Question

Express in the form of $\frac{p}{q}:0.\overline{38}+1.\overline{27}$.

Answer 1

$$\begin{gathered} \mathrm{Let}0.\overline{38}=x \\ 1.\overline{27}=y \end{gathered}$$
x = 0.3838... ...(i)
Multiply with 100 as there are 2 repeating digits after decimals
100x = 38.3838... ...(ii)
Subtracting (i) from (ii) we get
99x = 38
$\Rightarrow x=\frac{38}{99}$
Similarly, we take
y = 1.2727... ...(iii)
Multiply y with 100 as there are 2 repeating digits after decimal.
100y = 127.2727... ...(iv)
Subtract (iii) from (iv) we get
99y = 126
$$\begin{gathered} \Rightarrow y=\frac{126}{99} \\ \mathrm{Now},x+y=\frac{38}{99}+\frac{126}{99}=\frac{164}{99} \end{gathered}$$