There are three pipes-A, B & C. A & B are filling pipes and C is emptying pipe. Pipe-A alone can fill the tank in 1 hours & pipe-C is 20% more efficient than pipe. A If pipe-A & B working together fill the tank in `(75)/(2)` minutes. Then find in how much time pipe. A B & C working together can fill the tank?

Options

A. 6 hours
B. 4.5 hours
C. 2.5 hours
D. 3 hours

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the efficiencies of pipes A and C
- Pipe A can fill the tank in 1 hour, which means its efficiency is:
\[ \text{Efficiency of A} = \frac{1 \text{ tank}}{1 \text{ hour}} = 1 \text{ tank/hour} \]
- Since pipe C is 20% more efficient than pipe A, we calculate the efficiency of pipe C:
\[ \text{Efficiency of C} = 1 \text{ tank/hour} + 0.2 \times 1 \text{ tank/hour} = 1.2 \text{ tanks/hour} \]

### Step 2: Convert the time taken by A and B to fill the tank together
- Pipe A and B together fill the tank in \( \frac{75}{2} \) minutes. We convert this to hours:
\[ \frac{75}{2} \text{ minutes} = \frac{75}{120} \text{ hours} = \frac{5}{8} \text{ hours} \]

### Step 3: Calculate the combined efficiency of pipes A and B
- The total work done (filling one tank) can be expressed as:
\[ \text{Efficiency of A + B} = \frac{1 \text{ tank}}{\frac{5}{8} \text{ hours}} = \frac{8}{5} \text{ tanks/hour} \]

### Step 4: Find the efficiency of pipe B
- We know the efficiency of pipe A is 1 tank/hour. Therefore, the efficiency of pipe B can be calculated as:
\[ \text{Efficiency of B} = \text{Efficiency of A + B} - \text{Efficiency of A} = \frac{8}{5} - 1 = \frac{8}{5} - \frac{5}{5} = \frac{3}{5} \text{ tanks/hour} \]

### Step 5: Calculate the total efficiency of pipes A, B, and C together
- The total efficiency of A, B, and C is:
\[ \text{Total Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} - \text{Efficiency of C} \]
Substituting the values:
\[ \text{Total Efficiency} = 1 + \frac{3}{5} - 1.2 = 1 + 0.6 - 1.2 = 0.4 \text{ tanks/hour} \]

### Step 6: Calculate the time taken by A, B, and C together to fill the tank
- The time taken to fill the tank when all three pipes are working together is:
\[ \text{Time} = \frac{1 \text{ tank}}{\text{Total Efficiency}} = \frac{1}{0.4} = 2.5 \text{ hours} \]

### Final Answer
Thus, the time taken by pipes A, B, and C working together to fill the tank is **2.5 hours**.

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