Question

A teaching assistant at a university needs 28% acid solution for her class's lab experiment. There isn't any of this concentration in stock, but the lab has 50 liters of 22% acid solution, as well as a lot of 32% acid solution. How much of the 32% acid solution should the teaching assistant add to the 22% acid solution to obtain a solution with the desired concentration?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.
liters

Answer 1

To solve the problem, we need to find out how many liters of the 32% acid solution to add to the 50 liters of the 22% acid solution in order to create a final solution that is 28% acid.

Let \( x \) be the amount of 32% acid solution to be added in liters.

1. **Calculate the amount of acid in the 22% solution**:
\[
\text{Acid from 22% solution} = 0.22 \times 50 = 11 \text{ liters}
\]

2. **Calculate the amount of acid in the 32% solution**:
\[
\text{Acid from 32% solution} = 0.32 \times x \text{ liters}
\]

3. **Total acid in the final solution**:
The total amount of acid after adding \( x \) liters of the 32% solution will be:
\[
\text{Total acid} = 11 + 0.32x \text{ liters}
\]

4. **Total volume of the final solution**:
The total volume of the mixture will be:
\[
\text{Total volume} = 50 + x \text{ liters}
\]

5. **Set up the equation for desired concentration**:
We want the final concentration to be 28%. Therefore, we set up the equation:
\[
\frac{11 + 0.32x}{50 + x} = 0.28
\]

6. **Multiply both sides by \( 50 + x \) to eliminate the fraction**:
\[
11 + 0.32x = 0.28(50 + x)
\]

7. **Expand the right side**:
\[
11 + 0.32x = 14 + 0.28x
\]

8. **Combine like terms**:
Subtract \( 0.28x \) from both sides:
\[
11 + 0.32x - 0.28x = 14
\]
Simplifying gives:
\[
11 + 0.04x = 14
\]

9. **Isolate \( x \)**:
\[
0.04x = 14 - 11
\]
\[
0.04x = 3
\]
\[
x = \frac{3}{0.04} = 75
\]

Thus, the teaching assistant should add **75 liters** of the 32% acid solution to the 22% acid solution to obtain a 28% acid solution.