Question

Express $2.\overline{36}+0.\overline{23}$ as a fraction in simplest form.

Answer 1

Given: $2.\overline{36}+0.\overline{23}$
Let
$$\begin{gathered} x=2.\overline{36}...\left(\mathrm{i}\right) \\ y=0.\overline{23}...\left(\mathrm{ii}\right) \end{gathered}$$
First we take x and convert it into $\frac{p}{q}$
100x = 236.3636... ...(iii)
Subtracting (i) from (iii) we get
$$\begin{gathered} 99x=234 \\ \Rightarrow x=\frac{234}{99} \end{gathered}$$
Similarly, multiply y with 100 as there are 2 decimal places which are repeating themselves.
$100y=23.2323...$ ...(iv)
Subtracting (ii) from (iv) we get
$$\begin{gathered} 99y=23 \\ \Rightarrow y=\frac{23}{99} \end{gathered}$$
Adding x and y we get
$2.\overline{36}+0.\overline{23}$ = $x+y=\frac{234}{99}+\frac{23}{99}=\frac{257}{99}$