document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209574>Question 1209574</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A school teams contain 68 students 33 do field events 40 do track events 23 do swimming  14 do both field and track events  8 do both swimming and field events.  If 15 students do field event only and 10 do both swimming and track events how many students do<BR>\n" );
document.write( "1. Swimming only <BR>\n" );
document.write( "2. Track events only<BR>\n" );
document.write( "3. All three events<BR>\n" );
document.write( "Solve with venn diagram  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849543 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=849543\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "F = students who do field events<BR>\n" );
document.write( "T = students who do track events<BR>\n" );
document.write( "S = students who swim\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's a blank template for a Venn Diagram with 3 overlapped circles.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-4%2C4%2C4%2C4%29%2C%0D%0Aline%284%2C4%2C4%2C-4%29%2C%0D%0Aline%284%2C-4%2C-4%2C-4%29%2C%0D%0Aline%28-4%2C-4%2C-4%2C4%29%2C%0D%0A%0D%0Acircle%28-1%2C1%2C2%29%2Ccircle%281%2C1%2C2%29%2Ccircle%280%2C-0.7321%2C2%29%2C%0D%0A%0D%0Alocate%28-2.8%2C2.8%2CF%29%2C%0D%0Alocate%282.6%2C2.8%2CT%29%2C%0D%0Alocate%281%2C-2.6%2CS%29%2C%0D%0Alocate%28-2%2C1.8%2Ca%29%2C%0D%0Alocate%280%2C1.8%2Cb%29%2C%0D%0Alocate%282%2C1.8%2Cc%29%2C%0D%0Alocate%28-1.5%2C0%2Cd%29%2C%0D%0Alocate%280%2C0.4%2Ce%29%2C%0D%0Alocate%281.2%2C0%2Cf%29%2C%0D%0Alocate%280%2C-1.5%2Cg%29%2C%0D%0Alocate%283%2C-1.5%2Ch%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Letters a through h represent the 8 distinct regions.<BR>\n" );
document.write( "They are the subsets of people such that,<BR>\n" );
document.write( "a = number who do field events only<BR>\n" );
document.write( "b = number who do field and track, but not swim<BR>\n" );
document.write( "c = number who do track events only<BR>\n" );
document.write( "and so on.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's the info from the instructions presented as a numbered list.<ol><li>There are 68 students total</li><li>There are 33 students doing field events</li><li>There are 40 students doing track events</li><li>There are 23 students swimming</li><li>There are 14 students do both field and track events</li><li>There are 8 students doing both swimming and field events.</li><li>There are 15 students doing field events only</li><li>There are 10 doing both swimming and track events</li></ol>There are 8 bits of info to somehow match with the 8 letters a through h. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I'll refer to those facts as <font color=blue>fact (1)</font> through <font color=blue>fact (8)</font> in the order presented above.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font color=blue>fact (7)</font> leads immediately to a = 15.<BR>\n" );
document.write( "<font color=blue>fact (5)</font> tells us that b+e = 14 while <font color=blue>fact (6)</font> leads to d+e = 8<BR>\n" );
document.write( "Adding these equations straight down gives b+d+2e = 22 and rewrites to b+d = 22-2e\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then turn to <font color=blue>fact (2)</font><BR>\n" );
document.write( "a+b+d+e = everyone in circle F = 33 students<BR>\n" );
document.write( "a+b+d+e = 33<BR>\n" );
document.write( "15+b+d+e = 33 .......  plug in a = 15<BR>\n" );
document.write( "15+(22-2e)+e = 33 ....... replace b+d with 22-2e<BR>\n" );
document.write( "15+22-e = 33<BR>\n" );
document.write( "37-e = 33<BR>\n" );
document.write( "-e = 33-37<BR>\n" );
document.write( "-e = -4<BR>\n" );
document.write( "e = 4<BR>\n" );
document.write( "We have determined there are 4 students who performed at all 3 events.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then,<BR>\n" );
document.write( "b+e = 14 becomes b+4 = 14 which solves to b = 10<BR>\n" );
