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?

Sep 2, 2017

$0.4$ $\text{L}$ must be added.

Explanation:

We're asked to find the volume (in liters) of 100% fruit juice that must be added to $1$ $\text{L}$ 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

• ${C}_{\text{final}}$ and ${V}_{\text{final}}$ are the concentration and volume of the final solution. We're given that the final concentration must be 25%.

• ${C}_{\text{pure}}$ and ${V}_{\text{pure}}$ 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 $2$ $\text{L}$ respectively.

Plugging in all known values, we have

$0.25 \left({V}_{\text{final") = 1(V_"pure") + 0.10(2color(white)(l)"L}}\right)$

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

${V}_{\text{final" = V_"pure" + 2color(white)(l)"L}}$

We'll now plug this into the equation for ${V}_{\text{final}}$:

$0.25 \left({V}_{\text{pure" + 2color(white)(l)"L") = V_"pure" + 0.10(2color(white)(l)"L}}\right)$

Now, we just solve for the necessary volume, ${V}_{\text{pure}}$:

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 ${V}_{\text{pure}}$:

$0.25 + \left(0.3 \textcolor{w h i t e}{l} \text{L")/(V_"pure}\right) = 1$

$\left(0.3 \textcolor{w h i t e}{l} \text{L")/(V_"pure}\right) = 0.75$

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