Three pipes A, B and C can fill a tank in 6 hours, 9 hours and 12 hours respectively. B and C are opened for half an hour, then A is also opened. The time taken by the three pipes together to fill the remaining part of the tank is

Options

A. 3 hours
B. 2 hours
C. `2(1)/(2)` hours
D. `3(1)/(2)` hours

Answer

To solve the problem step-by-step, we need to determine the time taken by three pipes A, B, and C to fill a tank when they are opened in a specific sequence. Here's the detailed solution:

### Step 1: Determine the filling rates of each pipe
- Pipe A can fill the tank in 6 hours, so its rate is \( \frac{1}{6} \) of the tank per hour.
- Pipe B can fill the tank in 9 hours, so its rate is \( \frac{1}{9} \) of the tank per hour.
- Pipe C can fill the tank in 12 hours, so its rate is \( \frac{1}{12} \) of the tank per hour.

### Step 2: Calculate the combined rate of pipes B and C
- The combined rate of pipes B and C is:
\[ \text{Rate of B} + \text{Rate of C} = \frac{1}{9} + \frac{1}{12} \]
- To add these fractions, find a common denominator (which is 36):
\[ \frac{1}{9} = \frac{4}{36}, \quad \frac{1}{12} = \frac{3}{36} \]
- Therefore,
\[ \text{Combined rate of B and C} = \frac{4}{36} + \frac{3}{36} = \frac{7}{36} \]

### Step 3: Calculate the amount of tank filled by B and C in half an hour
- In half an hour, the amount filled by B and C is:
\[ \text{Amount filled} = \text{Rate} \times \text{Time} = \frac{7}{36} \times \frac{1}{2} = \frac{7}{72} \]

### Step 4: Calculate the remaining part of the tank
- The total capacity of the tank is 1 (whole tank), so the remaining part after B and C have worked for half an hour is:
\[ \text{Remaining part} = 1 - \frac{7}{72} = \frac{72}{72} - \frac{7}{72} = \frac{65}{72} \]

### Step 5: Calculate the combined rate of pipes A, B, and C
- The combined rate of pipes A, B, and C is:
\[ \text{Rate of A} + \text{Rate of B} + \text{Rate of C} = \frac{1}{6} + \frac{1}{9} + \frac{1}{12} \]
- Finding a common denominator (which is 36):
\[ \frac{1}{6} = \frac{6}{36}, \quad \frac{1}{9} = \frac{4}{36}, \quad \frac{1}{12} = \frac{3}{36} \]
- Therefore,
\[ \text{Combined rate of A, B, and C} = \frac{6}{36} + \frac{4}{36} + \frac{3}{36} = \frac{13}{36} \]

### Step 6: Calculate the time taken to fill the remaining part of the tank
- The time taken to fill the remaining part of the tank is given by:
\[ \text{Time} = \frac{\text{Remaining part}}{\text{Combined rate}} = \frac{\frac{65}{72}}{\frac{13}{36}} \]
- This can be simplified:
\[ \text{Time} = \frac{65}{72} \times \frac{36}{13} = \frac{65 \times 36}{72 \times 13} \]
- Simplifying further:
\[ = \frac{65 \times 36}{936} = \frac{65 \times 1}{26} = \frac{65}{26} = 2.5 \text{ hours} \]

### Final Answer
The time taken by the three pipes together to fill the remaining part of the tank is **2.5 hours**.

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