Two vessels A and B contain milk and water mixed in the ratio 4 : 3 and 2 : 3 respectively. The ratio in which these mixtures containing half milk and half water is = ?

Ratios are 4:3 (4+3=7) and 2:3 (2+3=5). So LCM of 7, 5 is 35.
Now, for a 35-liter mixture.
Let M=milk, W=water

Vessel A,
M : W = 4 : 3 (4+3=7) means in 35-liter mixture. Milk is 4 × $$\frac{{{\text{35}}}}{7}$$   =   20 liter and water is 3 × $$\frac{{{\text{35}}}}{7}$$   =   15 liter.

Vessel B,
M : W = 2 : 3 (2+3=5) means in 35-liter mixture. Milk is 2 × $$\frac{{{\text{35}}}}{5}$$   =  14 liter and water is $$\frac{{{\text{35}}}}{5}$$   =   21 liter.

For Vessel C,
M : W = 1 : 1 (required)
Let the ratio of Vessel A = x and Vessel B = y.
In-Vessel C, both milk and water will be in equal quantity. So,
Net quantity of milk= Net quantity of water
⇒ 20x + 14y = 15x + 21y
⇒ 5x = 7y
⇒ $$\frac{x}{y}$$ = $$\frac{7}{5}$$
So, x : y = 7 : 5
Vessel A : Vessel B = 7 : 5