Happy Birthday, SiS!

A year ago today I made the first two posts to Science in Suburbia. I know that, due to work/distractions, I haven’t posted consistently for the entire year, but it’s still my first blogiversary and I’m happy that I’ve turned up the posting frequency over the last couple of months.

Something else I’ve seen with increasing frequency over the last couple of months? “Jokes” about how taking algebra was stupid and useless because you never use any of it after high school.

Besides an annoyed grumble, my reaction to this type of statement is twofold:

1) You probably haven’t written a paper analyzing Shakespeare since high school either; why aren’t you complaining about that? (I’m not trying to devalue literature here, I just wanted to point out the inconsistency.)

2) You actually do use algebra on a fairly regular basis without ever calling it out by name; if you’re not using it, you might be paying too much for internet.

Say one internet plan has free installation and costs 20 dollars a month. A competitor’s plan has an installation fee of 125 dollars, but it only costs 15 dollars a month. Common sense tells us that the second plan will be cheaper in the long run, but what does “the long run” actually mean? As a former apartment dweller, I know that sometimes you don’t know where you’ll be living a year or two down the road; if it takes longer than that for the second plan to pay off, it’s not worth it. How do we figure out when the second plan becomes a better deal?

Algebra.

But let’s not stick in x and y just yet. “Putting the alphabet in math” throws people off; I can’t tell you how many Etsy products/attempts at internet memes feature sayings like “I was ok with math until they put the alphabet in it” or “I’m sick of looking for your x, solve your own problems.” Let’s take a second and define what our variables actually mean in this context.

Plan 1: total amount of money spent = 20 dollars/month * number of months

Plan 2: total amount of money spent = (15 dollars/month * number of months) + 125 dollars

A variable is just any number we don’t know the value of. We don’t know how much money we’ll spend, so let’s replace every instance of that unknown with y. We don’t know how many months we’ll have the internet, so let’s replace every instance of that unknown with x. This makes our equations look like this.

Plan 1: y = 20x

Plan 2: y = 15x + 125

Now we can use these equations- plus a little algebra- to figure out how long it takes Plan 2 to pay off. Again, common sense tells us that Plan 1 is cheaper in the short run and Plan 2 is cheaper in the long run; since the total cost of Plan 1 has to catch up with and then surpass the cost of Plan 2, there must be some amount of time that makes the cost of Plan 1 equal to the cost of Plan 2. (If you’re in a race, you can’t pass the person in first place without being right next to them at some point.) We have equations for the cost of each plan, so let’s set them equal to each other.

20x = 15x + 125

Since x is the number of months we have the internet, the value of x that makes this equation true is the number of months that makes the cost of each plan the same. We could just start throwing numbers in there until we find one that works, but solving this equation isn’t so bad if we remember the Ultimate Goal of Algebra and the Ultimate Rule of Algebra.

Ultimate Goal of Algebra: Get the variable by itself on one side of the equals sign.

Ultimate Rule of Algebra: Whatever you do one side of the equation you have to do to the other.

We’re starting here:

20x = 15x + 125

We only want x on one side of the equation, so let’s subtract 15x from both sides.

20x – 15x = 15x + 125  – 15x

5x = 125

Now let’s divide both sides by 5.

x = 25

If you have the internet for exactly 25 months, the two plans will cost the same. Plan 1 is cheaper if you have the internet for fewer than 25 months; Plan 2 is cheaper if you have the internet for more than 25 months. Apartment dwellers should probably opt for Plan 1.

We come to the same conclusion if we graph our equations.

The black line represents Plan 1, while the gray line represent Plan 2. The vertical axis is cost and the horizontal axis is number of months.
The black line represents Plan 1, while the gray line represents Plan 2. The vertical axis is cost, measured in dollars, and the horizontal axis is time, measured in months.

The graphs intersect at 25 months. Before that, Plan 1 costs less; after that, Plan 2 costs less.

So there’s the utilitarian side of algebra, proving there really is Math After High School.

For Pi Day we’ll look at the fun side of algebra- yes, it really does exist!

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