Question

universal set
sports
arts
76
36
24
14
a) there were a total of high school students that responded to the survey.
b) students play sports.
c) students participate in the arts.
d) students play sports and participate in the arts.
e) students play sports or participate in the arts.
f) students dont play sports and are not involved in the arts.
g) students participate in the arts, but do not play sports.
h) of students play sports.
i) of students play sports or participate in the arts.

Explanation:

Step1: Calculate total number of students (a)

Add all the numbers in the Venn - diagram: \(76 + 36+24 + 14=150\)

Step2: Calculate number of students who play sports (b)

Students who play sports are the sum of those only in sports and those in both sports and arts. So \(76 + 36=112\)

Step3: Calculate number of students who participate in arts (c)

Students who participate in arts are the sum of those only in arts and those in both sports and arts. So \(36+24 = 60\)

Step4: Calculate number of students who play sports and participate in arts (d)

The intersection of the two circles (sports and arts) gives this number. So it is \(36\)

Step5: Calculate number of students who play sports or participate in arts (e)

Use the formula \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). Here \(n(\text{sports}) = 112\), \(n(\text{arts})=60\), \(n(\text{sports}\cap\text{arts}) = 36\). So \(112+60 - 36=148\)

Step6: Calculate number of students who don't play sports and are not involved in arts (f)

This is the number outside the two circles (sports and arts). So it is \(14\)

Step7: Calculate number of students who participate in arts but do not play sports (g)

This is the part of the arts circle that does not overlap with the sports circle. So it is \(24\)

Step8: Calculate fraction of students who play sports (h)

The fraction is \(\frac{\text{Number of students who play sports}}{\text{Total number of students}}=\frac{112}{150}=\frac{56}{75}\)

Step9: Calculate fraction of students who play sports or participate in arts (i)

The fraction is \(\frac{\text{Number of students who play sports or participate in arts}}{\text{Total number of students}}=\frac{148}{150}=\frac{74}{75}\)

Answer:

a) \(150\)
b) \(112\)
c) \(60\)
d) \(36\)
e) \(148\)
f) \(14\)
g) \(24\)
h) \(\frac{112}{150}=\frac{56}{75}\)
i) \(\frac{148}{150}=\frac{74}{75}\)