document.write( "d+e = 8 becomes d+4 = 8 which solves to d = 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We have this so far<BR>\n" );
document.write( "a = 15<BR>\n" );
document.write( "b = 10<BR>\n" );
document.write( "d = 4<BR>\n" );
document.write( "e = 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's update the Venn Diagram to reflect this.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-4%2C4%2C4%2C4%29%2C%0D%0Aline%284%2C4%2C4%2C-4%29%2C%0D%0Aline%284%2C-4%2C-4%2C-4%29%2C%0D%0Aline%28-4%2C-4%2C-4%2C4%29%2C%0D%0A%0D%0Acircle%28-1%2C1%2C2%29%2Ccircle%281%2C1%2C2%29%2Ccircle%280%2C-0.7321%2C2%29%2C%0D%0A%0D%0Alocate%28-2.8%2C2.8%2CF%29%2C%0D%0Alocate%282.6%2C2.8%2CT%29%2C%0D%0Alocate%281%2C-2.6%2CS%29%2C%0D%0Alocate%28-2%2C1.8%2C15%29%2C%0D%0Alocate%280%2C1.8%2C10%29%2C%0D%0Alocate%282%2C1.8%2Cc%29%2C%0D%0Alocate%28-1.5%2C0%2C4%29%2C%0D%0Alocate%280%2C0.4%2C4%29%2C%0D%0Alocate%281.2%2C0%2Cf%29%2C%0D%0Alocate%280%2C-1.5%2Cg%29%2C%0D%0Alocate%283%2C-1.5%2Ch%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As a check so far, the numbers in circle F should add to 33 students who did field events.<BR>\n" );
document.write( "15+10+4+4 = 33<BR>\n" );
document.write( "So far so good.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now onto <font color=blue>fact (8)</font> which gives the equation 4+f = 10. It solves to f = 6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once you know f, you can determine g. <BR>\n" );
document.write( "This is because all of the values in circle S should add to 23. See <font color=blue>fact (4)</font>.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4+4+f+g = 23<BR>\n" );
document.write( "4+4+6+g = 23<BR>\n" );
document.write( "14+g = 23<BR>\n" );
document.write( "g = 23-14<BR>\n" );
document.write( "g = 9\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Similarly, we can determine the value of c when using <font color=blue>fact (3)</font>.<BR>\n" );
document.write( "10+c+4+f = 40<BR>\n" );
document.write( "10+c+4+6 = 40<BR>\n" );
document.write( "c+20 = 40<BR>\n" );
document.write( "c = 40-20<BR>\n" );
document.write( "c = 20\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "At this point we have the following<BR>\n" );
document.write( "a = 15<BR>\n" );
document.write( "b = 10<BR>\n" );
document.write( "c = 20<BR>\n" );
document.write( "d = 4<BR>\n" );
document.write( "e = 4<BR>\n" );
document.write( "f = 6<BR>\n" );
document.write( "g = 9<BR>\n" );
document.write( "These seven values add to 68<BR>\n" );
document.write( "<font color=blue>fact (1)</font> says that this is the total number of students, so each student does at least one of the three events mentioned.<BR>\n" );
document.write( "This means h = 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is the completed Venn Diagram <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-4%2C4%2C4%2C4%29%2C%0D%0Aline%284%2C4%2C4%2C-4%29%2C%0D%0Aline%284%2C-4%2C-4%2C-4%29%2C%0D%0Aline%28-4%2C-4%2C-4%2C4%29%2C%0D%0A%0D%0Acircle%28-1%2C1%2C2%29%2Ccircle%281%2C1%2C2%29%2Ccircle%280%2C-0.7321%2C2%29%2C%0D%0A%0D%0Alocate%28-2.8%2C2.8%2CF%29%2C%0D%0Alocate%282.6%2C2.8%2CT%29%2C%0D%0Alocate%281%2C-2.6%2CS%29%2C%0D%0Alocate%28-2%2C1.8%2C15%29%2C%0D%0Alocate%280%2C1.8%2C10%29%2C%0D%0Alocate%282%2C1.8%2C20%29%2C%0D%0Alocate%28-1.5%2C0%2C4%29%2C%0D%0Alocate%280%2C0.4%2C4%29%2C%0D%0Alocate%281.2%2C0%2C6%29%2C%0D%0Alocate%280%2C-1.5%2C9%29%2C%0D%0Alocate%283%2C-1.5%2C0%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answers:<BR>\n" );
document.write( "Only swimming = <font color=red>9</font><BR>\n" );
document.write( "Only track = <font color=red>20</font><BR>\n" );
document.write( "All three events = <font color=red>4</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "More practice with Venn Diagrams is found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/playground/test.faq.question.1209540.html\">here</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );