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?

1 Answer
Sep 2, 2017

0.40.4 "L"L must be added.

Explanation:

We're asked to find the volume (in liters) of 100%100% fruit juice that must be added to 11 "L"L of a 10%10% fruit juice mixture so that the final concentration is 25%25%.

To do this, we can use the following relationship:

C_"final"V_"final" = C_"pure"V_"pure" + C_(25%)V_(25%)CfinalVfinal=CpureVpure+C25%V25%

where

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

  • C_"pure"Cpure and V_"pure"Vpure are the concentration and volume of the pure solution. We'll say that a pure solution has a concentration of 11.

  • C_(25%)C25% and V_(25%)V25% are the concentration and volume of the 25%25% solution. We're given both of these quantities as 0.100.10 and 22 "L"L respectively.

Plugging in all known values, we have

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

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"Vfinal=Vpure+2lL

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

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

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

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

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

Divide all terms by V_"pure"Vpure:

0.25 + (0.3color(white)(l)"L")/(V_"pure") = 10.25+0.3lLVpure=1

(0.3color(white)(l)"L")/(V_"pure") = 0.750.3lLVpure=0.75

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