Time Between Two Clock Hand Angles

On a standard analog clock, there are two distinct times between 4:00 PM and 5:00 PM when the minute and hour hands of the clock form a $99^\circ$ angle. How many minutes apart are those two times? The answer is in the form $\frac{\alpha}{\beta}$ where $\alpha$ and $\beta$ are coprime positive integers. Input the value of $\alpha + \beta$.