Two pipes X and Y can full a cistern in 24 minutes and 32 minutes respectively. If both the pipes are opened together, then after how much time (In minutes) should Y be closed so that the tank is full in 18 minutes ?

Options

A. 10
B. 8
C. 6
D. 5

Answer

To solve the problem step by step, we need to determine how long pipe Y should remain open while both pipes X and Y are filling the cistern together, so that the tank is full in a total of 18 minutes.

### Step 1: Determine the filling rates of pipes X and Y.
- Pipe X can fill the cistern in 24 minutes. Therefore, its rate of filling is:
\[ \text{Rate of X} = \frac{1 \text{ cistern}}{24 \text{ minutes}} = \frac{1}{24} \text{ cistern/minute} \]
- Pipe Y can fill the cistern in 32 minutes. Therefore, its rate of filling is:
\[ \text{Rate of Y} = \frac{1 \text{ cistern}}{32 \text{ minutes}} = \frac{1}{32} \text{ cistern/minute} \]

### Step 2: Calculate the combined rate of pipes X and Y.
When both pipes are opened together, their combined rate of filling is:
\[ \text{Combined Rate} = \text{Rate of X} + \text{Rate of Y} = \frac{1}{24} + \frac{1}{32} \]
To add these fractions, we need a common denominator. The least common multiple of 24 and 32 is 96.
\[ \frac{1}{24} = \frac{4}{96}, \quad \frac{1}{32} = \frac{3}{96} \]
Thus,
\[ \text{Combined Rate} = \frac{4}{96} + \frac{3}{96} = \frac{7}{96} \text{ cistern/minute} \]

### Step 3: Determine the total work done in 18 minutes.
In 18 minutes, the total work done (the amount of the cistern filled) is:
\[ \text{Total Work} = \text{Combined Rate} \times \text{Time} = \frac{7}{96} \times 18 = \frac{126}{96} = \frac{21}{16} \text{ cistern} \]
This means that if both pipes worked together for 18 minutes, they would fill more than one cistern, which is not possible. Therefore, we need to find out how long Y should remain open.

### Step 4: Let Y be open for \( t \) minutes and X for 18 minutes.
Let \( t \) be the time in minutes that pipe Y is open. Then pipe X will be open for the entire 18 minutes.
- The work done by pipe X in 18 minutes:
\[ \text{Work by X} = \text{Rate of X} \times 18 = \frac{1}{24} \times 18 = \frac{3}{4} \text{ cistern} \]
- The work done by pipe Y in \( t \) minutes:
\[ \text{Work by Y} = \text{Rate of Y} \times t = \frac{1}{32} \times t \text{ cistern} \]

### Step 5: Set up the equation for the total work done.
The total work done by both pipes must equal 1 cistern:
\[ \frac{3}{4} + \frac{t}{32} = 1 \]

### Step 6: Solve for \( t \).
Subtract \( \frac{3}{4} \) from both sides:
\[ \frac{t}{32} = 1 - \frac{3}{4} = \frac{1}{4} \]
Now, multiply both sides by 32 to solve for \( t \):
\[ t = 32 \times \frac{1}{4} = 8 \text{ minutes} \]

### Conclusion
Pipe Y should be closed after **8 minutes**.