Two taps can fill a tank in 3 hours and 4 hours respectively and a waste pipe can empty it in 2 hours. When all the three are opened, in how long time the tank will be filled?

Options

A. 12 hour
B. 6 hour
C. 10 hour
D. 8 hour

Answer

To solve the problem of how long it will take to fill the tank when two taps and a waste pipe are opened, we can follow these steps:

### Step 1: Determine the rates of each tap and the waste pipe.
- **Tap A** fills the tank in 3 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{3 \text{ hours}} = \frac{1}{3} \text{ tanks per hour} \]

- **Tap B** fills the tank in 4 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{4 \text{ hours}} = \frac{1}{4} \text{ tanks per hour} \]

- **Waste Pipe C** empties the tank in 2 hours. Therefore, its rate is:
\[ \text{Rate of C} = \frac{1 \text{ tank}}{2 \text{ hours}} = \frac{1}{2} \text{ tanks per hour} \]

### Step 2: Combine the rates when all taps are opened.
When all three are opened, the effective rate of filling the tank is the sum of the rates of the filling taps minus the rate of the waste pipe:
\[ \text{Effective Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \]
Substituting the values we found:
\[ \text{Effective Rate} = \frac{1}{3} + \frac{1}{4} - \frac{1}{2} \]

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

Now we can combine:
\[ \text{Effective Rate} = \frac{4}{12} + \frac{3}{12} - \frac{6}{12} = \frac{4 + 3 - 6}{12} = \frac{1}{12} \text{ tanks per hour} \]

### Step 4: Calculate the time to fill the tank.
Using the formula:
\[ \text{Time} = \frac{\text{Volume}}{\text{Rate}} \]
Assuming the volume of the tank is 1 tank (for simplicity):
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{12} \text{ tanks per hour}} = 12 \text{ hours} \]

### Final Answer:
The tank will be filled in **12 hours**.

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