How Do You Figure Out the Equations for Word Problems?

Date: 04/09/2006 at 19:58:07
From: Marissa
Subject: How do you make equations out of any word problems?

I am so messed up with word problems.  I can solve the equations, I
just can't set them up correctly.  If someone gives me the right
equations I can do it.  For example:

Mrs Johnson is five times as old as her son.  Three years ago she was
eight times as old as her son then.  Find each of their ages.  I put
5s(-3 + 8s), but I'm soooo confused.  Please help!

Date: 04/09/2006 at 23:13:12
From: Doctor Peterson
Subject: Re: How do you make equations out of any word problems?

Hi, Marissa.

A lot of students have trouble with this; I see them all the time even
at the college level.  What you're having trouble with is translating
from English into algebra, and I think it helps a lot to actually
think of it as translating from one language to another.  To do that,
you start with a "dictionary".

We first need to define a variable; you seem to have done that, but
it's helpful to state your definition clearly:

  Age of son = s

Now, we have other phrases or quantities in the problem, and we can
write an expression for each phrase, gradually building up to a full
equation.  First, how old is Mrs. Johnson?

  Mrs. Johnson is five times as old as her son

means

  Age of Mrs. Johnson = 5s

Now we come to the big, complicated sentence:

  Three years ago she was eight times as old as her son then.

We can break this down into several parts; I'll rewrite it with an
emphasis on the numbers involved:

  her age 3 years ago = eight times her son's age 3 years ago

Now, we need to add these phrases to our dictionary:

  her age 3 years ago = 

  her son's age 3 years ago = 

  eight times her son's age 3 years ago = 

Can you start to see how to fill these in? Three years ago, both of
them were 3 years younger, which means we subtract 3 from their
current ages.  Then we can multiply by 8:

  her age 3 years ago = 5s - 3

  her son's age 3 years ago = s - 3

  eight times her son's age 3 years ago =  8(s - 3)

Now, the sentence just says that two of these quantities are equal:

  5s - 3 = 8(s - 3)

With experience, you can write an equation like this directly, but at
first it will be a good idea to take it very slowly and build it up
piece by piece this way.  I like to look at the overall meaning of a
sentence first (which is what a language translator does--he doesn't
just translate one word at a time, because different languages have
different word order).  By rewording the sentence to make its meaning
clearer, I was able to change it into algebra more easily.  But still
I had to take my time!

If you have any further questions, feel free to write back.  I'd like
to hear whether this helps, and perhaps see you try another example.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/