Bibhor takes 3 hours to fetch as much water as Ahmed can fetch in 2 hours. Deepak takes 5 hours to fetch as much water as Bibhor can fetch in 4 hours. A tank takes 20 hours to fill if all work together.
What time would Bibhor take to fill the tank alone?

Options

A. 50 h
B. 56 h
C. 77 h
D. 66 h

Answer

To solve the problem step by step, we need to determine the efficiencies of Bibhor, Ahmed, and Deepak, and then calculate how long it would take Bibhor to fill the tank alone.

### Step 1: Define the efficiencies
Let:
- Bibhor's efficiency = B (amount of water fetched per hour)
- Ahmed's efficiency = A (amount of water fetched per hour)
- Deepak's efficiency = D (amount of water fetched per hour)

### Step 2: Establish relationships from the problem
From the problem, we have two key relationships:
1. Bibhor takes 3 hours to fetch as much water as Ahmed can fetch in 2 hours:
\[ 3B = 2A \implies \frac{A}{B} = \frac{3}{2} \implies A = \frac{3}{2}B \]

2. Deepak takes 5 hours to fetch as much water as Bibhor can fetch in 4 hours:
\[ 5D = 4B \implies \frac{D}{B} = \frac{4}{5} \implies D = \frac{4}{5}B \]

### Step 3: Calculate total efficiency when all work together
Now, we can express the total efficiency of all three working together:
\[ \text{Total Efficiency} = A + B + D \]
Substituting the values of A and D in terms of B:
\[ \text{Total Efficiency} = \frac{3}{2}B + B + \frac{4}{5}B \]

### Step 4: Find a common denominator and simplify
To add these fractions, we need a common denominator. The least common multiple of 2 and 5 is 10.
- Convert \(\frac{3}{2}B\) to \(\frac{15}{10}B\)
- Convert \(B\) to \(\frac{10}{10}B\)
- Convert \(\frac{4}{5}B\) to \(\frac{8}{10}B\)

Now, we can add them:
\[ \text{Total Efficiency} = \frac{15}{10}B + \frac{10}{10}B + \frac{8}{10}B = \frac{33}{10}B \]

### Step 5: Use the total efficiency to find the total work
We know that the tank takes 20 hours to fill when all work together:
\[ \text{Total Work} = \text{Total Efficiency} \times \text{Time} \]
\[ \text{Total Work} = \left(\frac{33}{10}B\right) \times 20 = 66B \]

### Step 6: Calculate the time taken by Bibhor alone to fill the tank
If Bibhor works alone, the time taken by him to fill the tank can be calculated as:
\[ \text{Time taken by Bibhor} = \frac{\text{Total Work}}{B} = \frac{66B}{B} = 66 \text{ hours} \]

### Conclusion
Thus, Bibhor would take **66 hours** to fill the tank alone.

---