How many times both needles of a clock will be overlapping each other in a day?

Question

Accepted Answer

The question asks how many times both needles of a clock will overlap each other in a day. The correct answer, as given, is Option 1: 22 times. Let's explore why this is the case and why the other options are incorrect.

### Explanation for Option 1: 22 times

1. **Understanding Clock Overlaps:**
   - In a 12-hour period, the minute and hour hands of a clock overlap 11 times. This is because the hands start together at 12:00 and overlap approximately every 1 hour and 5 minutes thereafter.
   - To calculate this, consider that the minute hand completes a full circle (360 degrees) every hour, while the hour hand moves 30 degrees in the same time. The minute hand needs to gain 360 degrees on the hour hand to overlap again, which happens approximately every 65.45 minutes (or 1 hour and 5 minutes).

2. **Calculating for a Full Day:**
   - Since there are two 12-hour periods in a day (AM and PM), the hands will overlap 11 times in each period.
   - Therefore, in a full 24-hour day, the hands overlap $11 \times 2 = 22$ times.

### Explanation for Why Other Options are Incorrect

- **Option 2: 24 times**
  - This option suggests that the hands overlap every hour, which is incorrect. The overlap occurs approximately every 65.45 minutes, not every 60 minutes. Therefore, the total number of overlaps in a day is not 24.

- **Option 3: 12 times**
  - This option would imply that the hands overlap once every two hours, which is incorrect. As explained, the hands overlap 11 times in a 12-hour period, not 6 times.

- **Option 4: 11 times**
  - This option represents the number of overlaps in a single 12-hour period, not a full 24-hour day. Therefore, it does not account for the second 12-hour period in a day.

In conclusion, the correct answer is Option 1: 22 times, as the clock hands overlap 11 times in each 12-hour period, resulting in 22 overlaps in a full 24-hour day.