Venn Tutorial

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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

    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:

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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:

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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 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

    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

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    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