The milk and water in two vessels A and B are in the ratio 4 : 3 and 2 : 3 respectively. In what ratio, the liquids in both the vessels be mixed to obtain a new mixture in vessel C containing half milk and half water?

The correct option is 7 : 5.

To find the ratio in which the mixtures from vessel A and vessel B should be mixed, we can analyze the concentration of milk in each vessel and use the method of allegation or a step-by-step algebraic approach.

Step 1: Determine the fraction of milk in each vessel
In vessel A, the ratio of milk to water is 4 : 3.
Therefore, the fraction of milk in vessel A is:
$M_A=\frac{4}{4+3}=\frac{4}{7}$
In vessel B, the ratio of milk to water is 2 : 3.
Therefore, the fraction of milk in vessel B is:
$M_B=\frac{2}{2+3}=\frac{2}{5}$
In the final mixture in vessel C, the liquid is half milk and half water (ratio 1 : 1).
Therefore, the fraction of milk in vessel C is:
$M_C=\frac{1}{1+1}=\frac{1}{2}$

Step 2: Set up the mixture equation
Let us mix the liquids from vessel A and vessel B in the ratio x : y.
The total quantity of milk from both vessels divided by the total volume of the mixture must equal the fraction of milk in vessel C:
$\frac{\frac{4}{7}x+\frac{2}{5}y}{x+y}=\frac{1}{2}$

Step 3: Solve for the ratio x : y
Cross-multiply to simplify the equation:
$2\left(\frac{4}{7}x+\frac{2}{5}y\right)=x+y$
Distribute the 2 on the left side:
$\frac{8}{7}x+\frac{4}{5}y=x+y$
Subtract x from both sides and subtract 4/5 y from both sides to group the variables:
$\left(\frac{8}{7}-1\right)x=\left(1-\frac{4}{5}\right)y$
Simplify the coefficients:
$\frac{1}{7}x=\frac{1}{5}y$
Now, find the ratio of x to y:
$\frac{x}{y}=\frac{7}{5}$

Thus, the liquids in both vessels must be mixed in the ratio 7 : 5 to get a mixture containing equal parts of milk and water in vessel C.