Writing deliberate expressions to study and solve problems

Our Algebra 2 class has recently been studying rational expressions and functions.  One type of problem that can be solved using rational expressions is a mixing problem.  If you start off with 200mL of a 0.03% iodine solution, how much of a 0.2% solution must be added to end up with a solution that is 0.14% acid?

Finding the answer to this question involves much more than just solving the algebra, it requires developing the algebra.  Even a guess-and-check approach would require a thorough understanding of concentrations and fluency with writing expressions.  To study the problem, we must be deliberate about designating a variable and recognizing what each statement represents.

Our unknown quantity is the amount of 0.2% solution that must be added, so it makes sense to designate that quantity as x.  Now we set about writing some relevant expressions in terms of x.  Ultimately, we are interested in finding a concentration.  The expression for a concentration is the amount of the material divided by the amount of solution.  The amount of total solution is simple:  just our original 200 mL plus x mL of the new solution.  (200+x).

The numerator of our concentration expression will be a little more complicated.  It must represent the total amount of iodine.  We start with 0.03% of 200 mL. (0.0003 times 200)  We add 0.2% of x mL.  (0.002x).  So our concentration expression is ultimately

C = (0.002x + 0.0006)/(200+x)

Since our particular question gives us a target concentration, we can replace C with 0.0014 and solve for x.  The algebra is the easy part!

Finally, we can graph this equation and notice that there is a horizontal asymptote at y = 0.002.  As we add more and more of our concentrated solution, the concentration of the resulting solution will get closer and closer to 0.002, but never quite get there, thanks to our original 200 mL of diluted iodine.  One more applied example of how real situations can be modeled with relatively simple mathematical equations.

Dan Thurber, Math Teacher