Tap A can fill a water tank in 12 min, tap B will fill the same tank in 10 min and tap C can empty the tank in 6 min. If all the three taps are opened together, in how many minutes will the tank be completely filled up or emptied ?

Options

A. 15 min
B. 30 min
C. 45 min
D. 60 min

Answer

To solve the problem of how long it will take to fill or empty the tank when taps A, B, and C are opened together, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the rate of each tap:**
- Tap A fills the tank in 12 minutes. Therefore, in 1 minute, it fills \( \frac{1}{12} \) of the tank.
- Tap B fills the tank in 10 minutes. Therefore, in 1 minute, it fills \( \frac{1}{10} \) of the tank.
- Tap C empties the tank in 6 minutes. Therefore, in 1 minute, it empties \( \frac{1}{6} \) of the tank.

2. **Calculate the combined rate of all taps:**
- The combined rate when all taps are opened together is:
\[ \text{Rate of A} + \text{Rate of B} - \text{Rate of C} = \frac{1}{12} + \frac{1}{10} - \frac{1}{6} \]

3. **Find a common denominator:**
- The least common multiple (LCM) of 12, 10, and 6 is 60. We will convert each fraction to have a denominator of 60:
- \( \frac{1}{12} = \frac{5}{60} \)
- \( \frac{1}{10} = \frac{6}{60} \)
- \( \frac{1}{6} = \frac{10}{60} \)

4. **Combine the rates:**
- Now substituting the values into the equation:
\[ \frac{5}{60} + \frac{6}{60} - \frac{10}{60} = \frac{5 + 6 - 10}{60} = \frac{1}{60} \]

5. **Calculate the time to fill the tank:**
- The combined rate of \( \frac{1}{60} \) means that together, the taps fill \( \frac{1}{60} \) of the tank in 1 minute. Therefore, to fill the entire tank, it will take:
\[ \text{Time} = \frac{1}{\frac{1}{60}} = 60 \text{ minutes} \]

### Final Answer:
The tank will be completely filled in **60 minutes**.