How much does a watch gain per day if its hands coincide every 64 min

Options

A. 37 min
B. `32(8)/(11)`
C. 31 min
D. none of these

Answer

To determine how much a watch gains per day if its hands coincide every 64 minutes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
The hands of a watch coincide every 64 minutes. We need to find out how much time the watch gains in a day based on this information.

2. **Using the Formula**:
The formula to calculate the gain or loss in time is:
\[ \text{Gain or Loss} = \left( \frac{65 \frac{5}{11}}{\text{Given Minutes}} - 1 \right) \times 60 \times 24 \]
Here, the correct time for the hands to coincide is \( 65 \frac{5}{11} \) minutes.

3. **Convert the Correct Time**:
Convert \( 65 \frac{5}{11} \) into an improper fraction:
\[ 65 \frac{5}{11} = \frac{720 + 5}{11} = \frac{725}{11} \text{ minutes} \]

4. **Substituting Values**:
Substitute the values into the formula:
\[ \text{Gain or Loss} = \left( \frac{\frac{725}{11}}{64} - 1 \right) \times 60 \times 24 \]

5. **Calculating the Fraction**:
First, calculate \( \frac{725}{11} \div 64 \):
\[ \frac{725}{11} \times \frac{1}{64} = \frac{725}{704} \]

6. **Finding the Difference**:
Now, calculate:
\[ \frac{725}{704} - 1 = \frac{725 - 704}{704} = \frac{21}{704} \]

7. **Calculating Gain in a Day**:
Now substitute this back into the formula:
\[ \text{Gain or Loss} = \frac{21}{704} \times 60 \times 24 \]
First calculate \( 60 \times 24 = 1440 \):
\[ \text{Gain or Loss} = \frac{21 \times 1440}{704} \]

8. **Simplifying the Calculation**:
Calculate \( 21 \times 1440 = 30240 \):
\[ \text{Gain or Loss} = \frac{30240}{704} \]

9. **Final Calculation**:
Simplify \( \frac{30240}{704} \):
\[ 30240 \div 704 = 42.92 \text{ minutes (approximately)} \]

10. **Conclusion**:
Since the result is positive, the watch gains approximately 42.92 minutes in a day.

### Final Answer:
The watch gains approximately **42.92 minutes per day**.