Two taps A and B can fill a cistern is 18 minutes and 24 minutes respectively. If both taps are opened together, when the tap B must be turned off so that the cistern is filled in 12 minutes?

Let's solve this step by step.

### Step 1: Determine the rate of filling for each tap.
- Tap A can fill the cistern in 18 minutes, so the rate of Tap A is:
  \[
  \text{Rate of Tap A} = \frac{1}{18} \text{ of the cistern per minute.}
  \]
  
- Tap B can fill the cistern in 24 minutes, so the rate of Tap B is:
  \[
  \text{Rate of Tap B} = \frac{1}{24} \text{ of the cistern per minute.}
  \]

### Step 2: Determine the combined rate when both taps are open.
When both taps are open together, the combined rate of filling the cistern is:
\[
\text{Combined Rate} = \frac{1}{18} + \frac{1}{24}
\]
To combine these, find the least common denominator:
\[
\text{Combined Rate} = \frac{4 + 3}{72} = \frac{7}{72} \text{ of the cistern per minute.}
\]

### Step 3: Calculate the time when Tap B is turned off.
Let \( t \) be the time in minutes that both taps are open. After \( t \) minutes, Tap B is turned off, and Tap A continues to fill the cistern alone for the remaining \( 12 - t \) minutes.

The total amount of water filled in the cistern is given by:
\[
\text{Amount filled by Tap A in } t \text{ minutes} = t \times \frac{1}{18}
\]
\[
\text{Amount filled by Tap B in } t \text{ minutes} = t \times \frac{1}{24}
\]
\[
\text{Amount filled by Tap A alone in } (12 - t) \text{ minutes} = (12 - t) \times \frac{1}{18}
\]

Since the cistern is fully filled in 12 minutes:
\[
t \times \frac{1}{18} + t \times \frac{1}{24} + (12 - t) \times \frac{1}{18} = 1
\]

### Step 4: Simplify the equation.
Substitute the rates into the equation:
\[
t \times \left(\frac{1}{18} + \frac{1}{24}\right) + (12 - t) \times \frac{1}{18} = 1
\]
\[
t \times \frac{7}{72} + \frac{12 - t}{18} = 1
\]
Multiply through by 72 to eliminate the fractions:
\[
7t + 4(12 - t) = 72
\]
\[
7t + 48 - 4t = 72
\]
\[
3t = 24
\]
\[
t = 8
\]

### Step 5: Conclusion
Tap B must be turned off after 8 minutes.

**Would you like a further explanation or have any questions?**

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Here are 8 related questions:
1. What if Tap A could fill the cistern in 12 minutes instead of 18 minutes?
2. How would the solution change if Tap B's rate was doubled?
3. What happens if Tap B is left on for the full 12 minutes?
4. Can you generalize a formula for any two taps with different rates?
5. What if there were a third tap involved with a different rate?
6. How long would it take for Tap A alone to fill half the cistern?
7. What if the cistern had a leak that reduces its capacity by 10%?
8. How does the problem change if Tap B fills at a variable rate?

**Tip:** When solving problems involving rates, always check if the rates can be combined and simplified for easier calculation.

Learn how to optimize the turn-off time for Tap B to fill a cistern in 12 minutes. This problem involves calculating rates of work for taps A and B, and using linear equations to determine the optimal time. Suitable for high school students.

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