Section 2-4 Using Sets to Solve Problems

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Aim: How do we use Venn Diagrams to visualize survey results.

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Section 2-4 Using Sets to Solve Problems Objectives: Solve problems by using Venn diagrams.

Example 1: Solving a Problem by Using a Venn Diagram In 2008, there were 36 states that had some form of casino gambling in the state, 42 states that sold lottery tickets of some kind, and 34 states that had both casinos and lotteries. Draw a Venn diagram to represent the survey results, and find how many states have only casino gambling, how many states have only lotteries, and how many states have neither.

Extra Example 1 In an average year, Columbus, Ohio, has 163 days with some rain, 63 days with some snow, and 24 days with both. Draw a Venn diagram to represent these averages, and find how many days have only rain, only snow, and neither.

Example 2: Solving a Problem by Using a Venn Diagram In a survey published in the Journal of the American Academy of Dermatologists, 500 people were polled by random telephone dialing. Of these, 120 reported having a tattoo, 72 reported having a body piercing, and 41 had both. Draw a Venn diagram to represent these results and find out how many respondents have only tattoos, only body piercings, and neither.

Extra Example 2 According to an online survey on Howstuffworks.com, 12,595 people gave their thoughts on Coke versus Pepsi. Of these, 7,642 drink Coke, 5,619 drink Pepsi, and 1.856 drink both. Draw a Venn diagram to represent these results, and find out how many respondents drink only Coke, only Pepsi, and neither.

Example 3: Solving a Problem Using a Venn Diagram A survey of 300 first-year students at a large university was conducted to aid in scheduling for the following year. Responses indicated that 194 were taking a math class, 210 were taking an English class, and 170 were taking a speece course. In addition, 142 were taking both math and English, 111 were taking both English and speech, 91 were taking both math and speech, and 45 were taking all three. Draw a Venn diagram to represent these survey results.

Extra Example 3 An online music service surveyed 500 customers and found that 270 listen to hip-hop, 320 listen to rock, and 160 listen to country. In addition, 140 listen to both rock and hip-hop, 120 listen to rock and country, and 80 listen to hip-hop and country. Finally, 50 listen to all three. Draw a Venn diagram to represent the results of the survey.

Example 4 Out of 20 students taking a midterm exam, 15 answered the first of two bonus questions, 13 answered the second bonus question, and 2 answered neither bonus question. How many students answered both questions.

Homework P. 80 #2, 3, 4, 7, 8, 15