Let's work through the problem step by step to find the value of $X$, which represents the percentage of milk in solution B.

Step 1: Understand the problem

• Solution A has 10% milk.
• Solution B has $X\%$ milk.
• Solutions A and B are mixed in a ratio of 3:2 to form Solution C.
• Then, solutions A and C are mixed in equal quantities to form the final solution, which has 12.5% milk.

Step 2: Milk percentage in Solution C

When A and B are mixed in the ratio of 3:2, the milk content in Solution C can be calculated using the weighted average formula for mixtures:

$\text{Milk percentage in C} = \frac{3 \times 10\% + 2 \times X\%}{3 + 2}$

Simplifying this expression:

$\text{Milk percentage in C} = \frac{30 + 2X}{5} = 6 + \frac{2X}{5}$

Thus, the milk percentage in Solution C is $6 + \frac{2X}{5}$.

Step 3: Milk percentage in the final mixture

Now, Solution A and Solution C are mixed in equal quantities to form the final solution, which has 12.5% milk. Since they are mixed in equal amounts, the milk percentage in the final solution is the simple average of the milk percentages in Solution A and Solution C:

$\frac{10\% + \left(6 + \frac{2X}{5}\right)}{2} = 12.5\%$

Step 4: Set up the equation

We can now set up the equation:

$\frac{10 + \left(6 + \frac{2X}{5}\right)}{2} = 12.5$

Multiply both sides of the equation by 2 to eliminate the denominator:

$10 + 6 + \frac{2X}{5} = 25$

Simplify:

$16 + \frac{2X}{5} = 25$

Subtract 16 from both sides:

$\frac{2X}{5} = 9$

Step 5: Solve for $X$

Multiply both sides by 5:

$2X = 45$

Now divide by 2:

$X = 22.5$

Thus, the percentage of milk in Solution B is $\boxed{22.5\%}$.

---

Would you like more details or have any questions?

Related questions:

1. How do you calculate the weighted average in mixtures?
2. What would the final percentage of milk be if Solution B had 25% milk instead of 22.5%?
3. How does changing the mixing ratio of solutions A and B affect the final mixture?
4. What if solutions A and B were mixed in a ratio of 2:3 instead? How would the answer change?
5. Can the formula for weighted averages be used for other substances, like sugar in water?

Tip:

Always ensure that ratios are applied correctly when mixing solutions to get accurate results in percentage calculations.