A vessel is full of vinegar. 35L of vinegar is taken out & replaced by water. This process is repeated three more times. Find the final amount of vinegar in the vessel if at the end the ratio of vinegar and water becomes `625:671`.

Options

A. 210L
B. 70.6L
C. 175L
D. 101.27L

Answer

To solve the problem step by step, we will follow the process of calculating the final amount of vinegar in the vessel after the replacement with water is done four times.

### Step 1: Understand the Initial Conditions
The vessel is initially full of vinegar. Let's denote the initial volume of vinegar in the vessel as \( X \) liters.

### Step 2: Determine the Amount of Vinegar Removed
Each time, 35 liters of vinegar is taken out and replaced with water. This process is repeated 4 times.

### Step 3: Set Up the Ratio
After the four repetitions of the process, the ratio of vinegar to water is given as \( 625:671 \). The total parts in this ratio is \( 625 + 671 = 1296 \).

### Step 4: Calculate the Amount of Vinegar Remaining
The amount of vinegar remaining can be expressed in terms of the total volume of the vessel:
- Let \( V \) be the total volume of the vessel, which is equal to \( X \) liters.
- The amount of vinegar left after the process can be represented as:
\[ \text{Amount of vinegar left} = \frac{625}{1296} \times X \]

### Step 5: Apply the Repeated Process Formula
The formula for the remaining quantity of vinegar after removing a certain amount \( a \) (35 liters in this case) from a total quantity \( X \) after \( n \) repetitions is:
\[ \text{Remaining vinegar} = X \left(1 - \frac{a}{X}\right)^n \]
Here, \( n = 4 \) and \( a = 35 \).

### Step 6: Set Up the Equation
We can equate the two expressions for the amount of vinegar left:
\[ X \left(1 - \frac{35}{X}\right)^4 = \frac{625}{1296} \times X \]
Since \( X \) is not zero, we can divide both sides by \( X \):
\[ \left(1 - \frac{35}{X}\right)^4 = \frac{625}{1296} \]

### Step 7: Simplify the Equation
Taking the fourth root of both sides:
\[ 1 - \frac{35}{X} = \frac{5}{6} \]

### Step 8: Solve for \( X \)
Rearranging gives:
\[ \frac{35}{X} = 1 - \frac{5}{6} = \frac{1}{6} \]
Thus,
\[ X = 35 \times 6 = 210 \text{ liters} \]

### Step 9: Calculate the Final Amount of Vinegar
Now that we have \( X \), we can find the final amount of vinegar:
\[ \text{Final amount of vinegar} = \frac{625}{1296} \times 210 \]

### Step 10: Perform the Calculation
Calculating:
\[ \text{Final amount of vinegar} = \frac{625 \times 210}{1296} \]
Calculating \( 625 \times 210 = 131250 \):
\[ \text{Final amount of vinegar} = \frac{131250}{1296} \approx 101.27 \text{ liters} \]

### Final Answer
The final amount of vinegar in the vessel is approximately **101.27 liters**.

---