24 liters are drawn from a vessel full of wine, then filled with water. This operation is performed 3 times more. The ratio of quantity of wine now left in vessel to that of water is `16 : 65`. How much wine did vessel hold initially?

Options

A. 48
B. 72
C. 96
D. 120

Answer

To solve the problem step by step, we will use the information given and apply the concept of mixtures.

### Step 1: Understand the Problem
We have a vessel that is initially full of wine. We draw 24 liters of wine from it and replace it with water. This operation is repeated 3 more times (total of 4 times). After these operations, the ratio of the quantity of wine left to the quantity of water is 16:65.

### Step 2: Set Up the Initial Conditions
Let the initial quantity of wine in the vessel be \( x \) liters.

### Step 3: Determine the Total Quantity After Operations
After the operations, the total quantity of liquid in the vessel remains constant. The total quantity of liquid after the operations is \( x \) liters (since we are replacing the wine with water).

### Step 4: Use the Ratio Given
From the problem, we know that the final ratio of wine to water is 16:65. Therefore, the total parts of the mixture is \( 16 + 65 = 81 \) parts.

### Step 5: Express the Quantities of Wine and Water
Let the quantity of wine left in the vessel be \( \frac{16}{81} \times x \) and the quantity of water be \( \frac{65}{81} \times x \).

### Step 6: Apply the Formula for Repeated Operations
When we draw out 24 liters of wine, the remaining quantity of wine after each operation can be expressed as:
\[ \text{Remaining wine} = \left(1 - \frac{24}{x}\right)^n \times x \]
where \( n \) is the number of times the operation is performed. Since the operation is performed 4 times, we have:
\[ \text{Remaining wine} = \left(1 - \frac{24}{x}\right)^4 \times x \]

### Step 7: Set Up the Equation
We know that the remaining quantity of wine is also equal to \( \frac{16}{81} \times x \). Therefore, we can set up the equation:
\[ \left(1 - \frac{24}{x}\right)^4 \times x = \frac{16}{81} \times x \]
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ \left(1 - \frac{24}{x}\right)^4 = \frac{16}{81} \]

### Step 8: Solve for \( x \)
Taking the fourth root of both sides, we have:
\[ 1 - \frac{24}{x} = \frac{2}{3} \]
Now, rearranging gives:
\[ \frac{24}{x} = 1 - \frac{2}{3} = \frac{1}{3} \]
Cross-multiplying gives:
\[ 24 = \frac{x}{3} \]
Thus,
\[ x = 24 \times 3 = 72 \]

### Conclusion
The initial quantity of wine in the vessel was \( 72 \) liters.