Two pipes P and Q can fill a cistern in 12 and 15 min, respectively. If both are opened together and at the end of 3 min, the first is closed. How much longer will the cistern take to fill?

Options

A. 8 `(1)/(4)` min
B. `5 (3)/(4) ` min
C. 5 min
D. `8(1)/(2)` min

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the filling rates of pipes P and Q.
- Pipe P can fill the cistern in 12 minutes.
- Pipe Q can fill the cistern in 15 minutes.

**Filling rate of Pipe P:**
\[ \text{Rate of P} = \frac{1 \text{ cistern}}{12 \text{ min}} = \frac{60 \text{ units}}{12 \text{ min}} = 5 \text{ units/min} \]

**Filling rate of Pipe Q:**
\[ \text{Rate of Q} = \frac{1 \text{ cistern}}{15 \text{ min}} = \frac{60 \text{ units}}{15 \text{ min}} = 4 \text{ units/min} \]

### Step 2: Calculate the combined filling rate of both pipes.
\[ \text{Combined Rate} = \text{Rate of P} + \text{Rate of Q} = 5 \text{ units/min} + 4 \text{ units/min} = 9 \text{ units/min} \]

### Step 3: Calculate the amount filled in the first 3 minutes when both pipes are open.
\[ \text{Amount filled in 3 minutes} = \text{Combined Rate} \times 3 \text{ min} = 9 \text{ units/min} \times 3 \text{ min} = 27 \text{ units} \]

### Step 4: Determine the remaining volume to be filled.
- Total capacity of the cistern = 60 units.
- Amount filled in 3 minutes = 27 units.

\[ \text{Remaining volume} = 60 \text{ units} - 27 \text{ units} = 33 \text{ units} \]

### Step 5: Calculate the time required for Pipe Q to fill the remaining volume.
- Pipe Q fills at a rate of 4 units/min.

\[ \text{Time taken by Q to fill 33 units} = \frac{33 \text{ units}}{4 \text{ units/min}} = 8.25 \text{ min} = 8 \frac{1}{4} \text{ min} \]

### Final Answer:
The cistern will take an additional \( 8 \frac{1}{4} \) minutes to fill after Pipe P is closed.

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