Question

The value of $2.\overline{45}+0.\overline{36}$ is

(a) $\frac{67}{33}$

(b) $\frac{24}{11}$

(c) $\frac{31}{11}$

(d) $\frac{167}{110}$

Answer 1

Let $x=2.\overline{45}=2.4545........\left(1\right)$
Multiplying both sides by 100, we get
$100x=245.\overline{45}.....\left(2\right)$
Subtracting (1) from (2), we get
$$\begin{gathered} 100x-x=245.\overline{45}-2.\overline{45} \\ \Rightarrow 99x=245-2=243 \\ \Rightarrow x=\frac{243}{99} \\ \therefore 2.\overline{45}=\frac{243}{99} \end{gathered}$$
Let $y=0.\overline{36}=0.3636........\left(3\right)$
Multiplying both sides by 100, we get
$100y=36.\overline{36}.....\left(4\right)$
Subtracting (3) from (4), we get
$$\begin{gathered} 100y-y=36.\overline{36}-0.\overline{36} \\ \Rightarrow 99y=36 \\ \Rightarrow y=\frac{36}{99} \\ \therefore 0.\overline{36}=\frac{36}{99} \end{gathered}$$
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).