Right Angle Frequencies in an Analog Clock

Question

I need to find out how many times the hands of an analog clock make a perfect right angle between the times of 8:00 am to 3:05 pm. The second hand is not taken into account and note that the minute and hour hands move proportionally. Thank you so much!

Answer 1

See, that minute hand moves with velocity $v=\frac{11}{12}\left[\frac{rounds}{h}\right]$ relatively to the hour hand.

At 9:00 am minute hand is exactly $90^{\circ}$ 'before' the hour hand. Earlier it was in that position $\frac{12}{11} (>1)$ hour ago, so after 8:00 am it was the first time it was in that position. Until 3:05 pm the minute hand had made $\frac{73}{12}\cdot\frac{11}{12}=\frac{803}{144}\approx 5.6$ rounds (relatively to the hour hand), which means it passed the position $90^{\circ}$ before the hour hand 5 more times.

At 3:00 pm minute hand is exactly $90^{\circ}$ 'behind' the hour hand. Until 3:05 pm it surely didn't made full round, so it was the last time it was in that position. Since 8:00 am the minute hand passed $7\cdot \frac{11}{12}=\frac{77}{12} \approx 6.4$ rounds relatively to the hour hand, so it passed the position $90^{\circ}$ behind the hour hand 6 more times.

In conclusion we have $6+7=13$ times between 8:00 am and 3:05 pm, when two hands create a right angle.