You need a 50% alcohol solution. On hand, you have a 150 mL of a 45% alcohol mixture. You also have 65% alcohol mixture. How much of the 65% mixture will you need to add to obtain the desired solution?

To solve this problem, we will use the concept of mixing solutions and the method of solving systems of equations. Let \( x \) be the amount of 65% alcohol mixture to be added to the 150 mL of 45% alcohol mixture to obtain a 50% alcohol solution.

We will start by setting up the equation based on the volume and the alcohol content.

### Step-by-Step Solution:

1. **Define the variables:**
   - Let \( x \) be the volume in mL of the 65% alcohol mixture to be added.

2. **Set up the equation based on the total volume of the solution:**
   - The final volume of the solution will be \( 150 + x \) mL.

3. **Set up the equation based on the alcohol content:**
   - The amount of alcohol in the 45% mixture is \( 0.45 \times 150 \) mL.
   - The amount of alcohol in the 65% mixture is \( 0.65 \times x \) mL.
   - The total amount of alcohol in the final mixture should be \( 0.50 \times (150 + x) \) mL.

4. **Write the equation:**
   \[
   0.45 \times 150 + 0.65 \times x = 0.50 \times (150 + x)
   \]

5. **Simplify and solve for \( x \):**
   \[
   67.5 + 0.65x = 75 + 0.50x
   \]
   \[
   0.65x - 0.50x = 75 - 67.5
   \]
   \[
   0.15x = 7.5
   \]
   \[
   x = \frac{7.5}{0.15}
   \]
   \[
   x = 50
   \]

### Conclusion:
You need to add **50 mL** of the 65% alcohol mixture to the 150 mL of 45% alcohol mixture to obtain a 50% alcohol solution.

Do you want details or have any questions? Here are some related questions:

1. How do you derive the equation for mixing solutions?
2. What are other methods to solve such mixture problems?
3. Can this method be applied to other types of solutions, such as acids or bases?
4. What happens if you add too much of the 65% alcohol mixture?
5. How can you check if the final mixture has the desired concentration?
6. What are the practical applications of this type of calculation?
7. How do different concentrations affect the properties of a solution?
8. What are the limitations of using linear equations in mixture problems?

**Tip:** Always double-check your calculations by substituting the values back into the original equations to ensure they satisfy all conditions.

Learn how to mix alcohol solutions to obtain a 50% alcohol solution using 150 mL of 45% alcohol mixture and additional 65% alcohol mixture. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation