The minute hand of a clock overtakes the hour hand at intervals of 63 minutes of the correct time. How much does a clock gain or loss in a day

Answer

To solve the problem, we need to determine how much the clock gains or loses in a day based on the information that the minute hand overtakes the hour hand every 63 minutes of the correct time.

### Step-by-Step Solution:

1. **Understanding the Overtaking of Hands:**
The minute hand completes one full revolution (360 degrees) every 60 minutes, while the hour hand completes one full revolution every 12 hours (720 minutes). Therefore, the minute hand moves at a speed of 6 degrees per minute (360 degrees / 60 minutes), and the hour hand moves at a speed of 0.5 degrees per minute (360 degrees / 720 minutes).

2. **Relative Speed of the Hands:**
The relative speed of the minute hand with respect to the hour hand is:
\[ \text{Relative Speed} = \text{Speed of Minute Hand} - \text{Speed of Hour Hand} = 6 \text{ degrees/minute} - 0.5 \text{ degrees/minute} = 5.5 \text{ degrees/minute} \]

3. **Time Taken for Overtaking:**
The minute hand overtakes the hour hand every 63 minutes of the correct time. In this time, the minute hand moves:
\[ \text{Distance moved by Minute Hand} = 6 \text{ degrees/minute} \times 63 \text{ minutes} = 378 \text{ degrees} \]

4. **Calculating the Number of Overtakes in 12 Hours:**
In 12 hours (720 minutes), the number of times the minute hand overtakes the hour hand is:
\[ \text{Number of Overtakes} = \frac{720 \text{ minutes}}{63 \text{ minutes}} \approx 11.43 \]
Since the minute hand can only overtake the hour hand a whole number of times, it overtakes 11 times in 12 hours.

5. **Total Time Taken for Overtakes:**
The total time taken for these 11 overtakes is:
\[ \text{Total Time} = 11 \times 63 \text{ minutes} = 693 \text{ minutes} \]

6. **Calculating the Gain or Loss:**
The actual time for 12 hours is 720 minutes. However, the clock shows only 693 minutes for the same period. Therefore, the clock is slow by:
\[ \text{Loss} = 720 \text{ minutes} - 693 \text{ minutes} = 27 \text{ minutes} \]

7. **Calculating Daily Gain or Loss:**
Since this loss occurs in 12 hours, in 24 hours (which is double the time), the clock will lose:
\[ \text{Loss in 24 hours} = 2 \times 27 \text{ minutes} = 54 \text{ minutes} \]

### Final Answer:
The clock loses 54 minutes in a day.