Three pipes, A, B, C can fill an empty cistern in 2, 3 and 6 hours respectively. They are opened together. After what time should B be closed, so that the cistern gets filled in exactly 1 hr 15 min?

Options

A. 30 min
B. 20 min
C. 45 min
D. 15 min

Answer

To solve the problem step by step, we will first determine the rate at which each pipe fills the cistern, then calculate how long pipe B should remain open to ensure the cistern is filled in 1 hour and 15 minutes.

### Step 1: Determine the rates of each pipe
- Pipe A fills the cistern in 2 hours, so its rate is \( \frac{1}{2} \) cisterns per hour.
- Pipe B fills the cistern in 3 hours, so its rate is \( \frac{1}{3} \) cisterns per hour.
- Pipe C fills the cistern in 6 hours, so its rate is \( \frac{1}{6} \) cisterns per hour.

### Step 2: Calculate the combined rate of all three pipes
When all three pipes are open together, their combined rate is:
\[ \text{Combined rate} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 2, 3, and 6 is 6.
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{6} = \frac{1}{6} \]
Adding these together:
\[ \text{Combined rate} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1 \text{ cistern per hour} \]

### Step 3: Convert the total time to hours
The total time given is 1 hour and 15 minutes. We convert this to hours:
\[ 1 \text{ hour} + 15 \text{ minutes} = 1 + \frac{15}{60} = 1 + 0.25 = 1.25 \text{ hours} \]

### Step 4: Calculate the work done by A and C in 1.25 hours
Let \( t \) be the time in hours that pipe B is open. Then pipes A and C will be open for the full 1.25 hours, while pipe B will be open for \( t \) hours.

The work done by pipes A and C in 1.25 hours is:
\[ \text{Work by A} = \frac{1}{2} \times 1.25 = \frac{1.25}{2} = 0.625 \text{ cisterns} \]
\[ \text{Work by C} = \frac{1}{6} \times 1.25 = \frac{1.25}{6} \approx 0.2083 \text{ cisterns} \]

### Step 5: Calculate the total work done by A and C
Total work done by A and C together:
\[ \text{Total work by A and C} = 0.625 + 0.2083 \approx 0.8333 \text{ cisterns} \]

### Step 6: Determine the work done by B
The total work to fill the cistern is 1 cistern. Therefore, the work done by pipe B must be:
\[ \text{Work by B} = 1 - 0.8333 \approx 0.1667 \text{ cisterns} \]

### Step 7: Calculate the time B needs to be open
Since pipe B fills at a rate of \( \frac{1}{3} \) cisterns per hour, we can find the time \( t \) that pipe B needs to be open:
\[ \frac{1}{3} t = 0.1667 \]
\[ t = 0.1667 \times 3 \approx 0.5 \text{ hours} \]

### Step 8: Convert time to minutes
Convert \( t \) from hours to minutes:
\[ t = 0.5 \text{ hours} \times 60 \text{ minutes/hour} = 30 \text{ minutes} \]

### Final Answer
Pipe B should be closed after 30 minutes.

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