The document provides information and steps to complete a three circle Venn diagram representing survey results from a class of 40 students on their pet preferences of cats, dogs, and birds. It gives some values to place directly on the diagram and other information to help determine missing values. It works through placing the known values and using the additional information to logically determine the remaining unknown values step-by-step to fully complete the Venn diagram.

1. Image purchased by Passy’s World from istockphoto

2. Three Circle Venn Diagrams can take quite a bit of working out.
The steps to follow are generally these:
Get the Information from the Question that can go straight onto the
Venn Diagram and place it there.
Work through the remaining information, a bit at a time, to work out
each of the missing values.
Work from the centre of the diagram outwards when finding these
unknown values.
Check that all the numbers on the final diagram add up to the
E = everything grand total.
We will start by reading a completed three circle diagram.

3. Dance 20
14 5 Rock
8
2 16
5
RAP
Free ClipArt Images were obtained from Google Images

4. Question 1 – Total People
Dance 20
What is the Total
Number of People
represented in this
Diagram ?
14 5 ANSWER: Add up
all the numbers on
8 the diagram and
the total is 70 .
2 16
5
RAP

5. Question 2– Rock People
Danc
What is the Total
Number of People 20
e
who like Rock
music ?
ANSWER: Add up 14 5 Rock
all the numbers in
the Rock Circle, 8
and the total is 31 .
2 16
5
RAP

6. Question 3 – Probability
Dance 20
If one person is
chosen at random,
what are the odds
of picking a person
who likes Rock ?
14 5
ANSWER: 31 out
8 of 70 people liked
Rock, so odds are:
2 16 Pr(Rock) = 31 / 70
5 or 31 out of 70.
RAP
Rock

7. Question 4 – All Types
How many of the
People like all 20
Dance
three types of
music : Dance,
Rock, and Rap ?
14 5
ANSWER: 8 people
as shown in the 8
diagram’s centre,
where all three 2 16
circles overlap. 5
RAP
Rock

8. Question 5 – Probability
If a person is
chosen randomly,
Dance what are the odds
that they like all
three types of
20 music ?
ANSWER: 8 people
out of the total 70
14 5 liked all three, so
odds are 8 / 70.
8
2 16
5
RAP
Rock

9. A Class of 40 students completed a survey on what pets they like.
The choices were: Cats, Dogs, and Birds.
Everyone liked at least one pet.
10 students liked Cats and Birds but not dogs
6 students liked Cats and Dogs but not birds
2 students liked Dogs and Birds but not Cats
2 students liked all three pets
10 students liked Cats only
9 students liked Dogs only
1 student liked Birds only
Represent these results using a three circle Venn Diagram.
The type of Venn Diagram we need to use is shown on the next slide.

10. = everything
Birds
Cats
Cats Only & Birds Birds Only
Not Dogs
Cats
Birds
Dogs
Cats Cats & Dogs
Not Birds
Birds & Dogs
Not Cats
Dogs Only
Dogs

11. This three circle word problem is an easy one.
All of the number values for each section of the diagram
have been given to us in the question.
All we need to do is carefully put the number values onto
the Diagram.
We also need to check that all of the numbers add up to
the total of 40 students when we are finished.
The completed Venn Diagram is shown on the next slide.

12. = 40 students
Birds
10 10 1
2
Cats 6 2
9
Dogs

13. = 40 students
Birds
10 10 1
2
Cats 6 2
9
Dogs

14. This is a harder version of Problem One, where we are given less
information in the question text. This means that we will need to do
some working out steps to get to the final completed diagram.
“A Class of 40 students completed a survey on what pets they like.
The choices were: Cats, Dogs, and Birds.
Everyone liked at least one pet.
10 students liked Cats and Birds but not Dogs
2 students liked Dogs and Birds but not Cats
12 students liked Cats and Birds
8 students liked Cats and Dogs
All together, 28 students liked Cats, 19 students liked Dogs, and
15 students liked Birds.
Represent these results using a three circle Venn Diagram.”
The type of Venn Diagram we need is shown on the next slide.

