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 both a 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 both a 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 both a 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 us solve this problem. Dog only Cat only Both Dog and Cat Neither 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 both a 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 is important that the sum of the number of children in each of the four classifications does not exceed 29.

Tutorial: 

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 both a 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 only and… 15 children have a cat only, then based this scenario there would be already be 35 total children in the class. 20 + 15 = 35 Therefore, at least 6 children must have both a dog and a cat. 35 – 29 = 6 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 both a 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 dog and a cat, this means… 14 children have a dog only and 9 children have a cat only. 14 6 9

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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 both a 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 dog nor a cat. 29 14 6 11 0

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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 both a 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 dog and a cat, we can see that this number can increase from 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 both a 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 dog and a cat. 13 + 8 + 7 + 1 = 29 29 13 7 8 1

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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 both a 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 a dog and a cat. 12 + 7 + 8 + 2 = 29 29 12 8 7 2

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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 both a 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 children who can have both a dog and a cat is 15 children. That is, all children who have a cat also have a dog.

Tutorial: 

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 both a 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, the number of children who have both a dog and a cat must be… (D) At least 6 and at most 15

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