Two pipes can fill a cistern in 3 hours and 4 hours respectively and a waste pipe can empty it in 2 hours. If all the three pipes are kept open, then the cistern will be filled in :

Options

A. 5 hours
B. 8 hours
C. 10 hours
D. 12 hours

Answer

To solve the problem, we need to determine how long it will take for two filling pipes and one waste pipe to fill a cistern when all are opened simultaneously. Let's break it down step by step.

### Step 1: Determine the capacity of the cistern
Let's assume the capacity of the cistern is 12 units (this is a common multiple of the times taken by the pipes).

### Step 2: Calculate the filling rates of the pipes
1. **Pipe A** fills the cistern in 3 hours.
- In 1 hour, it fills \( \frac{12 \text{ units}}{3 \text{ hours}} = 4 \text{ units/hour} \).

2. **Pipe B** fills the cistern in 4 hours.
- In 1 hour, it fills \( \frac{12 \text{ units}}{4 \text{ hours}} = 3 \text{ units/hour} \).

3. **Pipe C** empties the cistern in 2 hours.
- In 1 hour, it empties \( \frac{12 \text{ units}}{2 \text{ hours}} = 6 \text{ units/hour} \) (this will be considered negative since it is emptying).

### Step 3: Calculate the net filling rate when all pipes are open
Now, we can find the net filling rate when all three pipes are opened:
- Net filling rate = Filling rate of A + Filling rate of B - Emptying rate of C
- Net filling rate = \( 4 \text{ units/hour} + 3 \text{ units/hour} - 6 \text{ units/hour} \)
- Net filling rate = \( 7 - 6 = 1 \text{ unit/hour} \)

### Step 4: Calculate the time taken to fill the cistern
To find the time taken to fill the cistern, we use the formula:
\[ \text{Time} = \frac{\text{Capacity of cistern}}{\text{Net filling rate}} \]
- Time = \( \frac{12 \text{ units}}{1 \text{ unit/hour}} = 12 \text{ hours} \)

Thus, the cistern will be filled in **12 hours**.

### Final Answer
The cistern will be filled in **12 hours**.

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