Mixing solutions of specified strength While reading one aptitude book, I came across this question:


  $35\%$ alcohol mixed with $60\%$ alcohol to get a $50\%$ alcohol. In what ratio were they mixed?


I spent 20 minutes on this question, but I couldn't even figure out what it means! Could anyone explain this to me?
Call the ratio you want to find r=(volume of 35% alcohol):(volume of 60% alcohol).  Suppose that there are r liters of the 35% alcohol (so there are 0.35 L of alcohol and 0.65 L of water), 1 liter of the 60% alcohol (so the ratio is r), and 1+r liters of the 50% alcohol.  From there, you can construct an equation about the alcohol (or about the water) and solve for r.
Given a liters of 35% alcohol solution, and b liters of 60% alcohol solution, what would be the alcohol ratio after we'll mix this two solutions?

The a liters are actually a*35/100 alcohol, and a*(1-(35/100)) water. Likewise the b liters contains b*60/100 alcohol and b*(1-(60/100)) water.

Mixing them together would give us a*35/100+b*60/100 alcohol and a*(1-(35/100))+b*(1-(60/100)) water. The question is for which a,b we have the ratio

((a*35+b*60)/100)/((a*65+b*40)/100) = 50% = 1/2


or

(a*35+b*60)/(a*65+b*40) = 1/2


I believe you can take it from here...