Question 364938
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.