Question

1. on a recent survey, students were asked if they ice skate, snowboard, or ski. the venn diagram below shows the results of the survey. a. how many students ski or snowboard? ______ b. how many students ice skate and ski? ____ c. how many students ice skate, ski, and snowboard? ____ d. how many students do not ski or ice skate? ____ e. how many students just snowboard? ____ f. how many students do not ice skate, ski, or snowboard? ____ g. how many students ice skate? ____ h. how many students do not snowboard?____ i. how many students took the survey? ______

Explanation:

Step1: Recall set - union formula for part a

For students who ski or snowboard, we use the formula \(n(S\cup N)=n(S)+n(N)-n(S\cap N)\). Here \(n(S) = 2 + 3+13\), \(n(N)=4 + 3+8 + 10\), and \(n(S\cap N)=3 + 8\). So \(n(S\cup N)=(2 + 3+13)+(4 + 3+8 + 10)-(3 + 8)=24 + 25 - 11=38\).

Step2: Identify intersection for part b

The number of students who ice - skate and ski is the number in the intersection of the ice - skate and ski circles, which is \(2+3 = 5\).

Step3: Identify triple - intersection for part c

The number of students who ice - skate, ski, and snowboard is the number in the triple - intersection of the three circles, which is \(3\).

Step4: Find non - ski and non - ice - skate for part d

The number of students who do not ski or ice - skate is the number of students in the snowboard - only part and the outside part. So it is \(10+6 = 16\).

Step5: Identify snowboard - only for part e

The number of students who just snowboard is \(10\).

Step6: Identify outside number for part f

The number of students who do not ice - skate, ski, or snowboard is \(6\).

Step7: Sum ice - skate parts for part g

The number of students who ice - skate is \(7+2 + 3+4=16\).

Step8: Sum non - snowboard parts for part h

The number of students who do not snowboard is \(7+2+13 + 6=28\).

Step9: Sum all parts for part i

The number of students who took the survey is \(7+2+13+4 + 3+8+10+6=53\).

Answer:

a. 38
b. 5
c. 3
d. 16
e. 10
f. 6
g. 16
h. 28
i. 53