The sum of

Let $x=0 . \overline{3}=0.333 \ldots \quad \ldots(1)$

Multiplying both sides by 10, we get

$10 x=3 . \overline{3} \quad \ldots . .(2)$

Subtracting (1) from (2), we get

$10 x-x=3 . \overline{3}-0 . \overline{3}$

$\Rightarrow 9 x=3$

$\Rightarrow x=\frac{3}{9}$

$\therefore 0 . \overline{3}=\frac{3}{9}$

Let $y=0 . \overline{4}=0.444 \ldots \quad \ldots(3)$

Multiplying both sides by 10, we get

$10 y=4 . \overline{4} \quad \ldots \ldots(4)$

Subtracting (3) from (4), we get

$10 y-y=4 \overline{4}-0 . \overline{4}$

$\Rightarrow 9 y=4$

$\Rightarrow y=\frac{4}{9}$

$\therefore 0 . \overline{4}=\frac{4}{9}$

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).