Pipe A and pipe B can fill a tank in 24 minutes and 28 minutes , respectively. IF both the pipes are opened simultaneously, then after how many minues. Should pipe B be closed such that the tank becomes full in 18 minuts ?

Answer

To solve the problem, we need to determine how long pipe B should remain open while both pipes A and B are filling the tank simultaneously, so that the tank is full in a total of 18 minutes.

### Step-by-Step Solution:

1. **Determine the rates of filling for each pipe:**
- Pipe A can fill the tank in 24 minutes. Therefore, the rate of pipe A is:
\[ \text{Rate of Pipe A} = \frac{1}{24} \text{ tanks per minute} \]
- Pipe B can fill the tank in 28 minutes. Therefore, the rate of pipe B is:
\[ \text{Rate of Pipe B} = \frac{1}{28} \text{ tanks per minute} \]

2. **Calculate the combined rate when both pipes are open:**
- The combined rate of both pipes A and B working together is:
\[ \text{Combined Rate} = \text{Rate of Pipe A} + \text{Rate of Pipe B} = \frac{1}{24} + \frac{1}{28} \]
- To add these fractions, we need a common denominator. The least common multiple of 24 and 28 is 168.
- Converting each rate:
\[ \frac{1}{24} = \frac{7}{168}, \quad \frac{1}{28} = \frac{6}{168} \]
- Therefore,
\[ \text{Combined Rate} = \frac{7}{168} + \frac{6}{168} = \frac{13}{168} \text{ tanks per minute} \]

3. **Set up the equation for the total filling time:**
- Let \( x \) be the time (in minutes) that both pipes are open. After that, pipe B will be closed, and pipe A will continue to fill the tank alone.
- The total time for filling the tank is 18 minutes:
\[ x + (18 - x) = 18 \]
- The amount of tank filled in \( x \) minutes by both pipes is:
\[ \text{Amount filled by both pipes} = \frac{13}{168} \cdot x \]
- The amount of tank filled by pipe A alone in the remaining time \( (18 - x) \) minutes is:
\[ \text{Amount filled by Pipe A} = \frac{1}{24} \cdot (18 - x) \]

4. **Set up the equation for the total tank filled:**
- The total amount filled must equal 1 tank:
\[ \frac{13}{168} x + \frac{1}{24} (18 - x) = 1 \]

5. **Solve the equation:**
- Convert \( \frac{1}{24} \) to a fraction with a denominator of 168:
\[ \frac{1}{24} = \frac{7}{168} \]
- Substitute into the equation:
\[ \frac{13}{168} x + \frac{7}{168} (18 - x) = 1 \]
- Multiply through by 168 to eliminate the denominator:
\[ 13x + 7(18 - x) = 168 \]
- Distribute:
\[ 13x + 126 - 7x = 168 \]
- Combine like terms:
\[ 6x + 126 = 168 \]
- Subtract 126 from both sides:
\[ 6x = 42 \]
- Divide by 6:
\[ x = 7 \]

6. **Conclusion:**
- Pipe B should be closed after 7 minutes.