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 $\frac{1}{999}$. 

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Updated on: 10-Oct-2022

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