How many times in a continuous period of 24 hours, are the 2 hands of an ordinary clock at right angle?

The minute hand completes 360° in 60 minutes, moving at 6° per minute.

The hour hand completes 360° in 12 hours (720 minutes), moving at 0.5° per minute.

The relative speed at which the minute hand gains on the hour hand:

$6° - 0.5° = 5.5°$ per minute

Starting from when the hands coincide (overlap), the minute hand continuously gains on the hour hand until they coincide again.

Time for minute hand to gain 360° on the hour hand:

$\dfrac{360°}{5.5°\text{ per minute}} = \dfrac{720}{11}$ minutes

This is the time between consecutive coincidences of the hands.

In a 12-hour period (720 minutes), the number of times the hands coincide:

$\dfrac{720}{\dfrac{720}{11}} = 11$ times

Between each pair of consecutive coincidences, the hands are perpendicular (at right angles) exactly twice:

• Once when they are 90° apart
• Once when they are 270° apart

In 12 hours, the number of right angles formed:

$11 \times 2 = 22$ times

In 24 hours, the number of right angles formed:

$22 \times 2 = 44$ times

Therefore, the hands of a clock are at right angles 44 times in a 24-hour period.