Question 1193384
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Let the beginning numbers in the halls be A, B, and C.<br>
In the first interval, 3/5 of A move to B (2/5 stay in A)<br>
# in A: (2/5)A
# in B: B+(3/5)A
# in C: still C<br>
In the second interval, 2/3 of those in B moved to C (1/3 remained in B)<br>
# in A: (2/5)A
# in B: (1/3)(B+(3/5)A) = (1/3)B+(1/5)A
# in C: C+(2/3)(B+(3/5)A) = C+(2/3)B+(2/5)A<br>
In the third interval, half of those in C left (half remained)<br>
# in A: (2/5)A
# in B: (1/3)B+(1/5)A
# in C: (1/2)C+(1/3)B+(1/5)A<br>
In the end, the number in B was twice the number in A:<br>
(1/3)B+(1/5)A=2((2/5)A)
(1/3)B=(3/5)A
B=(9/5)A<br>
And in the end, the number in C was twice the number in B:<br>
(1/2)C+(1/3)B+(1/5)A=2((1/3)B+(1/5)A)
(1/2)C=(1/3)B+(1/5)A=(1/3)((9/5)A)+(1/5)A=(3/5)A+(1/5)A=(4/5)A
C=(8/5)A<br>
The total number of students was 2200:<br>
A+B+C=2200
A+(9/5)A+(8/5)A=2200
(22/5)A=2200
A=500<br>
B=(9/5)A=900
C=(8/5)A=800<br>
ANSWER: The number in B at the start was 900<br>
CHECK:
start (A,B,C) = (500,900,800)
after first interval: (200,1200,800)
after second interval: (200,400,1600)
after third interval: (100,400,800)<br>