How many times do the hour and the minute hands of a clock overlap in 24 hours?
Step-by-step Solution:
1. The Basic Logic The minute hand moves faster than the hour hand. They overlap whenever the minute hand completes a full "lap" around the hour hand.
2. Overlaps in a 12-Hour Period - It might seem like the hands should overlap once every hour, for a total of 12 times. However, there's a catch.
- Between 11:00 and 1:00 (a two-hour window), the hands only overlap once, exactly at 12:00. The overlap that would typically occur in the 11 o'clock hour happens at the 12 o'clock mark instead.
- Because of this one "skipped" overlap, the hands coincide only 11 times in a 12-hour period.
3. Overlaps in a 24-Hour Period - A 24-hour day is simply two 12-hour cycles on a clock.
- Total overlaps = (Overlaps in 12 hours) × 2
- Total overlaps = 11 × 2 = 22 times.
Final Answer: The hour and minute hands overlap 22 times in 24 hours.