A cup of milk contains 3 parts of pure milk and 1 part of water. How much mixture must be withdrawn    and water substituted in order that the resulting mixture may be half milk and half water?

Let's solve this step by step:

1. Initially, the cup of milk contains 3 parts of pure milk and 1 part of water. So, the total parts in the cup are 3 (milk) + 1 (water) = 4 parts.

2. We want the resulting mixture to be half milk and half water. This means the ratio of milk to water should be 1:1.

3. Let's assume that we withdraw 'x' parts of the mixture from the cup. Since the mixture is 3 parts milk and 1 part water, the withdrawn mixture will also be in the same ratio. So, we withdraw 3x/4 parts of milk and x/4 parts of water.

4. After withdrawing, the amount of milk left in the cup is 3 - 3x/4 and the amount of water left is 1 - x/4.

5. We then substitute the withdrawn mixture with pure water. So, the amount of water in the cup becomes 1 - x/4 + x = 1 + 3x/4.

6. For the mixture to be half milk and half water, the amount of milk should be equal to the amount of water. So, we can set up the equation 3 - 3x/4 = 1 + 3x/4.

7. Solving this equation gives x = 2/3.

So, we must withdraw 2/3 parts of the mixture and substitute it with water to make the resulting mixture half milk and half water.