Question

The sum of $0.\overline{3}\mathrm{and}0.\overline{4}$ is
(a) $\frac{7}{10}$

(b) $\frac{7}{9}$

(c) $\frac{7}{11}$

(d) $\frac{7}{99}$

Answer 1

Let $x=0.\overline{3}=0.333........\left(1\right)$
Multiplying both sides by 10, we get
$10x=3.\overline{3}.....\left(2\right)$
Subtracting (1) from (2), we get
$$\begin{gathered} 10x-x=3.\overline{3}-0.\overline{3} \\ \Rightarrow 9x=3 \\ \Rightarrow x=\frac{3}{9} \\ \therefore 0.\overline{3}=\frac{3}{9} \end{gathered}$$
Let $y=0.\overline{4}=0.444........\left(3\right)$
Multiplying both sides by 10, we get
$10y=4.\overline{4}.....\left(4\right)$
Subtracting (3) from (4), we get
$$\begin{gathered} 10y-y=4.\overline{4}-0.\overline{4} \\ \Rightarrow 9y=4 \\ \Rightarrow y=\frac{4}{9} \\ \therefore 0.\overline{4}=\frac{4}{9} \end{gathered}$$
Sum of $0.\overline{3}$ and $0.\overline{4}$ = $0.\overline{3}+0.\overline{4}=\frac{3}{9}+\frac{4}{9}=\frac{7}{9}$
Hence, the correct answer is option (b).