3838 of all students at Social High are in all three of the following clubs: Albanian, Bardic, and Checkmate. 1212 of all students are in Albanian, 5858 are in Bardic, and 3434 are in Checkmate. If every student is in at least one club, what fraction of the student body is in exactly 2 clubs

1/8

Explanation:

Assuming there are say 40 students at Social high

No of students in all 3 clubs will be

3/8 x 40 = 15 students

The no of students in Albanian club will be

1/2 x 40 = 20 students

The number of students in Bardic club will be

5/8 x 40 = 25 students

The number of students in Checkmate club will be

3/4 x 40 = 30 students

Number of students in none of the club = 0

If we use B to denote the number of student in 2 clubs exactly, then using the knowledge of Sets and Venn Diagram we know that

Total number of students at social High = (Number of students at Albanian Club) + (Number of students at Bardic Club) + (Number of students at Checkmate Club) - (Number of Students in exactly two clubs) - 2 x (Total number of students in all 3 clubs) + (Number of student in none of the clubs)

40 = 20 + 25 + 30 - B - 2 (15) + 0

B = 75 - 40 - 30

B = 5

Hence the fraction of students in exactly 2 clubs will be = number of students in exactly 2 clubs/Total number of students at Social High

= 5/80

= 1/8

This Question is not correct.

The Correct Question is below

3/8 of all students at Social High are in all three of the following clubs: Albanian, Bardic, and Checkmate. 1/2 of all students are in Albanian, 5/8 are in Bardic, and 3/4 are in Checkmate. If every student is in at least one club, what fraction of the student body is in exactly 2 clubs?

(A) 1/8

(B) 1/4

(C) 3/8

(D) 1/2

(E) 5/8

Answer:

(A) 1/8

Explanation:

Let us assume that the total number of students at Social High is = 32

Therefore,

Since 3/8 of all students are in all 3 clubs,

The total number of students in the 3 clubs = (3:8) * 32 = 12

Since 1/2 of all students are in Albanian club,

The total number of students in the Albanian club = (1:2) * 32 = 16

Since 5/8 of all students are in Bardic club,

The total number of students in the Bardic club = (5:8) * 32 = 20

Since 3/4 of all students are in Checkmate club,

The total number of students in the Checkmate club = (3:4) * 32 = 24

From above information in the question, we know that zero (0) students are in none of the clubs.

Let X be given as the number of students in exactly two clubs, we can now use the equation:

Total number of Students at Social High = (Total number of students at Albanian club) + (Total number of students at Bardic club) + (Total number of students at Checkmate club) - (Total number of Students in exactly two clubs) - 2 (Total number of students

in all the 3 clubs) + none

32 = 16 + 20 + 24 - X - 2 (12) + 0

32 = 36 + 24 - X - 24

X = 36-32

X = 4

Therefore, the fraction of people in exactly two clubs = The total number of students in exactly two clubs / The total number of students at Social High

= 4/32

= 1/8

The fraction of students in exactly two club is 1/8.