document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150787>Question 1150787</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a rural community on wells and septic systems, which had been surrounded by new developments, 149 voted to have town water provided, 117 wanted a sewer system, and 15 people wanted to remain on wells and septic systems because they could not afford town water or sewer. If 188 people voted, how many wanted both town water and a sewer system? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772239 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=772239\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "x = number who voted for both town water and a town sewer system\r<BR>\n" );
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document.write( "A = number of people who want town water provided, but no town sewer<BR>\n" );
document.write( "A = 149 - x<BR>\n" );
document.write( "this is because there are 149 people who voted for town water and x who voted for both. Visually this is represented by the region in circle W but outside circle S (see the Venn diagram below)\r<BR>\n" );
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document.write( "B = number of people who voted for a town sewer system, but no town water<BR>\n" );
document.write( "B = 117 - x<BR>\n" );
document.write( "we have 117 who voted for sewer, and x for both, so 117-x just want sewer only<BR>\n" );
document.write( "visually this region B is inside circle S but outside circle W\r<BR>\n" );
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document.write( "C = number of people who voted for both town water and town sewer system<BR>\n" );
document.write( "C = x<BR>\n" );
document.write( "region C is inside both circles W and S at the same time\r<BR>\n" );
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document.write( "D = number of people who voted for neither system<BR>\n" );
document.write( "D = 15<BR>\n" );
document.write( "these are the people that want to remain on wells and septic systems<BR>\n" );
document.write( "region D is outside both circles\r<BR>\n" );
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document.write( "Venn Diagram<BR>\n" );
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document.write( "There are four distinct regions in the venn diagram Each region labeled A,B,C,D in blue. The count for each region is shown in red. The universal set in this case refers to the set of all voters in this town.\r<BR>\n" );
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document.write( "So we have<BR>\n" );
document.write( "A = 149-x<BR>\n" );
document.write( "B = 117-x<BR>\n" );
document.write( "C = x<BR>\n" );
document.write( "D = 15<BR>\n" );
document.write( "These four values must add to 188 as there were 188 people who voted overall.\r<BR>\n" );
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document.write( "A+B+C+D = total number of voters<BR>\n" );
document.write( "( 149-x ) + ( 117-x ) + ( x ) + ( 15 ) = 188<BR>\n" );
document.write( "(149+117+15)+(-x-x+x) = 188<BR>\n" );
document.write( "281-x = 188<BR>\n" );
document.write( "281-x-281 = 188-281<BR>\n" );
document.write( "-x = -93<BR>\n" );
document.write( "-x/(-1) = -93/(-1)<BR>\n" );
document.write( "x = 93\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>93 people</font> voted for both town water and a sewer system\r<BR>\n" );
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document.write( "Side note: plug this x value into the equations for A and B to find that...<BR>\n" );
document.write( "A = 149-x<BR>\n" );
document.write( "A = 149-93<BR>\n" );
document.write( "A = 56 people wanted town water but no town sewer<BR>\n" );
document.write( "and<BR>\n" );
document.write( "B = 117 - x<BR>\n" );
document.write( "B = 117 - 93<BR>\n" );
document.write( "B = 24 people wanted a town sewer system but no town water\r<BR>\n" );
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document.write( "As a check,<BR>\n" );
document.write( "A+B+C+D = 56+24+93+15 = 188<BR>\n" );
document.write( "and we get the proper total number of voters. <BR>\n" );
document.write( "This helps confirm the answer.<BR>\n" );
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