A vessel is full of refined oil. `1/4` of the refined oil is taken out and the vessel is filled with mustard oil. If the process is repeated 4 times and 10 liters of refined oil finally left in the vessel. What is the capacity of the vessel?

Options

A. `2660/81` litres
B. `2460/81` litres
C. `2560/81` litres
D. `2960/81` litres

Answer

To solve the problem, we need to determine the capacity of the vessel based on the information given about the refined oil and the mustard oil. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understand the Initial Condition**:
- Let the capacity of the vessel be \( C \) liters. Initially, the vessel is full of refined oil, so the amount of refined oil is \( C \) liters.

2. **Calculate the Amount of Oil Taken Out**:
- In the first operation, \( \frac{1}{4} \) of the refined oil is taken out. Therefore, the amount of refined oil taken out is:
\[ \text{Oil taken out} = \frac{1}{4}C \]
- The remaining refined oil after the first operation is:
\[ \text{Remaining refined oil} = C - \frac{1}{4}C = \frac{3}{4}C \]

3. **Fill the Vessel with Mustard Oil**:
- After taking out \( \frac{1}{4}C \), the vessel is filled with mustard oil. The total volume in the vessel remains \( C \) liters.

4. **Repeat the Process**:
- This process is repeated 4 times. Each time, \( \frac{1}{4} \) of the remaining refined oil is taken out. We can express the remaining refined oil after \( n \) operations using the formula:
\[ R_n = C \left(1 - \frac{1}{4}\right)^n = C \left(\frac{3}{4}\right)^n \]
- Here, \( R_n \) is the remaining refined oil after \( n \) operations.

5. **Substituting the Values**:
- After 4 operations, we know that \( R_4 = 10 \) liters (the amount of refined oil left). Therefore:
\[ R_4 = C \left(\frac{3}{4}\right)^4 \]
- Setting this equal to 10 liters gives us:
\[ C \left(\frac{3}{4}\right)^4 = 10 \]

6. **Calculate \( \left(\frac{3}{4}\right)^4 \)**:
- Calculate \( \left(\frac{3}{4}\right)^4 \):
\[ \left(\frac{3}{4}\right)^4 = \frac{3^4}{4^4} = \frac{81}{256} \]

7. **Substituting Back to Find \( C \)**:
- Now substitute this back into the equation:
\[ C \cdot \frac{81}{256} = 10 \]
- Solving for \( C \):
\[ C = 10 \cdot \frac{256}{81} = \frac{2560}{81} \]

8. **Final Calculation**:
- Calculate \( C \):
\[ C \approx 31.6 \text{ liters} \]

### Conclusion:
The capacity of the vessel is approximately \( 31.6 \) liters.