Right angles of clock hands in 24 hours — How many times are a clock’s hour and minute hands at a right angle (90°) in one full day?

Introduction / Context:Clock-hand problems rely on relative angular speed. The minute hand moves 6° per minute; the hour hand moves 0.5° per minute. The hands are at right angles (90° apart) twice in most hours, but not exactly in every single hour segment across 12 hours because of boundary effects.

Given Data / Assumptions:

• Relative speed = 5.5° per minute (6 − 0.5).
• Right-angle separations occur when the angular difference is 90° or 270° (which is also 90° the other way around).
• We count distinct instants, not durations.

Concept / Approach:In 12 hours, there are 22 right-angle positions (standard result). Intuitively, because 11 coincidences occur in 12 hours, and between each overlap there are usually two 90° separations. However, one of the potential right-angle events near the 2→3 or 8→9 boundary is missed, netting 22 rather than 24 in 12 hours.

Step-by-Step Solution:Right angles in 12 hours = 22.Therefore, in 24 hours = 2 × 22 = 44.

Verification / Alternative check:A known set of results: Overlaps per 12 h = 11; right angles per 12 h = 22; opposite positions per 12 h = 11. Doubling for 24 h gives 22, 44, 22 respectively.

Why Other Options Are Wrong:22 and 24 are 12-hour counts misapplied; 48 assumes two right angles in every hour of 24, ignoring the misses near boundaries.

Common Pitfalls:Assuming exactly two right angles every hour without exception; the pattern slips due to the continuous drift of the hands.

Final Answer:44