How many times do the hands of a clock coincide in a day?

### Concept & Logic The hands of a clock coincide when the angle between them is $0^\circ$. In every 12 hours, the hands coincide 11 times. This happens because between 11:00 and 1:00, they coincide only once, exactly at 12:00. ### Step-by-Step Solution 1. **Half-day analysis:** In a 12-hour period, the minute hand completes 12 revolutions while the hour hand completes 1. The minute hand overtakes the hour hand exactly 11 times. 2. **Full-day calculation:** A day has 24 hours. Since they coincide 11 times in 12 hours, in 24 hours they will coincide $11 \times 2 = 22$ times. ### Exam Strategy & Shortcut This is a standard clock fact that should be memorized: - Coincide ($0^\circ$): 22 times a day. - Opposite ($180^\circ$): 22 times a day. - Straight line ($0^\circ$ or $180^\circ$): 44 times a day. - Right angles ($90^\circ$): 44 times a day. ### Common Pitfall A very common error is to think they coincide 24 times a day (once every hour). Students forget that between 11 and 1, they only meet once at 12 o'clock. ### Final Answer Therefore, the correct answer is **22**.