Express 2.36¯¯¯¯+0.23¯¯¯¯

Solution:

Given: $2 . \overline{36}+0 . \overline{23}$

Let 

$x=2 . \overline{36} \quad \ldots(\mathrm{i})$

$\begin{array}{ll}y=0 . \overline{23} & \ldots \text { (ii) }\end{array}$

First we take $x$ and convert it into $\frac{p}{q}$

100x = 236.3636...        ...(iii)
Subtracting (i) from (iii) we get

$99 x=234$

$\Rightarrow x=\frac{234}{99}$

Similarly, multiply y with 100 as there are 2 decimal places which are repeating themselves. 

$100 y=23.2323 \ldots$(iv)

Subtracting (ii) from (iv) we get

$99 y=23$

$\Rightarrow y=\frac{23}{99}$

Adding x and y we get

$2 . \overline{36}+0 . \overline{23}=x+y=\frac{234}{99}+\frac{23}{99}=\frac{257}{99}$