x is directly proportional to y means the following:
x = ky, where k is a constant.
In the problem above, let n = the number of occupants, v = the variable expense, and c = the constant expense.
Total expenses = v + c.
Since the variable expense is directly proportional to the number of occupants, we get:
v = kn.
Thus:
Total expenses = kn + c, where n is the number of occupants, and k and c are constants.
200 occupants:
Total expenses = kn + c.
1300 = 200k + c.
250 occupants:
Total expenses = kn + c.
1600 = 250k + c.
Subtracting the first equation from the second, we get:
1600-1300 = (250k + c) - (200k + c)
300 = 50k
k = 6.
Substituting k=6 into 1300 = 200k + c, we get:
1300 = 200*6 = c
c = 100.
Total expenses for 300 inmates:
Total = kn + c = 6*300 + 100 = 1900.
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