Question

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?

Answer 1

Hint:Percentage of a solution tells us its concentration. Here $50%$ ethyl alcohol means 50 parts of ethyl alcohol is dissolved in 100 parts of solution by the given unit.

Complete step-by-step answer:We have been given a lotion that contains $50%$ v/v ethyl alcohol. Now to 6 fluid ounces of this lotion we have to add water, to reduce its alcoholic character by $75%$.
For this we have to calculate the concentration of $50%$ ethyl alcohol in 6 fluid ounce of this lotion. This will be calculated by:
Concentration of ethyl alcohol in 6 fl. oz of solution = $\dfrac{amount\,of\,ethyl\,alcohol}{amount\,of\,solution}\times $ 6 fl.oz solution
Concentration of ethyl alcohol in 6 fl. oz of solution = $\dfrac{50\,fl.\,oz\,\,ethyl\,alcohol}{100\,fl.\,oz\,\,solution}\times $ 6 fl.oz solution
Concentration of ethyl alcohol in 6 fl. oz of solution = 3 fl. oz ethyl alcohol
So, 3 fluid ounce of $50%$ v/v ethyl alcohol is present in 6 fluid ounce of lotion.
Now for the solution to be nonalcoholic by$75%$, only $25%$alcohol should remain. For this we have to add up some extra volume of solution in 6 fluid ounce of lotion to reduce the alcoholic character by$75%$. To calculate the amount of extra water which has to be added, we have to use the concentration formula in which we have to add a volume of ‘x’ to 6 fluid ounce of solution volume to get the amount of extra water added to make it $25%$. This is calculated as:
Percentage of alcohol, $25%$= $\dfrac{\,concentration\,of\,\,ethyl\,alcohol}{volume\,of\,solution}\times $100
Percentage of alcohol, $25%$= $\dfrac{3\,fl.\,oz\,\,ethyl\,alcohol}{(6+x)fl.\,oz\,\,solution}\times $100
Through this we can get ‘x’ which is the volume of water added to make it nonalcoholic as:
\[3\times 100=25\times (6+x)\]
\[\Rightarrow 300=150+25x\]
\[\Rightarrow x=\dfrac{150}{25}\]
x= 6 fluid ounce
Hence, 6 fl. oz of water should be added to 6 fl. oz of lotion to make it $75%$ nonalcoholic.

Note:Here the unit of volume is taken in fluid ounce throughout so the answer will be derived in this unit only. Value of water ‘x’ is added to the solution volume to dilute it, as by diluting, it will be made non alcoholic.