Interpreting Intersections

Date: 07/09/2002 at 00:57:24
From: Jay Bunag
Subject: Intersection of Sets

Hi Dr. Math,

I have been struggling with this Math problem from my brother. The 
problem is written below.

Given that there are 100 people in the universal set. 10 people 
drink Coke, 15 drink Sprite, and 20 drink Pepsi.  How many people 
are drinking both Coke and Pepsi at the same time?  Coke and Sprite? 
Sprite and Pepsi?  All three drinks? 

I tried getting the percentages and multiplying them to get the 
intersections. Is this the right way to do it?

Thanks,
Jay

Date: 07/09/2002 at 12:55:23
From: Doctor Peterson
Subject: Re: Intersection of Sets

Hi, Jay.

Is there some extra information you left out?

First, there probably are NO people drinking both Coke and Pepsi at 
the same time; that would require two straws and peculiar taste. I 
suspect you really mean, how many people LIKE (that is, are 
_willing_ to drink) both Coke and Sprite. But there is no way of 
knowing that without further information. 

There might be 10 who drink ONLY Coke, 15 who drink ONLY Sprite, and 
20 who drink ONLY Pepsi (and 55 who drink nothing):

     __10__  ____15____  ______20______  ____________55____________
    /      \/          \/              \/                          \
    | Coke  |  Sprite   |     Pepsi     |         nothing           |
    \______/\__________/\______________/\__________________________/

Or there may be 10 who drink ANYTHING, 5 who drink anything EXCEPT 
Coke, and 5 more who drink ONLY Pepsi, while 80 drink nothing:

     ______20______  _____________________80______________________
    /    Pepsi     \/                                             \
   | ____15____     |                                              |
   |/  Sprite  \    |                                              |
   | __10__     |   |                                              |
   |/ Coke \    |   |                                              |
   |\______/    |   |                                              |
   |\__________/    |                                              |
    \______________/\_____________________________________________/

Or there could be any combination in between. So we just don't have 
enough information.
      
If you have any further questions, feel free to write back.

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