A vessel contains milk and water in the ratio 3:2. The volume of the

The problem isn't very tough if your understand your mixtures well.
Work with each statement one by one.

Step 1:
"A vessel contains milk and water in the ratio 3:2. The volume of the contents is increased by 50% by adding water to this."

You have milk:water = 3:2
volume of this solution:volume of water added = 2:1 (Since volume is increased by 50%)
Let's focus on milk. Solution 1 with 3/5 milk is mixed with solution 2, which has no milk, in the ratio 2:1
Fraction of milk in final mixture of step 1 = [(3/5)*2 + 0*1]/(2+1) = 2/5
Say the volume of this final mixture is V litres.

Step 2:
"From this solution 30 litres of the mixture is withdrawn and then replaced with water. "

This just means that (V - 30) lt of our final mixture above is mixed with 30 lt of water.

Step 3:
"The resultant ratio of milk to water in the final solution is 3:7. "

This tells us that when we mix V-30 lt of mixture in which milk is 2/5 with 30 lt water, we get fraction of milk = 3/10

(V - 30)/30 = (0 - 3/10)/(3/10 - 2/5)
We get V = 120 lt

Since V was the volume of the final mixture in step 1, the volume of solution 1 in step 1 must have been 80 lt (because when you increase it by 50% i.e. 40 lt, you get 120 lt)

The original volume of the solution = 80 lt