https://file.aiquickdraw.com/mathbot/file/1729939015299602omubx.jpg

To solve this problem, let's break it down step-by-step.

### Step 1: Understand Each Pipe's Work Rate
- Pipe 1 fills the cistern in 3 hours, so its work rate is:
  \[
  \frac{1}{3} \text{ of the cistern per hour}
  \]

- Pipe 2 fills the cistern in 4 hours, so its work rate is:
  \[
  \frac{1}{4} \text{ of the cistern per hour}
  \]

- Pipe 3 can drain the cistern in 8 hours, so its work rate is:
  \[
  -\frac{1}{8} \text{ of the cistern per hour (since it's draining)}
  \]

### Step 2: Calculate the Net Work Rate
When all three pipes are open, the net rate is:
\[
\frac{1}{3} + \frac{1}{4} - \frac{1}{8}
\]

To find this net rate, we first find a common denominator:
- The least common multiple of 3, 4, and 8 is 24.

So, converting each fraction to have a denominator of 24:
\[
\frac{1}{3} = \frac{8}{24}, \quad \frac{1}{4} = \frac{6}{24}, \quad \text{and} \quad -\frac{1}{8} = -\frac{3}{24}
\]

Thus, the net rate is:
\[
\frac{8}{24} + \frac{6}{24} - \frac{3}{24} = \frac{11}{24} \text{ of the cistern per hour}
\]

### Step 3: Determine How Much of the Cistern Needs to Be Filled
The cistern was already \(\frac{1}{12}\) full. Therefore, the remaining part to fill is:
\[
1 - \frac{1}{12} = \frac{12 - 1}{12} = \frac{11}{12}
\]

### Step 4: Calculate the Time Required to Fill the Remaining Part
Since the net rate of filling is \(\frac{11}{24}\) per hour, we set up the equation:
\[
\frac{11}{24} \times t = \frac{11}{12}
\]

Solving for \(t\):
\[
t = \frac{\frac{11}{12}}{\frac{11}{24}} = \frac{11}{12} \times \frac{24}{11} = 2 \text{ hours}
\]

### Answer
It will take **2 hours** for the cistern to be completely full.

Would you like more details or have any questions?

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Here are 5 related questions for further practice:
1. What if the third pipe drained the cistern in 6 hours instead of 8?
2. How long would it take if the cistern were initially empty and only the two filling pipes were used?
3. If Pipe 1’s capacity were reduced, how would it affect the filling time?
4. How would the result change if the cistern was initially half full?
5. What if a fourth pipe was added that could fill the cistern in 6 hours?

**Tip:** When dealing with multiple rates, always combine them by finding a common denominator for easier calculations.

Two pipes, when working one at a time, can fill a cistern in 3 hours and 4 hours, respectively, while a third pipe can drain the cistern empty in 8 hours. All three pipes were opened together when the cistern was 1/12 full. How long did it take for the cistern to be completely full?

Solve a work rate problem involving two pipes filling a cistern and one pipe draining it, with the cistern initially 1/12 full. Learn step-by-step to find the time required for the cistern to be completely full. Suitable for Grades 8-10.

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