Directions (58-60): Read the information carefully and answer the following question given below. A right circular cylindrical vessel was filled with milk and water in the ratio 4: 5. If x ml of mixture was sold and replaced with x ml of pure milk , then the ratio of milk to water in the final mixture becomes 57:60.The quantity of mixture sold is 108 ml less than the initial quantity of mixture in the vessel.

Find the quantity of milk in the final mixture.

The correct answer is 57 ml.

Let's break down the problem and solve it step-by-step.

Step 1: Understand the initial state and the variables.
Let the initial quantity of the mixture in the right circular cylindrical vessel be $V$ ml.
The initial ratio of milk to water is $4:5$.
So, the initial quantity of milk = $\frac{4}{9}V$ ml, and the initial quantity of water = $\frac{5}{9}V$ ml.

Step 2: Relate the sold quantity $x$ to the initial quantity $V$.
The problem states that the quantity of mixture sold, $x$, is $108$ ml less than the initial quantity of mixture $V$.
Therefore, we can write:
$x=V-108$
or equivalently, $V=x+108$.

Step 3: Analyze the mixture after removing $x$ ml.
When $x$ ml of the mixture is sold, the remaining mixture is $V-x=108$ ml.
Since the ratio of milk to water remains the same in the remaining mixture, the components in the remaining $108$ ml are:
Milk in remaining mixture = $\frac{4}{9}\times 108=48$ ml.
Water in remaining mixture = $\frac{5}{9}\times 108=60$ ml.

Step 4: Analyze the mixture after replacing with $x$ ml of pure milk.
Now, $x$ ml of pure milk is added to the remaining mixture.
The final quantity of milk becomes:
$\text{Final Milk}=48+x$ ml.
The final quantity of water remains unchanged because only pure milk was added:
$\text{Final Water}=60$ ml.

Step 5: Use the final ratio to find $x$.
The final ratio of milk to water is given as $57:60$.
So, we set up the proportion:
$\frac{48+x}{60}=\frac{57}{60}$
By equating the numerators (since the denominators are both 60):
$48+x=57$
$x=57-48=9$ ml.

Step 6: Find the quantity of milk in the final mixture.
The final quantity of milk is represented by the numerator in our ratio equation, which is:
$48+x=48+9=57$ ml.
This matches the ratio value directly because the denominator is $60$ ml. Therefore, the quantity of milk in the final mixture is indeed $57$ ml.