document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=63420>Question 63420</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A survey of 50 students found that 30 fad cats, 25 had dogs, 5 had white mice, 16 had both dogs and cats, 4 had both dogs and mice, 2 had both cats and mice, and only 1 had all three kinds of pets. How many students had no pets of these types?\r<BR>\n" );
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document.write( "can u please tell me how you found this answer </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #44134 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20054)</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=44134\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">A survey of 50 students found that 30 had cats, 25 had dogs, 5 had mice, \r\n" );
document.write( "16 had both dogs and cats, 4 had both dogs and mice, 2 had both cats and mice,\r\n" );
document.write( "and only 1 had all three kinds of pets. How many students had no pets of these\r\n" );
document.write( "types? \r\n" );
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document.write( "can u please tell me how you found this answer.\r\n" );
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document.write( "You are supposed to use a Venn diagram. But I can't draw a\r\n" );
document.write( "Venn diagram on here, so I'll have to use algebra instead.\r\n" );
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document.write( "Let X = the number who had no pets of these types\r\n" );
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document.write( "Let C = the number that had cats only, but no dogs or mice.\r\n" );
document.write( "Let D = the number that had dogs only, but no cats or mice.\r\n" );
document.write( "Let M = the number that had mice only, but no cats or dogs.\r\n" );
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document.write( "Let CD = the number that had cats and dogs only, but no mice.\r\n" );
document.write( "Let CM = the number that had cats and mice only, but no dogs.\r\n" );
document.write( "Let DM = the number that had dogs and mice only, but no cats.\r\n" );
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document.write( "Let CDM = the number that had all three kinds of animals.\r\n" );
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document.write( ">>...50 students...<<\r\n" );
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document.write( "Translation:\r\n" );
document.write( "eq. 1:    C + D + M + CD + CM + DM + CDM + X = 50 \r\n" );
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document.write( ">>...30 had cats...<<\r\n" );
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document.write( "Translation:\r\n" );
document.write( "eq. 2:    C + CD + CM + CDM = 30\r\n" );
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document.write( ">>...25 had dogs...<<\r\n" );
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document.write( "Translation:\r\n" );
document.write( "eq. 3:    D + CD + DM + CDM = 25\r\n" );
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document.write( ">>...5 had mice...<<\r\n" );
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document.write( "Translation:\r\n" );
document.write( "eq. 4:    M + CM + DM + CDM = 5\r\n" );
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document.write( ">>...16 had both dogs and cats...<<\r\n" );
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document.write( "Translation:\r\n" );
document.write( "eq. 5:    CD + CDM = 16\r\n" );
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document.write( ">>...4 had both dogs and mice...<< \r\n" );
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document.write( "Translation:\r\n" );
document.write( "eq. 6:    DM + CDM = 4 \r\n" );
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document.write( ">>...2 had both cats and mice...<< \r\n" );
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document.write( "Translation:\r\n" );
document.write( "eq. 7:    CM + CDM = 2\r\n" );
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document.write( ">>...only 1 had all three kinds of pets...<< \r\n" );
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document.write( "Translation:\r\n" );
document.write( "eq. 8:    CDM = 1\r\n" );
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document.write( "So we have 8 equations in 8 unknowns\r\n" );
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document.write( "eq. 1:    C + D + M + CD + CM + DM + CDM + X = 50 \r\n" );
document.write( "eq. 2:    C + CD + CM + CDM = 30\r\n" );
document.write( "eq. 3:    D + CD + DM + CDM = 25\r\n" );
document.write( "eq. 4:    M + CM + DM + CDM = 5\r\n" );
document.write( "eq. 5:    CD + CDM = 16\r\n" );
document.write( "eq. 6:    DM + CDM = 4 \r\n" );
document.write( "eq. 7:    CM + CDM = 2\r\n" );
document.write( "eq. 8:    CDM = 1\r\n" );
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document.write( "Use equation 8 to substitute 1 for CDM in the other 7 equations,\r\n" );
document.write( "and you have:\r\n" );
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document.write( "eq. 1:    C + D + M + CD + CM + DM + X = 49 \r\n" );
document.write( "eq. 2:    C + CD + CM = 29\r\n" );
document.write( "eq. 3:    D + CD + DM = 24\r\n" );
document.write( "eq. 4:    M + CM + DM = 4\r\n" );
document.write( "eq. 5:    CD = 15\r\n" );
document.write( "eq. 6:    DM = 3 \r\n" );
document.write( "eq. 7:    CM = 1\r\n" );
document.write( "eq. 8:    CDM = 1\r\n" );
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document.write( "Use equation 7 to substitute 1 for CM in eqs. 4, 2 and 1,\r\n" );
document.write( "and you have:\r\n" );
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document.write( "eq. 1:    C + D + M + CD + DM + X = 48 \r\n" );
document.write( "eq. 2:    C + CD = 28\r\n" );
document.write( "eq. 3:    D + CD + DM = 24\r\n" );
document.write( "eq. 4:    M + DM = 3\r\n" );
document.write( "eq. 5:    CD = 15\r\n" );
document.write( "eq. 6:    DM = 3 \r\n" );
document.write( "eq. 7:    CM = 1\r\n" );
document.write( "eq. 8:    CDM = 1\r\n" );
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document.write( "Use equation 6 to substitute 3 for DM in eqs. 4, 3, and 1,\r\n" );
document.write( "and you have:\r\n" );
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document.write( "eq. 1:    C + D + M + CD + X = 45 \r\n" );
document.write( "eq. 2:    C + CD = 28\r\n" );
document.write( "eq. 3:    D + CD = 21\r\n" );
document.write( "eq. 4:    M = 0\r\n" );
document.write( "eq. 5:    CD = 15\r\n" );
document.write( "eq. 6:    DM = 3 \r\n" );
document.write( "eq. 7:    CM = 1\r\n" );
document.write( "eq. 8:    CDM = 1\r\n" );
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document.write( "Use equation 5 to substitute 15 for CD in eqs. 3, 2, and 1,\r\n" );
document.write( "and you have:\r\n" );
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document.write( "eq. 1:    C + D + M + X = 30 \r\n" );
document.write( "eq. 2:    C = 13\r\n" );
document.write( "eq. 3:    D = 6\r\n" );
document.write( "eq. 4:    M = 0\r\n" );
document.write( "eq. 5:    CD = 15\r\n" );
document.write( "eq. 6:    DM = 3 \r\n" );
document.write( "eq. 7:    CM = 1\r\n" );
document.write( "eq. 8:    CDM = 1\r\n" );
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document.write( "Use equations 4, 3 and 2 to substitute 0 for M,\r\n" );
document.write( "6 for D, and 13 for C in eq. 1, and you have\r\n" );
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document.write( " C + D + M + X = 30\r\n" );
document.write( "13 + 6 + 0 + X = 30\r\n" );
document.write( "        19 + X = 30\r\n" );
document.write( "             X = 11\r\n" );
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document.write( "That's the answer.\r\n" );
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document.write( "Edwin</PRE> </SPAN>\n" );
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