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.

If 3x ml mixture was sold and x/3 ml of milk & 2x ml of water is added in the vessel, then how much volume of vessel is filled in cm cube?

The correct option is 111.

Step-by-Step Explanation:

Let the initial volume of the mixture in the right circular cylindrical vessel be $V$ ml.
The initial ratio of milk to water is given as $4:5$.
Therefore, the initial quantities of milk and water in the vessel are:
Initial Milk = $\frac{4}{9}V$
Initial Water = $\frac{5}{9}V$

When $x$ ml of the mixture is sold, the quantities of milk and water removed from the vessel are in the same ratio ($4:5$):
Milk removed = $\frac{4}{9}x$
Water removed = $\frac{5}{9}x$

After selling $x$ ml of the mixture, $x$ ml of pure milk is added to the vessel. The new quantities of milk and water in the vessel become:
New Milk = $\frac{4}{9}V-\frac{4}{9}x+x=\frac{4}{9}V+\frac{5}{9}x$
New Water = $\frac{5}{9}V-\frac{5}{9}x$

According to the problem, the ratio of milk to water in the final mixture is $57:60$, which simplifies to $19:20$.
We can set up the equation as follows:

$\frac{\frac{4}{9}V+\frac{5}{9}x}{\frac{5}{9}V-\frac{5}{9}x}=\frac{19}{20}$

Multiplying both the numerator and the denominator by 9 simplifies the expression to:
$\frac{4V+5x}{5V-5x}=\frac{19}{20}$

Cross-multiplying to solve for $V$ in terms of $x$:
$20(4V+5x)=19(5V-5x)$
$80V+100x=95V-95x$
$195x=15V$
$V=13x$

We are also given that the quantity of mixture sold ($x$ ml) is 108 ml less than the initial quantity of mixture in the vessel ($V$ ml):
$V-x=108$

Substituting $V=13x$ into the equation:
$13x-x=108$
$12x=108$
$x=9$ ml

Thus, the initial volume of the mixture in the vessel is:
$V=13\times 9=117$ ml

Now, let us calculate the volume of the vessel filled after the second set of operations:
1. 3x ml of the mixture is sold:
Volume sold = $3\times 9=27$ ml.
Remaining volume = $117-27=90$ ml.
2. $\frac{x}{3}$ ml of milk is added:
Milk added = $\frac{9}{3}=3$ ml.
3. 2x ml of water is added:
Water added = $2\times 9=18$ ml.

The total volume of the mixture now in the vessel is:
$\text{Final Volume}=90+3+18=111$ ml.

Since $1\text{ml}=1{\text{cm}}^3$, the volume of the vessel filled is $111{\text{cm}}^3$.