Let $a, b$ be positive integers such that $a + b = 10$. Let $\frac{p}{q}$ be the difference between the maximum and minimum possible values of $\frac{1}{a} + \frac{1}{b}$, where $p$ and $q$ are relatively prime positive integers. Compute $p + q$.
Let $a, b$ be positive integers such that $a + b = 10$. Let $\frac{p}{q}$ be the difference between the maximum and minimum possible values of $\frac{1}{a} + \frac{1}{b}$, where $p$ and $q$ are relatively prime positive integers. Compute $p + q$.