Two pipe M and N can fill a tank in `8(1)/(3)` min and `12(1)/(2)` min respectively and an outler pipe can drain 162 litre water in 1 minute. If tank is full, all the three pipes are opened then tank will be empty in 4 minute then capacity of tank will be –

Options

A. 916 litre
B. 300 litre
C. 360 litre
D. None of these

Answer

To solve the problem step by step, we need to determine the capacity of the tank based on the information given about the filling and draining pipes.

### Step 1: Convert the times into improper fractions
- Pipe M fills the tank in \(8 \frac{1}{3}\) minutes, which is \( \frac{25}{3} \) minutes.
- Pipe N fills the tank in \(12 \frac{1}{2}\) minutes, which is \( \frac{25}{2} \) minutes.

### Step 2: Calculate the filling rates of pipes M and N
- The filling rate of pipe M is:
\[ \text{Rate of M} = \frac{1 \text{ tank}}{\frac{25}{3} \text{ minutes}} = \frac{3}{25} \text{ tanks per minute} \]

- The filling rate of pipe N is:
\[ \text{Rate of N} = \frac{1 \text{ tank}}{\frac{25}{2} \text{ minutes}} = \frac{2}{25} \text{ tanks per minute} \]

### Step 3: Calculate the combined filling rate of pipes M and N
- The combined filling rate of pipes M and N is:
\[ \text{Combined Rate} = \text{Rate of M} + \text{Rate of N} = \frac{3}{25} + \frac{2}{25} = \frac{5}{25} = \frac{1}{5} \text{ tanks per minute} \]

### Step 4: Determine the draining rate of the outlet pipe
- The outlet pipe drains 162 liters per minute.
- Let the capacity of the tank be \(C\) liters. The draining rate in terms of tanks is:
\[ \text{Draining Rate} = \frac{162}{C} \text{ tanks per minute} \]

### Step 5: Set up the equation for when all pipes are opened
- When all three pipes are opened, the net rate of filling/draining is:
\[ \text{Net Rate} = \text{Combined Rate} - \text{Draining Rate} = \frac{1}{5} - \frac{162}{C} \]
- According to the problem, the tank will be empty in 4 minutes, which means the net rate must equal \(-\frac{1}{4}\) tanks per minute (since it is emptying):
\[ \frac{1}{5} - \frac{162}{C} = -\frac{1}{4} \]

### Step 6: Solve for C
- Rearranging the equation gives:
\[ \frac{1}{5} + \frac{1}{4} = \frac{162}{C} \]
- Finding a common denominator (20):
\[ \frac{4}{20} + \frac{5}{20} = \frac{162}{C} \]
\[ \frac{9}{20} = \frac{162}{C} \]
- Cross-multiplying gives:
\[ 9C = 3240 \]
- Thus, solving for \(C\):
\[ C = \frac{3240}{9} = 360 \text{ liters} \]

### Conclusion
The capacity of the tank is **360 liters**.

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