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 answer is 111.

Let us break down the problem step-by-step to find the volume of the vessel filled in the final state.

Step 1: Understand the initial state and variables
Let the initial volume of the mixture in the right circular cylindrical vessel be $V$ ml.
The initial ratio of milk to water in the mixture is 4:5.
Therefore, the initial volume of milk is:

$\text{Milk}=\frac{4}{9}V$

and the initial volume of water is:

$\text{Water}=\frac{5}{9}V$

Step 2: Account for the first replacement operation
When $x$ ml of the mixture is sold, the ratio of milk to water in the remaining mixture remains 4:5.
The volume of the remaining mixture is $V-x$ ml.
Thus, the remaining quantities are:
Remaining Milk =

$\frac{4}{9}(V-x)$

Remaining Water =

$\frac{5}{9}(V-x)$

Now, the mixture is replaced with $x$ ml of pure milk.
New quantity of Milk =

$\frac{4}{9}(V-x)+x=\frac{4}{9}V+\frac{5}{9}x$

New quantity of Water =

$\frac{5}{9}(V-x)=\frac{5}{9}V-\frac{5}{9}x$

Step 3: Relate variables using the final ratio
The ratio of milk to water in this final mixture becomes 57:60.

$\frac{\frac{4}{9}V+\frac{5}{9}x}{\frac{5}{9}V-\frac{5}{9}x}=\frac{57}{60}$

Multiplying the numerator and the denominator on the left side by 9, and simplifying the ratio on the right side:

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

Cross-multiplying:

$20(4V+5x)=19(5V-5x)$

$80V+100x=95V-95x$

$195x=15V$

$V=13x$

Step 4: Solve for the quantities
We are given that the quantity sold ($x$) is 108 ml less than the initial quantity of mixture ($V$):

$x=V-108$

Substitute $V=13x$ into the equation:

$x=13x-108$

$12x=108$

$x=9\text{ml}$

Thus, the initial volume $V$ of the mixture is:

$V=13\left(9\right)=117\text{ml}$

Step 5: Calculate the final volume in the new scenario
In the second scenario:
- $3x$ ml of mixture is sold.
- $\frac{x}{3}$ ml of milk is added.
- $2x$ ml of water is added.
The final volume of the mixture filled in the vessel, $V_{\text{final}}$, is:

$V_{\text{final}}=V-3x+\frac{x}{3}+2x$

$V_{\text{final}}=V-x+\frac{x}{3}=V-\frac{2x}{3}$

Substitute the values $V=117$ and $x=9$:

$V_{\text{final}}=117-\frac{2\left(9\right)}{3}$

$V_{\text{final}}=117-6=111\text{ml}$

Since $1\text{ml}=1{\text{cm}}^3$, the total volume of the vessel filled is 111 cm³.