document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1154733>Question 1154733</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A group of 2193 students were surveyed about the courses they were taking at their college with the following results:\r<BR>\n" );
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document.write( "1192 students said they were taking Dance.<BR>\n" );
document.write( "1143 students said they were taking English.<BR>\n" );
document.write( "1199 students said they were taking Math.<BR>\n" );
document.write( "583 students said they were taking Dance and English.<BR>\n" );
document.write( "697 students said they were taking English and Math.<BR>\n" );
document.write( "646 students said they were taking Dance and Math.<BR>\n" );
document.write( "372 students said they were taking all three courses.\r<BR>\n" );
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document.write( "a) Fill in the following Venn Diagram with the cardinality of each region.<BR>\n" );
document.write( "DanceEnglishMathI.<BR>\n" );
document.write( "335<BR>\n" );
document.write( "II.<BR>\n" );
document.write( "211<BR>\n" );
document.write( "III.<BR>\n" );
document.write( "235<BR>\n" );
document.write( "IV.<BR>\n" );
document.write( "274<BR>\n" );
document.write( "V.<BR>\n" );
document.write( "372<BR>\n" );
document.write( "VI.<BR>\n" );
document.write( "325<BR>\n" );
document.write( "VII.<BR>\n" );
document.write( "228<BR>\n" );
document.write( "VIII.<BR>\n" );
document.write( "213\r<BR>\n" );
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document.write( "a) How many students took Dance or Math, but not English?\r<BR>\n" );
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document.write( "b) How many students took Dance, English, or Math?\r<BR>\n" );
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document.write( "c) How many students took Dance & English or took English & Math?\r<BR>\n" );
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document.write( "d) How many students took English and Math, but not Dance?\r<BR>\n" );
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document.write( "e) How many students took Dance or didn't take English?\r<BR>\n" );
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document.write( "f) How many students took none of the courses? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #777250 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=777250\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A group of 2193 students were surveyed about the courses they were taking at<BR>\n" );
document.write( "their college with the following results: <BR>\n" );
document.write( "1192 students said they were taking Dance.<BR>\n" );
document.write( "1143 students said they were taking English.<BR>\n" );
document.write( "1199 students said they were taking Math.<BR>\n" );
document.write( "583 students said they were taking Dance and English.<BR>\n" );
document.write( "697 students said they were taking English and Math.<BR>\n" );
document.write( "646 students said they were taking Dance and Math.<BR>\n" );
document.write( "372 students said they were taking all three courses.<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=%3F%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=%3F%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=%3F%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=%3F%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=%3F%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=%3F%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=%3F%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=%3F%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "Venn diagram clues are nearly always give in the reverse order\r\n" );
document.write( "from the order in which you use them, so we take them in the\r\n" );
document.write( "reverse order and begin with the last clue and end with the first\r\n" );
document.write( "clue:\r\n" );
document.write( "</PRE>372 students said they were taking all three courses.<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So V=372. We fill that in:\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=%3F%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=%3F%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=%3F%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=%3F%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=%3F%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=%3F%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=%3F%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=372%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "</PRE>646 students said they were taking Dance and Math.<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So IV+V=646, so substituting V=372, IV=274. We fill that in:\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=%3F%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=%3F%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=%3F%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=%3F%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=274%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=%3F%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=%3F%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=372%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "</PRE>697 students said they were taking English and Math.<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So V+VI=646, so substituting V=372, VI=325. Fill that in:\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=%3F%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=%3F%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=%3F%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=325%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=274%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=%3F%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=%3F%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=372%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "</PRE>583 students said they were taking Dance and English.<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So II+V=583, so substituting V=372, we get II=211. We fill that in:\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=%3F%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=%3F%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=%3F%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=325%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=274%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=211%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=%3F%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=372%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "</PRE>1199 students said they were taking Math.<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So IV+V+VI+VII=n(M)=1199 and substituting for IV,V and VI gives VII=228.\r\n" );
