An after shave lotion contains 50% ethyl alcohol. How much water must be added to 6 fluid ounces of this lotion to reduce it to one which is 75% nonalcoholic?

Aug 5, 2015

You need to add 6 fl. oz. of water.

Explanation:

So, your solution is 50% "v/v", which means that you get 50 lf. oz of ethanol for every 100 lf. oz of solution.

This means that your 6-fl. oz sample will contain

6color(red)(cancel(color(black)("fl. oz. solution"))) * "50 fl. oz ethanol"/(100color(red)(cancel(color(black)("fl. oz. solution")))) = "3 fl. oz ethanol"

Now, your target solution must be 75% nonalcoholic, which means that it must have a percent concentration by volume of 25%.

The important thing to ralize here is tha the volume of ethanol will be the same for both your starting solution, and for your target solution.

The only thing that will change will be the volume of solution.

This means that you can write

"3 fl. oz ethanol"/((6 + x)"fl. oz. solution") * 100 = 25%, where

$x$ - the volume of water you need to add to the 50% v/v solution.

Rearrange this equation and solve for $x$ to get

$3 \cdot 100 = 25 \cdot \left(6 + x\right)$

$300 = 150 + 25 x \implies x = \frac{150}{25} = \textcolor{g r e e n}{\text{6 fl. oz.}}$

SImply put, if you add enough water to double the volume of the initial solution, the percent concetration of ethanol will go from 50 to 25%.