If Pipes A and B can fill a tank in 15 min and 20 mins respectively and pipe C empties the tank in 12 mins. What will be the time taken by A, B and C together to fill the tank completely?

Options

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

Answer

To solve the problem, we need to find the combined efficiency of pipes A, B, and C, and then determine how long it will take for them to fill the tank together.

### Step-by-Step Solution:

1. **Determine the efficiency of each pipe:**
- Pipe A can fill the tank in 15 minutes. Therefore, its efficiency is:
\[ \text{Efficiency of A} = \frac{1}{15} \text{ tanks per minute} \]
- Pipe B can fill the tank in 20 minutes. Therefore, its efficiency is:
\[ \text{Efficiency of B} = \frac{1}{20} \text{ tanks per minute} \]
- Pipe C can empty the tank in 12 minutes. Therefore, its efficiency (as it is emptying) is:
\[ \text{Efficiency of C} = -\frac{1}{12} \text{ tanks per minute} \]

2. **Calculate the combined efficiency of A, B, and C:**
- The combined efficiency is the sum of the efficiencies of A, B, and C:
\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} + \text{Efficiency of C} \]
\[ = \frac{1}{15} + \frac{1}{20} - \frac{1}{12} \]

3. **Find a common denominator to add the fractions:**
- The least common multiple (LCM) of 15, 20, and 12 is 60. We will convert each fraction:
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
- Now substituting these values into the combined efficiency:
\[ \text{Combined Efficiency} = \frac{4}{60} + \frac{3}{60} - \frac{5}{60} = \frac{4 + 3 - 5}{60} = \frac{2}{60} = \frac{1}{30} \text{ tanks per minute} \]

4. **Calculate the time taken to fill the tank:**
- If the combined efficiency is \(\frac{1}{30}\) tanks per minute, then the time taken to fill 1 tank is the reciprocal of the efficiency:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{30} \text{ tanks per minute}} = 30 \text{ minutes} \]

### Final Answer:
The time taken by pipes A, B, and C together to fill the tank completely is **30 minutes**.

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