How many times minute-hand coincides with hour-hand in 12 hours?

Answer

To find out how many times the minute hand coincides with the hour hand in 12 hours, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Clock Movement**:
- The minute hand completes one full revolution (360 degrees) every hour, while the hour hand moves 30 degrees (1/12 of a full circle) every hour.

2. **Relative Speed**:
- The minute hand moves at a speed of 360 degrees per hour (or 6 degrees per minute).
- The hour hand moves at a speed of 30 degrees per hour (or 0.5 degrees per minute).
- The relative speed of the minute hand with respect to the hour hand is \(6 - 0.5 = 5.5\) degrees per minute.

3. **Time to Coincide**:
- To find the time taken for the minute hand to catch up to the hour hand, we can use the formula:
\[ \text{Time} = \frac{\text{Angle Difference}}{\text{Relative Speed}} \]
- Initially, at 12:00, both hands are at 0 degrees. After that, the minute hand needs to catch up to the hour hand, which is moving away.

4. **Calculating Coincidences**:
- The minute hand will coincide with the hour hand every time it completes a full cycle of 360 degrees relative to the hour hand.
- The time taken for the minute hand to catch up to the hour hand can be calculated as:
\[ \text{Time to coincide} = \frac{360 \text{ degrees}}{5.5 \text{ degrees/min}} \approx 65.45 \text{ minutes} \]
- This means that approximately every 65.45 minutes, the minute hand coincides with the hour hand.

5. **Counting Coincidences in 12 Hours**:
- In 12 hours, there are \(12 \times 60 = 720\) minutes.
- The number of times the minute hand coincides with the hour hand in 720 minutes is:
\[ \text{Number of Coincidences} = \frac{720 \text{ minutes}}{65.45 \text{ minutes}} \approx 11 \]

6. **Final Answer**:
- Therefore, the minute hand coincides with the hour hand **11 times** in 12 hours.

### Summary:
The minute hand coincides with the hour hand 11 times in a 12-hour period.