A pipe can fill a cistern in 12 min and another pipe can fill it in 15 min, but a third pipe can empty it in 6 min. The first two pipes are kept open for 5 min in the beginning and then the third pipe is also opened. Time taken to empty the cistern is

Options

A. 38 min
B. 22 min
C. 42 min
D. 45min

Answer

To solve the problem step by step, we will analyze the filling and emptying rates of the pipes involved.

### Step 1: Determine the rates of the pipes
1. **First Pipe**: Fills the cistern in 12 minutes.
- Rate = \( \frac{60 \text{ units}}{12 \text{ min}} = 5 \text{ units/min} \)

2. **Second Pipe**: Fills the cistern in 15 minutes.
- Rate = \( \frac{60 \text{ units}}{15 \text{ min}} = 4 \text{ units/min} \)

3. **Third Pipe**: Empties the cistern in 6 minutes.
- Rate = \( -\frac{60 \text{ units}}{6 \text{ min}} = -10 \text{ units/min} \)

### Step 2: Calculate the total units filled in the first 5 minutes
- In 1 minute, the first two pipes together fill:
- \( 5 \text{ units/min} + 4 \text{ units/min} = 9 \text{ units/min} \)

- In 5 minutes, they will fill:
- \( 9 \text{ units/min} \times 5 \text{ min} = 45 \text{ units} \)

### Step 3: Calculate the remaining units after 5 minutes
- After 5 minutes, the cistern has:
- \( 60 \text{ units} - 45 \text{ units} = 15 \text{ units} \) remaining to be filled.

### Step 4: Determine the net rate when all three pipes are open
- When all three pipes are open, the net rate is:
- \( 5 \text{ units/min} + 4 \text{ units/min} - 10 \text{ units/min} = -1 \text{ unit/min} \)

### Step 5: Calculate the time taken to empty the remaining units
- Since the net rate is -1 unit/min, it means the cistern is emptying at a rate of 1 unit per minute.
- To empty the remaining 15 units:
- Time = \( \frac{15 \text{ units}}{1 \text{ unit/min}} = 15 \text{ minutes} \)

### Step 6: Total time taken to empty the cistern
- The total time taken to empty the cistern after the first 5 minutes is:
- \( 5 \text{ minutes} + 15 \text{ minutes} = 20 \text{ minutes} \)

Thus, the total time taken to empty the cistern after the third pipe is opened is **20 minutes**.

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