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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


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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

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


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Dogs and Cats



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:


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Dogs and Cats



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:


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Dogs and Cats



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


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Dogs and Cats



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


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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


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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


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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


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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


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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


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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

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