Show that $1.272727 \ldots = 1.\overline{27}$ can be expressed in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \neq 0$.

(C) Let $x = 1.272727 \ldots$
Since two digits are repeating,we multiply $x$ by $100$ to get:
$100x = 127.2727 \ldots$
We can write this as:
$100x = 126 + 1.272727 \ldots$
Since $x = 1.272727 \ldots$,we substitute $x$ into the equation:
$100x = 126 + x$
Subtracting $x$ from both sides:
$100x - x = 126$
$99x = 126$
$x = \frac{126}{99}$
Dividing both numerator and denominator by their greatest common divisor,$9$:
$x = \frac{14}{11}$
Thus,$1.\overline{27} = \frac{14}{11}$,where $p = 14$ and $q = 11$ are integers and $q \neq 0$.