Algebraic Expressions - In the Real World

In the Real World

I Like Practical Stuff; Why Should I Care?

Hey, we like practical stuff, too! The great thing is that all of these formulas can be applied to real-world situations. That's good news if you live in the real world.

One thing the geometrical formulas are useful for is figuring out what amounts of materials are needed for projects such as carpeting floors.

Sample Problem

How much carpet is needed to cover the floor shown in the picture below?

The area of the floor is the area of the rectangle plus the area of the square:

A_rectangle = 10(20) = 200 ft²

A_square = 5² = 25 ft²

A_total = 200 + 25 = 225 ft²

By the way, we think that 5 × 5 area on the right would make a splendid breakfast nook. Just something to consider.

The unit conversion formulas are useful when traveling or reading information from countries that use different, or "silly," units.

The simplification of algebraic expressions will be insanely useful for later material. It's a simplify-now-understand-why-later sort of deal. Getting rid of parentheses and combining like terms often makes formulas easier to work with, especially formulas that we'll use a lot.

• I Like Abstract Stuff; Why Should I Care?

  
  Wait, we thought you just said you liked...oh my. So fickle.

  That's okay, there's also plenty of abstract to go around if that's the sort of thing you're into.

  When we talked about the language of math, we were talking in particular about the language of algebra. Each different area of mathematics has its own dialect with its own symbols, and there are tons of different areas of mathematics.

  Sometimes mathematicians who work in different branches don't understand each other's symbols, and sometimes different areas of math have different meanings for the same word. It's not that strange, really, when you think about it. In England, "pants" means "underwear," and why shouldn't "normal" mean different things to different mathematicians? All we can hope is that most of them are wearing normal pants.

  All areas of math share an underlying idea of a "proof." As do all big-time lawyers, but that's neither here nor there. In math, we start with a collection of statements called a hypothesis (ex: "We hypothesize that you'll read the rest of this paragraph"), and from there we determine that some other statement, called the conclusion, must be true (ex: "You'll enjoy this paragraph so much that you'll move quickly and eagerly onto the next").

  The area of mathematical logic studies reasoning itself. Since it's necessary to make precise statements in order to get good proofs (what is a "side" of a circle?), logic is also concerned with language. Logic has "terms" and "formulas" as well, although they aren't the same as terms and formulas in algebra.

  While we're at it, politicians have terms and formulas, too. They spend a certain number of "terms" in office, and an example of a formula they may use is 8(10,000v) = w, in which the v stands for votes, the 8 indicates ballot-stuffing by 10,000 of the voters, and the w represents an election win.

  Most of the geometric formulas we've shown you were understandable...that is, we were able to explain where the formulas came from. The exceptions were the formulas for the circumference and area of a circle. These are reeeeal hard to explain because they involve π, which people have been trying to understand for millennia. If you ever have a spare millennium, drop us a line and we'll tell you all about it.

  Seriously though, you'll be able to understand the formula for the circumference of a circle when we get to geometry. By the time we get to calculus, you'll be able to pick apart the formula for the area of a circle. Sure, mathematicians have already done that, but knowing that someone has already climbed a mountain won't stop you from reaching the top, will it? If you're not the adventurous type, please don't answer that.

  So far we've only looked at geometric formulas for two-dimensional shapes. As the dimensions become higher, objects become harder to visualize. Three dimensions aren't too bad—a rectangular box is an example of a 3D object—but what about 4 dimensions? Fifteen? Could you have an object with infinitely many dimensions? Even if you did, where in the world would you store it?

• How to Solve a Math Problem

  We've been doing this solving thing all along; this part is a little refresher. If you've been performing different steps or doing them in a backwards order...oopsies.

  There are only three steps to solving a math problem.

  1. Figure out what the problem is asking.
  2. Solve the problem.
  3. Check the answer.

  Finito. That's it.

  Sample Problem

  If M denotes miles and K denotes kilometers, here are the conversion rates between M and K:

  K = 1.609M
  M = 0.6214K

  The distance from Jenny's house to her parents' house is 500 miles. In fact, that was the most attractive selling point. How many kilometers is this?

  Let's work through the steps.

  1. Figure out what the problem is asking.

  We want to find a number in kilometers that represents the same distance as 500 miles.

  2. Solve the problem.

  We need to use one of the two given formulas, and we need to be careful to use the right one. Since the problem gave a number of miles and we want a number of kilometers, we need to use the formula K = 1.609M.

  We substitute 500 for M, and evaluate to get K = 1.609(500) = 804.5 kilometers.

  For Jenny, that sounds even better than 500 miles. She'll need to remember to mention this 804.5 kilometers thing to her mother the next time she explains why she can't make it for Thanksgiving.

  3. Check the answer.

  First of all, is the answer reasonable? We didn't get a ridiculous number of kilometers or get an answer in furlongs, did we?

  We want a distance, so we know the answer had better not be negative. Yep, 804.5 isn't negative, which is reassuring. The number of kilometers it takes to travel a certain distance is bigger than the number of miles it takes to travel that distance—this is something you eventually get a feel for if you pay enough attention to maps—so the answer should be bigger than 500. Yep again, 804.5 is bigger than 500, which is also reassuring. Doesn't it feel good to be this reassured?

  Is the answer right?

  If it's right, then when we substitute 804.5 for kilometers in the second formula, we should calculate 500 miles.

  0.6214(804.5) = 499.9163

  That's about 500. Since most of these conversion formulas aren't exact, we can be fairly confident that we have the right answer. Jenny had better not let her mom see that 499.9163 number, though. Talk about giving her unnecessary ammunition.