In a survey of 289 college students, it is found that
65 like Brussels sprouts,
93 like broccoli,
55 like cauliflower,
28 like both Brussels sprouts and broccoli,
24 like both Brussels sprouts and cauliflower,
20 like both broccoli and cauliflower, and
13 of the students like all three vegetables.
Answer each of the following using a Venn diagram:
a) How many of the 289 college students do not like any of these three vegetables?
b) How many like broccoli only?
c) How many like broccoli AND cauliflower but not Brussels sprouts?
d) How many like neither Brussels sprouts nor cauliflower?
2345
174
dyr
48454
•1
•93
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To answer each of these questions using a Venn diagram, we need to visualize the relationships between the three groups: those who like Brussels sprouts, those who like broccoli, and those who like cauliflower.
First, let's draw a Venn diagram with three overlapping circles representing each vegetable: Brussels sprouts, broccoli, and cauliflower. The overlapping regions will represent the number of students who like multiple vegetables.
a) To find the number of college students who do not like any of these three vegetables, we need to find the students who are not in any of the three circles. To do this, we subtract the total number of students who like at least one vegetable from the total number of college students.
Total number of college students who do not like any vegetable = Total number of college students - (Number of students who like Brussels sprouts + Number of students who like broccoli + Number of students who like cauliflower)
Total number of college students = 289
Number of students who like Brussels sprouts = 65
Number of students who like broccoli = 93
Number of students who like cauliflower = 55
Number of students who like all three vegetables = 13
Number of students who like at least one vegetable = (Number of students who like Brussels sprouts + Number of students who like broccoli + Number of students who like cauliflower) - (Number of students who like all three vegetables + Number of students who like exactly two vegetables)
Number of students who like at least one vegetable = (65 + 93 + 55) - (13 + 28 + 24 + 20)
Number of students who like at least one vegetable = 191
Total number of college students who do not like any vegetable = 289 - 191 = 98
Therefore, 98 college students do not like any of these three vegetables.
b) To find the number of college students who like broccoli only, we need to find the students who like broccoli but not Brussels sprouts or cauliflower. We can find this by subtracting the number of students who like both broccoli and another vegetable from the number of students who like broccoli.
Number of students who like broccoli only = Number of students who like broccoli - (Number of students who like broccoli and Brussels sprouts + Number of students who like broccoli and cauliflower + Number of students who like all three vegetables)
Number of students who like broccoli only = 93 - (28 + 20 + 13) = 32
Therefore, 32 college students like broccoli only.
c) To find the number of college students who like both broccoli and cauliflower but not Brussels sprouts, we need to find the students who like both broccoli and cauliflower, but not Brussels sprouts. We can find this by subtracting the number of students who like all three vegetables from the number of students who like both broccoli and cauliflower.
Number of students who like both broccoli and cauliflower but not Brussels sprouts = Number of students who like both broccoli and cauliflower - Number of students who like all three vegetables
Number of students who like both broccoli and cauliflower but not Brussels sprouts = 20 - 13 = 7
Therefore, 7 college students like both broccoli and cauliflower but not Brussels sprouts.
d) To find the number of college students who like neither Brussels sprouts nor cauliflower, we need to find the students who do not like either Brussels sprouts or cauliflower. We can find this by subtracting the number of students who like at least one of the two vegetables from the total number of students.
Number of students who like neither Brussels sprouts nor cauliflower = Total number of college students - (Number of students who like Brussels sprouts + Number of students who like cauliflower - Number of students who like both Brussels sprouts and cauliflower - Number of students who like all three vegetables)
Number of students who like neither Brussels sprouts nor cauliflower = 289 - (65 + 55 - 24 - 13) = 156
Therefore, 156 college students do not like either Brussels sprouts or cauliflower.