Question

A research chemist needs 22 liters of a 25% acid solution to perform a test. The supplier for the laboratory only shipped a 10% acid solution and a 30% acid solution. How many liters of the 10% acid solution and the 30% acid solution should the chemist mix?
Write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. Do not round.
liters of the 10% acid solution
liters of the 30% acid solution

Answer 1

Let \( x \) be the number of liters of the 10% acid solution, and let \( y \) be the number of liters of the 30% acid solution.

We have two main conditions to satisfy:

1. The total volume of the solution:
\[
x + y = 22 \quad (1)
\]

2. The total amount of acid in the solution:
The 10% solution contributes \( 0.10x \) liters of acid, and the 30% solution contributes \( 0.30y \) liters of acid. We need the total acid to equal 25% of 22 liters, which is:
\[
0.25 \times 22 = 5.5 \text{ liters of acid}
\]
This gives us the second equation:
\[
0.10x + 0.30y = 5.5 \quad (2)
\]

Now we can solve these equations simultaneously. From equation (1), we can express \( y \) in terms of \( x \):
\[
y = 22 - x
\]

Now, substitute \( y \) into equation (2):
\[
0.10x + 0.30(22 - x) = 5.5
\]
Expanding this:
\[
0.10x + 6.6 - 0.30x = 5.5
\]
Combining like terms:
\[
-0.20x + 6.6 = 5.5
\]
Subtracting 6.6 from both sides:
\[
-0.20x = 5.5 - 6.6
\]
\[
-0.20x = -1.1
\]
Dividing both sides by -0.20:
\[
x = \frac{-1.1}{-0.20} = 5.5
\]

Now substituting \( x \) back into the equation for \( y \):
\[
y = 22 - 5.5 = 16.5
\]

Thus, the chemist should mix:
- **5.5 liters of the 10% acid solution**
- **16.5 liters of the 30% acid solution**