Alternate Solution Path to Venn Diagram Problem

Date: 05/12/2009 at 16:55:36
From: Autumm
Subject: word problem-finding a missing number

There are 30 students in a math class.  Twelve belong to the computer
club, and eight belong to the photography club.  Three belong to both
clubs.  How many belong to neither club?

I know that the correct answer (according to the book) is 13.  I can't
come up with that answer.  I have either 10 or 7 as my answers, not
knowing if the "both club" members are included already in the listed
number of members.

This question is listed in a section introducing proportions.  I 
can't figure out a way to come up with the correct answer or use a 
proportion to solve it.  Sometimes when I know the correct answer I 
can work backwards to solve it, but I can't do that with this 
problem.  It is driving me CRAZY!

Date: 05/12/2009 at 17:02:59
From: Doctor Ian
Subject: Re: word problem-finding a missing number

Hi Autumn,

This sort of problem is often solved using a Venn diagram.  You can
read about them at these locations:

  Wolfram Mathworld: Venn Diagram
    http://mathworld.wolfram.com/VennDiagram.html 

  Venn Diagram: Fast Food Restaurants
    http://mathforum.org/library/drmath/view/60772.html 

But here's another way to organize the data you have and solve the
problem.  Suppose we make a table like this:

               photography      ~photography
            +---------------+-----------------+
   computer |               |                 |
            +---------------+-----------------+
  ~computer |               |                 |
            +---------------+-----------------+

(I'm using "~" as an abbreviation for "not".)

Can we agree that everyone in the class must be in EXACTLY one of
these boxes? 

We're told that there are 30 students.  So the numbers in the boxes
have to add up to 30. 

We're told that 3 students belong to both clubs:

               photography      ~photography
            +---------------+-----------------+
   computer |      3        |                 |
            +---------------+-----------------+
  ~computer |               |                 |
            +---------------+-----------------+

Now, if 12 students belong to the computer club, and we've already
accounted for 3 of them, there must be 9 who are in the computer club,
but NOT the photography club, right? 

               photography      ~photography
            +---------------+-----------------+
   computer |      3        |       9         |  <- These have to
            +---------------+-----------------+     add up to 12
  ~computer |               |                 |
            +---------------+-----------------+

Can you use similar reasoning to fill in the lower left box, given
that 8 students belong to the photography club? 

And once you know that, you know how many belong to neither club.

Does this make sense? 

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

Date: 05/12/2009 at 21:05:44
From: Autumm
Subject: Thank you (word problem-finding a missing number)

Thank you so much.  This makes perfect sense and now I feel silly.  I
was so stuck on the numbers that I couldn't think beyond to get over
the hump.  Great visual and clues!  I appreciate your help.