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How Many Stools? How Many Tables?

Date: 12/13/2002 at 09:12:12
From: Shakyra Aughtry
Subject: I don't know how to do math

In their spare time, the Buffalo Bills operations staff members build 
3-legged stools and 4-legged tables.  Last month they used 72 legs to 
build 3 more stools than tables.  How many stools and how many tables 
did they build?


Date: 12/13/2002 at 12:03:15
From: Doctor Ian
Subject: Re: I don't know how to do math

Hi Shakyra,

A lot of people who can't 'do math' have trouble because they think of
math as something that's supposed to be different from regular
thinking. But it's not, really. 

Let's look at this problem, for example. If we know the number of
stools, then we also know the number of tables, and vice versa. Do you 
see why? If the number of tables is 5, then the number of stools has 
to be 3 more than that, or 8. Or we can go in the other direction: If 
the number of stools is 5, the number of tables has to be 3 less than 
that, or 2.  

That's not really 'math', is it? It's just thinking about what the
problem says.  

If you know how many stools and tables they made, then you can figure
out how many legs they used:

  number of    number of                 number of
  legs used  = stools     * 3     +      tables     * 4

           
               numbers of                number of
             = stools     * 3     +     (stools    - 3) * 4

So it's easy to go from the number of stools (or tables) to the number
of legs.  But the problem tells you the number of legs, and asks you
to work out the number of stools (or tables). 

I'm going to use the letter S to represent the number of stools, just
so I don't have to keep writing so much:

   72 = S*3 + (S - 3)*4

Now you need to find a value of S that makes the equation true. 
Believe it or not, one way to do it is by guessing. Suppose I guess
that the number of stools is 5.  Then I get

  72 = 5*3 + (S - 3)*4

     = 15 + 3*4

     = 15 + 12

     = 27

which is too small. It's also kind of a drag to evaluate a guess, so
here's about the only place where I'll use some 'math', and that's to
simplify the expression:

     S*3 + (S - 3)*4

I'll start by applying the distributive property, 

  a(b + c) = ab + ac      or        a(b - c) = ab - ac

to get rid of the parentheses. Why do I want to do that? Because I'd
like to get all my S's together, if possible, and I have at least one
trapped inside the parentheses. To get it out, I have to use the
distributive property:

  = S*3 + (S*4 - 3*4)

  = S*3 + S*4 - 12

Now I can use the distributive property again, to combine the terms
with S's:

  = S*3 + S*4 - 12

  = S(3 + 4) - 12

  = 7*S - 12

This is much easier to deal with. So my equation is now

  72 = 7*S - 12

At this point, we can guess again:

  S = 8       72 = 7*8 - 12

                 = 56 - 12

                 = 44             (too small)


  S = 20      72 = 7*20 - 12

                 = 140 - 12

                 = 128            (too large)

So whatever the number of stools is, it's greater than 8, and smaller
than 20.   

And we can keep guessing, or we can try to think our way to an answer.  
Suppose the equation says

  72 = something - 12

When we put it this way, it seems pretty clear that 'something' must
be 84.  Does that make sense?  But the 'something' is just 7 * S, so
we can say 

  84 = 7 * S

So now we can find the number of stools directly, and subtract 3 from
that to get the number of tables. 

Were you able to follow all of this?  If not, please write back and
tell me the first part that didn't make sense.
 
The important point to keep in mind is that when you're trying to
solve a problem, even if you can't think of 'the right way to solve
it', that's no reason to give up. There are lots of ways to solve any
problem, and if you have to count on your fingers, or draw pictures,
or use pennies and nickels to represent stools and tables, then that's
perfectly fine. As you get better at using simple methods, you'll
naturally start looking around for more complicated ones.  (It's a
little like when you were a small child, and you got tired of
crawling, so you started trying to walk. But you had to crawl before
you could walk!) 

Some methods are faster than others, but _any_ method that solves the
problem is 'the right one', if you understand how it works.  

I hope this helps!

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Algebra
Middle School Word Problems

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