Question

$0.\overline{3}+0.\overline{4}$ is equal to _________.

Answer 1

$$\begin{gathered} \mathrm{Let}x=0.3333........\left(1\right) \\ \mathrm{Multiply}\left(1\right)\mathrm{by}10\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ 10x=3.3333........\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ 10x-x=3.333....-0.333.... \\ \Rightarrow 9x=3 \\ \Rightarrow x=\frac{3}{9}...\left(3\right) \\ \mathrm{Let}y=0.4444........\left(4\right) \\ \mathrm{Multiply}\left(1\right)\mathrm{by}10\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ 10y=4.4444........\left(5\right) \\ \mathrm{Subtracting}\left(4\right)\mathrm{from}\left(5\right),\mathrm{we}\mathrm{get} \\ 10y-y=4.4444....-0.4444.... \\ \Rightarrow 9y=4 \\ \Rightarrow y=\frac{4}{9}...\left(6\right) \\ \mathrm{Now}, \\ 0.\overline{3}+0.\overline{4}=x+y \\ =\frac{3}{9}+\frac{4}{9}\left(\mathrm{From}\left(3\right)\mathrm{and}\left(6\right)\right) \\ =\frac{7}{9} \end{gathered}$$

Hence, $0.\overline{3}+0.\overline{4}$ is equal to $\underline{\frac{7}{9}}$.