How many times in a 24-hour day are the hour and minute hands of a clock at right angles (i.e., 90° apart)?

Introduction / Context:Clock hands become perpendicular twice in most hour intervals, but there are exceptions due to the hands’ continuous motion and varying relative angular speed. Counting must consider the full 12-hour cycle and then double for 24 hours.

Given Data / Assumptions:

• In 12 hours, the hands are perpendicular 22 times.
• Across 24 hours, the pattern repeats exactly twice.

Concept / Approach:The relative angle θ between the minute and hour hands satisfies |30h − 5.5m| = 90 at the instants of right angles. Solving across the 12-hour span yields 22 solutions; doubling gives the 24-hour count.

Step-by-Step Solution:

Verification / Alternative check:Empirical enumeration or a plotted relative-angle function in [0, 12h) shows 22 crossings at ±90°; the next 12 hours replicate.

Why Other Options Are Wrong:48 overcounts; 24 or 12 substantially undercount and would correspond to much sparser events than observed.

Common Pitfalls:Assuming “twice per hour” uniformly (which would suggest 24) without handling skipped instances near 2–4 o’clock transitions.

Final Answer:44.