zisis wrote:
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
A student asked me to solve this without using a formula, so here we go....
Let's assign a nice value to the TOTAL number of students.
So we need a number that will work well with all of the fractions in the question.
Let's say there are
16 students in total
3/8 of 16 = 6. So there are 6 students and all three clubs.
1/2 of 16 = 8. So, there are 8 students in in Albanian
5/8 of 16 = 10. So, there are 10 students in in Bardic
3/4 of 16 = 8. So, there are 12 students in in Checkmate
We get the following:
What fraction of the student body is in exactly 2 clubs?Let a, b and c = the number of students in exactly 2 clubs
So, our goal is to find the SUM a + b + c
Since there is a total of 8 students in Albanian, we know that 2 - a - b = the number of students that are in Albanian only (this way the total number of students in Albanian adds to 8)
Since there is a total of 10 students in Bardic, we know that 4 - b - c = the number of students that are in Bardic only (this way the total number of students in Bardic adds to 10)
Since there is a total of 12 students in Checkmate, we know that 6 - a - c = the number of students that are in Checkmate only (this way the total number of students in Checkmate adds to 10)
We get:
Since there are 16 students all together, the sum of all the expressions in our diagram must add to 16.
In other words: (2 - a - b) + b + (4 - b - c) + 6 + a + c + (6 - a - c) = 16
Simplify to get: 18 - a - b - c = 16
Subtract 16 from both sides to get: 2 - a - b - c = 0
Rearrange to get: a + b + c = 2
In other words, 2 of the 16 students are in exactly two clubs.
2/16 = 1/8
Answer: A
Cheers,
Brent
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