Question

Express each of the following decimals in the form $\frac{p}{q}$ :

(i) 0.$\overline{4}$ (ii) $0.\overline{37}$

Answer 1

(i) 0.$\overline{4}$

Let $x=0.\overline{4}$ = 0.4444.... (i)
10 x = 4.4444... (ii)

Subtracting (i) from (ii) we get, 9x = 4
$\Rightarrow$ x = $\frac{4}{9}$

$\therefore 0.\overline{4}=\frac{4}{9}$

(ii) $0.\overline{37}$

Let x = $0.\overline{37}$ = 0.373737.... (i)

100 x = 37.373737... (ii)

Subtracting (i) from (ii) 99 x = 37
$\Rightarrow$ x = $\frac{37}{99}$

$\therefore$ $0.\overline{37}$ = $\frac{37}{99}$