Question

The value of 0.111.... + 0.222..... is

• A. 0.444...
• B. 0.111....
• C. 0.151515....
• D. 0.333....

Answer 1

The correct option is D 0.333....

Given operation is 0.111... + 0.222...
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, 0.222... = 2 $\times$ 0.111....
$\because$ from (1), 0.111.... = $\frac{1}{9}$
$\Rightarrow \mathrm{0.222...}=2\times \frac{1}{9}$
$=\frac{2}{9}$
$\therefore \mathrm{0.111...}+\mathrm{0.222...}=\frac{1}{9}+\frac{2}{9}$
$=\frac{3}{9}$
$\frac{3}{9}=3\times \frac{1}{9}$
$\because \frac{1}{9}=\mathrm{0.111.....}$ [from eq(1)]
$\Rightarrow \frac{3}{9}=3\times \mathrm{0.111...}=\mathrm{0.333....}$
$\therefore \mathrm{0.111...}+\mathrm{0.222...}=\mathrm{0.333....}$