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

Concept: The hands of a clock move at constant angular speeds — the minute hand at 6° per minute and the hour hand at 0.5° per minute. So the minute hand gains on the hour hand at a constant relative rate of 6 − 0.5 = 5.5° per minute. Because this relative gain increases steadily, every relative alignment between the two hands — including the hands being exactly opposite (180° apart) — recurs at a fixed time interval within any 12-hour cycle.

1. Time between two consecutive opposite alignments = 360° ÷ 5.5° per minute = 720/11 minutes (about 65 5/11 minutes).

2. Number of such alignments possible in one 12-hour cycle (720 minutes) = 720 ÷ (720/11) = 11.

3. This matches the classic exception in the hour-by-hour pattern: between the 5 o'clock and 7 o'clock marks the hands are opposite only once, exactly at 6:00, instead of once per hour-to-hour interval — which is exactly what keeps the 12-hour count at 11 rather than 12.

4. A full day is made up of two 12-hour cycles (00:00 to 12:00, and 12:00 to 24:00), so the total number of opposite alignments in a day = 11 × 2 = 22.

Cross-check: by the same relative-speed argument, the hands also coincide (0° apart) exactly 11 times in every 12-hour cycle. Both counts come from dividing the same 12-hour span by the same 720/11-minute cycle, which confirms 11 per half-day and 22 opposite alignments across the full 24-hour day.