Are you one of those people who used to zone out during algebra class because you thought you’d never have a use it in real life” Maybe you scraped by with a C average, or you possibly managed to get a B because you copied your best friend’s homework. If so, then you are one of the Algebra Allergic. Guess what” The Doctor is in, and we’re here to make it all better. We’re not saying that we can completely clear up your hives, but we can surely keep you from slipping into anaphylactic shock.
Now don’t worry. We’re not going to talk about irrational or transcendental numbers or crazy stuff like that, so you can start breathing normally again. We are only going to address the most basic of algebraic problems: having one unknown number. You probably already use some of these techniques, but you just may not know that you are doing it. If you understand what you are really doing, then it will make the more complicated problems down the line seem much less so.
Let’s look at our first example, shall we”
We want to paint the four walls of a room, but we don’t know how much paint to buy. Let’s say the room is rectangular, and it’s 20 foot (commonly written as 20’) wide, 24’ wide and 12’ tall. For the purposes of simplicity, let’s assume it has no doors and no windows. This is one big, boring, creepy room, but it works for our purposes. We want to paint this freakish room with a fresh coat of Goth black paint because we feel like it. So how do we estimate how much paint to buy” Let’s break it on down:
We need to figure out the surface area of the walls, i.e. the amount of surface we need to cover with paint. (Yes, we’re slipping into a little geometry here, but it’s the most basic of the basic.) To figure out an area of a rectangle – which is what each wall is – we multiply the distances of the sides (Note: Square feet will be designated as “sf”.):
This 1056sf of surface area needs to be covered with the Goth black paint. Now that we’re finished with the geometry, let’s get to the basic algebra…
Most estimates say that one gallon of paint will cover 250 sf to 400 sf (we’ll say 325 sf) on the first coat and about 400 sf to 600 sf (we’ll say 500 sf) on the subsequent coats. Let’s assume that we’re applying our black paint over white paint, so we need major coverage: say 3 coats. Here we go (Note: Our unknown we’re going to designate as “a”.):
So you need 7.47 gallons of Goth black paint, which you would then round up to 8 gallons because the paint store isn’t going to mix you 7.47 gallons of paint.
Let’s take another example of having one unknown… when you are looking for a ratio or a proportion. The basic relationship of a ratio is represented as:
Percentages are a common example of how you could use this concept.
Say that local fire code says that only 15% of a wall along a particular exit corridor can be glass. You are helping your client remodel a portion of his space, which currently has a door is 3’ wide and 9’ tall, and a sidelight of the same dimensions. For budgetary reasons, these have to remain. Therefore, you have:
In order to comply with the code, how long can the wall be” First you must figure out the surface area. (Back to geometry again.) This is where the ratio comes in (Note: Keep in mind that 15% is 15 out of 100):
The easiest thing to do here is to cross-multiply, which gives you:
So then you isolate the unknown “a”:
Therefore, if your ceiling height is 10’, you that wall can be 36’ long because…
Now let’s say that you need to reverse this process because your client has changed his mind (which clients often do). In this version of the remodel, the wall length must remain, but your client wants to change the current solid wood door to glass and hopefully add a sidelight. The wall is 25’ long and the ceiling height is (still) 10’ high, so you have 250sf of wall. If the code says you can only have up to 15% of glass, then…
We do our groovy cross-multiplication, which gives us:
We then you isolate the unknown “a”:
The door needs to be 3’ wide to meet accessibility requirements, and for our own aesthetic preferences, we, as the designer, have decided to make the sidelight 1/2 the width of the door, i.e. 1.5’ wide. The client wants the tallest door he can have, so how do we figure out how tall that can be” With algebra, of course! Here’s how you set it up:
Since 9.375’ (or 9’-4.5”) is not a standard door height, and we don’t want to mess with custom products because of budget constraints, so we would probably round that off to a 9’ height for both the door and the sidelight. This height puts us comfortably under the 15% maximum, so the Fire Marshal will pass us, our client will love us, and we can sleep comfortably tonight.
See” That wasn’t as hard as you thought, was it” Believe us – you need to understand these basic principles. You may not use them every day, but you will use them more often than you think.