A cistern can be filled up in 4 h by an inlet A. An outlet B can empty the cistern in 8 h. If both A and B are opened simultaneously, then after how much time will the cistern get filled?

Options

A. 5h
B. 7h
C. 8h
D. 6h

Answer

To solve the problem step by step, we need to determine how much of the cistern is filled by the inlet A and how much is emptied by the outlet B when both are open simultaneously.

### Step 1: Determine the filling rate of inlet A
Inlet A can fill the cistern in 4 hours. Therefore, in 1 hour, it fills:
\[ \text{Filling rate of A} = \frac{1}{4} \text{ of the cistern} \]

### Step 2: Determine the emptying rate of outlet B
Outlet B can empty the cistern in 8 hours. Therefore, in 1 hour, it empties:
\[ \text{Emptying rate of B} = \frac{1}{8} \text{ of the cistern} \]

### Step 3: Calculate the net rate when both A and B are open
When both A and B are open, the net effect on the cistern's filling is the filling rate of A minus the emptying rate of B:
\[ \text{Net rate} = \text{Filling rate of A} - \text{Emptying rate of B} = \frac{1}{4} - \frac{1}{8} \]

To perform this calculation, we need a common denominator. The least common multiple of 4 and 8 is 8. Thus, we convert \(\frac{1}{4}\) to eighths:
\[ \frac{1}{4} = \frac{2}{8} \]
Now we can calculate the net rate:
\[ \text{Net rate} = \frac{2}{8} - \frac{1}{8} = \frac{1}{8} \text{ of the cistern per hour} \]

### Step 4: Determine the time to fill the cistern
Since the net rate of filling the cistern when both A and B are open is \(\frac{1}{8}\) of the cistern per hour, we can find the time taken to fill the entire cistern:
\[ \text{Time to fill the cistern} = \frac{1 \text{ (full cistern)}}{\frac{1}{8} \text{ (cistern per hour)}} = 8 \text{ hours} \]

### Final Answer
Thus, the cistern will be filled in **8 hours** when both inlet A and outlet B are opened simultaneously.

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