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,
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
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"
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" 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".
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.
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"
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 me:
chewed on: 15
– 2 – 1 – x = 12 – x
– 2 – 1 – x = 9 – x
There were a total of
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 University.
There is also a software
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.