document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1078801>Question 1078801</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 25 students in a class. Of these, 14 speak Spanish, 12 speak French, 6 speak French and<BR>\n" );
document.write( "Spanish, 5 speak German and Spanish, 2 speak all three languages. Each of the 6 who speak German<BR>\n" );
document.write( "speaks another one of these languages as well.\r<BR>\n" );
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document.write( "(a) How many speak both French and German?<BR>\n" );
document.write( "(b) How many speak none of these three languages?<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 #693279 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=693279\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let s = number of students who speak spanish<BR>\n" );
document.write( "let f = number of students who speak french<BR>\n" );
document.write( "let g = number of students who speak german.\r<BR>\n" );
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document.write( "let sf = number of students who speak spanish and french<BR>\n" );
document.write( "let sg = number of students who speak spanish and german<BR>\n" );
document.write( "let fg = number of students who speak french and german\r<BR>\n" );
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document.write( "let sfg = number of students who speak spanish and french and german.\r<BR>\n" );
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document.write( "all of these categories include subtending categories.\r<BR>\n" );
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document.write( "this means that:\r<BR>\n" );
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document.write( "s includes sf and sg and sfg<BR>\n" );
document.write( "f includes sf and fg and sfg<BR>\n" );
document.write( "g includes sg and fg and sfg\r<BR>\n" );
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document.write( "you want to purify these categories so that they do not include any subtending categories.\r<BR>\n" );
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document.write( "these pure categories will have an o (for only) letter added to the end of them.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "sfgo = sfg\r<BR>\n" );
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document.write( "sfo = sf - sfgo<BR>\n" );
document.write( "sgo = sg - sfgo<BR>\n" );
document.write( "fgo = fg - sfgo\r<BR>\n" );
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document.write( "so = s - sfo - sgo - sgfo<BR>\n" );
document.write( "fo = f - sfo - fgo - sgfo<BR>\n" );
document.write( "go = g - sgo - fgo - sgfo\r<BR>\n" );
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document.write( "your formula for t will be:\r<BR>\n" );
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document.write( "t = so + fo + go + sfo + sgo + fgo + sfgo + n\r<BR>\n" );
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document.write( "t is the total which is equal to 25<BR>\n" );
document.write( "n represent the students who aren't in each category.\r<BR>\n" );
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document.write( "you start with:\r<BR>\n" );
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document.write( "s = 14<BR>\n" );
document.write( "f = 12<BR>\n" );
document.write( "g = 6<BR>\n" );
document.write( "sf = 6<BR>\n" );
document.write( "sg = 5<BR>\n" );
document.write( "fg = x<BR>\n" );
document.write( "sfg = 2<BR>\n" );
document.write( "n = y\r<BR>\n" );
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document.write( "fg = x because we don't know what that number is.<BR>\n" );
document.write( "n = y because we don't know what that number is either.\r<BR>\n" );
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document.write( "first you want to find sfgo.\r<BR>\n" );
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document.write( "sfgo = sfg = 2\r<BR>\n" );
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document.write( "next you want to find sfo\r<BR>\n" );
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document.write( "sfo = sf - sfgo = 6 - 2 = 4\r<BR>\n" );
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document.write( "next you want to find sgo\r<BR>\n" );
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document.write( "sgo = sg - sfgo = 5 - 2 = 3\r<BR>\n" );
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document.write( "next you ant to find fgo\r<BR>\n" );
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document.write( "fgo = fg - sfg = x - 2 = x-2\r<BR>\n" );
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document.write( "so far you have:\r<BR>\n" );
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document.write( "s = 14<BR>\n" );
document.write( "f = 12<BR>\n" );
document.write( "g = 6<BR>\n" );
document.write( "sf = 6<BR>\n" );
document.write( "sfo = 4<BR>\n" );
document.write( "sg = 5<BR>\n" );
document.write( "sgo = 3<BR>\n" );
document.write( "fg = x<BR>\n" );
document.write( "fgo = x-2<BR>\n" );
document.write( "sft = 2<BR>\n" );
document.write( "sfgo = 2\r<BR>\n" );
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document.write( "now you want to find so.\r<BR>\n" );
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document.write( "so = s - sfo - sgo - sgfo = 14 - 4 - 3 - 2 = 5\r<BR>\n" );
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document.write( "now you want to find fo.\r<BR>\n" );
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document.write( "fo = f - sfo - fgo - sgfo = 12 - 4 - (x-2) - 2 = 8-x\r<BR>\n" );
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document.write( "now you want to find go.\r<BR>\n" );
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document.write( "go = g - sgo - fgo - sfgo = 6 - 3 - (x-2) - 2 = 3-x\r<BR>\n" );
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document.write( "so far you have:\r<BR>\n" );
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document.write( "s = 14<BR>\n" );
document.write( "so = 5<BR>\n" );
document.write( "f = 12<BR>\n" );
document.write( "fo = 8-x<BR>\n" );
document.write( "g = 6<BR>\n" );
document.write( "go = 3-x<BR>\n" );
document.write( "sf = 6<BR>\n" );
document.write( "sfo = 4<BR>\n" );
document.write( "sg = 5<BR>\n" );
document.write( "sgo = 3<BR>\n" );
document.write( "fg = x<BR>\n" );
document.write( "fgo = x-2<BR>\n" );
document.write( "sft = 2<BR>\n" );
document.write( "sfgo = 2\r<BR>\n" );
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document.write( "you know that go must be equal to 0 because you are told that each of the 6 students who speak german speak another language.\r<BR>\n" );
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document.write( "sincego = 3-x, this means that x must be equal to 3 because 3-3 = 0\r<BR>\n" );
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document.write( "assuming that x = 3, you get:\r<BR>\n" );
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document.write( "s = 14<BR>\n" );
document.write( "so = 5<BR>\n" );
document.write( "f = 12<BR>\n" );
document.write( "fo = 8-3 = 5<BR>\n" );
document.write( "g = 6<BR>\n" );
document.write( "go = 3-3 = 0<BR>\n" );
document.write( "sf = 6<BR>\n" );
document.write( "sfo = 4<BR>\n" );
document.write( "sg = 5<BR>\n" );
document.write( "sgo = 3<BR>\n" );
document.write( "fg = x<BR>\n" );
document.write( "fgo = 3-2 = 1<BR>\n" );
document.write( "sft = 2<BR>\n" );
document.write( "sfgo = 2\r<BR>\n" );
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document.write( "to simplify these results, you end up with:\r<BR>\n" );
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document.write( "s = 14<BR>\n" );
document.write( "so = 5<BR>\n" );
document.write( "f = 12<BR>\n" );
document.write( "fo = 5<BR>\n" );
document.write( "g = 6<BR>\n" );
document.write( "go = 0<BR>\n" );
document.write( "sf = 6<BR>\n" );
document.write( "sfo = 4<BR>\n" );
document.write( "sg = 5<BR>\n" );
document.write( "sgo = 3<BR>\n" );
document.write( "fg = x<BR>\n" );
document.write( "fgo = 1<BR>\n" );
document.write( "sft = 2<BR>\n" );
document.write( "sfgo = 2\r<BR>\n" );
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document.write( "if you let k = the total number of students and if you let n = the number students who don't speak any of the 3 languages, your formula will become:\r<BR>\n" );
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document.write( "k = so + fo + go + sfo + sgo + fgo + sgf + n\r<BR>\n" );
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document.write( "since the total number of students is 25, then k = 25.\r<BR>\n" );
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document.write( "replacing all the variable with their respective numbers, you get:\r<BR>\n" );
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document.write( "25 = 5 + 5 + 0 + 4 + 3 + 1 + 2 + n\r<BR>\n" );
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document.write( "n is still these because we don't know what that number is yet.\r<BR>\n" );
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document.write( "simplifying you get:\r<BR>\n" );
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document.write( "25 = 20 + n\r<BR>\n" );
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document.write( "solve for n to get n = 5\r<BR>\n" );
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document.write( "you wanted to know the number of students who speak german and french plus the number of students who don't speak any of those languages.\r<BR>\n" );
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document.write( "that means you want fgo plus sfgo plus n\r<BR>\n" );
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document.write( "fgo = 1<BR>\n" );
document.write( "sfgo = 2<BR>\n" );
document.write( "n = 5\r<BR>\n" );
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document.write( "fg is equal to fgo + sfgo, therefore fg = 3\r<BR>\n" );
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document.write( "the number of students who speak french and german is 3.\r<BR>\n" );
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document.write( "the number of students who do not speak any of those 3 languages is 5.\r<BR>\n" );
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document.write( "that's my somewhat educated guess.\r<BR>\n" );
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document.write( "i'm reasonably sure it's correct.\r<BR>\n" );
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document.write( "how do i know?\r<BR>\n" );
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document.write( "i worked the problem back from the solution and came up with the original numbers of the problem.\r<BR>\n" );
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document.write( "you were originally given that s = 14, f = 12, g = 6\r<BR>\n" );
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document.write( "the table i drew up is shown below:\r<BR>\n" );
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document.write( "                        s               f              g\r\n" );
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document.write( "s only                  5               0              0\r\n" );
document.write( "f only                  0               5              0\r\n" );
document.write( "g only                  0               0              0\r\n" );
document.write( "sf only                 4               4              0\r\n" );
document.write( "sg only                 3               0              3\r\n" );
document.write( "fg only                 0               1              1\r\n" );
document.write( "sgf only                2               2              2\r\n" );
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document.write( "total                   14              12             6\r\n" );
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document.write( "since those are the numbers for s, f, and g originally given, i'm reasonably certain that i got it right.\r<BR>\n" );
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document.write( "consequently, i'll go with:\r<BR>\n" );
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document.write( "number of students who speak german and french = 3<BR>\n" );
document.write( "number of students who don't speak any of the three languages = 5\r<BR>\n" );
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