Let's solve this problem step by step.

Initial Setup:

You have a solution with 75% milk. Let the initial volume of the solution be 100 units for simplicity.

• Initially, there is $75\%$ of milk, so the volume of milk is: $75\ \text{units of milk} \text{ and } 25\ \text{units of water}.$

Step 1: First removal and replacement

• 20% of the solution is removed. The total volume removed is $20\%$ of 100 units, which equals: $20\ \text{units}.$

  Out of this, $75\%$ is milk and $25\%$ is water. So, the removed quantities are: $\text{Milk removed} = 75\%\ \text{of}\ 20\ = 15\ \text{units}.$ $\text{Water removed} = 25\%\ \text{of}\ 20\ = 5\ \text{units}.$

• After removal, the remaining amounts of milk and water are: $\text{Milk remaining} = 75 - 15 = 60\ \text{units},$ $\text{Water remaining} = 25 - 5 = 20\ \text{units}.$

• Replacement: The 20 units removed are replaced entirely by milk. So, the new volume of milk becomes: $\text{Milk after replacement} = 60 + 20 = 80\ \text{units}.$ The total solution is still 100 units, so the water remains at 20 units.

Step 2: Second removal and replacement

• Again, 20% of the solution is removed. This is 20 units, which contains the new percentage of milk and water:

  • The percentage of milk after the first replacement is: $\text{Percentage of milk} = \frac{80}{100} = 80\%.$
  • So, in the 20 units removed: $\text{Milk removed} = 80\%\ \text{of}\ 20 = 16\ \text{units},$ $\text{Water removed} = 20\%\ \text{of}\ 20 = 4\ \text{units}.$

• After the second removal, the remaining amounts of milk and water are: $\text{Milk remaining} = 80 - 16 = 64\ \text{units},$ $\text{Water remaining} = 20 - 4 = 16\ \text{units}.$

• Replacement: The 20 units removed are again replaced by milk. So, the new volume of milk becomes: $\text{Milk after replacement} = 64 + 20 = 84\ \text{units}.$ The total solution is still 100 units, with the water remaining at 16 units.

Final Answer:

The total amount of milk is 84 units out of 100, so the percentage of milk in the solution is: $\frac{84}{100} \times 100 = 84\%.$

Thus, the percentage of milk in the solution after the two replacements is 84%.

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Would you like further details or have any questions?

Related Questions:

1. How would the result change if only 10% of the solution were removed and replaced?
2. What would happen if the initial percentage of milk was 50%?
3. How does the process differ if water is added instead of milk?
4. Can you generalize the formula for $n$ repetitions of this process?
5. What is the effect of removing and replacing different percentages each time?

Tip:

Always start with simple values (like 100 units) when dealing with percentages to make the calculations easier.