Two pipes S and T can fill a tank in 3 hours and 6 hours respectively. Pipe U can alone empty the same tank in 4 hours. If all three pipes are opened together, then in how many hours will the tank be filled?

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem step by step, we will first determine the rates at which each pipe fills or empties the tank, and then combine these rates to find out how long it will take to fill the tank when all three pipes are opened together.

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

- Pipe T can fill the tank in 6 hours. Therefore, its rate of filling is:
\[ \text{Rate of T} = \frac{1 \text{ tank}}{6 \text{ hours}} = \frac{1}{6} \text{ tank/hour} \]

### Step 2: Determine the emptying rate of pipe U
- Pipe U can empty the tank in 4 hours. Therefore, its rate of emptying is:
\[ \text{Rate of U} = \frac{1 \text{ tank}}{4 \text{ hours}} = \frac{1}{4} \text{ tank/hour} \]
Since it is emptying the tank, we will consider this as a negative rate:
\[ \text{Rate of U} = -\frac{1}{4} \text{ tank/hour} \]

### Step 3: Combine the rates of all three pipes
Now, we will combine the rates of pipes S, T, and U to find the net rate when all three are opened together:
\[ \text{Net Rate} = \text{Rate of S} + \text{Rate of T} + \text{Rate of U} \]
Substituting the values we found:
\[ \text{Net Rate} = \frac{1}{3} + \frac{1}{6} - \frac{1}{4} \]

### Step 4: Find a common denominator and calculate the net rate
The least common multiple of 3, 6, and 4 is 12. We will convert each fraction:
\[ \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{6} = \frac{2}{12}, \quad \text{and} \quad -\frac{1}{4} = -\frac{3}{12} \]
Now, substituting these values:
\[ \text{Net Rate} = \frac{4}{12} + \frac{2}{12} - \frac{3}{12} = \frac{4 + 2 - 3}{12} = \frac{3}{12} = \frac{1}{4} \text{ tank/hour} \]

### Step 5: Calculate the time to fill the tank
To find the time taken to fill the tank, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Net Rate}} = \frac{1 \text{ tank}}{\frac{1}{4} \text{ tank/hour}} = 4 \text{ hours} \]

### Final Answer
Thus, if all three pipes are opened together, the tank will be filled in **4 hours**.

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