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Venn Diagram - Choose One of Three Options

Date: 01/24/99 at 06:44:09
From: Remi
Subject: Venn Diagrams

In a computer science class of 150, it was found that when students 
filled out their option forms each student chose at least one option 
from the list:

   A) Database Design
   B) System Building
   C) Computer Methods

When the registers for each of the options were assembled, there were 
102 names on the database design option, 70 on the system building 
option, and 40 on computer methods. By cross-checking between the lists 
in pairs, it was found that 25 were doing database design and system 
building, 27 were taking database design and computer methods, and 30 
were studying computer methods and system building.

A) Find out how many students were taking all three options by using 
a Venn diagram and all the information provided.

B) Use your completed Venn diagram to find out how many chose formal 
methods only.

I drew 3 circles into which I put all the numbers of the students who 
were doing two courses. For example 30 were doing CM and SB, 27 were 
doing DD and CM, 25 were doing DD and SB. I then used those numbers 
compared with the first set of numbers to find out what those numbers 
would be. For example in the DD circle, there were the numbers 25 and 
27 in half circles (as they were combined with other courses), and I 
knew that the overall total for DD was 102. Therefore, I put 50 in the 
circle as 25 + 27 + 50 = 102. I did this for the SB circle too - but I 
am not sure what goes in the CM circle, as the overall total for that 
circle is 40, but with the combined figures already there it totals 57. 
Do I do an x-17 in there? Or do I do a 3, as that would bring the 
overall student total to 150, as it should be.

Obviously I cannot answer either of the two questions, as I am unsure 
what goes in each circle.


Date: 01/24/99 at 08:28:00
From: Doctor Anthony
Subject: Re: Venn Diagrams

Let D = Design, S = System Building, and C = Computer methods.

Draw the Venn diagram as described and let x = the number who study all 
three. Then the rest of the overlap between D and S is 25-x, between D 
and C is 27-x, and between S and C is 30-x. Thus:

  those in D only = 102 - 25 - (27-x) = 50+x
  those in S only = 70 - 25 - (30-x)  = 15+x
  those in C only = 40 - 27 - (30-x)  = -17+x

From this we can see that x > 17, but to find x we now use the fact 
that the total of all students is 150.

That is:

   102 + 15+x + 30-x -17+x = 150
                   130 + x = 150
                         x = 20  

And so 20 were taking all three subjects.           

Since the Venn diagram is now complete, you can answer any question. 
For example,
   Those taking D only are 50+x = 70
   Those taking S only are 15+x = 35
   Those taking C only are -17+x = 3

- Doctor Anthony, The Math Forum   
Associated Topics:
High School Discrete Mathematics
High School Logic
High School Sets

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