Question

A survey of 115 college students was taken to determine the musical styles they liked. Of​ those, 40 students listened to​ rock, 41 to​ classical, and 52 to jazz.​ Also, 25 students listened to rock and​ jazz, 19 to rock and​ classical, and 27 to classical and jazz.​ Finally, 13 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn Diagram to answer the following questions.

a. How many listened to only rock​ music?
n​(only ​rock)
b. How many listened to classical and​ jazz, but not​ rock?
n​(classical and​ jazz, not ​rock)
c. How many listened to classical or​ jazz, but not​ rock?
n​(classical or​ jazz, not ​rock)
d. How many listened to music in exactly one of the musical​ styles?
n​(exactly one ​style)
e. How many listened to music in exactly two of the musical​ styles?
n​(exactly two ​styles)
f. How many did not listen to any of the musical​ styles?
n​(none)

Answer 1

• a. There are 9 students listened to only rock music.
• b. There are 14 students listened to classical and jazz, but not rock.
• c. There are 67 students listened to classical or jazz, but not rock.
• d. There are 27 students listened to music in exactly one of the musical styles.
• e. There are 32 students listened to music in exactly two of the musical styles.
• f. There are 8 students did not listen to any of the musical styles.

Step-by-step explanation:

a. To find the number of students who listened to only rock music, we can subtract the number of students who listened to rock and other styles from the total number of students who listened to rock.

• Total students who listened to rock: 40
• Students who listened to rock and jazz: 25
• Students who listened to rock and classical: 19
• Students who listened to all three styles: 13

To find the number of students who listened to only rock:

40 - (25 + 19 - 13) = 40 - 31 = 9

Therefore, 9 students listened to only rock music.

b. To find the number of students who listened to classical and jazz, but not rock, we can subtract the number of students who listened to all three styles from the total number of students who listened to both classical and jazz.

• Students who listened to both classical and jazz: 27
• Students who listened to all three styles: 13
• 27 - 13 = 14

Therefore, 14 students listened to classical and jazz, but not rock.

c. To find the number of students who listened to classical or jazz, but not rock, we can add the number of students who listened to each style individually and subtract the number of students who listened to all three styles.

• Students who listened to classical: 41
• Students who listened to jazz: 52
• Students who listened to all three styles: 13
• (41 - 13) + (52 - 13) = 28 + 39 = 67

Therefore, 67 students listened to classical or jazz, but not rock.

d. To find the number of students who listened to music in exactly one of the musical styles, we can add together the number of students who listened to each style individually (rock, classical, jazz) and subtract the number of students who listened to more than one style.

• Students who listened to only rock: 9
• Students who listened to only classical: 6
• Students who listened to only jazz: 12
• 9 + 6 + 12 = 27

Therefore, 27 students listened to music in exactly one of the musical styles.

e. To find the number of students who listened to music in exactly two of the musical styles, we can add together the number of students who listened to each combination of two styles (rock and classical, rock and jazz, classical and jazz) and subtract the number of students who listened to all three styles.

• Students who listened to rock and classical: 6
• Students who listened to rock and jazz: 12
• Students who listened to classical and jazz: 14

• 6 + 12 + 14 = 32

Therefore, 32 students listened to music in exactly two of the musical styles.

f. To find the number of students who did not listen to any of the musical styles, we can subtract the total number of students who listened to at least one style from the total number of students surveyed.

• Total students surveyed: 115

• Students who listened to at least one style: 40 + 41 + 52 - 13 = 120 - 13 = 107
• 115 - 107 = 8

Therefore, 8 students did not listen to any of the musical styles.