Question

Doreen Schmidt is a chemist. She needs to prepare 28 ounces of a 11​% hydrochloric acid solution. Find the amount of 14​% solution and the amount of 7​% solution she should mix to get this solution.

Answer 1

To solve this problem, we can use the method of systems of equations. Let \( x \) be the amount of the 14% hydrochloric acid solution and \( y \) be the amount of the 7% hydrochloric acid solution. We need to satisfy the following two conditions:

1. The total amount of solution must be 28 ounces:
\[
x + y = 28
\]

2. The total amount of hydrochloric acid must equal to the desired concentration in the final mixture:
\[
0.14x + 0.07y = 0.11 \times 28
\]

Calculating the right side of the second equation:
\[
0.11 \times 28 = 3.08 \text{ ounces of hydrochloric acid}
\]

So, our two equations are:
1. \( x + y = 28 \)
2. \( 0.14x + 0.07y = 3.08 \)

Now we will solve these equations step-by-step.

From the first equation, we can express \( y \) in terms of \( x \):
\[
y = 28 - x
\]

Now substitute this expression for \( y \) into the second equation:
\[
0.14x + 0.07(28 - x) = 3.08
\]

Distributing \( 0.07 \):
\[
0.14x + 1.96 - 0.07x = 3.08
\]

Combining like terms:
\[
0.07x + 1.96 = 3.08
\]

Subtracting \( 1.96 \) from both sides:
\[
0.07x = 3.08 - 1.96
\]
\[
0.07x = 1.12
\]

Now, divide by \( 0.07 \):
\[
x = \frac{1.12}{0.07} \approx 16
\]

Now, substituting \( x \) back to find \( y \):
\[
y = 28 - x = 28 - 16 = 12
\]

Thus, the amounts of the solutions are:
- Amount of 14% solution: \( x \approx 16 \) ounces
- Amount of 7% solution: \( y \approx 12 \) ounces

In summary, Doreen should mix approximately **16 ounces of the 14% hydrochloric acid solution** and **12 ounces of the 7% hydrochloric acid solution** to prepare 28 ounces of an 11% hydrochloric acid solution.