How many times in a day, are the hands of a clock in straight line but…

Concept: The minute hand gains on the hour hand at a fixed relative speed. Every time this relative gain completes a full 360°, the hands pass through exactly one moment of coincidence and exactly one moment of being exactly opposite (180° apart). Counting these 360° cycles over a day gives the count of opposite-direction alignments.

Application:

1. Minute hand speed = 6° per minute; hour hand speed = 0.5° per minute, so the minute hand gains on the hour hand at 6 - 0.5 = 5.5° per minute.

2. In 12 hours (720 minutes), the total relative gain = 720 × 5.5 = 3960°.

3. Number of complete 360° cycles in 12 hours = 3960° ÷ 360° = 11.

4. Each 360° cycle contains exactly one moment where the hands are 180° apart (opposite direction), so the hands are opposite 11 times in 12 hours.

5. A day has two 12-hour halves, so the hands are opposite 2 × 11 = 22 times in 24 hours.

Cross-check: This matches the standard clock-hands result for a 24-hour day: the hands coincide 22 times and are opposite 22 times (once per 360° relative cycle each), while they are perpendicular 44 times (twice per cycle, since 90° apart occurs on both the way up to and down from 180°). The counts are internally consistent.