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
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?
3 answers
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.
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.
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?
2 answers