document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=364938>Question 364938</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #260163 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=260163\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> s = number of people in semi-private rooms.<BR>\n" );
document.write( "p = number of people in private rooms.\r<BR>\n" );
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document.write( "the total revenues when the hospital is full is equal to 200*p + 150*s.\r<BR>\n" );
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document.write( "since the total revenue when the hospital is full is 17,400, this means that the equation for total revenue becomes:\r<BR>\n" );
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document.write( "200*p + 150*s = 17,400\r<BR>\n" );
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document.write( "that's one of your equations.\r<BR>\n" );
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document.write( "when the hospital is full, there are 18 more people in semi-private rooms than in private rooms.\r<BR>\n" );
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document.write( "the number of people in private rooms is equal to p.\r<BR>\n" );
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document.write( "the number of people in semi-private rooms is equal to s.\r<BR>\n" );
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document.write( "this statement is saying that when the hospital is full:\r<BR>\n" );
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document.write( "s = p + 18\r<BR>\n" );
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document.write( "you now have 2 equations that can be solved simultaneously to get your answer.\r<BR>\n" );
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document.write( "those equations are:\r<BR>\n" );
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document.write( "s = p + 18<BR>\n" );
document.write( "200*p + 150*s = 17,400\r<BR>\n" );
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document.write( "substitute for s in the second equation to get:\r<BR>\n" );
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document.write( "200*p + 150*(p + 18) = 17,400\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "200*p + 150*p + 18*150 = 17,400\r<BR>\n" );
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document.write( "simplify further to get:\r<BR>\n" );
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document.write( "200*p + 150*p + 2,700 = 17,400\r<BR>\n" );
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document.write( "subtract 2,700 from both sides of the equation to get:\r<BR>\n" );
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document.write( "200*p + 150*p = 17,400 - 2,700\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "350*p = 14,700\r<BR>\n" );
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document.write( "divide both sides by 350 to get:\r<BR>\n" );
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document.write( "p = 42\r<BR>\n" );
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document.write( "since we know that s = p + 18, then:\r<BR>\n" );
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document.write( "s = 42 + 18 = 60\r<BR>\n" );
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document.write( "we have:\r<BR>\n" );
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document.write( "p = 42<BR>\n" );
document.write( "s = 60\r<BR>\n" );
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document.write( "in our revenue equation, we can substitute these values to get:\r<BR>\n" );
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document.write( "42*200 + 60*150 = 17,400 which becomes:\r<BR>\n" );
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document.write( "8,400 + 9,000 = 17,400 which becomes:\r<BR>\n" );
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document.write( "17,400 = 17,400 which is true, confirming that the numbers for p and s are good.\r<BR>\n" );
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document.write( "we now know the number of people in private rooms and the number of people in semi-private rooms.\r<BR>\n" );
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document.write( "p = 42 is the number of people in private rooms.<BR>\n" );
document.write( "s = 60 is the number of people in semi-private rooms.\r<BR>\n" );
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document.write( "the question asked is how many rooms of each type does the hospital have?\r<BR>\n" );
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document.write( "since there is 1 person per private room, then the number of private rooms is equal to 42 / 1 = 42.\r<BR>\n" );
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document.write( "since there are 2 persons per semi-private room, then the number of semi-private rooms is equal to 60 / 2 = 30.\r<BR>\n" );
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document.write( "the hospital has 42 private rooms and 30 semi-private rooms.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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