To solve the problem, we need to analyze the statements provided and determine whether they give us enough information to answer the question: "In how many days can men A, B, and C together finish the same piece of work?"
### Step-by-Step Solution:
1. **Understanding the Statements**:
- **Statement I**:
- A and B can finish the work in 6 days.
- B and C can finish the work in 12 days.
- C and A can finish the work in 10 days.
- **Statement II**:
- The time taken by A alone to finish the work is 24 days less than the time taken by C alone.
2. **Converting Statements into Equations**:
- From Statement I, we can derive the work rates:
- Let the total work be 1 unit.
- The work done by A and B together in one day = \( \frac{1}{6} \) units.
- The work done by B and C together in one day = \( \frac{1}{12} \) units.
- The work done by C and A together in one day = \( \frac{1}{10} \) units.
3. **Setting Up the Equations**:
- Let the work done by A in one day = A, by B = B, and by C = C.
- From the statements, we can write:
- \( A + B = \frac{1}{6} \) (1)
- \( B + C = \frac{1}{12} \) (2)
- \( C + A = \frac{1}{10} \) (3)
4. **Adding the Equations**:
- Adding equations (1), (2), and (3):
\[
(A + B) + (B + C) + (C + A) = \frac{1}{6} + \frac{1}{12} + \frac{1}{10}
\]
- This simplifies to:
\[
2A + 2B + 2C = \frac{1}{6} + \frac{1}{12} + \frac{1}{10}
\]
5. **Finding a Common Denominator**:
- The least common multiple (LCM) of 6, 12, and 10 is 60.
- Converting the fractions:
\[
\frac{1}{6} = \frac{10}{60}, \quad \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{10} = \frac{6}{60}
\]
- Adding these gives:
\[
\frac{10 + 5 + 6}{60} = \frac{21}{60}
\]
6. **Solving for A + B + C**:
- Therefore:
\[
2(A + B + C) = \frac{21}{60}
\]
- Dividing by 2:
\[
A + B + C = \frac{21}{120} = \frac{7}{40}
\]
7. **Finding the Total Time**:
- The work done by A, B, and C together in one day is \( \frac{7}{40} \).
- Therefore, the time taken by A, B, and C together to finish the work is:
\[
\text{Total time} = \frac{1}{\frac{7}{40}} = \frac{40}{7} \text{ days}
\]
8. **Conclusion from Statement I**:
- Statement I alone is sufficient to answer the question.
9. **Analyzing Statement II**:
- Statement II gives a relationship between A and C but does not provide enough information about B.
- Therefore, Statement II alone is not sufficient to answer the question.
### Final Answer:
- The data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient.
