A small school has 100 students who occupy three classrooms: A, B and C. After the first period of the school day, half of the students in room A move to room B, a fifth of the students in room B move to room C, and one-third of the students in room C move to room A. However, the total number of students in each room is the same for both periods. How many students occupy each room? a+b+c = 100
1/2 a + 1/3 c = a
4/5 b + 1/2 a = b
2/3 c + 1/5 c = c
then,
a=20, b=50, c=30 This is not math.
a = students in room A
b = students in room B
c = students in room C
a + b + c = 100
Half of the students in room A go to room B
In room A remains one/2 students.
A third of the students in room C move to room A
The total number of students in each room is the same for both periods therefore:
a/2 + c/3 = a
Subtract a/2 from both sides
c / 3 = a – a / 2
c/3 = a/2
cross multiply
3 a = 2 c
Divide both sides by 3
a = 2 / 3c
After the first period, a fifth of the students in room B move to room C.
In room B remain 4/5 students.
A third of the students in room C move to room A
The total number of students in each room is the same for both periods therefore:
4 / 5 b + c / 3 = b
Subtract 4 / 5 b from both sides
c / 3 = b – 4 / 5 b = 5 / 5 b – 4 / 5 b
c/3 = b/5
cross multiply
3g = 5c
Divide both sides by 3
b = 5 / 3c
Put these values into the equation:
a + b + c = 100
2 / 3 c + 5 / 3 c + c = 100
2 / 3t + 5t / 3 + 3 / 3t = 100
10 c / 3 = 100
cross multiply
10c = 300
Divide both sides by 10
c = 30
a = 2 / 3c
one = 2 ∙ 30 / 3 = 60 / 3
one = 20
b = 5 / 3c
b = 5 ∙ 30 / 3 = 150 / 3
b=50
Check the result.
There are 20 students in room A
When half of the students leave room A, in room A will be 20 – 10 = 10
10 students left.
A third of the students in Room C move to Room A.
c / 3 = 30 / 30 = 10 students from room C move to room A.
a / 2 + c / 3 = 10 + 10 = 20 = one
When a fifth of the students leave room B, in room B will be 50 – 50 / 5 = 50 – 10 = 40
40 students left.
Half of the students in room A go to room B.
a / 2 = 20 / 2 = 10 students in room A go to room B.
4 / 5 b + a / 2 = 40 + 10 = 50 = b
When a third of the students in room C move to room A, in room C there will be 30 – 30 / 3 = 30 – 10 = 20
20 students left.
A fifth of the students in room B go to room C.
b / 5 = 50 / 50 = 10
2 / 3 c + b / 5 = 20 + 10 = 30 = c
Thus during the move, 10 students leave each room and 10 students enter each room.