After 4 o'clock, how many minutes elapse between the first and second times that the hour and minute hands form a 38-degree angle? Express your answer as a mixed number.

Answer 1

At 4:00, the angle between the hour and minute hands is $120^\circ$. If they are going to form an angle of $38^\circ$, the minute hand must rotate $120^\circ - 38^\circ = 82^\circ$ more than the hour hand. Each time the minute hand moves $360^\circ$, the hour hand moves $30^\circ$. Therefore, the minute hand must move an additional $2\frac{2}{3}$ times around the clock face before it forms an angle of $38^\circ$ with the hour hand. (Note that the hour hand moves $\frac{2}{3}^\circ$ in the time it takes for the minute hand to move $1^\circ$.) Since the minute hand moves $360^\circ$ in $60$ minutes, it takes $\frac{2}{3}$ of a minute for the minute hand to move $\frac{2}{3}^\circ$. Thus, it takes $2\frac{2}{3} \times \frac{2}{3} = \frac{8}{9}$ of a minute for the minute hand to move $2\frac{2}{3}$ times around the clock face. So the answer is $\frac{2}{3} + \frac{8}{9} = \frac{6}{9} + \frac{8}{9} = \boxed{1\frac{2}{9}}$ minutes.