Answer: 40 L of 80 percent solution and 60 L of 30 percent solution
Step-by-step explanation:
The total mixture of 100 L of apple juice of [tex]50\%[/tex] solution must contain:
[tex]80\%[/tex] solution of apple juice ---> Let's call it [tex]A[/tex]
[tex]30\%[/tex] solution of apple juice ---> Let's call it [tex]B[/tex]
So, we can set these values in a table taking into account [tex]80\%=0.8[/tex], [tex]30\%=0.3[/tex] and [tex]50\%=0.5[/tex]:
[tex]\left[\begin{array}{ccccc}&apple-juice (L)&Percent&Total\\80\% Juice&x&0.8&0.8 x\\30\% juice&y&0.3&0.3 y\\Mixture&x+y=100&0.5&0.5(100)\end{array}\right] [/tex]
Now, with the information of the [tex]Total[/tex] column we can write the first equation:
[tex]0.8 x + 0.3 y= 0.5 (100)[/tex] (1)
And with the information of the [tex]apple-juice[/tex] column, the second equation:
[tex]x + y=100[/tex] (2)
At this point we are able to calculate how many litters have [tex]x[/tex] and [tex]y[/tex].
Isolating [tex]y[/tex] from (2):
[tex]y=100-x[/tex] (3)
Substituting (3) in (1):
[tex]0.8 x + 0.3 (100-x)= 0.5 (100)[/tex] (4)
[tex]0.8 x + 30 - 0.3x= 50[/tex]
[tex]x=40[/tex] (5) 40 L of solution A
Substituting (5) in (2):
[tex]40 + y=100[/tex] (6)
[tex]y=60[/tex] (7) 60 L of solution B
Therefore, the cafeteria worker needs to mix 40 L of solution A with 60 L of solution B.
