Question

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

• A. $\frac{407}{3300}$
• B. $\frac{37}{3000}$
• C. $\frac{4}{3300}$
• D. $\frac{1343}{10989}$

Answer 1

Answer: (A)

$\frac{407}{3300}$

$0.\overline{1}+0.0\overline{1}+0.00\overline{1}$

Let $x=\mathrm{0.11111...}+\mathrm{0.01111...}+\mathrm{0.00111....}$

$\Rightarrow x=\mathrm{0.12333...}$

$\therefore x=\mathrm{0.123333...}$

$\Rightarrow 100x=\mathrm{12.333...}$

$\Rightarrow 100x-x=\mathrm{12.333333...}-\mathrm{0.123333...}$

$\Rightarrow 99x=\mathrm{12.210000....}$

$\Rightarrow 99x=12.21$

$\Rightarrow x=\frac{12.21}{99}$

$\Rightarrow x=\frac{1221}{9900}$

$\therefore x=\frac{407}{3300}$