Answer
The answer is: 320 mL of the 30% alcohol solution and 480 mL of the 80% alcohol solution.
Work Step by Step
We assign the variables:
x = volume of solution A (30% alcohol)
y = volume of solution B (80% alcohol)
Step 1:
Find the equations that represent the problem.
We are mixing both solutions to create a new one:
30% solution + 80% solution = 60% solution -> Eq. 1
We can get the second equation form the wording in the problem.
From "a solution that is 30% alcohol",
30% solution $=0.3x$ -> Eq. 2
We can get the third equation form the wording in the problem.
From "a solution that is 80% alcohol",
80% solution $=0.8y$ -> Eq. 3
We can get the fourth equation form the wording in the problem.
From "to make 800 mL of a solution that is 60% alcohol",
60% solution $=0.6(800)=480$ -> Eq. 4
Remember the volume from the new solution has to equal the sum of both volumes of the initial solutions, so
$x+y=800$
If we solve for x, we get:
$x=800-y$ -> Eq. 5
Step 2:
Solve the system of equations using the substitution method,
-> Substitute Eq. 2, Eq. 3 and Eq. 4 into Eq. 1
$0.3x+0.8y=480$ -> Eq. 6
->Substitute Eq. 5 into Eq. 6
$0.3(800-y)+0.8y=480$
$240-0.3y+0.8y=480$
$0.5y=480-240$
$0.5y=240$
$y=\frac{240}{0.5}$
$y=480$
-> Substitute the value for $y$ into Eq. 5
$x=800-480$
$x=320$
Step 3:
The answer is: 320 mL of the 30% alcohol solution and 480 mL of the 80% alcohol solution.
