Question

The value of $0.\overline{1}+0.0\overline{1}+0.00\overline{1}$ is equal to

• A. $\frac{343}{900}$
• B. $\frac{37}{300}$
• C. $\frac{4}{33}$
• D. $\frac{1343}{10989}$

Answer 1

Answer: (B)

$\frac{37}{300}$

Let $x=0.\overline{1}\Rightarrow 10x=1.\overline{1}$

$10x-x=1\Rightarrow x=1/9...\left(I\right)$

$y=0.0\overline{1}=0.01+0.001+0.0001+...$

$=\frac{1}{100}+\frac{1}{1000}+\frac{1}{10000}+...$

$=\frac{1}{100}(1+\frac{1}{10}+\frac{1}{100}+...)$

$=\frac{1}{100}\times \frac{1}{1-\frac{1}{10}}=\frac{1}{90}...\left(II\right)$

$z=0.00\overline{1}=0.001+0.0001+0.00001+...$

$=0.001(1+\frac{1}{10}+\frac{1}{100}+...)$

$=\frac{1}{1000}\times \frac{1}{1-\frac{1}{10}}=\frac{1}{1000}\times \frac{10}{9}=\frac{1}{900}...\left(III\right)$

From (1),(2),(3) $0.\overline{1}+0.0\overline{1}+0.00\overline{1}=\frac{1}{9}+\frac{1}{90}+\frac{1}{900}=\frac{111}{900}=\frac{37}{300}$

$\therefore$ option B is correct