A small school has 130 students who occupy three classrooms: A, B, and C. After the first period of the school day, half the students in room A move to room B, one-fifth of the students in room B move to room C, and one-third of the students in room C move to room A. Nevertheless, the total number of students in each room is the same for both periods. How many students occupy each room?

Answer 1

to start,
a+b+c = 100

After the room changes, we have

a/2 + c/3 = a
4b/5 + a/2 = b
2c/3 + b/5 = c

or,

a/2 = c/3
a/2 = b/5
b/5 = c/3 = a/2

so, substituting in,

a + 5a/2 + 3a/2 = 100
2a + 5a + 3a = 200
a = 20
b = 50
c = 30
during the move, 10 students move to each room.