Question 364938: Hospital capacity. When Memorial Hospital is filled to capacity, it has 18 more people in semiprivate rooms (two patients to a room) than in private rooms. The room rates are $200 per day for a private room and $150 per day for a semiprivate room. If the total receipts for rooms is $17,400 per day when all are full, then how many rooms of each type does the hospital have?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! s = number of people in semi-private rooms.
p = number of people in private rooms.
the total revenues when the hospital is full is equal to 200*p + 150*s.
since the total revenue when the hospital is full is 17,400, this means that the equation for total revenue becomes:
200*p + 150*s = 17,400
that's one of your equations.
when the hospital is full, there are 18 more people in semi-private rooms than in private rooms.
the number of people in private rooms is equal to p.
the number of people in semi-private rooms is equal to s.
this statement is saying that when the hospital is full:
s = p + 18
you now have 2 equations that can be solved simultaneously to get your answer.
those equations are:
s = p + 18
200*p + 150*s = 17,400
substitute for s in the second equation to get:
200*p + 150*(p + 18) = 17,400
simplify to get:
200*p + 150*p + 18*150 = 17,400
simplify further to get:
200*p + 150*p + 2,700 = 17,400
subtract 2,700 from both sides of the equation to get:
200*p + 150*p = 17,400 - 2,700
combine like terms to get:
350*p = 14,700
divide both sides by 350 to get:
p = 42
since we know that s = p + 18, then:
s = 42 + 18 = 60
we have:
p = 42
s = 60
in our revenue equation, we can substitute these values to get:
42*200 + 60*150 = 17,400 which becomes:
8,400 + 9,000 = 17,400 which becomes:
17,400 = 17,400 which is true, confirming that the numbers for p and s are good.
we now know the number of people in private rooms and the number of people in semi-private rooms.
p = 42 is the number of people in private rooms.
s = 60 is the number of people in semi-private rooms.
the question asked is how many rooms of each type does the hospital have?
since there is 1 person per private room, then the number of private rooms is equal to 42 / 1 = 42.
since there are 2 persons per semi-private room, then the number of semi-private rooms is equal to 60 / 2 = 30.
the hospital has 42 private rooms and 30 semi-private rooms.
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