Pipe A basically used as inlet pipe and pipe B is used as outlet pipe. Pipes A and B both are opened simultaneously, all the time. When pipe A fills the tank and B empty the tank, it will take double the time than when both the pipe, fill the tank. When pipe B is used for fillng the tank, its efficiency remains constant. What is the ratio of efficiency of pipe A and pipe B respectively?

Options

A. `3: 1`
B. `5:2`
C. `1 : 3`
D. `3:2`

Answer

To solve the problem, we need to establish the relationship between the efficiencies of Pipe A and Pipe B based on the information provided. Let's break it down step by step.

### Step 1: Define Variables
Let the efficiency of Pipe A be \( a \) (units of tank filled per hour) and the efficiency of Pipe B be \( b \) (units of tank emptied per hour).

### Step 2: Establish Relationships
1. When both pipes are opened, Pipe A fills the tank while Pipe B empties it. The net efficiency when both are open is \( a - b \).
2. When only Pipe A is filling the tank, the time taken to fill the tank is \( \frac{1}{a} \) hours.
3. When both pipes are open, the time taken to fill the tank is \( \frac{1}{a - b} \) hours.
4. According to the problem, the time taken when both pipes are open (filling and emptying) is double the time taken when only Pipe A is filling. Thus, we can write the equation:
\[ \frac{1}{a - b} = 2 \cdot \frac{1}{a} \]

### Step 3: Solve the Equation
Now, we can solve the equation derived from the relationship:
\[ \frac{1}{a - b} = \frac{2}{a} \]
Cross-multiplying gives:
\[ a = 2(a - b) \]
Expanding this, we get:
\[ a = 2a - 2b \]
Rearranging the terms, we find:
\[ 2b = a \quad \text{or} \quad \frac{a}{b} = 2 \]

### Step 4: Determine the Ratio
From the equation \( \frac{a}{b} = 2 \), we can express the ratio of the efficiencies of Pipe A to Pipe B:
\[ \text{Ratio of efficiency of Pipe A to Pipe B} = 2:1 \]

### Conclusion
Thus, the ratio of the efficiency of Pipe A to Pipe B is \( 2:1 \).

---