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Setting up Proportions

Date: 03/11/2010 at 22:18:41
From: allie
Subject: Advice on word problems

I've been having a hard time answering word problems. Say there's 10
notebooks and you bought 3 for $1.89. What is the cost of 10 
notebooks?

On a test, I got the answer wrong. I'll show you what I did wrong. I
THINK it went like this:

   10 over $1.89 = say the variable is n over $1.89

I find setting up difficult. I don't understand anything about setting
up word problems. I've always had problems with this. I'm not sure why,
though.

If you can just give me advice on how to set up word problems or tips
or even a very good website, that will be much appreciated. I have
another test Very Soon.

I'm in the 7th grade in smart math.

Thank you,
Allie



Date: 03/11/2010 at 23:29:04
From: Doctor Peterson
Subject: Re: Advice on word problems

Hi, Allie.

I think what you most need at the moment is some advice about working
with proportions, because that is where you are going wrong. Perhaps
out of this I'll have some more general comments to make that would
apply to other kinds of word problems.

This kind of proportion word problem depends on setting up the
proportion properly. The key is that a proper proportion looks like a
table of data, where each row is consistent and each column is 
consistent.

In this case, you have costs, and you have numbers of books; and you
have two cost/number pairs. A table of the data might look like this:

            what you       what the
             bought      question asks
            --------    --------------
  number:      3             10

  cost($):   1.89             ?

Each column represents one case given in the problem: what you
actually bought, or what you are asking about. Each row represents one
kind of quantity: the number of books bought, or the cost of the 
books.

The proportion has to work the same way: the numerators have to be
corresponding things, and the denominators have to be corresponding
things, and each ratio has to go together.

So the proportion can be written directly from the table:

    3     10    (number)
  ---- = ----
  1.89     x    (cost)

Do you see the connection to my table? The first ratio is two numbers
that go together, and so is the second. In each case, the numerator is
the number and the denominator is the cost.

Now, this isn't the only way to write the proportion; it's just the
way my table turned out. Possibly, a teacher would prefer that you make
it so each ratio compares two numbers of the same kind (cost or
number), and the numerators correspond to one another, and the
denominators correspond to one another. It might look like this:

  (cost) (number)

   1.89     3     (what you bought)
   ---- = ----
     x     10     (what the question asks)

So here we are saying that the ratio of costs is equal to the ratio of
numbers; and each ratio has the same order (bought : question). 

Either of these (and several others) will work just as well; in fact,
if you solve them by cross-products, you'll have the same equation to
solve:

  3x = 1.89*10

Now let's look at what you did, which I think you said was this:

   10      n
  ---- = ----
  1.89   1.89

If that's what you meant, it's pretty clearly wrong because the 3
doesn't show up anywhere, and 1.89 is in there twice.

But let's suppose you wrote this:

   10      n
  ---- = ----
  1.89     3

You can check it for consistency by looking at each "row" and each
"column." Let's just replace each number with what it means:

  number in question   cost in question
  ------------------ = ------------------
  cost as bought       number bought

Do you see what's wrong? The left has number over cost, but the right
has cost over number. The numerators go together and the denominators
go together, at least! If you make this check on your work and find
it's wrong, just rearrange the numbers so they do line up right, and
then check it again. Once it's set up right, you're ready to solve the
proportion.

Does that help? Try a few example problems using this way of thinking,
and show me your work on a few more if you'd like me to check them.

I see one big principle that applies to any problem solving,
especially of an algebraic type: You have to begin by looking for
relationships among the quantities in a problem, and make sure that
what you write algebraically represents those relationships
accurately. Making a table is just one way to help yourself focus on
that idea.

Finally, here is a link to our FAQ on Word Problems, which may give
you some other useful ideas:

  Word problems
    http://mathforum.org/dr.math/faq/faq.word.problems.html 

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Ratio and Proportion

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