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"" ")|)`