https://file.aiquickdraw.com/mathbot/file/1727176192166harnqkp2.jpg

The problem in the image deals with mixing two solutions of milk and water in vessels \( X \) and \( Y \). 

- Vessel \( X \) contains \( b \, \text{ml} \) of a mixture that has \( b\% \) milk.
- Vessel \( Y \) contains \( a \, \text{ml} \) of a mixture that has \( a\% \) milk.
  
The question is asking for the milk concentration in the resulting mixture when the contents of both vessels are combined.

### Solution:
When you mix two solutions, the concentration of the final mixture can be found by calculating the weighted average of the two concentrations based on their volumes.

Let’s calculate the total amount of milk from each vessel:
- From vessel \( X \): \( b \times \frac{b}{100} = \frac{b^2}{100} \) ml of milk.
- From vessel \( Y \): \( a \times \frac{a}{100} = \frac{a^2}{100} \) ml of milk.

Now, the total amount of milk in the final mixture is:
\[
\text{Total Milk} = \frac{b^2}{100} + \frac{a^2}{100}
\]

The total volume of the mixture is \( a + b \) ml.

So, the concentration of milk in the final mixture is:
\[
\text{Milk Concentration} = \frac{\frac{b^2}{100} + \frac{a^2}{100}}{a + b} \times 100
\]
Simplifying this:
\[
\text{Milk Concentration} = \frac{b^2 + a^2}{a + b} \%
\]

This is different from the simple average given in the choices \( \frac{a+b}{2} \), which is why the correct answer is likely **d) none of these**.

Would you like further clarification, or to dive into a specific part of this solution?

### Relative Questions:
1. How is the concentration of a solution determined when combining mixtures?
2. What is the mathematical difference between a simple average and a weighted average?
3. Why is the correct answer not the average of the two percentages, \( \frac{a+b}{2} \)?
4. How does the concept of weighted averages apply to real-world problems like mixtures?
5. How can the formula for weighted averages be extended to more than two mixtures?

### Tip:
Always remember that when mixing solutions of different concentrations, the final concentration depends on the volume of each solution, not just the percentages.

Vessels X and Y have mixtures of milk and water. X has b ml of mixture with b% milk. Y has a ml of mixture with a% milk. What will be the milk concentration when the contents of both vessels are mixed completely?

Learn how to solve a mixture problem involving the milk concentration from two vessels with different milk percentages. We explain the concept of weighted averages and provide step-by-step calculations for determining the final concentration when mixing solutions.

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