Show that $0.2353535 \ldots = 0.2 \overline{35}$ can be expressed in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \neq 0$.

(N/A) Let $x = 0.2 \overline{35}$.
Here,the digit $2$ does not repeat,but the block $35$ repeats.
Since two digits are repeating,we multiply $x$ by $100$ to get:
$100x = 23.53535 \ldots$
We can write this as:
$100x = 23.3 + 0.23535 \ldots$
Since $x = 0.23535 \ldots$,we substitute $x$ into the equation:
$100x = 23.3 + x$
Subtract $x$ from both sides:
$99x = 23.3$
Convert the decimal to a fraction:
$99x = \frac{233}{10}$
Divide by $99$:
$x = \frac{233}{990}$
Thus,$0.2 \overline{35} = \frac{233}{990}$.