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Question: An after shave lotion contains \(50%\) ethyl alcohol. How much water must be added to 6 fluid ounces...

An after shave lotion contains 5050% ethyl alcohol. How much water must be added to 6 fluid ounces of this lotion to reduce it to one which is 7575% nonalcoholic?

Explanation

Solution

Percentage of a solution tells us its concentration. Here 5050% 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 5050% v/v ethyl alcohol. Now to 6 fluid ounces of this lotion we have to add water, to reduce its alcoholic character by 7575%.
For this we have to calculate the concentration of 5050% ethyl alcohol in 6 fluid ounce of this lotion. This will be calculated by:
Concentration of ethyl alcohol in 6 fl. oz of solution = amount of ethyl alcoholamount 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 = 50 fl. oz  ethyl alcohol100 fl. oz  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 5050% v/v ethyl alcohol is present in 6 fluid ounce of lotion.
Now for the solution to be nonalcoholic by7575%, only 2525%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 by7575%. 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 2525%. This is calculated as:
Percentage of alcohol, 2525%=  concentration of  ethyl alcoholvolume of solution×\dfrac{\,concentration\,of\,\,ethyl\,alcohol}{volume\,of\,solution}\times 100
Percentage of alcohol, 2525%= 3 fl. oz  ethyl alcohol(6+x)fl. oz  solution×\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:
3×100=25×(6+x)3\times 100=25\times (6+x)
⇒300=150+25x\Rightarrow 300=150+25x
⇒x=15025\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 7575% 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.