Anne, Benne and Cenne are three friends. Anne and Benne are twins. Benne takes 2 days more than Cenne to complete the work. If Anne starts a work and 3 days later Benne joins him, then .the work gets completed in 3 more days. Working together Anne, Benne and Cenne can complete thrice the original work in 6 days. In how many days Benne can complete twice the original work with double the efficiency working alone?

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem step by step, let's denote the time taken by Cenne to complete the work as \( C \) days. Therefore, Benne takes \( C + 2 \) days to complete the same work.

1. **Understanding the Work Rates**:
- The work rate of Cenne is \( \frac{1}{C} \) (work done per day).
- The work rate of Benne is \( \frac{1}{C + 2} \).
- The work rate of Anne (who is a twin of Benne) is also \( \frac{1}{C + 2} \).

2. **Setting Up the First Equation**:
- Anne works alone for 3 days, completing \( \frac{3}{C + 2} \) of the work.
- Then Benne joins Anne, and they work together for 3 more days. The total work done in these 3 days is \( 3 \left( \frac{1}{C + 2} + \frac{1}{C + 2} \right) = \frac{3 \times 2}{C + 2} = \frac{6}{C + 2} \).
- The total work done is \( \frac{3}{C + 2} + \frac{6}{C + 2} = \frac{9}{C + 2} \).
- This total work equals 1 (the whole work), so we have:
\[ \frac{9}{C + 2} = 1 \]
- From this, we can solve for \( C \):
\[ 9 = C + 2 \implies C = 7 \]

3. **Finding Benne's Time to Complete the Work**:
- Since \( C = 7 \), Benne's time to complete the work is:
\[ C + 2 = 7 + 2 = 9 \text{ days} \]

4. **Working Together**:
- Now we know the rates:
- Anne's rate: \( \frac{1}{9} \)
- Benne's rate: \( \frac{1}{9} \)
- Cenne's rate: \( \frac{1}{7} \)
- Together, they can complete:
\[ \frac{1}{9} + \frac{1}{9} + \frac{1}{7} = \frac{2}{9} + \frac{1}{7} \]
- To combine these fractions, find a common denominator (63):
\[ \frac{2 \times 7}{63} + \frac{1 \times 9}{63} = \frac{14 + 9}{63} = \frac{23}{63} \]
- Thus, they can complete \( \frac{23}{63} \) of the work in one day.

5. **Calculating Time for Thrice the Work**:
- To complete thrice the original work, they will take:
\[ \text{Time} = \frac{3}{\frac{23}{63}} = 3 \times \frac{63}{23} = \frac{189}{23} \approx 8.22 \text{ days} \]

6. **Benne's Efficiency**:
- If Benne works alone at double efficiency, his rate becomes \( 2 \times \frac{1}{9} = \frac{2}{9} \).
- To complete twice the original work:
\[ \text{Time} = \frac{2}{\frac{2}{9}} = 2 \times 9 = 18 \text{ days} \]

### Final Answer:
Benne can complete twice the original work with double the efficiency in **18 days**.