document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=191874>Question 191874</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780321501080/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I am taking my first online algebra course and it scares the heck out of me.  Can someone please help me with a few problems I missed on a quiz. I am lost!  Here is the first one:<BR>\n" );
document.write( "In a survey of 100 consumers, 33 indicated that they were going to buy a new car, 18 said they were going to buy a new refrigerator, and 34 said they were going to buy a new washer. Of these, 7 were going to buy both a car and a refrigerator, 15 were going to buy a car and a washer, and 9 were going to buy a washer and a refrigerator. Three consumers indicated that they were going to buy all three items. And she wants a diagram too! Help me please.\r<BR>\n" );
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document.write( "Construct a Venn diagram, label your diagram clearly.  <BR>\n" );
document.write( "Use your diagram to answer the following questions:\r<BR>\n" );
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document.write( "(a)	How many were going to buy only a car? <BR>\n" );
document.write( "(b)	How many were going to buy only a washer? <BR>\n" );
document.write( "(c)	How many were going to buy only a refrigerator<BR>\n" );
document.write( "(d)	How many were going to buy a car and a washer but not a refrigerator? <BR>\n" );
document.write( "(e)	How many were going to buy none of these items? \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #143983 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=143983\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "To start things off, let's draw three blank overlapping circles and label them \"car\", \"refrigerator\", and \"washer\". Now draw a large rectangle that surrounds the three circles and label this rectangle U (for the universal set).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/9step1.png\">\r<BR>\n" );
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document.write( "Since \"Three consumers indicated that they were going to buy all three items\", this means that the value \"3\" goes in the direct center of the overlapping circles.\r<BR>\n" );
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document.write( "Now because \"9 were going to buy a washer and a refrigerator\", this means that <i>some</i> of those 9 people might have bought all three. So to find the ones who ONLY bought a washer and a refrigerator (not all three), we need to subtract 3 from 9 to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-3=6 ALIGN=MIDDLE  ALT=\"9-3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this tells us that 6 people bought a washer and a refrigerator (and not all three). So place this value in the overlapping region between the circles labeled \"refrigerator\" and \"washer\".\r<BR>\n" );
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document.write( "Also, since \"15 were going to buy a car and a washer\" and we just want to find those who ONLY bought a a car and a washer (not all three), we just subtract 3 from 15 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-3=12 ALIGN=MIDDLE  ALT=\"15-3=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So 12 people bought a car and a washer (not all three). Now place this value in the region between the \"car\" and \"washer\" circles.\r<BR>\n" );
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document.write( "Since \"7 were going t o buy both a car and a refrigerator\", and we're only interested in those who ONLY bought the two (not all three), we subtract 3 from 7 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-3=4 ALIGN=MIDDLE  ALT=\"7-3=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So there are 4 people who bought a car and a refrigerator (not all three). Stick this value in the region between the \"car\" and \"refrigerator\" circles.\r<BR>\n" );
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document.write( "So we should have the following so far:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/9step2.png\">\r<BR>\n" );
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document.write( "Now because \"34 said they were going to buy a new washer\" and we're only interested in the number who ONLY bought a washer (not a car or refrigerator), we need to subtract the number who bought all three (3 people), the number who bought a car and washer (12 people), and the number of people who bought a refrigerator and washer (6 people) from 34 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=34-3-12-6=13 ALIGN=MIDDLE  ALT=\"34-3-12-6=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So there are 13 people who ONLY bought a washer. Place this value in the \"washer\" circle (and not in any overlapping regions)\r<BR>\n" );
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document.write( "If the method above is a bit cumbersome, then let's try another method in finding the number of people who ONLY bought a car:\r<BR>\n" );
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document.write( "Since 12 people bought a car and a washer, 4 people bought a car and a refrigerator, and 3 people bought all three, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2B4%2B3=18 ALIGN=MIDDLE  ALT=\"12%2B4%2B3=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > people bought AT LEAST 2 items (where one of the items is a car). Since we ONLY want the car, we need to subtract this from 33 (the number people who bought a car and maybe something else) to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33-18=15 ALIGN=MIDDLE  ALT=\"33-18=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So 15 people ONLY bought a car. Place this value in the \"car\" circle (and not in any overlapping regions)\r<BR>\n" );
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document.write( "Because 4 people bought a refrigerator and a car, 6 people bought a refrigerator and a washer, and 3 bought all three, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B6%2B3=13 ALIGN=MIDDLE  ALT=\"4%2B6%2B3=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > people bought AT LEAST two items in which one item is a refrigerator. Subtract this value from 18 (the number who bought a refrigerator and maybe something more) to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18-13=5 ALIGN=MIDDLE  ALT=\"18-13=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So 5 people bought ONLY a refrigerator. Place this value in the \"refrigerator\" circle (and not in any overlapping regions)\r<BR>\n" );
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document.write( "Finally, add up EVERY element that is in every region to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B6%2B12%2B4%2B13%2B15%2B5=58 ALIGN=MIDDLE  ALT=\"3%2B6%2B12%2B4%2B13%2B15%2B5=58\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So 58 people bought AT LEAST one item (either a car, refrigerator, or washer). Subtract this value from the number who participated (100 people) to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100-58=42 ALIGN=MIDDLE  ALT=\"100-58=42\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So 42 people did NOT buy any of the items listed. Place this value outside the three circles but inside the set U (the rectangle).\r<BR>\n" );
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document.write( "So we should now have the completed Venn diagram:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/9step3.png\">\r<BR>\n" );
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document.write( "Now let's answer the questions:\r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "How many were going to buy only a car?\r<BR>\n" );
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document.write( "Looking at the diagram (and going over our previous work), we get 15 people only bought a car.\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "How many were going to buy only a washer?\r<BR>\n" );
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document.write( "Similarly, we see from the Venn diagram that 13 people only bought a washer.\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "How many were going to buy only a refrigerator?\r<BR>\n" );
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document.write( "From the diagram, we see that 5 people only bought a refrigerator.\r<BR>\n" );
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document.write( "d) How many were going to buy a car and a washer but not a refrigerator?\r<BR>\n" );
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document.write( "From the diagram, we see that the number that lies in the region between the \"car\" and \"washer\" is 12. So 12 people bought a car and a washer but not a refrigerator.\r<BR>\n" );
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document.write( "e) How many were going to buy none of these items?\r<BR>\n" );
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document.write( "Since the value 42 is outside all of the circles, this means that 42 people did not buy any of these items.\r<BR>\n" );
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