An inlet pipe takes 8 hoursto fill a tank. An outlet pipe takes 12 hours to empty it. If both pipes are opened simultaneously, in how many hours will the tank be filled?

Options

A. 36
B. 8
C. 24
D. 12

Answer

To solve the problem of how long it will take to fill the tank when both the inlet and outlet pipes are open simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Rate of Filling by the Inlet Pipe:**
- The inlet pipe fills the tank in 8 hours. Therefore, its rate of filling is:
\[ \text{Rate of Inlet Pipe} = \frac{1 \text{ tank}}{8 \text{ hours}} = \frac{1}{8} \text{ tanks per hour} \]

2. **Determine the Rate of Emptying by the Outlet Pipe:**
- The outlet pipe empties the tank in 12 hours. Therefore, its rate of emptying is:
\[ \text{Rate of Outlet Pipe} = \frac{1 \text{ tank}}{12 \text{ hours}} = \frac{1}{12} \text{ tanks per hour} \]

3. **Calculate the Net Rate When Both Pipes are Open:**
- When both pipes are open, the net rate of filling the tank is the rate of the inlet pipe minus the rate of the outlet pipe:
\[ \text{Net Rate} = \text{Rate of Inlet Pipe} - \text{Rate of Outlet Pipe} = \frac{1}{8} - \frac{1}{12} \]

4. **Find a Common Denominator:**
- The least common multiple of 8 and 12 is 24. We can rewrite the rates with a common denominator:
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
- Therefore, the net rate becomes:
\[ \text{Net Rate} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24} \text{ tanks per hour} \]

5. **Calculate the Time to Fill the Tank:**
- To find the time taken to fill one tank at the net rate of \(\frac{1}{24}\) tanks per hour, we take the reciprocal:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{24} \text{ tanks per hour}} = 24 \text{ hours} \]

### Final Answer:
The tank will be filled in **24 hours** when both pipes are opened simultaneously.