Two pipes A and B can fill a tank in 60 min. and 75 min. respectively. There is also awaste pipe in the tank (cistern). When all the three are opened, the cistern get filled in 50 minutes. How long will the waste pipe take to empty the full cistern.

Options

A. 90 min.
B. 150 min.
C. 80 min.
D. 100 min.

Answer

To solve the problem step by step, we will first determine the rates at which pipes A and B fill the tank, then calculate the rate of the waste pipe, and finally find out how long it takes for the waste pipe to empty the tank.

### Step 1: Determine the rates of pipes A and B
- Pipe A fills the tank in 60 minutes.
- Pipe B fills the tank in 75 minutes.

To find the rate at which each pipe fills the tank, we can use the formula:
\[ \text{Rate} = \frac{\text{Volume}}{\text{Time}} \]

Assuming the volume of the tank is 300 liters (a common multiple of 60 and 75 for easier calculations):
- Rate of A:
\[ \text{Rate of A} = \frac{300 \text{ liters}}{60 \text{ minutes}} = 5 \text{ liters/minute} \]

- Rate of B:
\[ \text{Rate of B} = \frac{300 \text{ liters}}{75 \text{ minutes}} = 4 \text{ liters/minute} \]

### Step 2: Determine the combined rate when all pipes are open
Let the rate of the waste pipe be \( X \) liters/minute. When all three pipes are open, the effective rate of filling the tank is:
\[ \text{Effective Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \]
\[ \text{Effective Rate} = 5 + 4 - X = 9 - X \text{ liters/minute} \]

### Step 3: Use the effective rate to find the rate of the waste pipe
When all three pipes are open, the tank gets filled in 50 minutes. Therefore, the effective rate can also be expressed as:
\[ \text{Effective Rate} = \frac{\text{Volume}}{\text{Time}} = \frac{300 \text{ liters}}{50 \text{ minutes}} = 6 \text{ liters/minute} \]

Now we can set up the equation:
\[ 9 - X = 6 \]

### Step 4: Solve for \( X \)
Rearranging the equation gives:
\[ X = 9 - 6 = 3 \text{ liters/minute} \]

This means the waste pipe empties the tank at a rate of 3 liters per minute.

### Step 5: Calculate the time taken by the waste pipe to empty the full tank
To find the time taken by the waste pipe to empty the full tank, we can use the formula:
\[ \text{Time} = \frac{\text{Volume}}{\text{Rate}} \]
\[ \text{Time} = \frac{300 \text{ liters}}{3 \text{ liters/minute}} = 100 \text{ minutes} \]

### Final Answer
The waste pipe will take **100 minutes** to empty the full cistern.

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