A pipe can fill a cistern in 8 minutes and a waste pipe can empty 6 kilo liters water in one minute. If both pipes are opened together and the empty cistern can be filled in 20 minutes. What is the capacity of the cistern?

Options

A. 100 kilo liter
B. 80 kilo liter
C. 120 kilo liter
D. 160 kilo liter

Answer

To solve the problem step by step, we will analyze the information given and apply the relevant formulas.

### Step 1: Understand the Rates of the Pipes
- The filling pipe (let's call it Pipe A) can fill the cistern in 8 minutes. Therefore, its rate of filling is:
\[ \text{Rate of Pipe A} = \frac{1 \text{ cistern}}{8 \text{ minutes}} = \frac{1}{8} \text{ cistern/minute} \]

- The waste pipe (let's call it Pipe W) can empty 6 kiloliters of water in one minute. Thus, its rate of emptying is:
\[ \text{Rate of Pipe W} = -6 \text{ kiloliters/minute} \]

### Step 2: Set Up the Equation for Both Pipes Working Together
When both pipes are opened together, the net rate of filling the cistern is:
\[ \text{Net Rate} = \text{Rate of Pipe A} - \text{Rate of Pipe W} \]
Converting the rate of Pipe A to kiloliters, we have:
\[ \text{Rate of Pipe A} = \frac{1 \text{ cistern}}{8 \text{ minutes}} = \frac{1 \text{ kiloliter}}{8 \text{ minutes}} \text{ (assuming the cistern's capacity is in kiloliters)} \]

If the capacity of the cistern is \( V \) kiloliters, then:
\[ \text{Net Rate} = \frac{V}{8} - 6 \]

### Step 3: Determine the Time Taken to Fill the Cistern
According to the problem, when both pipes are opened together, the empty cistern can be filled in 20 minutes. Therefore, we can write:
\[ \text{Net Rate} \times \text{Time} = \text{Capacity} \]
Substituting the values we have:
\[ \left(\frac{V}{8} - 6\right) \times 20 = V \]

### Step 4: Solve the Equation
Expanding the equation:
\[ 20 \left(\frac{V}{8} - 6\right) = V \]
\[ \frac{20V}{8} - 120 = V \]
\[ \frac{5V}{2} - 120 = V \]

Now, rearranging the equation:
\[ \frac{5V}{2} - V = 120 \]
\[ \frac{5V - 2V}{2} = 120 \]
\[ \frac{3V}{2} = 120 \]

Multiplying both sides by 2:
\[ 3V = 240 \]
Dividing by 3:
\[ V = 80 \text{ kiloliters} \]

### Conclusion
The capacity of the cistern is **80 kiloliters**.