.7777.... - 4.7777.....$
$90x = 43$
$x = \frac{43}{90}$
Therefore,
$0. 4\overline{7}$ in $\frac{p}{q}$ form is $\frac{43}{90}$. 
(c) $0. \overline{001}$
Let $x = 0.001001001....$
Multiply both sides by 10.
$10x = 10(0.001001001....)$
$10x = 0.01001001.....$
Multiply both sides by 100.
$100x = 100(0.001001001....)$
$100x = 0.1001001.....$
Multiply both sides by 1000.
$1000x = 100(0.001001001....)$
$1000x = 1.001001001.....$
Therefore,
$1000x-x = 1.001001001.... - 0.001001001.....$
$999x = 1$
$x = \frac{1}{999}$
Therefore,
$0.\overline{001}$ in $\frac{p}{q}$ form is $\frac{1}{999}$.