Two pipes A and B can fill a tank in 12 and 15 minutes respectively. A third pipe C can empty it in 10 minutes. How long will it take to fill in the tank if all pipes are opened simultaneously ?

Options

A. 20 minutes
B. 30 minutes
C. 40 minutes
D. 25 minutes

Answer

To solve the problem of how long it will take to fill the tank when all three pipes A, B, and C are opened simultaneously, we can follow these steps:

### Step 1: Determine the filling rates of pipes A and B
- Pipe A can fill the tank in 12 minutes. Therefore, its filling rate is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{12 \text{ minutes}} = \frac{1}{12} \text{ tanks per minute} \]

- Pipe B can fill the tank in 15 minutes. Therefore, its filling rate is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{15 \text{ minutes}} = \frac{1}{15} \text{ tanks per minute} \]

### Step 2: Determine the emptying rate of pipe C
- Pipe C can empty the tank in 10 minutes. Therefore, its emptying rate is:
\[ \text{Rate of C} = \frac{1 \text{ tank}}{10 \text{ minutes}} = \frac{1}{10} \text{ tanks per minute} \]

### Step 3: Calculate the net rate when all pipes are opened
- The net filling rate when all pipes are opened simultaneously is the sum of the filling rates of A and B minus the emptying rate of C:
\[ \text{Net Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \]
\[ \text{Net Rate} = \frac{1}{12} + \frac{1}{15} - \frac{1}{10} \]

### Step 4: Find a common denominator and calculate the net rate
- The least common multiple (LCM) of 12, 15, and 10 is 60. Now we convert each rate to have a denominator of 60:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{10} = \frac{6}{60} \]
- Substitute these values into the net rate equation:
\[ \text{Net Rate} = \frac{5}{60} + \frac{4}{60} - \frac{6}{60} = \frac{5 + 4 - 6}{60} = \frac{3}{60} = \frac{1}{20} \text{ tanks per minute} \]

### Step 5: Calculate the time to fill the tank
- Since the net rate is \(\frac{1}{20}\) tanks per minute, it will take:
\[ \text{Time} = \frac{1 \text{ tank}}{\text{Net Rate}} = \frac{1 \text{ tank}}{\frac{1}{20} \text{ tanks per minute}} = 20 \text{ minutes} \]

### Final Answer
Thus, it will take **20 minutes** to fill the tank when all three pipes are opened simultaneously.

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