A water tank has two pipes. The empty tank is filled in 12 min by the 1st and the, full tank is emptied by the 2nd in 20 min. The time required to fill the 1/2 full tank when both pipes are in action, is?

Options

A. 16 min
B. 15 min
C. 20 min
D. 30 min

Answer

To solve the problem step by step, we will analyze the situation involving the two pipes filling and emptying the tank.

### Step 1: Determine the filling and emptying rates of the pipes.
- The first pipe fills the tank in 12 minutes. Therefore, its rate of filling is:
\[ \text{Rate of Pipe 1} = \frac{1 \text{ tank}}{12 \text{ min}} = \frac{1}{12} \text{ tanks/min} \]

- The second pipe empties the tank in 20 minutes. Therefore, its rate of emptying is:
\[ \text{Rate of Pipe 2} = \frac{1 \text{ tank}}{20 \text{ min}} = \frac{1}{20} \text{ tanks/min} \]

### Step 2: Calculate the combined rate when both pipes are in action.
- Since the second pipe is emptying the tank, we subtract its rate from the first pipe's rate:
\[ \text{Combined Rate} = \text{Rate of Pipe 1} - \text{Rate of Pipe 2} = \frac{1}{12} - \frac{1}{20} \]

- To perform this subtraction, we need a common denominator. The least common multiple of 12 and 20 is 60. Thus, we convert the rates:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{20} = \frac{3}{60} \]

- Now we can subtract:
\[ \text{Combined Rate} = \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30} \text{ tanks/min} \]

### Step 3: Determine the time required to fill half the tank.
- We need to fill half the tank, which is \( \frac{1}{2} \) tank. Using the combined rate:
\[ \text{Time} = \frac{\text{Amount of work}}{\text{Rate}} = \frac{\frac{1}{2} \text{ tank}}{\frac{1}{30} \text{ tanks/min}} = \frac{1}{2} \times 30 = 15 \text{ minutes} \]

### Conclusion:
The time required to fill half the tank when both pipes are in action is **15 minutes**.

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