Question

Express each of the following decimals in the form $\frac{p}{q}$ :
$\left(i\right)0.\overline{6}$ $\left(ii\right)0.\overline{585}$

Answer 1

$\left(i\right)\text{Let x}=0.\overline{6}$
then x = 0.666... ... (i)
Here, we have only one repeating digit, So, we multiply both sides of (i) by 10 to get
10 x = 6.66… ....(ii)
Subtracting (i) from (ii), we get
10 x – x = (6.66…) – (0.66…)
$\Rightarrow 9x=6$ $\Rightarrow x=\frac{6}{9}$
$\Rightarrow x=\frac{2}{3}$ Hence $0.\overline{6}=\frac{2}{3}$

$\left(i\right)\text{Let x}=0.\overline{585}$
$\Rightarrow x=\mathrm{0.585585585...}$ ....(i)
Here, we have three repeating digits after the decimal point. so, we multiply both sides of (i) by ${10}^3=1000$ to get
1000 x = 585.585585… ....(ii)
Subtracting (i) from (ii), we get
1000x – x = (585.585585...) – (0.585585585...)
1000x – x = 585
$\Rightarrow 999x=585$
$\Rightarrow x=\frac{585}{999}=\frac{65}{111}$