Pipe A can fill a tank in 12 hours and pipe B can empty the tank in 18 hours. Both pipes are opened at 6 a.m. and after some time pipe B is closed and the tank is full at 8 p.m. At what time was the pipe B closed ?

Options

A. 10 a.m.
B. 8 a.m.
C. 9 a.m.
D. 11 a.m.

Explanation

To solve the problem step by step, we need to analyze the information given and set up an equation based on the rates of work for both pipes.

### Step 1: Determine the rates of work for both pipes.
- Pipe A can fill the tank in 12 hours. Therefore, its rate of work is:
\[ \text{Rate of Pipe A} = \frac{1}{12} \text{ tank/hour} \]
- Pipe B can empty the tank in 18 hours. Therefore, its rate of work is:
\[ \text{Rate of Pipe B} = \frac{1}{18} \text{ tank/hour} \]

### Step 2: Calculate the total time taken to fill the tank.
- The pipes are opened at 6 a.m. and the tank is full at 8 p.m. This means the total time taken to fill the tank is:
\[ 8 \text{ p.m.} - 6 \text{ a.m.} = 14 \text{ hours} \]

### Step 3: Let \( x \) be the time (in hours) that Pipe B is open.
- Pipe A is open for the entire 14 hours.
- Pipe B is open for \( x \) hours.

### Step 4: Set up the equation based on the work done.
- The work done by Pipe A in 14 hours is:
\[ \text{Work by Pipe A} = 14 \times \frac{1}{12} = \frac{14}{12} = \frac{7}{6} \text{ tanks} \]
- The work done by Pipe B in \( x \) hours is:
\[ \text{Work by Pipe B} = x \times \frac{1}{18} = \frac{x}{18} \text{ tanks} \]
- Since the tank is full, the total work done is equal to 1 tank. Therefore, we can set up the equation:
\[ \frac{7}{6} - \frac{x}{18} = 1 \]

### Step 5: Solve the equation for \( x \).
- Rearranging the equation gives:
\[ \frac{7}{6} - 1 = \frac{x}{18} \]
\[ \frac{1}{6} = \frac{x}{18} \]
- Cross-multiplying gives:
\[ 1 \times 18 = 6 \times x \]
\[ 18 = 6x \]
\[ x = \frac{18}{6} = 3 \]

### Step 6: Determine the time when Pipe B was closed.
- Since Pipe B was open for \( x = 3 \) hours and it was opened at 6 a.m., it was closed at:
\[ 6 \text{ a.m.} + 3 \text{ hours} = 9 \text{ a.m.} \]

### Final Answer:
Pipe B was closed at **9 a.m.**.

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