Question

Express the following in the form $\frac{p}{q}$, wher $p$ and $q$ are integers and $q\ne 0.$
(i) $0.\overline{6}$
(ii) $0.4\overline{7}$
(iii) $0.\overline{001}$

Answer 1

(i) Let $x=0.666$ ...............(1)

Multiply by $10$ we get

$10x=6.666$ .....................(2)

Substract (1) from (2), we get

$9x=6\Rightarrow x=\frac{6}{9}=\frac{2}{3}$

Here $p=2$ or $q=3$.

(ii) Let $x=0.47777$ ...............(1)

Multiply by $10$ we get

$10x=4.7777$ .....................(2)

Substract (1) from(2), we get

$9x=4.3\Rightarrow x=\frac{4.3}{9}=\frac{43}{90}$

Here $p=43$ or $q=90$.

(iii) Let $x=0.001001001$ ...............(1)

Multiply by $1000$ we get

$1000x=1.001001$ .....................(2)

Substract (1) from (2), we get

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

Here $p=1$ or $q=999$.