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 step-by-step to find the final quantity of milk in the mixture.

Step 1: Understand the remaining mixture quantity
Let the initial volume of the mixture in the vessel be V ml, and the quantity of the mixture sold and replaced be x ml.
We are given that the quantity of mixture sold (x) is 108 ml less than the initial quantity of mixture (V):

$x=V-108$

Rearranging this equation gives the volume of the remaining mixture after selling x ml:

$V-x=108\text{ml}$

Step 2: Find the quantities of milk and water in the remaining mixture
When a part of the mixture is sold, the ratio of milk to water in the remaining mixture remains unchanged at the initial ratio of 4:5.
Now, we calculate the individual quantities of milk and water in the remaining 108 ml mixture:

Quantity of milk in remaining mixture = $108\times \frac{4}{4+5}=108\times \frac{4}{9}=48\text{ml}$

Quantity of water in remaining mixture = $108\times \frac{5}{4+5}=108\times \frac{5}{9}=60\text{ml}$

Step 3: Analyze the addition of pure milk
Next, x ml of pure milk is added to the remaining mixture to replace the sold amount. Since only pure milk is added, the quantity of water remains constant at 60 ml in the final mixture.
The final ratio of milk to water is given as 57:60.

Let the quantity of milk in the final mixture be M. We can write the ratio as:

$\frac{\text{Quantity of Milk}}{\text{Quantity of Water}}=\frac{57}{60}$

Substituting the constant quantity of water (60 ml) into the ratio:

$\frac{M}{60}=\frac{57}{60}$

Solving for M:

$M=57\text{ml}$

Therefore, the quantity of milk in the final mixture is 57 ml.