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?

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.