Two pipes A and B can fill a cistern in 30 min and 40 min respectively. Both the pipes are turned on simultaneously. When should the second pipe be closed if the cistern is to be filled in 24 min.

Options

A. 6 min
B. 8 min
C. 10 min
D. 12 min

Answer

To solve the problem step-by-step, we need to determine how long the second pipe (B) should remain open in order to fill the cistern in 24 minutes when both pipes A and B are working together initially.

### Step 1: Determine the rates of the pipes
- Pipe A fills the cistern in 30 minutes, so its rate is:
\[ \text{Rate of A} = \frac{1}{30} \text{ cisterns per minute} \]
- Pipe B fills the cistern in 40 minutes, so its rate is:
\[ \text{Rate of B} = \frac{1}{40} \text{ cisterns per minute} \]

### Step 2: Calculate the combined rate of both pipes
When both pipes are open, their combined rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{30} + \frac{1}{40} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 30 and 40 is 120.
\[ \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{40} = \frac{3}{120} \]
Thus,
\[ \text{Combined Rate} = \frac{4}{120} + \frac{3}{120} = \frac{7}{120} \text{ cisterns per minute} \]

### Step 3: Set up the equation for the total work done
Let \( x \) be the time (in minutes) that both pipes are open. After \( x \) minutes, pipe B is closed, and pipe A continues to fill the cistern alone for the remaining time, which is \( 24 - x \) minutes.

The work done by both pipes in \( x \) minutes is:
\[ \text{Work by A and B} = \frac{7}{120} \times x \]
The work done by pipe A alone for the remaining time is:
\[ \text{Work by A alone} = \frac{1}{30} \times (24 - x) \]

### Step 4: Set up the equation for total work
The total work done by both pipes must equal 1 full cistern:
\[ \frac{7}{120} x + \frac{1}{30} (24 - x) = 1 \]

### Step 5: Solve the equation
First, we convert \(\frac{1}{30}\) to a fraction with a denominator of 120:
\[ \frac{1}{30} = \frac{4}{120} \]
Now, substituting this back into the equation:
\[ \frac{7}{120} x + \frac{4}{120} (24 - x) = 1 \]
Multiplying through by 120 to eliminate the denominator:
\[ 7x + 4(24 - x) = 120 \]
Expanding and simplifying:
\[ 7x + 96 - 4x = 120 \]
\[ 3x + 96 = 120 \]
\[ 3x = 120 - 96 \]
\[ 3x = 24 \]
\[ x = 8 \]

### Step 6: Conclusion
The second pipe (B) should be closed after 8 minutes.