document.write( "We fill that in:\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=%3F%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=%3F%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=228%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=325%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=274%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=211%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=%3F%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=372%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "</PRE>1143 students said they were taking English.<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So II+III+V+VI=n(E)=1199 and substituting for II,V and VI gives III=235.\r\n" );
document.write( "We fill that in:\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=%3F%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=%3F%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=228%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=325%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=274%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=211%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=235%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=372%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "</PRE>1192 students said they were taking Dance.<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So I+II+IV+V=n(D)=1199 and substituting for II,IV and V gives I=335.\r\n" );
document.write( "So fill that in:\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=335%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=%3F%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=228%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=325%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=274%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=211%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=235%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=372%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "</PRE>A group of 2193 students were surveyed about the courses they were taking....<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So I+II+III+IV+V+VI+VII+VIII=n(U)=n(universal set)=2193, and substituting for\r\n" );
document.write( "I,II,III,IV,V,VI and VII gives VIII=213.  So fill that in (the number who\r\n" );
document.write( "took none of the courses):\r\n" );
document.write( "\r\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28450%2C450%2C-4%2C4%2C-5%2C4%2C%0D%0Arectangle%28-4%2C-3.5%2C4%2C4%29%2C%0D%0Acircle%280%2C-.5%2C2%29%2C%0D%0Alocate%28-2.3%2C2%2C%22I=335%22%29%2C%0D%0Alocate%28-3.5%2C-2%2C%22VIII=213%22%29%2C%0D%0Alocate%280%2C-2.7%2CM%29%2C%0D%0Alocate%28-.45%2C-1%2C%22VII=228%22%29%2C%0D%0Alocate%28.6%2C.4%2C%22VI=325%22%29%2C+%0D%0Acircle%28sqrt%282%29%2Csqrt%282%29%2C2%29%2C+%0D%0Alocate%28-3.5%2C2.5%2CD%29%2C%0D%0Acircle%28-sqrt%282%29%2Csqrt%282%29%2C2%29%2C%0D%0Alocate%283.5%2C2.5%2CE%29%2C%0D%0Alocate%28-1.5%2C.5%2C%22IV=274%22%29%2C%0D%0Alocate%28-.4%2C2.3%2C%22II=211%22%29%2C%0D%0Alocate%281.8%2C2%2C%22III=235%22%29%2C%0D%0Alocate%28-.4%2C1.1%2C%22V=372%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
document.write( "</PRE>a) How many students took Dance or Math, but not English?<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "They were I,IV and VII, or 335+274+228=837\r\n" );
document.write( "</PRE>b) How many students took Dance, English, or Math?<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "That's all 2193 except those 213 in VIII who took none of the courses.\r\n" );
document.write( "\r\n" );
document.write( "2193-VIII = 2193-213 = 1980\r\n" );
document.write( "</PRE>c) How many students took Dance & English or took English & Math?<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Those that took Dance & English were II+V = 211+372\r\n" );
document.write( "Those that took English & Math were V+VI = 372+325\r\n" );
document.write( "But WE MUST BE CAREFUL HERE!!! If we added all those together we\r\n" );
document.write( "would be counting the 372 TWICE! So we must add it only once, so the\r\n" );
document.write( "answer here is II+V+VI = 211+372+325 = 908\r\n" );
document.write( "</PRE>d) How many students took English and Math, but not Dance?<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"> \r\n" );
document.write( "\r\n" );
document.write( "That's VI, which is 325 \r\n" );
document.write( "</PRE>e) How many students took Dance or didn't take English?<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"> \r\n" );
document.write( "Those that took Dance were given as 1192 which was I+II+IV+V\r\n" );
document.write( "Those that didn't take English were I+IV+VII+VIII.\r\n" );
document.write( "But AGAIN WE MUST BE CAREFUL HERE!!! If we added all those together we\r\n" );
document.write( "would be counting the I and IV TWICE! So we must add those only once, so the\r\n" );
document.write( "answer here is I+II+IV+V+VII+VIII = 335+211+274+372+228+213 = 1633    \r\n" );
document.write( "</PRE>f) How many students took none of the courses?<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "We've already answered that.  It's VIII or 213.\r\n" );
document.write( "\r\n" );
document.write( "Edwin</PRE> \r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );