A pipe P can fill a tank in 12 min and another pipe R can fill it in 15 min .but the 3rd pipe M can empty it in 6 min. The 1st two pipes P and R are kept open for double the 2.5 min in the beginning and then the 3rd pipe is also opened. In what time is the tank emptied?

Options

A. 30 min
B. 25 min
C. 45 min
D. 35 min

Answer

To solve the problem step by step, we need to analyze the rates at which the pipes fill and empty the tank.

### Step 1: Determine the filling and emptying rates of the pipes.
- Pipe P fills the tank in 12 minutes. Therefore, its rate is:
\[ \text{Rate of P} = \frac{1 \text{ tank}}{12 \text{ min}} = \frac{60 \text{ units}}{12 \text{ min}} = 5 \text{ units/min} \]

- Pipe R fills the tank in 15 minutes. Therefore, its rate is:
\[ \text{Rate of R} = \frac{1 \text{ tank}}{15 \text{ min}} = \frac{60 \text{ units}}{15 \text{ min}} = 4 \text{ units/min} \]

- Pipe M empties the tank in 6 minutes. Therefore, its rate is:
\[ \text{Rate of M} = -\frac{1 \text{ tank}}{6 \text{ min}} = -\frac{60 \text{ units}}{6 \text{ min}} = -10 \text{ units/min} \]

### Step 2: Calculate the total units filled by pipes P and R in the initial time.
- Pipes P and R are open for double the time of 2.5 minutes, which is:
\[ 2 \times 2.5 = 5 \text{ minutes} \]

- In 5 minutes, the total units filled by both pipes is:
\[ \text{Total units filled} = (\text{Rate of P} + \text{Rate of R}) \times \text{time} \]
\[ = (5 + 4) \times 5 = 9 \times 5 = 45 \text{ units} \]

### Step 3: Determine the combined rate when all three pipes are open.
- When all three pipes are open, the combined rate is:
\[ \text{Combined rate} = \text{Rate of P} + \text{Rate of R} + \text{Rate of M} \]
\[ = 5 + 4 - 10 = -1 \text{ unit/min} \]

### Step 4: Calculate the time taken to empty the tank after the third pipe is opened.
- Since 45 units were filled in the first 5 minutes, we need to find out how long it will take to empty these 45 units at the rate of -1 unit/min:
\[ \text{Time to empty} = \frac{\text{Total units}}{\text{Rate}} = \frac{45 \text{ units}}{1 \text{ unit/min}} = 45 \text{ minutes} \]

### Final Answer:
The tank will be emptied in **45 minutes** after the third pipe is opened.

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