The problem on the screen asks you to solve a mixture problem involving alcohol solutions. Specifically, you are given:

• 480 mL of a 5% alcohol solution.
• A 90% alcohol mixture is also available.

You need to determine how much of the 90% alcohol mixture should be added to obtain a 10% alcohol solution.

Let’s define the variables:

• Let $x$ be the amount (in mL) of the 90% alcohol solution to add.

Step-by-Step Solution:

1. Alcohol in the 5% solution: The amount of pure alcohol in the 480 mL of the 5% solution is: $0.05 \times 480 = 24 \text{ mL of alcohol.}$

2. Alcohol in the 90% solution: The amount of pure alcohol in $x$ mL of the 90% solution is: $0.90 \times x = 0.90x \text{ mL of alcohol.}$

3. Final amount of alcohol in the mixture: After mixing, the total volume of the solution will be $480 + x$ mL. The desired concentration is 10%, so the total amount of alcohol should be: $0.10 \times (480 + x) = 48 + 0.10x \text{ mL of alcohol.}$

4. Set up the equation: The amount of alcohol from both the 5% solution and the 90% solution should add up to the total amount of alcohol in the final solution: $24 + 0.90x = 48 + 0.10x$

5. Solve for $x$: Subtract $0.10x$ from both sides: $24 + 0.80x = 48$ Subtract 24 from both sides: $0.80x = 24$ Divide both sides by 0.80: $x = \frac{24}{0.80} = 30 \text{ mL.}$

Final Answer:

You need to add 30 mL of the 90% alcohol solution to obtain the desired mixture.

Would you like more details or have any questions about the steps? Here are 5 related questions to extend the learning:

1. What if the original solution had a different concentration, say 7%? How would the equation change?
2. How would the solution change if you needed to end with 15% alcohol instead of 10%?
3. Can this problem be solved using a system of linear equations? How?
4. How does the total volume affect the final alcohol concentration?
5. What if the available alcohol concentrations were 50% and 90% instead of 5% and 90%?

Tip: Always ensure that the units (mL, percentage) are consistent throughout the calculations to avoid errors.