Lisa will make punch that is 25% fruit juice by adding pure fruit juice to a 2-liter mixture that is 10% pure fruit juice. How many liters of pure fruit juice does she need to add?

We're asked to find the volume (in liters) of #100%# fruit juice that must be added to #1# #"L"# of a #10%# fruit juice mixture so that the final concentration is #25%#.

To do this, we can use the following relationship:

#C_"final"V_"final" = C_"pure"V_"pure" + C_(25%)V_(25%)#

where

• #C_"final"# and #V_"final"# are the concentration and volume of the final solution. We're given that the final concentration must be #25%#.

• #C_"pure"# and #V_"pure"# are the concentration and volume of the pure solution. We'll say that a pure solution has a concentration of #1#.

• #C_(25%)# and #V_(25%)# are the concentration and volume of the #25%# solution. We're given both of these quantities as #0.10# and #2# #"L"# respectively.

Plugging in all known values, we have

#0.25(V_"final") = 1(V_"pure") + 0.10(2color(white)(l)"L")#

Volumes here are going to be additive; that is, the final volume will be the sum of the volumes of the two components:

#V_"final" = V_"pure" + 2color(white)(l)"L"#

We'll now plug this into the equation for #V_"final"#:

#0.25(V_"pure" + 2color(white)(l)"L") = V_"pure" + 0.10(2color(white)(l)"L")#

Now, we just solve for the necessary volume, #V_"pure"#:

#0.25(V_"pure") + 0.5color(white)(l)"L" = V_"pure" + 0.2color(white)(l)"L"#

#0.25(V_"pure") + 0.3color(white)(l)"L" = V_"pure"#

Divide all terms by #V_"pure"#:

#0.25 + (0.3color(white)(l)"L")/(V_"pure") = 1#

#(0.3color(white)(l)"L")/(V_"pure") = 0.75#

#color(red)(ulbar(|stackrel(" ")(" "V_"pure" = 0.4color(white)(l)"L"" ")|)#