Question:
Express in the form of $\frac{p}{q}: 0 . \overline{38}+1 . \overline{27}$.
Solution:
Let $0 . \overline{38}=x$
$1 . \overline{27}=y$
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
$\Rightarrow y=\frac{126}{99}$
Now, $x+y=\frac{38}{99}+\frac{126}{99}=\frac{164}{99}$