Two pipes P and Q can fill a cistern in 15 minutes and 20 minutes respectively. Both are opened together, but at the end of 5 minutes, the pipe P is turned off. How long will the pipe Q take to fill the cistern ?

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the rates of pipes P and Q
- Pipe P can fill the cistern in 15 minutes.
- Pipe Q can fill the cistern in 20 minutes.

To find the rate of each pipe, we can use the formula:
\[ \text{Rate} = \frac{\text{Total Volume}}{\text{Time}} \]
Assuming the total volume of the cistern is 60 cubic units (for simplicity).

- Rate of Pipe P:
\[ \text{Rate of P} = \frac{60 \text{ cubic units}}{15 \text{ minutes}} = 4 \text{ cubic units per minute} \]

- Rate of Pipe Q:
\[ \text{Rate of Q} = \frac{60 \text{ cubic units}}{20 \text{ minutes}} = 3 \text{ cubic units per minute} \]

### Step 2: Calculate the combined rate when both pipes are open
When both pipes are opened together, their rates add up:
\[ \text{Combined Rate} = \text{Rate of P} + \text{Rate of Q} = 4 + 3 = 7 \text{ cubic units per minute} \]

### Step 3: Calculate the volume filled in the first 5 minutes
Now, we need to find out how much volume is filled when both pipes are open for the first 5 minutes:
\[ \text{Volume filled in 5 minutes} = \text{Combined Rate} \times \text{Time} = 7 \text{ cubic units per minute} \times 5 \text{ minutes} = 35 \text{ cubic units} \]

### Step 4: Calculate the remaining volume to be filled
The total volume of the cistern is 60 cubic units, and after 5 minutes, 35 cubic units have been filled. Therefore, the remaining volume is:
\[ \text{Volume left} = 60 - 35 = 25 \text{ cubic units} \]

### Step 5: Determine how long Pipe Q will take to fill the remaining volume
Since Pipe P is turned off after 5 minutes, only Pipe Q will fill the remaining volume. We can calculate the time it takes for Pipe Q to fill the remaining 25 cubic units:
\[ \text{Time taken by Q} = \frac{\text{Volume left}}{\text{Rate of Q}} = \frac{25 \text{ cubic units}}{3 \text{ cubic units per minute}} \approx 8.33 \text{ minutes} \]

### Final Answer
Thus, the total time taken by Pipe Q to fill the remaining volume is approximately 8 minutes and 20 seconds (8 minutes and 1/3 of a minute).

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