There's been a lot of talk lately about teaching algebra in schools. Some people believe that algebra should be scrapped from the curriculum, or at least severely curtailed. Books get written. Mathematicians get irritated. Stalemate.
Most of the arguments against algebra come down to the utilitarian: the old "When am I ever going to use this?" that teachers of every discipline abhor. And it's true that if you ask me to give you an explicit example of how you'll use the quadratic formula in your daily life I will struggle. I will retreat into standard arguments about how mathematical thinking is important for intellectual development, etc. and that algebra actually has practical applications involving ratios and scaling.
But today, a gift fell into my lap: an actual, solve for x algebra application in daily life. A gentleman from South Florida called my office (I was out) and left a message and also sent an email detailing his dilemma. He is on the board of his community's homeowners association and they needed to decide on the monthly assessment each homeowner would be required to pay to the association for general upkeep. There are two kinds of houses, small and large, 13 of the former and 18 of the latter. The HOA agreement stipulates that the large homes will pay 1.8 times what the small homes are assessed. The board determines the annual budget, let's say $50,000. How much is each homeowner responsible for?
Here's what they did. They formula used was (1 + 1.8)x equals $50,000 for the total annual assessment. Then 2.8x = $50,000 and so x = $17,857. Dividing that by the 13 small houses yields a monthly payment of $114.47. The remaining $32,143 is divided among the 18 large houses to yield a monthly payment of $148.81. But wait, 148.81/114.47 = 1.3, not 1.8. What went wrong?
Well, even though I had just finished one meeting and had another one in 20 minutes I couldn't help myself. This would have worked ok if there had been an equal number of small and large houses. What the equation above does is find an amount x that the group of small houses could be assessed collectively. But that assumes that the number of small houses equals the number of large houses. Since there are more large houses, the share assigned to them gets spread around more and so the result is a payment of 1.3 times the small house payment. Not what we want.
So, how to fix this? Let x be the amount a small house will be assessed for the year and let y be the amount for a large house. Then 13x + 18y = 50,000. We also want y = 1.8x. We then get the equation 13x + 18(1.8x) = 45.4x = 50,000. Solving this we get x = $1,101.32 per year, or $91.78 per month for a small house and y = $165.20 monthly for a large house. All is right with the world (and the lawyers).
Keep this example in your back pocket for the next time someone says that algebra is not useful in everyday life. I know I sure will.
