How many times are an hour hand and a minute hand of a clock at right angles during their motion from 1.00 p.m. to 10.00 p.m.?

A) 16
B) 18
C) 20
D) 22

The hands of a clock are at right angles twice every hour, but with two important exceptions. In any 12-hour period, they are at right angles 22 times (not 24). This happens because the right angles that form exactly at 3:00 and 9:00 are shared between the adjacent hours.

• Between 2:00 and 4:00, they form right angles 3 times (approx. 2:27, 3:00, and 3:32) instead of 4.
• Between 8:00 and 10:00, they form right angles 3 times (approx. 8:27, 9:00, and 9:32) instead of 4.

The time from 1:00 p.m. to 10:00 p.m. is exactly 9 hours. A common mistake is to simply multiply 9 × 2 = 18 right angles. However, this interval fully contains both the 2:00–4:00 and 8:00–10:00 windows where the exceptions occur. Therefore, we must subtract 1 for each exception:

Total = 18 - 2 = 16 times.