What is the total number of students at the school?

Question

Two-thirds of the students at a school are in choir, and one-half of the students are in orchestra. Every student at the school is in either choir or orchestra, and 24 of them are in both. What is the total number of students at the school?

Answer 1

There are N students in the school...

there are (2/3)N-24 students in choir only

there are (1/2)N-24 students in orchestra only

combining them:

(2/3)N - 24 + (1/2)N-24 + 24 = N

[(2/3)+(1/2)]N - 24 = N

(7/6)N - 24 = N

(1/6)N - 24 = 0

(1/6)N = 24

N = 24*6 = 144

There are 144 students in the school...

(2/3)*144 - 24 = 72 are in choir only

(1/2)*144 - 24 = 48 are in orchestra only

24 are in both

Yes, the total is 144

144 students