15. = everything
Birds
Cats
Cats Only & Birds Birds Only
Not Dogs
Cats
Birds
Dogs
Cats Cats & Dogs
Not Birds
Birds & Dogs
Not Cats
Dogs Only
Dogs

16. Let’s look carefully at the information we have been given
“A Class of 40 students” - This is the E = everything total and can go on diagram now
“Everyone liked at least one pet” – There is nobody outside the circles
“10 students liked Cats and Birds but not Dogs” – This can go on the diagram now
“12 students liked Cats and Birds” – This can help us work out values later
“ 8 students liked Cats and Dogs” – This can help us work out values later
2 students liked Dogs and Birds but not Cats - This can go on the diagram now
All together, 28 students liked Cats, 19 students liked Dogs, and 15 liked Birds.
- These three values are the Totals of each of the circles on our diagram.
We will use these later to help work out the number values for:
“Cats Only”, “Dogs Only” and “Birds Only”
The next slide shows our Venn Diagram so far.

17. = 40 students
Birds
Cats Only 10 Birds Only Total
Cats
15
Birds
Dogs
Cats Cats & Dogs
Not Birds
2
Total
28 Dogs Only
Dogs Total 19

18. Let’s now work on this piece of information:
“ 12 students liked Cats and Birds”
This tells us that :
the Cats and Birds Not Dogs part of the diagram
+
the Cats and Birds and Dogs part in the centre
=
A total of 12 items.
( We only care that the students liked Cats and Birds,
and we do not care whether or not they liked Dogs. )
The next slide highlights these areas on our Venn Diagram .

19. = 40 students
Birds
Cats Only 10 LIKES CATS AND BIRDS
Cats Question says - In total
there are 12 people
Birds who like Cats and Birds.
Dogs
Cats 2 This means that the
number of people who
like “Cats and Birds and
Dogs” in the very centre
of our diagram works
out as the value of “2”.
Dogs Only
Dogs

20. = 40 students
Birds
Cats Only 10 Birds Only
2
Cats Cats & Dogs
Not Birds
2
Dogs Only
Dogs

21. = 40 students
Birds
WORKING OUT BIRDS ONLY
Cats Only 10 Birds Only
Total
We know the Total of
the Birds Circle is 15.
= ? 15
This means that 2
Cats
10 + 2 + 2 + ? = 15
2
For this to be true
“Birds Only” needs
to equal 1.
Dogs Only
Dogs

22. = 40 students
Birds
Cats Only 10 Total
1
15
2
Cats Cats & Dogs
Not Birds
2
Dogs Only
Dogs

23. Let’s now work on this piece of information:
“ 8 students liked Cats and Dogs”
This tells us that :
the Cats and Dogs Not Birds part of the diagram
+
the Cats and Dogs and Birds part in the centre
=
A total of 8 items.
( We only care that the students liked Cats and Dogs,
and we do not care whether or not they liked Birds. )
The next slide highlights these areas on our Venn Diagram .

24. = 40 students
Birds
Cats Only 10 1 LIKES CATS AND DOGS
Question says - In total
2 there are 8 people who
like Cats and Dogs.
Cats Cats & Dogs
Not Birds 2 This means that People
who like “Cats and Dogs
and Not Birds” needs to
equal 6.
2 + 6 = 8 Total for the
Dogs Only yellow Cats and Dogs area.
Dogs

25. = 40 students
Birds
Cats Only 10 1
2
Cats 6 2
Dogs Only
Dogs

26. = 40 students
Birds
Cats Only
=?
10 WORKING OUT CATS ONLY
1
We know the Total of
the Cats Circle is 28.
2 This means that
Cats 6 2
10 + 2 + 6 + ? = 28
For this to be true
Total “Cats Only” needs
to equal 10.
28 Dogs Only
Dogs

27. = 40 students
Birds
10 10 1
2
Cats 6 2
Dogs Only
Dogs

28. = 40 students
Birds
WORKING OUT DOGS ONLY
10 10 1
We know the Total of
the Dogs Circle is 19.
This means that 2
Cats
6 + 2 + 2 + ? = 19
6 2
For this to be true
“Dogs Only” needs
to equal 9.
Dogs Only
=?
Dogs Total 19

29. = 40 students
Birds
10 10 1
2
Cats 6 2
9
Dogs

30. = 40 students
Birds
10 10 1
2
Cats 6 2
9
Dogs

31. Three Circle Venn Diagrams can take quite a bit of working out.
The steps to follow are generally these:
Get the Information from the Question that can go straight onto the
Venn Diagram and place it there.
Work through the remaining information, a bit at a time, to work out
each of the missing values.
Often we work from the centre of the diagram outwards, when finding
these unknown values.
Check that all the numbers on the final diagram add up to the
E = everything grand total.

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