Two pipes A and B can fill an empty tank in 10 hours and 15 hours respectively. Pipe C alone can empty the completely filled tank in 12 hours. First both pipes A and B are opened and after 5 hours pipe C is also opened. What is the total time (in hours) in which the tank will be filled?

Options

A. 7
B. 5
C. 9
D. 8

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the filling rates of pipes A, B, and C.
- Pipe A can fill the tank in 10 hours, so its rate is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{10 \text{ hours}} = \frac{1}{10} \text{ tanks per hour} \]
- Pipe B can fill the tank in 15 hours, so its rate is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{15 \text{ hours}} = \frac{1}{15} \text{ tanks per hour} \]
- Pipe C can empty the tank in 12 hours, so its rate is:
\[ \text{Rate of C} = -\frac{1 \text{ tank}}{12 \text{ hours}} = -\frac{1}{12} \text{ tanks per hour} \]

### Step 2: Calculate the combined rate of pipes A and B.
- The combined rate of A and B when both are opened is:
\[ \text{Combined rate of A and B} = \frac{1}{10} + \frac{1}{15} \]
- To add these fractions, find a common denominator (which is 30):
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
\[ \text{Combined rate of A and B} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ tanks per hour} \]

### Step 3: Calculate the amount of tank filled by A and B in 5 hours.
- In 5 hours, the amount filled by A and B is:
\[ \text{Amount filled in 5 hours} = \text{Rate of A and B} \times \text{Time} = \frac{1}{6} \times 5 = \frac{5}{6} \text{ of the tank} \]

### Step 4: Determine the remaining amount of the tank.
- The remaining amount of the tank to be filled is:
\[ \text{Remaining amount} = 1 - \frac{5}{6} = \frac{1}{6} \text{ of the tank} \]

### Step 5: Calculate the combined rate of A, B, and C when C is opened.
- When pipe C is opened, the combined rate of A, B, and C is:
\[ \text{Combined rate of A, B, and C} = \frac{1}{10} + \frac{1}{15} - \frac{1}{12} \]
- Finding a common denominator (which is 60):
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad -\frac{1}{12} = -\frac{5}{60} \]
\[ \text{Combined rate} = \frac{6}{60} + \frac{4}{60} - \frac{5}{60} = \frac{5}{60} = \frac{1}{12} \text{ tanks per hour} \]

### Step 6: Calculate the time taken to fill the remaining \(\frac{1}{6}\) of the tank.
- The time taken to fill the remaining \(\frac{1}{6}\) of the tank is:
\[ \text{Time} = \frac{\text{Remaining amount}}{\text{Combined rate}} = \frac{\frac{1}{6}}{\frac{1}{12}} = \frac{1}{6} \times \frac{12}{1} = 2 \text{ hours} \]

### Step 7: Calculate the total time taken to fill the tank.
- The total time taken is:
\[ \text{Total time} = \text{Time with A and B} + \text{Time with A, B, and C} = 5 + 2 = 7 \text{ hours} \]

### Final Answer:
The total time in which the tank will be filled is **7 hours**.

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