Translating Words into Math

Date: 8 Mar 1995 22:51:36 -0500
From: Anonymous
Subject: Algebra.

Dear Dr. Math,
     I need help with my Algebra.  We are doing story problems.  
I DON'T want you to do them for me.  I just need help on
understanding them so that I can solve them by myself.  I am 
stuck on one particular problem.

The number is decreased by 1/3 of the number.  The result is 62.  
Find the number.  Please help me. 

                                 Brittany I

Date: Thu, 09 Mar 1995 12:57:09 +0000
From: Dr. Math
Subject: Re: Algebra.

Hello there!

I think the hardest thing about doing these problems is translating
sentences into mathematical equations.  Basically, you have to pick 
them apart, piece by piece, translating one part at a time until what 
you have left is an equation that you can solve using Algebra.  So 
let's dig in.

Actually, I'll make up a new problem that's similar to yours, and I'll
lead you through the translation process on that one.  Here it is:  

A number is increased by 3/4 of the number, and the result is 28.  
What is the number?

One of the first things we want to do is to replace "the number" by 
a variable.  You can call it anything you want, and in this case I'll 
call it x.  So we go through the sentence and wherever we see the 
phrase "the number" or "itself" or something that obviously refers 
to the number we're talking about, we replace that by an x.  So we get:

x is increased by 3/4 of x, and the result is 28.  What is x?

Also, one of the first things you want to do in these sentences is 
find out where the equal sign goes.  Typically, you'll look for the 
verb "is," or "the result is," or if they're really throwing it out at 
your face, "is equal to."  In our problem, we find our equal sign in 
"and the result is":

x increased by 3/4 of x = 28. 
(I changed the grammar just a teency bit)

Now what does 3/4 of x mean?  It means we multiply 3/4 times x.  
So now we have

x increased by 3/4 * x = 28.

Now there's only one thing left.  We need to figure out what 
"increased by" means.  What would I mean if we were baking 
cookies, and I said to you "increase the amount of chocolate 
chips by two cups"?  I'd mean that you would add two extra 
cups of chocolate chips to the batter, i.e. ADD two cups, and 
we'd both have a stomach ache.  So the phrase "increased by" is
probably going to mean "plus."

x + 3/4 * x = 28.

And now we have a bona fide Algebraic sentence.  Notice the 
period at the end.  Usually people don't write those, but I 
wanted to show you that even when we write down something 
as weird as "x + 3/4 x = 28", it can always be read as a real 
English sentence, with punctuation at the end and everything.

So see if you can apply the same techniques to solve your 
problem.  If you still have trouble, write back!

-Ken "Dr." Math