Answer
A total of 6 qts of water must be added.
Work Step by Step
We assign the variable:
x = quarts of pure water (0% broth)
Step 1:
Find the equations that represent the problem.
We are mixing both solutions to create a new one, but it has to be in the same terms. Pure water means 0% broth, so
60% broth + 0% broth= 40% broth -> Eq. 1
We can get the second equation form the wording in the problem.
From "3 gal of soup that is 60% chicken broth ", we get
60% broth $=0.6(3gal)=1.8gal$
But the exercise is asking for quarts as units. Remember 1 gallon = 4quarts.
60% broth $=1.8gal*\frac{4qts}{1gal}$
60% broth $=7.2qts$ -> Eq. 2
We can get the third equation form the wording in the problem.
From "to make a solution that is 60% alcohol",
40% broth $=0.4(x+12)$ -> Eq. 3
(Remember the volume from the new solution has to equal the sum of both volumes of the initial solutions.)
Step 2:
Solve the system of equations using the substitution method,
-> Substitute Eq. 2 and Eq. 3 into Eq. 1
$7.2+0=0.4(x+12)$ -> 0% broth = 0
$7.2=0.4(x+12)$
$7.2=4.8+0.4x$
$0.4x=7.2-4.8$
$0.4x=2.4$
$x=\frac{2.4}{0.4}$
$x=6$
Step 3:
A total of 6 qts of water must be added.
