document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1123090>Question 1123090</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A survey of a group of 113 tourists was taken in St. Louis. The survey showed the following:<BR>\n" );
document.write( "62 of the tourists plan to visit Gateway Arch;<BR>\n" );
document.write( "48 plan to visit the zoo;<BR>\n" );
document.write( "11 plan to visit the Art Museum and the zoo, but not the Gateway Arch;<BR>\n" );
document.write( "14 plan to visit the Art Museum and the Gateway Arch, but not the zoo;<BR>\n" );
document.write( "16 plan to visit the Gateway Arch and the zoo, but not the Art Museum;<BR>\n" );
document.write( "8 plan to visit the Art Museum, the zoo, and the Gateway Arch;<BR>\n" );
document.write( "14 plan to visit none of the three places.\r<BR>\n" );
document.write( "\n" );
document.write( "How many plan to visit the Art Museum only </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #739350 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=739350\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "Break down what we know into 8 different facts<BR>\n" );
document.write( "<font color=\"blue\">Fact 1</font>: 113 tourists total</li><BR>\n" );
document.write( "<font color=\"blue\">Fact 2</font>: 62 of the tourists plan to visit Gateway Arch</li><BR>\n" );
document.write( "<font color=\"blue\">Fact 3</font>: 48 plan to visit the zoo</li><BR>\n" );
document.write( "<font color=\"blue\">Fact 4</font>: 11 plan to visit the Art Museum and the zoo, but not the Gateway Arch</li><BR>\n" );
document.write( "<font color=\"blue\">Fact 5</font>: 14 plan to visit the Art Museum and the Gateway Arch, but not the zoo</li><BR>\n" );
document.write( "<font color=\"blue\">Fact 6</font>: 16 plan to visit the Gateway Arch and the zoo, but not the Art Museum</li><BR>\n" );
document.write( "<font color=\"blue\">Fact 7</font>: 8 plan to visit the Art Museum, the zoo, and the Gateway Arch</li><BR>\n" );
document.write( "<font color=\"blue\">Fact 8</font>: 14 plan to visit none of the three places</li><BR>\n" );
document.write( "we'll use these facts to help set up the problem (venn diagram) and ultimately to answer the question.<BR>\n" );
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document.write( "Draw a venn diagram with 3 overlapping circles. The three circles will be located inside a rectangle. The rectangle represents all of the tourists, which we'll call set T. The numbers in circle A represent those who plan to visit the Art Museum. The numbers in circle G is for the Gateway Arch. Finally, those in circle Z are aiming to visit the zoo<BR>\n" );
document.write( "A = Art Museum<BR>\n" );
document.write( "G = Gateway Arch<BR>\n" );
document.write( "Z = Zoo<BR>\n" );
document.write( "T = set of all tourists<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/FRC0BTG.png\"><BR>\n" );
document.write( "We have 3 overlapping circles so there will be 2^3 = 8 different regions. I'll label the regions <font color=\"red\">R1</font>  through <font color=\"red\">R8</font><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/7ubADhm.png\"><BR>\n" );
document.write( "Definition of each region:<BR>\n" );
document.write( "<font color=\"red\">R1</font> = region inside circle A but outside of the other circles<BR>\n" );
document.write( "<font color=\"red\">R2</font> = region inside circle A and circle G but not inside circle Z<BR>\n" );
document.write( "<font color=\"red\">R3</font> = region inside circle G but outside of the other circles<BR>\n" );
document.write( "<font color=\"red\">R4</font> = region inside circle A and circle Z but not inside circle G<BR>\n" );
document.write( "<font color=\"red\">R5</font> = region inside all three circles<BR>\n" );
document.write( "<font color=\"red\">R6</font> = region inside circle G and circle Z but not inside circle A<BR>\n" );
document.write( "<font color=\"red\">R7</font> = region inside circle Z but outside of the other circles<BR>\n" );
document.write( "<font color=\"red\">R8</font> = region outside all of the circles\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The expanded definitions (based on the context of the problem) would be:<BR>\n" );
document.write( "<font color=\"red\">R1</font> = set of people who want to see the art museum only (no arch, no zoo)<BR>\n" );
document.write( "<font color=\"red\">R2</font> = people who want to see the art museum and gateway arch, but not the zoo<BR>\n" );
