The questions given below contain two statement numbered I and II giving certain data. You have to decide whether the data given in the statements are sufficient to answer the question. Give answer.
In how many days can A, B and C together finish a piece or work?
I. A and B together can finish the same piece of work in `8 2/3` days. B and C together can finish the same piece of work in `13 1/3` days. C and A together can finish the same piece of work in `11 3/7` days.
II. The time taken by A alone to finish the same piece of works is 24 days less than the time taken by C alone to finish the same piece of work.

Options

A. if statement I alone is sufficient but statement II alone is not sufficient.
B. if statement II alone is sufficient but statement I alone is not sufficient.
C. if either statement I or II is sufficient
D. If both statements I and II together are not sufficient.

Answer

To solve the problem, we need to analyze the information given in both statements and determine if they provide enough data to find out how many days A, B, and C can finish a piece of work together.

### Step-by-Step Solution:

1. **Understanding the Statements**:
- **Statement I**:
- A and B together can finish the work in \(8 \frac{2}{3}\) days.
- B and C together can finish the work in \(13 \frac{1}{3}\) days.
- C and A together can finish the work in \(11 \frac{3}{7}\) 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. **Convert Mixed Numbers to Improper Fractions**:
- Convert the days into improper fractions:
- \(8 \frac{2}{3} = \frac{26}{3}\)
- \(13 \frac{1}{3} = \frac{40}{3}\)
- \(11 \frac{3}{7} = \frac{80}{7}\)

3. **Calculate Work Done Per Day**:
- The work done by A and B together in one day is:
\[ A + B = \frac{1}{\frac{26}{3}} = \frac{3}{26} \]
- The work done by B and C together in one day is:
\[ B + C = \frac{1}{\frac{40}{3}} = \frac{3}{40} \]
- The work done by C and A together in one day is:
\[ C + A = \frac{1}{\frac{80}{7}} = \frac{7}{80} \]

4. **Setting Up the Equations**:
- Let the work done by A, B, and C in one day be \(a\), \(b\), and \(c\) respectively.
- From the statements, we can set up the following equations:
\[ a + b = \frac{3}{26} \]
\[ b + c = \frac{3}{40} \]
\[ c + a = \frac{7}{80} \]

5. **Adding the Equations**:
- Adding all three equations:
\[ (a + b) + (b + c) + (c + a) = \frac{3}{26} + \frac{3}{40} + \frac{7}{80} \]
- This simplifies to:
\[ 2(a + b + c) = \frac{3}{26} + \frac{3}{40} + \frac{7}{80} \]

6. **Finding a Common Denominator**:
- The least common multiple of 26, 40, and 80 is 520.
- Convert each fraction:
- \(\frac{3}{26} = \frac{60}{520}\)
- \(\frac{3}{40} = \frac{39}{520}\)
- \(\frac{7}{80} = \frac{45.5}{520}\)

7. **Calculate Total Work**:
- Adding these fractions gives:
\[ 2(a + b + c) = \frac{60 + 39 + 45.5}{520} = \frac{144.5}{520} \]
- Thus:
\[ a + b + c = \frac{144.5}{1040} \]

8. **Finding Days to Complete Work**:
- The total work done by A, B, and C together in one day is \(a + b + c\).
- The number of days to complete the work is the reciprocal of this value.

### Conclusion:
From Statement I alone, we can find the total work done by A, B, and C together and thus determine how many days they will take to finish the work.

**Answer**: Statement I alone is sufficient to answer the question, while Statement II alone is not sufficient.