Suppose that you have a supply of a 32% solution of alcohol and a 78% solution of alcohol. How many quarts of each should be mixed to produce 24 quarts that is 50% alcohol?
Define T: quarts of the 32% solution and S: quarts of the 78% solution, to be used.
Since the total is 24 quarts, we have T + S = 24.
The content of alcohol in the mix will be: 0.32 T + 0.78 S, and it should yield 50% of the 24 quarts, i.e. 12 quarts, so
0.32 T + 0.78 S = 12
Now we have two linear equations and two unknowns. Solving this system you get S=9.39 and T=14.61 (approximately).
Hope this helps!