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Venn Diagram Word Problems (page 4 of 4)

Sections: Introduction, Logic and set notation, Set-notation exercises, Venn word problems


Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance:

Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.

    a) If five students are in both classes, how many students are in neither class?
    b) How many are in either class?
    c) What is the probability that a randomly-chosen student from this group is taking only the Chemistry class?

    There are two classifications in this universe: English students and Chemistry students.

    First I'll draw my universe for the forty students, with two overlapping circles labelled with the total in each:

    Venn diagram with 'English' and 'Chemistry' circles

    Since five students are taking both classes, I'll put "5" in the overlap:

    the overlap contains a '5'

    I've now accounted for five of the 14 English students, leaving nine students taking English but not Chemistry, so I'll put "9" in the "English only" part of the "English" circle:

    the English-only part of the 'English' circle contains '9'

    I've also accounted for five of the 29 Chemistry students, leaving 24 students taking Chemistry but not English, so I'll put "24" in the "Chemistry only" part of the "Chemistry" circle:

    the Chemistry-only part of the 'Chemistry' circle contains '24'

    This tells me that a total of 9 + 5 + 24 = 38 students are in either English or Chemistry (or both).  This leaves two students unaccounted for, so they must be the ones taking neither class.

    there is a '2' inside the universe but outside the circles

    From this populated Venn diagram, I can get the answers to the questions.

      Two students are taking neither class.
      There are
      38 students in at least one of the classes.
      There is a
      24/40 = 0.6 = 60% probability that a randomly-chosen student in this group is taking Chemistry but not English.

  • Suppose I discovered that my cat had a taste for the adorable little geckoes that live in the bushes and vines in my yard, back when I lived in Arizona. In one month, suppose he deposited the following on my carpet: six gray geckoes, twelve geckoes that had dropped their tails in an effort to escape capture, and fifteen geckoes that he'd chewed on a little. Only one of the geckoes was gray, chewed on, and tailless; two were gray and tailless but not chewed on; two were gray and chewed on but not tailless. If there were a total of 24 geckoes left on my carpet that month, and all of the geckoes were at least one of "gray", "tailless", and "chewed on", how many were tailless and chewed on but not gray?
  • If I work through this step-by-step, using what I've been given, I can figure out what I need in order to answer the question. This is a problem that takes some time and a few steps to solve.

    There was one gecko that was gray, tailless, and chewed on, so I'll draw my Venn diagram with three overlapping circles, and put "1" in the center overlap:

    three circles, with '1' in the center overlap

    Two were gray and tailless but not chewed on, so "2" goes in the rest of the overlap between "gray" and "tailless".

    '2' is in the remaining overlap between 'gray' and 'tailless'

    Two were gray and chewed on but not tailless, so "2" goes in the rest of the overlap between "gray" and "chewed-on".

    '2' is placed in the remaining overlap between 'gray' and 'chewed-on'

    Since a total of six were gray, and since 2 + 1 + 2 = 5 have already been accounted for, this tells me that there was only one left that was only gray.

    '1' is placed in the remainder of the 'gray' circle

      
    This leaves me needing to know how many were tailless and chewed on but not gray, which is what the problem asks for. Since I don't know how many were only chewed on or only tailless, I cannot yet figure out the answer.
       

    I'll let "x" stand for this unknown number of tailless, chewed-on geckoes.

    'x' is placed in the remainder of the intersection of 'tailless' and 'chewed-on'

    I do know the total number of chewed geckoes (15) and the total number of tailless geckoes (12). This gives me:

      only chewed on:  15 2 1 x = 12 x

    '12 – x' is placed in the diagram

      only tailless:  12 2 1 x = 9 x

    '9 – x' is placed in the diagram

    There were a total of 24 geckoes for the month, so adding up all the sections of the diagram's circles gives me:

      1 + 2 + 1 + 2 + x + (12 x) + (9 x) = 27 x = 24

    Solving, I get that x = 3.  Copyright Elizabeth Stapel 1999-2009 All Rights Reserved

      Three geckoes were tailless and chewed on but not gray.

(Advisory: No geckoes or cats were injured during the production of the above word problem.)


For lots of word-problem examples to work on (plus explanations of the answers), try this page provided by the Oswego City (New York) School District, or this page provided by Joe Kahlig of Texas A&M University. There is also a software package (DOS-based) available through the Math Archives which can give you lots of practice with the set-theory aspect of Venn diagrams. The program is not hard to use, but you should definitely read the instructions first.

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Cite this article as:

Stapel, Elizabeth. "Venn Diagram Word Problems." Purplemath. Available from
    
http://www.purplemath.com/modules/venndiag4.htm. Accessed
 

 



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