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 $x=\mathrm{2.4545...}-----\left(1\right)$

Multiplying both sides by 100, we get

$100x=\mathrm{245.45...}-----\left(2\right)$

Subtracting (1) from (2), we get

$100x-x=245.45-2.45\Rightarrow 99x=245-2=243\Rightarrow x=\frac{243}{99}\therefore 2.\overline{45}=\frac{243}{99}$

Let $y=\mathrm{0.3636...}-----\left(3\right)$

Multiplying both sides by 100, we get

$100y=\mathrm{36.36..}-----\left(4\right)$

Subtracting (3) from (4), we get

$100y-y=36.36-0.36\Rightarrow 99y=36\Rightarrow y=\frac{36}{99}\therefore 0.\overline{36}=\frac{36}{99}$

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 the option (c).