Answer:
48
Explanation:
The given parameters are;
The activities in which the students of the school participate = 1) Debate, 2) Music, and 3) Community services
The number of students that do all three activities = 3
The number of students that do two or more activities = 21
The number of the students that participate in music = Half of the students
The ratio of the students that take part in debate, music, and community service = 5 : 4 : 3
Let x represent a common number factor of participants in each of the given three activities, we have;
5·x participate in debate, A
4·x participate in music, B
3·x participate in community service, C
We get;
The sum of the students in the school = A∪B∪C = A + B + C - A∩B - A∩C - B∩C + A∩B∩C
Given that the number of students that do two or ore activities = 21, we have;
(A∩B)∪(A∩C)∪(B∩C) = 21
However, we have;
(A∩B)∪(A∩C)∪(B∩C) = (A∩B) + (A∩C) + (B∩C) - A∩B∩C - B∩A∩C - C∩B∩A + A∩B∩C
∴ 21 = (A∩B) + (A∩C) + (B∩C) - 3 - 3 - 3 + 3
(A∩B) + (A∩C) + (B∩C) = 21 + 3 + 3 + 3 - 3 = 21 + 6 = 27
∴ -(A∩B) - (A∩C) - (B∩C) = -27
Therefore;
A + B + C - A∩B - A∩C - B∩C + A∩B∩C = 5·x + 4·x + 3·x - 27 + 3
5·x + 4·x + 3·x - 27 + 3 = 12·x - 24
The total number of students = A∪B∪C = 12·x - 24
The number of students participating in music, B = 4·x = Half the total number of students = (12·x - 24)/2
∴ 4·x = (12·x - 24)/2
8·x = 12·x - 24
12·x - 8·x = 24
4·x = 24
x = 24/4 = 6
The total number of students at the school = 12·x - 24 = 12 × 6 - 24 = 48