document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180850>Question 1180850</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A group of faculty members at a small college operate a carpool to three kinds of activities; baseball games, the opera, and the theatre. Suppose there are 86 families in the carpool and that in a given month,<BR>\n" );
document.write( "11 families attend none of the activities 33 families go to the baseball games 35 families go to the opera<BR>\n" );
document.write( "39 families go to the theatre<BR>\n" );
document.write( "14 families go to just baseball games 17 families go to just the theatre<BR>\n" );
document.write( "15 families go to just opera<BR>\n" );
document.write( "a. How many families go to all three activities?<BR>\n" );
document.write( "b. How many families go to baseball games and the opera but not to the theatre?<BR>\n" );
document.write( "c. How many families go to at least two activities? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #810707 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=810707\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Draw a Venn diagram to follow the analysis below....<br><BR>\n" );
document.write( "Let x be the number who go to all three activities<br><BR>\n" );
document.write( "let a be the number who go to baseball games and the opera but not the theater<BR>\n" );
document.write( "let b be the number who go to baseball games and the theater but not the opera<BR>\n" );
document.write( "let c be the number who go to the opera and the theater but not to baseball games<br><BR>\n" );
document.write( "Then the given numbers give us these equations:<br><BR>\n" );
document.write( "33 go to baseball games: 14+a+b+x=33; a+b+x=19 [1]<BR>\n" );
document.write( "35 go to the opera: 15+a+c+x=35; a+c+x=20 [2]<BR>\n" );
document.write( "39 go to the theater: 17+b+c+x=39; b+c+x=22 [3]<br><BR>\n" );
document.write( "From [1] and [2] we can conclude<BR>\n" );
document.write( "b=a+2 [4]<BR>\n" );
document.write( "c=a+3 [5]<br><BR>\n" );
document.write( "86-11=75 go to at least one of the three activities:<BR>\n" );
document.write( "(14+15+17)+(a+a+2+a+3)+x=75; 3a+x=24 [6]<br><BR>\n" );
document.write( "Again, 35 go to the opera:<BR>\n" );
document.write( "15+a+a+3+x=35; 2a+x=17 [7]<br><BR>\n" );
document.write( "Then [6] and [7] give us<BR>\n" );
document.write( "a=7<BR>\n" );
document.write( "x=3<br><BR>\n" );
document.write( "From which we last get<BR>\n" );
document.write( "b=9<BR>\n" );
document.write( "c=10<br><BR>\n" );
document.write( "ANSWERS:<BR>\n" );
document.write( "a. go to all three: x=3<BR>\n" );
document.write( "b. go to baseball games and the opera but not the theater: a=7<BR>\n" );
document.write( "c. go to at least two activities: a+b+c+x=29<br><BR>\n" );
document.write( "Of course, answer c can also be obtained as (number who go to at least one activity) minus (number who go to only one of the three activities):<BR>\n" );
document.write( "75-(14+15+17)=29<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );