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.

ice skate
7 2 13
3 4 8
10
6
snowboard
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: Find students who ski or snowboard

Use the principle of inclusion - exclusion for the ski and snowboard sets. Add the number of students in each set and subtract the number of students in their intersection. Ski: \(13 + 2+3\), Snowboard: \(10 + 4+3\), Intersection: \(3 + 8\). So, \((13 + 2+3)+(10 + 4+3)-(3 + 8)=24\).

Step2: Find students who ice - skate and ski

Look at the intersection of ice - skate and ski circles, which is \(2 + 3=5\).

Step3: Find students who ice - skate, ski, and snowboard

The number in the common intersection of all three circles is \(3\).

Step4: Find students who do not ski or ice - skate

First, find the number of students who ski or ice - skate. Ice - skate: \(7+2 + 4+3\), Ski: \(13+2 + 3+8\), Intersection: \(2 + 3\). Ski or Ice - skate: \((7+2 + 4+3)+(13+2 + 3+8)-(2 + 3)=34\). Total students (assuming the sum of all regions in the diagram including the outside region) is \(7+2+13+4+3+8+10+6 = 53\). So, students who do not ski or ice - skate is \(10 + 6=16\).

Step5: Find students who just snowboard

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

Step6: Find students who do not ice - skate, ski, or snowboard

The number outside all three circles is \(6\).

Step7: Find students who ice - skate

Add the numbers in the ice - skate circle: \(7+2 + 4+3=16\).

Step8: Find students who do not snowboard

First, find the number of students who snowboard: \(10+4 + 3+8 = 25\). Total students is \(53\). So, students who do not snowboard is \(7+2+13+6=28\).

Step9: Find students who took the survey

Sum all the numbers in the diagram including the outside region: \(7+2+13+4+3+8+10+6 = 53\).

Answer:

a. 24
b. 5
c. 3
d. 16
e. 10
f. 6
g. 16
h. 28
i. 53