Question

A small school has 100 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

a+b+c = 100
1/2 a + 1/3 c = a
4/5 b + 1/2 a = b
2/3 c + 1/5 b = c
so,
a=20, b=50, c=30

Answer 2

That is not calculus.

a = students in room A

b = students in room B

c = students in room C

a + b + c = 100

Half the students in room A move to room B

In room A remains a / 2 students.

One-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 so:

a / 2 + c / 3 = a

Subtract a / 2 to both sides

c / 3 = a - a / 2

c / 3 = a / 2

Cross multiply

3 a = 2 c

Divide both sides by 3

a = 2 / 3 c


After the first period one-fifth of the students in room B move to room C.

In room B remains 4 / 5 students.

One-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 so:

4 / 5 b + c / 3 = b

Subtract 4 / 5 b to toth sides

c / 3 = b - 4 / 5 b = 5 / 5 b - 4 / 5 b

c / 3 = b / 5

Cross multiply

3 b = 5 c

Divide both sides by 3

b = 5 / 3 c

Put this values in equation:

a + b + c = 100

2 / 3 c + 5 / 3 c + c = 100

2 / 3 c + 5 c / 3 + 3 / 3 c = 100

10 c / 3 = 100

Cross multiply

10 c = 300

Divide both sides by 10

c = 30

a = 2 / 3 c

a = 2 ∙ 30 / 3 = 60 / 3

a = 20

b = 5 / 3 c

b = 5 ∙ 30 / 3 = 150 / 3

b = 50

Check fresult.

There are 20 students in Room A

When one half students leaves room A, in room A will be 20 - 10 = 10

10 students left.

One-third of the students in room C move to room A.

c / 3 = 30 / 30 = 10 students in room C move to room A.

a / 2 + c / 3 = 10 + 10 = 20 = a

When one-fifth students leaves room B, in room B will be 50 - 50 / 5 = 50 - 10 = 40

40 students left.

Half the students in room A move to room B.

a / 2 = 20 / 2 = 10 students in room A move to room B.

4 / 5 b + a / 2 = 40 + 10 = 50 = b

When one-third students in room C move to room A, in room C will be 30 - 30 / 3 = 30 - 10 = 20

20 students left.

One-fifth of the students in room B move to room C.

b / 5 = 50 / 50 = 10

2 / 3 c + b / 5 = 20 + 10 = 30 = c

So during the move, 10 students leave each room and 10 students come in each room.

Answer 3

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?