Vessel A contains 150 liters mixture of milk and water in which water is 40%. ‘x’ liters mixture is taken out (quantity of water in ‘x’ liters is 12 liters) and poured into vessel B which already contain ‘x’ liters water . Find the difference between quantity (in liters) of milk and water in resultant mixture of vessel B.

The correct option is 24.

Let's solve the problem step-by-step to find the difference between the quantity of milk and water in the resultant mixture of vessel B.

Step 1: Understand the composition of the mixture in Vessel A
Vessel A contains a 150-liter mixture of milk and water.
The percentage of water in this mixture is 40%.
Therefore, the percentage of milk in the mixture is:
$100\%-40\%=60\%$

Step 2: Determine the value of 'x'
We are told that $x$ liters of the mixture is taken out of Vessel A.
Since the mixture is homogeneous, the taken-out portion ($x$ liters) will also contain milk and water in the same ratio as Vessel A (i.e., 40% water and 60% milk).
We are given that the quantity of water in $x$ liters is 12 liters.
Thus, 40% of $x$ is equal to 12:
$0.40\times x=12$
Solving for $x$:
$x=\frac{12}{0.40}=30\text{liters}$

Step 3: Find the quantities of milk and water in the portion transferred to Vessel B
Since the total portion taken out is $x=30$ liters and the quantity of water in it is 12 liters, the quantity of milk in this portion is:
$\text{Milk in}x\text{liters}=30-12=18\text{liters}$

Step 4: Determine the quantities of milk and water in the resultant mixture of Vessel B
Vessel B already contains $x$ liters of water, which is 30 liters of water.
When the $x$ liters of mixture from Vessel A (containing 18 liters of milk and 12 liters of water) is poured into Vessel B, the total quantities in Vessel B become:
$\text{Total Water in B}=\text{Initial water in B}+\text{Water from transferred mixture}$
$\text{Total Water in B}=30+12=42\text{liters}$

Since there was no milk initially in Vessel B:
$\text{Total Milk in B}=18\text{liters}$

Step 5: Find the difference between the quantities of milk and water in Vessel B
The difference between the quantity of water and milk in the resultant mixture in Vessel B is:
$\text{Difference}=42-18=24\text{liters}$