Two pipes, P and Q can fill a cistern in 12 and 15 minutes respectively. If both are opened together and at the end of 3 minutes, the first is closed, how much longer will the cistern take to fill?
दो नलियाँ P तथा एक टंकी को क्रमश: 12 तथा 15 मिनटों में भर सकती हैं। यदि उन दोनों को एक साथ खोल दिया जाए और 3 मिनटों बाद पहली को बन्द कर दिया जाए, तो टंकी को भरने में कितना समय ज्यादा लगेगा?

Options

A. A)`8 1/4` minute
B. B)`8 3/4` minute
C. C)5 mintute
D. D)`8 1/2` minute

Explanation

To solve the problem, we need to determine how long it will take to fill the cistern after the first pipe (P) is closed. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the filling rates of pipes P and Q
- Pipe P can fill the cistern in 12 minutes. Therefore, its filling rate is:
\[ \text{Rate of P} = \frac{1 \text{ cistern}}{12 \text{ minutes}} = \frac{1}{12} \text{ cistern per minute} \]
- Pipe Q can fill the cistern in 15 minutes. Therefore, its filling rate is:
\[ \text{Rate of Q} = \frac{1 \text{ cistern}}{15 \text{ minutes}} = \frac{1}{15} \text{ cistern per minute} \]

### Step 2: Calculate the combined filling rate when both pipes are open
- When both pipes P and Q are opened together, their combined filling rate is:
\[ \text{Combined Rate} = \text{Rate of P} + \text{Rate of Q} = \frac{1}{12} + \frac{1}{15} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 15 is 60.
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
- Thus,
\[ \text{Combined Rate} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \text{ cistern per minute} \]

### Step 3: Calculate the amount of cistern filled in 3 minutes
- In 3 minutes, the amount of the cistern filled by both pipes is:
\[ \text{Amount filled} = \text{Combined Rate} \times \text{Time} = \frac{3}{20} \times 3 = \frac{9}{20} \text{ cistern} \]

### Step 4: Determine the remaining volume of the cistern
- The total volume of the cistern is 1 (or 20/20). The remaining volume after 3 minutes is:
\[ \text{Remaining volume} = 1 - \frac{9}{20} = \frac{11}{20} \text{ cistern} \]

### Step 5: Calculate the time taken by pipe Q to fill the remaining volume
- Now, only pipe Q is open, which fills at a rate of \(\frac{1}{15}\) cistern per minute.
- To find the time taken to fill the remaining \(\frac{11}{20}\) cistern, we use the formula:
\[ \text{Time} = \frac{\text{Remaining Volume}}{\text{Rate of Q}} = \frac{\frac{11}{20}}{\frac{1}{15}} = \frac{11}{20} \times 15 = \frac{11 \times 15}{20} = \frac{165}{20} = 8.25 \text{ minutes} \]

### Final Answer
The total time taken to fill the cistern after closing pipe P is 8.25 minutes.

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