Let's solve the problem step by step.

Step 1: Determine the rate of each tap

• Tap A fills the cistern in 18 minutes, so its rate of filling is $\frac{1}{18}$ of the cistern per minute.
• Tap B fills the cistern in 24 minutes, so its rate of filling is $\frac{1}{24}$ of the cistern per minute.

Step 2: Calculate the combined rate of both taps

When both taps are open together, their combined rate is: $\text{Combined rate} = \frac{1}{18} + \frac{1}{24}$

Finding the least common denominator (LCD) for 18 and 24, which is 72: $\text{Combined rate} = \frac{4}{72} + \frac{3}{72} = \frac{7}{72}$

So, the combined rate is $\frac{7}{72}$ of the cistern per minute.

Step 3: Determine the time when tap B must be turned off

We want the cistern to be filled in 12 minutes. Let $t$ be the time in minutes that both taps are open, and let the remaining time be $12 - t$ minutes during which only tap A is open.

In 12 minutes, the entire cistern is filled, so: $\text{(Rate of both taps for t minutes) + (Rate of tap A for remaining time)} = 1 \text{ (the full cistern)}$

This equation is: $\left(\frac{7}{72} \times t\right) + \left(\frac{1}{18} \times (12 - t)\right) = 1$

Step 4: Solve the equation

$\frac{7t}{72} + \frac{12 - t}{18} = 1$

First, simplify $\frac{12 - t}{18}$: $\frac{12 - t}{18} = \frac{2(12 - t)}{36} = \frac{24 - 2t}{72}$

Now the equation becomes: $\frac{7t}{72} + \frac{24 - 2t}{72} = 1$

Combine the terms: $\frac{7t + 24 - 2t}{72} = 1$

$\frac{5t + 24}{72} = 1$

Multiply both sides by 72: $5t + 24 = 72$

Subtract 24 from both sides: $5t = 48$

Divide by 5: $t = 9.6 \text{ minutes}$

Conclusion

Tap B must be turned off after 9.6 minutes to ensure the cistern is filled in exactly 12 minutes.

Do you have any further questions or need more details?

Related Questions:

1. What happens if tap B is turned off after 8 minutes instead of 9.6 minutes?
2. How long will it take to fill the cistern if only tap A is open?
3. What fraction of the cistern is filled by tap A alone in 12 minutes?
4. If the taps are opened in succession, how long will it take to fill the cistern?
5. What is the total water flow from both taps in 12 minutes if both taps are never closed?
6. How does the fill time change if tap A’s rate increases by 10%?
7. What would happen if the cistern size doubled but the tap rates remained the same?
8. If tap B were slower, taking 30 minutes to fill the cistern, when would it need to be turned off?

Tip: Using equations to represent problems with varying conditions can simplify solving complex scenarios.