To determine the amount of each solution needed to produce the desired mixture, we need to set up and solve a system of equations based on the information given.
1. Define Variables:
- Let [tex]\( x \)[/tex] be the amount of the 25% acid solution.
- Let [tex]\( y \)[/tex] be the amount of the 65% acid solution.
2. Set Up Equations:
- The total volume of the mixture is 80 liters, so:
[tex]\[
x + y = 80
\][/tex]
- The total amount of acid in the mixture is 45% of 80 liters. The equation for the total acid content is:
[tex]\[
0.25x + 0.65y = 0.45 \times 80
\][/tex]
3. Solve the System of Equations:
- The first equation is:
[tex]\[
x + y = 80
\][/tex]
- The second equation, converting 0.45 * 80, we get:
[tex]\[
0.25x + 0.65y = 36
\][/tex]
4. Isolate one variable:
From the first equation, solve for [tex]\( y \)[/tex]:
[tex]\[
y = 80 - x
\][/tex]
5. Substitute [tex]\( y \)[/tex] in the second equation:
Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[
0.25x + 0.65(80 - x) = 36
\][/tex]
6. Simplify and Solve for [tex]\( x \)[/tex]:
[tex]\[
0.25x + 52 - 0.65x = 36
\][/tex]
[tex]\[
52 - 36 = 0.4x
\][/tex]
[tex]\[
16 = 0.4x
\][/tex]
[tex]\[
x = 40
\][/tex]
7. Find [tex]\( y \)[/tex] using the value of [tex]\( x \)[/tex]:
Substitute [tex]\( x = 40 \)[/tex] back into the equation [tex]\( y = 80 - x \)[/tex]:
[tex]\[
y = 80 - 40
\][/tex]
[tex]\[
y = 40
\][/tex]
So, to produce 80 liters of a 45% acid solution, you need:
- 40 liters of a 25% acid solution
- 40 liters of a 65% acid solution
