(Not Just) A Geometry Word Problem

If you’re new here, you may want to subscribe to my RSS feed. Thanks for visiting!

A reader recently sent in this problem:

Help, how do you solve this???

The area of a rectangle is 624cm².
The base is 8 less than 5 times the height.
What is the perimeter?

I can not find out how to do it on line. I have a number of similar problems to solve
thanks.

- A. Reader.

Professor Homunculus replies:

Hi, Reader,

I must say that it is a good thing that it is not taught online. Specific problems should never be shown online. That would be show-and-tell, not teaching.

What you need to learn is the concepts behind the problems, then you’ll be able to crack all problems that are similar. One of the concepts is a very interesting and important part of algebra. 

Here’s a way to start figuring out how to solve your problem:

First, you have to know the formula to find the area of a rectangle, using the base and the height.

Google it. I won’t be putting up a lesson on it, because it is in about every math book and website there is. Just keep looking for one that makes sense to you. They exist.

Call the base “b,” and the height “h“.
Now you have to make a little equation, showing the relationship of one to the other, using only one variable (either b or h).

You know the base is 8 less than 5 times the height. So the equation could be b = 5h - 8.

Now write the formula for the area of a rectangle. For the height, put in h. For the base, put in 5h – 8.
For the area, well, you know what the area is (you told it to me). So put that in.

Then solve the equation you have just created by substituting those things for the base, height, and area in the formula.

You will then know the base and the height of your rectangle. Add all the sides together, and you’ll have your perimeter.

So far it sounds pretty straighforward, right?

Here’s where you need to understand a very important math concept. It is quadratic equations. This is pretty much the hint you were looking for. 

Look at the problem once more:

b= 5h - 8

624 = b * h

Substitute 5h - 8 for b in the above equation, and get:

624 = h(5h-8)

Multiply (5h-8) by h and get:

624 = 5h²- 8h

Subtract 624 from both sides.

0 = 5h² - 8h - 624

Does this look familiar?
If not, switch the terms on each side of the equals sign, and look at it this way:

5h² - 8h - 624 = 0

If this still does not look familar, see if this form of the equation rings a bell:

ax² + bx + c = 0

If you do not recognize this as a quadratic equation, then you are in over your head. You really have to know what quadratic equations are, and how to solve them, to be able to solve this kind of question consistently. There are empirical methods, with trial and error, that will help you solve some of these kinds of problems (that’s the way I’d do this one, intuitively) but you cannot count on them to solve all solvable equations of this kind.

Fear not! Quadratics are not miserably hard (they are just usually taught that way.)
Unfortunately, this is not the simplest form of quadratic. That’s because a is larger than 1 and c is pretty large. You probably won’t be able to solve this by factoring (unless you have a lot of time on your hands), but you will be able to do it with the quadratic formula. There is a good lesson on exactly how to solve this at:
http://www.purplemath.com/modules/solvquad4.htm

Solving quadratics is not within the scope of MathMojo at this time, but being able to see how to approach a problem which involves quadratics is. Now that you (maybe) see how this problem can be approached, go get the details of how to solve quadratic equations

Warning:

If you do not understand how we approached the problem, so far, do not waste your time by thinking you just need some formula for each equation, and that someone should tell you what formula to use. You must understand, by yourself, which formula to use. Math isn’t about parroting things back (although, admittedly, stupid math tests are.)

Learn about the equations, practice a lot of given examples, then slowly learn to recognize how to use them for examples which you, yourself, have noticed fit the pattern. Anything else is not only a waste of time, it makes people think they understand things that they don’t. That is how the world tends to get into the messed up shape it is in.

One way to avoid problems is to get a good book that explains things well (I can recommend “Algebra the Easy Way”) and another one that’s full of good examples (with answers to at least some of them in the back. Go to a library and find one) and practice until you get that “AHA!” feeling (you know - the one where the light bulb goes off in your head.)

There is more than one way to solve quadratic equations

With quadratics, you should learn at least three ways to solve them - one with the formula, one with factoring, and at least one other (with a graphing calculator, maybe. This is one of the few times that MathMojo will suggest using a calculator, but only after you have learned how to solve quadratic equations by factoring, and with the quadratic formula.) There many ways to solve quadratics equations. Learning only one means you haven’t really learned any.

Why word problems sometimes seem hard to crack

The big problem with word problems is that teachers often ask you to solve them without making sure you understand the formulas first. When you google the formula for finding the area of a rectangle using the base and the height, make sure you understand it. Don’t just memorize it. If the site doesn’t explain it in a way that makes sense, google further.

Look at it, make a sketch of what you think it is trying to say, using graph paper. Make sure you understand why the formula works. If your teacher won’t help you with it, get him fired. (I’m only just partially kidding, here).

And of course, make sure you thoroughly know what the terms base, height, area and perimeter mean, and how they are measured (in units like inches or centimeters, etc, or in units like square inches or centimeters, etc. )

Of course, the same goes for understanding quadratic equations. Someday I’d like to make a nice, interactive lesson about them for MathMojo, but I’m so behind in simple things, like lessons in advanced addition and multiplication, that I know it will be a long time from now. 

Anything further I would tell you about it wouldn’t be help. It would be show-and-tell, and that is a way not to learn.

Please let me know how you did.

Yours truly,   

Professor Homunculus

P.S. Remember, respect your brain at all times. You can learn to tackle stuff like this.