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. If five students are in both classes,
how many students are in neither class? How many are in either class? 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:
Since five students are taking both classes,
I'll put "5" in the overlap:
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:
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"
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.
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:
Two were gray and tailless but not chewed
on, so "2" goes in the rest of the overlap between "gray"
Two were gray and chewed on but not tailless,
so "2" goes in the rest of the overlap between "gray"
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.
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.
I do know the total number of chewed geckoes
(15) and the total number of tailless geckoes (12). This gives
only chewed on: 15 – 2 – 1 – x = 12 – x
only tailless: 12 – 2 – 1 – x = 9 – x
There were a total of 24 geckoes for the
month, so adding up all the sections of the diagram's circles gives me:
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,
page provided by Joe Kahlig of Texas A&M
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.