Section 5

1. Section 5 Applications of Sets

2. Example: Toothpaste Taste Test • A drug company is considering manufacturing a new toothpaste. They are considering two flavors, regular and mint. • In a sample of 120 people, it was found that 74 liked the regular, 62 liked the mint, and 35 liked both types. • How many liked only the regular flavor? • How many liked either one or the other or both? • How many people did not like either flavor?

3. Begin by setting up a Venn diagram with sets A (regular flavor) and B (mint flavor). Since some people liked both flavors, the sets will overlap and the number who liked both with be placed in region II. 35 people liked both flavors. U 35 II A(Regular) B(Mint) Solution

4. Next, region I will refer to those who liked only the regular and region III will refer to those who liked only the mint. In order to get the number of people in each region, find the difference between all the people who liked each toothpaste and those who liked both. I: 74 – 35 = 39 III: 62 – 35 = 27 U 39 regular only 27 mint only A(Regular) B(Mint) Solution (continued) 35 both II I III

5. Solution (continued) • “One or the other or both” represents the UNION of the two sets. Therefore, 39 + 27 + 35 = 101 people who liked one or the other or both.

6. 19 liked U ne i ther 35 74 - 35 = 3 9 62 - 35 = 27 both L iked regular L iked mint only only A(Regular) B(Mint) Solution (continued) • Take the total number of people in the entire sample and subtract the number who liked one or the other or both. 120-101 = 19 people did not like either flavor. I I I I I I

7. Example: Ice Cream Flavors • The Ice Cream Shoppe surveyed its customers about their preferences of three ice cream flavors. The results of the survey showed the following number of people liked the respective flavor(s):

8. Using a Venn diagram, determine how many people completed the survey.

9. Solution 31 65 35 39 18 17 +12 217