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?

Question

Olivia Anton · Accepted Answer

#0.4# #"L"# must be added.

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%#. We can use the following relationship to accomplish this: #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. After entering every known value, we have #0.25(V_"final") = 1(V_"pure") + 0.10(2color(white)(l)"L")# Here, the volumes will be additive, meaning that the total volume will equal the sum of the volumes of the two parts: #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"" ")|)#

Noah Aldrich · Answer

undefined Lisa needs to add 3 liters of pure fruit juice to make the punch 25% fruit juice.