Two taps can fill a tank in 3 and 4 hours respectivel and an another tap can empty it in 2 hours. If all three taps are opened simultaneously then in how much time the tank will be filled?
दो नल एक हौज को क्रमशः 3 तथा 4 घण्टे में भर सकते हैं तथा एक निकास नल से 2 घण्टे में खाली कर सकता है। यदि तीनों नल खोल दिए जाए हौज कितने समय में भरेगा?

Options

A. 5 hours/5घण्टे
B. 8 hours/8 घण्टे
C. 10 hours /10 घण्टे
D. 12 hours/12 घण्टे

Explanation

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

### Step 1: Determine the filling rates of the taps
- Tap A can fill the tank in 3 hours. Therefore, its rate is:
\[ \text{Rate of Tap A} = \frac{1}{3} \text{ tank/hour} \]

- Tap B can fill the tank in 4 hours. Therefore, its rate is:
\[ \text{Rate of Tap B} = \frac{1}{4} \text{ tank/hour} \]

- Tap C can empty the tank in 2 hours. Therefore, its rate (as a negative contribution) is:
\[ \text{Rate of Tap C} = -\frac{1}{2} \text{ tank/hour} \]

### Step 2: Combine the rates of all taps
When all three taps are opened simultaneously, their combined rate is:
\[ \text{Combined Rate} = \text{Rate of Tap A} + \text{Rate of Tap B} + \text{Rate of Tap C} \]
Substituting the rates we found:
\[ \text{Combined Rate} = \frac{1}{3} + \frac{1}{4} - \frac{1}{2} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 3, 4, and 2 is 12. We will convert each rate to have a denominator of 12:
- For Tap A:
\[ \frac{1}{3} = \frac{4}{12} \]
- For Tap B:
\[ \frac{1}{4} = \frac{3}{12} \]
- For Tap C:
\[ -\frac{1}{2} = -\frac{6}{12} \]

### Step 4: Add the rates
Now we can add the rates:
\[ \text{Combined Rate} = \frac{4}{12} + \frac{3}{12} - \frac{6}{12} = \frac{4 + 3 - 6}{12} = \frac{1}{12} \text{ tank/hour} \]

### Step 5: Calculate the time to fill the tank
To find the time taken to fill the tank, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{1}{12}} = 12 \text{ hours} \]

### Conclusion
Thus, if all three taps are opened simultaneously, the tank will be filled in **12 hours**.

---