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8/2/2019 Venn Tutorial
1/12
Dogs and Cats
If a class of 29 children, each of 20 children has a dog and
each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5
(C) Exactly 6(D) At least 6 and at most 15
Tutorial
8/2/2019 Venn Tutorial
2/12
Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
Each of the 29 children in this class can be classified as
one and only one of the following:Dog only
Cat only
Both Dog and Cat
Neither Dog nor Cat
8/2/2019 Venn Tutorial
3/12
Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
Using this classification we can
construct a Venn Diagram to help ussolve this problem.
Dog only
Cat only
Both Dog and CatNeither Dog nor Cat
Total children:
Neither:
Dog: Cat:
Both:
8/2/2019 Venn Tutorial
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Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
For this problem, it isimportant that the sum ofthe number of children ineach of the fourclassifications does not
exceed 29.
Total children:29
Neither:
Dog: Cat:
Both:
8/2/2019 Venn Tutorial
5/12
Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
If 20 children have a dog onlyand
15 children have a cat only,
then based this scenario therewould be already be 35 totalchildren in the class.
20 + 15 = 35
Therefore, at least 6 children musthave both a dog and a cat.
35 29 = 6
Total children:
Neither:
Dog: Cat:
Both:
20 15
35
6
8/2/2019 Venn Tutorial
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Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
If six children have both a dogand a cat, this means
14 children have a dog only and
9 children have a cat only.
Total children:
Neither:
Dog: Cat:
Both:
14
6
9
8/2/2019 Venn Tutorial
7/12Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
In this scenario, a total of 29
children are accounted for.14 + 11 + 6 = 29
This means that there are no
children who have neither a dognor a cat.
Total children:
Neither:
Dog: Cat:
Both:
29
14
6
11
0
8/2/2019 Venn Tutorial
8/12Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
By changing the number of
children who have both a dogand a cat, we can see thatthis number can increasefrom 6.
Total children:
Neither:
Dog: Cat:
Both:
29
14
6
11
0
8/2/2019 Venn Tutorial
9/12Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
For example, a total of 7
children can have both a dogand a cat.
13 + 8 + 7 + 1 = 29
Total children:
Neither:
Dog: Cat:
Both:
29
13
7
8
1
8/2/2019 Venn Tutorial
10/12Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
As another example, a total of
8 children can have both adog and a cat.
12 + 7 + 8 + 2 = 29
Total children:
Neither:
Dog: Cat:
Both:
29
12
8
7
2
8/2/2019 Venn Tutorial
11/12Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
Since 15 students have a cat, the
maximum number of childrenwho can have both a dog anda cat is 15 children.
That is, all children who have a
cat also have a dog.
Total children:
Neither:
Dog: Cat:
Both:
29
5
15
0
9
8/2/2019 Venn Tutorial
12/12Dogs and Cats
If a class of 29 children, each of 20 children has a dog and each of 15 has a cat. How many of the children have botha dog and cat?
(A) None of the children necessarily has both.(B) Exactly 5(C) Exactly 6(D) At least 6 and at most 15
Tutorial
Returning to the original problem, thenumber of children who haveboth a dog and a cat must be
(D) At least 6 and at most 15
Total children:
Neither:
Dog: Cat:
Both:
29
5
15
0
9
Total children:
Neither:
Dog: Cat:
Both:
29
14
6
11
0