document.write( "<font color=\"red\">R3</font> = set of people who want to see the gateway arch only (no art museum, no zoo)<BR>\n" );
document.write( "<font color=\"red\">R4</font> = people who want to see the art museum and zoo, but not the art museum<BR>\n" );
document.write( "<font color=\"red\">R5</font> = set of people who want to see all three locations<BR>\n" );
document.write( "<font color=\"red\">R6</font> = people who want to see the gateway arch and zoo, but not the art museum<BR>\n" );
document.write( "<font color=\"red\">R7</font> = set of people who want to see the zoo only (no art museum, no arch)<BR>\n" );
document.write( "<font color=\"red\">R8</font> = set of people who don't want to see any of the three locations<BR>\n" );
document.write( "All regions <font color=\"red\">R1</font>  through <font color=\"red\">R8</font> are inside the rectangle T, to represent the set of all tourists<BR>\n" );
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document.write( "Now follow these steps (refer to the blue color coded facts that I wrote above at the top of the problem)<BR>\n" );
document.write( "<TABLE  border=1>\n" );
document.write( "<tr><th>Number</th><th>Instruction</th><th>Reason</th></tr><tr><TD >1</TD><TD >Replace <font color=\"red\">R8</font> with 14</TD><TD ><font color=\"blue\">Fact 8</font></TD></tr><tr><TD >2</TD><TD >Replace <font color=\"red\">R5</font> with 8</TD><TD ><font color=\"blue\">Fact 7</font></TD></tr><tr><TD >3</TD><TD >Replace <font color=\"red\">R2</font> with 14</TD><TD ><font color=\"blue\">Fact 5</font></TD></tr><tr><TD >4</TD><TD >Replace <font color=\"red\">R4</font> with 11</TD><TD ><font color=\"blue\">Fact 4</font></TD></tr><tr><TD >5</TD><TD >Replace <font color=\"red\">R6</font> with 16</TD><TD ><font color=\"blue\">Fact 6</font></TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "After completing those steps, you should have this venn diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/ktZ2dI4.png\"><BR>\n" );
document.write( "Since we have 14, 8 and 16 people in the \"G\" circle, this means that 14+8+16 = 38 people plan to go to the Gateway Arch in some way. This isn't the full amount of people that want to go to the Gateway Arch since <font color=\"blue\">Fact 2</font> states the total count for the Gateway Arch is actually 62. So we're short 62-38 = 24 people. This value 24 will replace <font color=\"red\">R3</font>. The value 24 represents those who only want to see the gateway arch (and nothing else).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Using similar logic, we have 11+8+16 = 33 people so far in the \"Zoo\" circle, but according to <font color=\"blue\">Fact 3</font>, that grand total for people who want to go to the Zoo should be 48. We're short 48-33 = 15 people, so this value (15) will go in region <font color=\"red\">R7</font>. The 15 is inside circle Z but outside the other circles, meaning that 15 people only want to see the zoo (and nothing else).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/755y1Vc.png\"><BR>\n" );
document.write( "The only region left to fill in is <font color=\"red\">R1</font>, which is the number of people who only want to go to the art museum (skipping out on the gateway arch and not going to the zoo either).<BR>\n" );
document.write( "From <font color=\"blue\">Fact 1</font>, we know that 113 tourists were surveyed. If x represents the value that replaces <font color=\"red\">R1</font>, then we can add up all the numbers, set it equal to 113 and solve for x<BR>\n" );
document.write( "x+14+24+11+8+16+15+14 = 113<BR>\n" );
document.write( "x+102 = 113<BR>\n" );
document.write( "x+102-102 = 113-102<BR>\n" );
document.write( "x = 11<BR>\n" );
document.write( "So 11 will replace <font color=\"red\">R1</font><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/yaqNtfh.png\"><BR>\n" );
document.write( "-------------------------------------------------------------------------------------------------------------------------------<BR>\n" );
document.write( "We now have the completed venn diagram. Now onto the question<BR>\n" );
document.write( "Question: How many plan to visit the Art Museum only?<BR>\n" );
document.write( "Answer: 11<BR>\n" );
document.write( "This is simply the value that is in region A but outside the other circles. Once you have the full completed venn diagram, it's simply a matter of locating the proper region and writing down the value. In this case, the proper region is <font color=\"red\">R1</font>.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );