Gregg O. answered • 09/04/15

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Here, I'm assuming a volume/volume solution (the math is equivalent for a mass/volume solution, but the units are different). It's helpful to understand the meaning of a % solution as a fraction. An 84% volume/volume solution means we have (.84 gallons alchohol)/(1 gallon of solution). Another way to think of it is (gallons of alcohol)/(gallons of solution) = .84.

Our target is an 84% solution. Since we're adding solutions together, we can think of this as (total gallons of alcohol)/(total gallons of solution) = .84. We'll let x stand for the gallons of 74% solution.

What are the total gallons of solution? Well, we start of with 5 gallons of the 90% solution, and x gallons of the 74% solution. So, total gallons of solution is x + 5.

To figure out the total amount of alcohol, we can use the formula

gallons of alcohol = (gallons of solution) * (gallons of alcohol/1 gallon of solution) = (gallons of solution)*(percentage of solution).

What is the total amount of alcohol? For the 90% solution, we have (gallons of solution)*(amount of alcohol/1 gallon of solution) which equals 5*.90, or 4.5 gallons of alcohol. For the 74% solution, we have (gallons of solution)*(amount of alcohol/1 gallon of solution), which is (x)(.74) = .74x. So the total amount of alcohol is .74x + 4.5.

Since the (total gallons of alcohol)/(total gallons of solution) = .84, we have

(.74x + 4.5)/(x+5) = .84.

It's all algebra from there.