Problem Solving with a Venn Diagram

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Problem Solving with a Venn Diagram
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2
Try to solve this.
In a class of 30 students, 21 belong to a
sports team, 16 belong to the band and 4
belong to neither. How many students belong
to both the band and a sports team?
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3
One method is to use a Venn Diagram.
We will use a rectangle for the whole class,
a circle for those that belong to a sports team,
and another circle for those that belong to the
band.
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Where are the 4 students that belong to neither a
sports team nor a band?
Where are the students that belong to both a
sports team and a band?
The section that we labeled Both is called the
intersection of Sports and Band.
Neither
Both
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Lets put some numbers in.
Problem In a class of 30 students, 21 belong
to a sports team, 16 belong to the band and
4 belong to neither. How many students belong
to both the band and a sports team?
We can see from the Venn Diagram that some of the
21 students who belong to a sports team also
belong to the band, and some of the 16 students
who belong to the band also belong to the sports
team. The problem is to find out how many are in
this intersection called Both.
(30)
(4)
Neither
(16)
(21)
Both
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How many students belong to either a sports team,
the band or both? That is how many students are
in the circles below.
There are 30 students in the class, four of which
do not belong to either a sports team or the
band. So 26 must belong in the circles.
(30)
How many of these are in the section labeled
Both?
(4)
Neither
(16)
(21)
Both
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26 students belong in the combined circles. But
when we add 21 sports members plus 16 band
members, we get 37 students. This is 11 (37
26) extra students. These 11 are the students
who belong to BOTH a sports team and the band.
This means that 21 11 10 students belong ONLY
to a sports team.
And that 16 11 5 students belong ONLY to the
band.
(30)
(4)
Neither
(16)
(21)
Both
10
5
11
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Lets make sure that our students add up. We
have 11 who belong to BOTH a sports team and the
band plus 10 who belong ONLY to a sports team.
These add up to the 21 who belong to a sports
team. We have 11 who belong to BOTH the band and
a sports team and 5 who belong ONLY to the band.
These add up to the 16 who belong to the band.
(30)
(4)
Neither
(16)
(21)
Both
10
5
11
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Lets make sure that the whole class add up.
Continuing our count, we have 11 who belong to
BOTH a sports team and the band, 10 who belong
ONLY to a sports team, 5 who belong ONLY to the
band, and 4 who belong to neither. 11 10 5
4 30 which is the total class.
(30)
(4)
Neither
(16)
(21)
Both
10
5
11
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You will see some similar problems in your
exercises. Please try using a Venn diagram when
your solve them.
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