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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!

 

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