In a container contains mixture of milk to water in the ratio of 3:2. When 75% of the mixture is taken out and 40 liters of mixture of milk and water added in which 40% is milk, then in the resultant mixture milk becomes $44\frac{4}{9}\%$ of total mixture. Find the quantity of milk in initial mixture.

The correct option is $\frac{192}{7}$ liters.

Step 1: Define the initial quantities of milk and water.
Let the initial total quantity of the mixture be $5x$ liters.
Given the ratio of milk to water in the initial container is 3 : 2:

Quantity of Milk = $3x$ liters

Quantity of Water = $2x$ liters

Step 2: Determine the remaining quantity after taking out 75% of the mixture.
When 75% of the mixture is removed, 25% (or $\frac{1}{4}$) of the mixture remains. Since a mixture is homogeneous, the proportion of milk and water in the remaining mixture stays the same.

Remaining Milk = $25\%$ of $3x=\frac{3}{4}x$ liters

Remaining Water = $25\%$ of $2x=\frac{2}{4}x$ liters

Remaining Total Mixture = $25\%$ of $5x=\frac{5}{4}x$ liters

Step 3: Calculate the milk and water added.
A 40-liter mixture is added to the container, in which 40% is milk.

Milk Added = $40\%$ of $40=\frac{40}{100}\times 40=16$ liters

Water Added = $40-16=24$ liters

Step 4: Formulate the equations for the resultant mixture.
Now, let us find the new total quantities of milk and the entire mixture:

Total Milk in resultant mixture = $\frac{3}{4}x+16$ liters

Total Resultant Mixture = $\frac{5}{4}x+40$ liters

Step 5: Use the given percentage of milk in the final mixture.
We are given that milk forms $44\frac{4}{9}\%$ of the total mixture.
Converting $44\frac{4}{9}\%$ to a fraction:

$44\frac{4}{9}\%=\frac{400}{9}\%=\frac{400}{9\times 100}=\frac{4}{9}$

Therefore, the ratio of milk to total mixture is:

$\frac{\frac{3}{4}x+16}{\frac{5}{4}x+40}=\frac{4}{9}$

Step 6: Solve for x.
Cross-multiplying both sides:

$9\times \left(\frac{3}{4}x+16\right)=4\times \left(\frac{5}{4}x+40\right)$

$\frac{27}{4}x+144=5x+160$

$\frac{27}{4}x-5x=160-144$

$\frac{27x-20x}{4}=16$

$\frac{7x}{4}=16$

$7x=64$

$x=\frac{64}{7}$

Step 7: Calculate the initial quantity of milk.
Initial milk quantity was defined as $3x$:

Quantity of initial milk = $3\times \frac{64}{7}=\frac{192}{7}$ liters