A vessel is filled with liquid, 3 parts of which are water and 5 parts of syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

Given:

• Initial mixture: 3 parts water and 5 parts syrup → Total = 8 parts
• We want final mixture: 4 parts water and 4 parts syrup (i.e., half water and half syrup)
• Let x = fraction of the mixture drawn off and replaced with water

Step 1: Focus on syrup (since only water is added)

After removing x fraction of the mixture, syrup left = 5 × (1 - x)

Step 2: Final syrup = 4 parts (since 50% of 8)

5 × (1 - x) = 4

Step 3: Solve for x

5 - 5x = 4
=> 5x = 1
=> x = 1/5

Answer: 1/5 of the mixture must be drawn off and replaced with water.