Venn Diagram - Fast Food Restaurants

Date: 05/09/2002 at 01:42:52
From: Tim
Subject: kind of probability, I think it needs a Venn diagram

The following data was obtained from the fast food restaurants in a 
city:

 13 served hamburgers
  8 served roast beef sandwiches
 10 served pizza
  5 served hamburgers and roast beef sandwiches
  3 served hamburgers and pizza
  2 served roast beef sandwiches and pizza
  1 served hamburgers, roast beef sandwiches and pizza
  5 served none of the three foods

How many fast food restaurants are there in the city?

Thank you.
Tim

Date: 05/09/2002 at 09:40:59
From: Doctor Rick
Subject: Re: kind of probability, I think it needs a Venn diagram

Hi, Tim.

I wouldn't call this probability; yes, it calls for a Venn diagram. 
There are three intersecting sets: restaurants that serve hamburgers, 
restaurants that serve roast beef, and restaurants that serve pizza. 
Draw the 3 circles (I'll use rectangles because they are easier to 
draw with the keyboard):

  +----------------------+
  |H                     |
  |           +----------+--------------+
  |           |b         |             R|
  |           |          |              |
  |     +-----+----------+------+       |
  |     |a    |d         |      |       |
  |     |     |          |      |       |
  +-----+-----+----------+      |       |
        |     |                c|       |
        |     +-----------------+-------+
        |                       |
        |P                      |
        +-----------------------+

Now, I have a problem with this sort of problem: it is not clear to 
me whether, for instance, "3 served hamburgers and pizza" means that 
these 3 restaurants served hamburgers and pizza but NOT roast beef 
sandwiches, or whether it is only commenting on whether they served 
hamburgers and whether they served pizza, ignoring any other foods.

If it means the former (they didn't serve roast beef), then this 
number goes in the section I marked a, and you have enough 
information to put a number in each section of the Venn diagram 
(including outside all the circles). The total number of restaurants 
is then simply the sum of all the numbers. That's too simple, so 
applying my "meta-problem-solving" skills (figuring out what a 
problem is *supposed* to be based on surmises about its purpose), I 
think that the 3 restaurants serving hamburgers and pizza is meant to 
be the number in the intersection of H and P (that is, the sum of the 
numbers in sections I marked a and d).

Using the information given, you can write a set of equations such as

  a + d = 3

You'll find that if you work your way up the list of data, you'll 
always have enough information to determine the number in one of the 
sections. You may not even need to write the equations, once you get 
the idea.

Once you have solved for all the numbers that go in the individual 
sections, you can add them up, as I said above, to get the total 
number of restaurants.

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