Converting Repeating Decimal $0.\overline{13}$ to Fraction

To convert the repeating decimal $0.\overline{13}$ into a fraction $\frac{p}{q}$, follow these steps:

1. Let the given number be $x$. So, $x = 0.\overline{13}$.

   This means $x = 0.131313...$

2. Since there are two repeating digits ('13'), multiply the equation by $100$ (which is $10^2$):

   $100x = 100 \times 0.131313...$

   $100x = 13.131313...$

3. Subtract the original equation ($x = 0.131313...$) from the new equation ($100x = 13.131313...$):

   $100x - x = (13.131313...) - (0.131313...)$

   $99x = 13$

4. Solve for $x$ by dividing both sides by $99$:

   $x = \frac{13}{99}$

Therefore, the number $0.\overline{13}$ in the form of $\frac{p}{q}$ is $\frac{13}{99}$. This corresponds to Option 3.