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 be taken by them to fill the tank if they fill the tank in the following manner :
Deepak and Bibhor fill together in `1^(st)` hr.
Bibhor and Ahmed together in `2^(nd)` hr.
And Ahmed and Deepak together in `3^(rd)` hr.

Options

A. 45 h
B. 35 h
C. 40 h
D. 30 h

Answer

To solve the problem step by step, we will first determine the work rates of Bibhor, Ahmed, and Deepak based on the information given. Then we will calculate how much work they can do together in the specified manner.

### Step 1: Determine the work rates of each person

1. **Bibhor's and Ahmed's work rate:**
- Bibhor takes 3 hours to fetch as much water as Ahmed can fetch in 2 hours.
- Let the amount of water fetched by Ahmed in 2 hours be \( A \).
- Therefore, in 3 hours, Bibhor fetches \( A \).
- This gives us the ratio of their work rates:
\[ \frac{A}{B} = \frac{2}{3} \implies A = \frac{2}{3}B \]

2. **Deepak's and Bibhor's work rate:**
- Deepak takes 5 hours to fetch as much water as Bibhor can fetch in 4 hours.
- Let the amount of water fetched by Bibhor in 4 hours be \( B \).
- Therefore, in 5 hours, Deepak fetches \( B \).
- This gives us the ratio of their work rates:
\[ \frac{D}{B} = \frac{4}{5} \implies D = \frac{4}{5}B \]

### Step 2: Express all work rates in terms of Bibhor's work rate

Now, we have:
- Ahmed's work rate: \( A = \frac{2}{3}B \)
- Deepak's work rate: \( D = \frac{4}{5}B \)

### Step 3: Find a common base for work rates

To find a common base, we can express all work rates in terms of \( B \):
- Let \( B = 15 \) (for easier calculations):
- \( A = \frac{2}{3} \times 15 = 10 \)
- \( D = \frac{4}{5} \times 15 = 12 \)

Thus, the work rates are:
- Ahmed (A) = 10 units/hour
- Bibhor (B) = 15 units/hour
- Deepak (D) = 12 units/hour

### Step 4: Calculate the total work to fill the tank

If all three work together, their combined work rate is:
\[ A + B + D = 10 + 15 + 12 = 37 \text{ units/hour} \]
Given that they can fill the tank in 20 hours, the total work required to fill the tank is:
\[ \text{Total Work} = \text{Work Rate} \times \text{Time} = 37 \times 20 = 740 \text{ units} \]

### Step 5: Calculate the work done in the specified manner

1. **In the 1st hour (Deepak and Bibhor):**
\[ \text{Work} = D + B = 12 + 15 = 27 \text{ units} \]

2. **In the 2nd hour (Bibhor and Ahmed):**
\[ \text{Work} = B + A = 15 + 10 = 25 \text{ units} \]

3. **In the 3rd hour (Ahmed and Deepak):**
\[ \text{Work} = A + D = 10 + 12 = 22 \text{ units} \]

### Step 6: Total work done in 3 hours

Total work done in 3 hours:
\[ 27 + 25 + 22 = 74 \text{ units} \]

### Step 7: Calculate the total time to fill the tank

To find the total time to fill the tank:
\[ \text{Total Work} = 740 \text{ units} \]
The number of complete cycles of 3 hours:
\[ \text{Number of cycles} = \frac{740}{74} = 10 \]
Thus, the total time taken is:
\[ \text{Total Time} = 10 \times 3 = 30 \text{ hours} \]

### Final Answer
The time taken by them to fill the tank is **30 hours**.