## A Student’s Aha Moment

Posted by mark schwartz on October 12, 2016

__Introduction__

One more example of how an aggregation of student’s imaginations led the class, including the instructor, to apply Algebra to Algebra. A type of problem which frequents texts in the U.S. is the ‘mixture’ problem. Mixing solutions of different concentrations to get a third with a desired concentration; mixing different kinds of candy or nuts to get a mixture to sell at a certain price, or getting a return on investing in two accounts at different interest rates (and other applications).

__The Story__

Gus has on hand a 5% alcohol solution and a 20% alcohol solution. He needs 30 liters of a 10% alcohol solution. How many liters of each solution should he mix together to obtain the 30 liters?

The classical solution is to write the equation .05x + .20(30 – x) = .10 (30). Solving the equation, the outcome is 20 liters at 5% and 10 liters at 20%.

As this type of problem was discussed, one student asked “can __15+x__ and __15-x__ be used, since they add to the total of 30 and this seems easier?” I wasn’t certain if it would work nor why it seemed easier, but we explored the idea. The student presented the problem on the board as 5(15 + x) + 20(15 – x) = 10 (30). Not only did he re-craft the unknowns, but he used whole numbers, not decimals. When asked why, he simply stated that we would be getting rid of the decimals anyway. I noted that ‘getting rid of’ is not a mathematical operation, but clearly it works. The answer to this equation is -5, and some students believed this solution to be awkward for two reasons. First because x = -5, and a negative quantity doesn’t make sense and second, because finding x doesn’t finish the problem. Recall that the unknowns in the equation are __15+x__ and __15–x__, so another step is required to come to the correct answer of 20 liters at 5% and 10 liters at 20%. Then the question came: “Isn’t there another way to do this?”

At this point, I introduced “Alligation”, a procedure described in detail in a post in this blog titled *Mixing it up with Alligation. *I won’t go into detail about the procedure, so look it up if you’re interested. It’s a very different approach which was popular in the 1800s but doesn’t seem to be in any current texts.

We did discuss several other ideas and it was an enjoyable session in which the class actually reported having fun doing Algebra!

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