Question

Find the value of 0.111.... + 0.333.....

Answer 1

Given operation is 0.111... + 0.333....
Let '$x$' be 0.111....
Multilpy $x$ with 10.
$\Rightarrow$ $10x$ = 1.11....
= 1 + 0.111....
= 1 + $x$ $(\because x=\mathrm{0.111....})$
$\Rightarrow$ $9x$ = 1
$\Rightarrow$ $x$ = $\frac{1}{9}$
$\therefore$ 0.111.... = $\frac{1}{9}$...............(1)

Now we can express 0.3 as 3 $\times$ 0.111....
$\because$ from (1), 0.111.... = $\frac{1}{9}$
$\Rightarrow \mathrm{0.333...}=3\times \frac{1}{9}$
$=\frac{3}{9}$
$\therefore$ 0.333.... = $\frac{1}{3}$

Now, 0.111... + 0.333.... = $\frac{1}{9}+\frac{1}{3}$
= $\frac{4}{9}$
$\frac{4}{9}=4\times \frac{1}{9}$
$\because \frac{1}{9}=\mathrm{0.111.....}$ [from eq(1)]
$\Rightarrow \frac{4}{9}=4\times \mathrm{0.111...}=\mathrm{0.444....}$

$\therefore$ 0.111... + 0.333.... = 0.444...