The value of

Solution:

Let $x=2 . \overline{45}=2.4545 \ldots \quad \ldots(1)$

Multiplying both sides by 100, we get

$100 x=245 . \overline{45} \quad \ldots \ldots(2)$

Subtracting (1) from (2), we get

$100 x-x=245 . \overline{45}-2 . \overline{45}$

$\Rightarrow 99 x=245-2=243$

$\Rightarrow x=\frac{243}{99}$

$\therefore 2 . \overline{45}=\frac{243}{99}$

Let $y=0 . \overline{36}=0.3636 \ldots . \quad \ldots .(3)$

Multiplying both sides by 100, we get

$100 y=36 . \overline{36} \quad \ldots \ldots(4)$

Subtracting (3) from (4), we get

$100 y-y=36 . \overline{36}-0 . \overline{36}$

$\Rightarrow 99 y=36$

$\Rightarrow y=\frac{36}{99}$

$\therefore 0 . \overline{36}=\frac{36}{99}$

So, $2 . \overline{45}+0 . \overline{36}=\frac{243}{99}+\frac{36}{99}=\frac{243+36}{99}=\frac{279}{99}=\frac{31}{11}$

Hence, the correct answer is option (c).