The following questions are accompanied by two statements I and II. You have to determine which statements(s) is/are sufficient/ necessary to answer the questions.
If A, B and C complete a work together in `3 (9)/(13)` hours, then find time taken by C alone to complete the work?
I. Efficiency of B is 25% less than A and efficiency of C is two times of B.
II. Time taken by A & B alone to complete the work is 12 hours and 16 hours respectively.

Options

A. Only statement I is sufficient to give answer of the question
B. Only statement Ilis Sufficient to give answer of the question
C. Statements I & II together are sufficient to give answer of the question
D. Either statement I or II is sufficient to give answer of question

Answer

To solve the problem, we need to determine how long C takes to complete the work alone, given the combined work time of A, B, and C together and the information provided in the two statements.

### Step-by-Step Solution:

1. **Understanding the Combined Work Time**:
- A, B, and C together complete the work in \(3 \frac{9}{13}\) hours.
- Convert this mixed fraction into an improper fraction:
\[ 3 \frac{9}{13} = \frac{3 \times 13 + 9}{13} = \frac{39 + 9}{13} = \frac{48}{13} \text{ hours} \]

2. **Finding the Work Rate**:
- The work rate (efficiency) of A, B, and C together can be calculated as:
\[ \text{Work Rate} = \frac{1 \text{ work}}{\frac{48}{13} \text{ hours}} = \frac{13}{48} \text{ work per hour} \]

3. **Analyzing Statement I**:
- Statement I states that the efficiency of B is 25% less than A and the efficiency of C is twice that of B.
- Let the efficiency of A be \(x\). Then:
- Efficiency of B = \(x - 0.25x = 0.75x\)
- Efficiency of C = \(2 \times 0.75x = 1.5x\)
- Therefore, the combined efficiency of A, B, and C is:
\[ x + 0.75x + 1.5x = 3.25x \]
- Setting this equal to the combined work rate:
\[ 3.25x = \frac{13}{48} \]
- Solving for \(x\):
\[ x = \frac{13}{48 \times 3.25} = \frac{13}{156} = \frac{1}{12} \]
- Thus, the efficiency of A is \(\frac{1}{12}\), B is \(\frac{3}{12} = \frac{1}{4}\), and C is \(\frac{1.5}{12} = \frac{1}{8}\).

4. **Finding Time Taken by C**:
- The time taken by C to complete the work alone is the reciprocal of C's efficiency:
\[ \text{Time taken by C} = \frac{1}{\frac{1}{8}} = 8 \text{ hours} \]

5. **Analyzing Statement II**:
- Statement II states that A takes 12 hours and B takes 16 hours to complete the work alone.
- Therefore, their efficiencies are:
- Efficiency of A = \(\frac{1}{12}\)
- Efficiency of B = \(\frac{1}{16}\)
- The combined efficiency of A and B is:
\[ \frac{1}{12} + \frac{1}{16} = \frac{4 + 3}{48} = \frac{7}{48} \]
- Since A, B, and C together have a combined efficiency of \(\frac{13}{48}\), we can find C's efficiency:
\[ \text{Efficiency of C} = \frac{13}{48} - \frac{7}{48} = \frac{6}{48} = \frac{1}{8} \]
- Thus, the time taken by C to complete the work alone is:
\[ \text{Time taken by C} = \frac{1}{\frac{1}{8}} = 8 \text{ hours} \]

### Conclusion:
Both statements I and II are sufficient to determine the time taken by C alone to complete the work, which is 8 hours.