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**Part 1 Module 3 Survey Problems and Venn diagrams**

EXAMPLE 1.3.1 A survey of 64 informed consumers revealed the following information: 45 believe that Elvis is still alive 49 believe that they have been abducted by space aliens 42 believe both of these things 1. How many believe Elvis is still alive or believe that they have been abducted by space aliens? 2. How many believe neither of these things?

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Solution A survey of 64 informed consumers revealed the following information: 45 believe that Elvis is still alive 49 believe that they have been abducted by space aliens 42 believe both of these things We will use a Venn diagram to organize the information. Let E represent the set of those who believe Elvis is still alive. Let A represent the set of those who believe they have been abducted by aliens. The universe (U) in this case is the set of 64 people who were surveyed.

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Solution, page 2 A survey of 64 informed consumers revealed the following information: 45 believe that Elvis is still alive 49 believe that they have been abducted by space aliens 42 believe both of these things

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Solution, page 3 49 believe that they have been abducted by space aliens

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Solution, page 4 45 believe that Elvis is still alive

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Solution, page 5 64 people were surveyed.

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Solution, page 6 1. How many believe Elvis is alive or believe that they have been abducted by aliens? The Venn diagram shows that the answer is = 52

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**Solution, page 7 2. How many believe neither of these things?**

The Venn diagram shows that the answer is 12.

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Exercise #2 99 grumpy old people were surveyed regarding things they dislike. 66 dislike puppies 34 dislike kitties 32 dislike children 10 dislike puppies and dislike kitties and dislike children 28 dislike puppies and dislike children 8 dislike kitties and don't dislike children and don't dislike puppies 25 dislike puppies and dislike kitties How many don't dislike puppies and don't dislike kitties and don't dislike children? A B C D E. None of these

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Solution We will use a three-circle Venn diagram to organize the information. Let P represent those who dislike puppies, K represent those who dislike kitties, and C represent those who dislike children. We will start with this fact: 10 dislike puppies and dislike kitties and dislike children

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Solution, page 2 Next: 8 dislike kitties and don't dislike children and don't dislike puppies

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**Solution, page 3 Next: 25 dislike puppies and dislike kitties**

Because 25 people dislike both puppies and kitties, the part of the diagram where those circles intersect must contain a total of 25 people. Ten of the 25 have already been accounted for. This causes the number 15 to appear in the region of the diagram where P and K intersect without intersecting C.

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**Solution, page 4 28 dislike puppies and dislike children Next:**

Because 28 people dislike both puppies and children, the part of the diagram where those circles intersect must contain a total of 28 people. Ten of the 28 have already been accounted for. This causes the number 18 to appear in the region where P and C intersect without intersecting K.

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**Solution, page 5 66 dislike puppies Next:**

Because 66 people dislike puppies, the four regions of circle P must contain a total of 66 people. This causes the number 23 to appear in the region that is inside P but outside the other two circles.

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**Solution, page 6 34 dislike kitties Next:**

Because 34 people dislike kitties, the four regions of circle K must contain a total of 34 people. This causes the number 1 to appear in the region where K and C intersect without intersecting P.

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**Solution, page 7 32 dislike children Next:**

Because 32 people dislike children, the four regions of circle C must contain a total of 32 people. This causes the number 3 to appear in the region that is inside C but outside the other two circles.

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**Solution, page 8 99 grumpy old people were surveyed. Next:**

Because 99 people were surveyed, the eight numbers in the diagram must add up to 99. This allows us to find the number that occupies the region outside of all three circles. Call this number x. We add the seven numbers on the diagram, then subtract the sum from 99. x = 99 – ( ) x = 99 – 77 = 21.

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Solution, page 9 Now that the diagram is filled in, we use it to read the answer to the question “How many don't dislike puppies and don't dislike kitties and don't dislike children?” The correct choice is D. 21.

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**Guidelines for 3-circle survey problems**

1. If a description names all three categories, then that description is pointing at exactly one of the eight regions of the diagram, so the accompanying number does not get “split up.” This is illustrated in the diagram below. Assume that a number of Neanderthals were surveyed regarding their favorite possessions. Set S represents those who own a stick, R represents those who own a rock, and B represents those who own a bone.

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**Guidelines for 3-circle survey problems**

2. If a description names the conjunction of two categories, then that description is pointing at the intersection of two circles, so the accompanying number gets “split up” among two regions of the diagram.

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**Guidelines for 3-circle survey problems**

3. If a description names just one category, then that description is pointing at an entire circle, so the accompanying number gets “split up” among four regions of that circle.

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**Guidelines for 3-circle survey problems**

4. “And” means “intersection,” or means “union.”

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Exercise #2 A number of young people were surveyed about their friends. The results are summarized in the Venn diagram below. How many have friends on MyFace.com and BookFace.com? A B C D. None of these

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**Solution #2 How many have friends on MyFace.com and BookFace.com?**

If a person has friends on both MyFace.com and BookFace.com, then that person is in the intersection of circles M and B. There are a total of 21 people in that part of the universe.

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Exercise #3 A number of young people were surveyed about their friends. The results are summarized in the Venn diagram below. How many have exactly one of the three kinds of friends? A B C D. None of these

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**Solution #3 How many have exactly one of the three kinds of friends?**

The diagram shows that 15 people have only MyFace friends, 17 people have only BookFace friends, and 1 person has only actual friends. = 33 people who have exactly one kind of friend.

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Exercise #4 A number of young people were surveyed about their friends. The results are summarized in the Venn diagram below. How many have actual, real friends, but don’t have friends on Bookface.com? A B C D. None of these

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Solution #4 How many have actual, real friends, but don’t have friends on Bookface.com? A person who satisfies this description must be simultaneously inside circle A yet outside circle B. The diagram shows a total of 2 people in those two regions.

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Exercise #5 A number of young people were surveyed about their friends. The results are summarized in the Venn diagram below. How many have actual real friends, or have friends on MyFace.com? A B C D. None of these

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Solution #5 How many have actual real friends, or have friends on MyFace.com? This description is referring to the union of circles A and M. We must add all six numbers that appear in A, M or both. = 41 The correct choice is “D. None of these”

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