document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151779>Question 1151779</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a survey of 100 student the number of student studying various language where find to be: Arabic-28, french-30, yoruba-42, Arabic and yoruba-10, Arabic and french-8, french and yoruba-5, all the three language-3<BR>\n" );
document.write( "(1). How many student where studying no language<BR>\n" );
document.write( "(2). How many student had yoruba as their only language<BR>\n" );
document.write( "(3). How many student study french if and only if study yoruba </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773627 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=773627\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "A = set of people who study Arabic<BR>\n" );
document.write( "F = set of people who study French<BR>\n" );
document.write( "Y = set of people who study Yoruba<BR>\n" );
document.write( "U = universal set = set of everyone surveyed (100 people total)\r<BR>\n" );
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document.write( "Draw a Venn Diagram of 3 circles representing the three sets of language classes.<BR>\n" );
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document.write( "Take note of the 8 distinct regions which I've marked in blue<BR>\n" );
document.write( "Region (a) = set of people taking Arabic only<BR>\n" );
document.write( "Region (b) = set of people taking Arabic and French only<BR>\n" );
document.write( "Region (c) = set of people taking French only<BR>\n" );
document.write( "Region (d) = set of people taking Arabic and Yoruba only<BR>\n" );
document.write( "Region (e) = set of people taking all three languages<BR>\n" );
document.write( "Region (f) = set of people taking French and Yoruba only<BR>\n" );
document.write( "Region (g) = set of people taking Yoruba only<BR>\n" );
document.write( "Region (h) = set of people taking neither of the three classes mentioned\r<BR>\n" );
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document.write( "We have 3 people studying all three languages, so that goes in the very center region (e).\r<BR>\n" );
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document.write( "There are 10 people taking Arabic and Yoruba, so that leaves 10-3 = 7 who are taking Arabic and Yoruba only (and not French). This goes in region (d)\r<BR>\n" );
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document.write( "There are 8 people taking Arabic and French, leaving 8-3 = 5 people taking these two languages only (not Yoruba). This goes in region (b).\r<BR>\n" );
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document.write( "Then we have a '2' in region (f) because there are 5 people studying French and Yoruba, so 5-3 = 2 are just studying these two languages (not Arabic).\r<BR>\n" );
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document.write( "We have this partially filled out Venn Diagram so far<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width = \"35%\" src = \"https://i.imgur.com/6WsUhwZ.png\"><BR>\n" );
document.write( "We know that 28 students are studying Arabic. So far we have 5, 3, and 7 in regions (b), (e) and (d). The total so far is 5+3+7 = 15. That leaves 28-15 = 13 who study Arabic only (and not French nor Yoruba). Write \"13\" in region (a).\r<BR>\n" );
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document.write( "Then there are 30 French students. Subtract off the values that are in circle F, and we have 30-5-3-2 = 20 students who are studying French only (neither Arabic nor Yoruba). Write this in region (c).\r<BR>\n" );
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document.write( "Next up there are 42 students in a Yoruba language class. Subtract off the values in circle Y: 42-7-3-2 = 30. Write \"30\" in region (g) to indicate we have 30 students who are only taking this language class (and neither French nor Arabic)\r<BR>\n" );
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document.write( "Lastly, we add all of the values we have filled out so far in regions (a) through (g)<BR>\n" );
document.write( "13+5+20+7+3+2+30 = 80<BR>\n" );
document.write( "There are 80 students who are taking at least one language class mentioned here (ie they are taking one or more of the three classes)\r<BR>\n" );
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document.write( "However, there were 100 people surveyed as part of the universal set. So we have 100-80 = 20 people taking neither of the three language courses. Write this in region (h)\r<BR>\n" );
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document.write( "This is the fully completed Venn Diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width = \"35%\" src = \"https://i.imgur.com/BNPVxLJ.png\"><BR>\n" );
document.write( "The region labels are optional, but I find them helpful to be able to describe the diagram values.\r<BR>\n" );
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document.write( "Once you have the Venn Diagram, the questions are fairly straight forward\r<BR>\n" );
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document.write( "Question (1). How many student where studying no language<BR>\n" );
document.write( "Answer: 20 students (see region (h))\r<BR>\n" );
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document.write( "Question (2). How many student had Yoruba as their only language<BR>\n" );
document.write( "Answer: 30 students (see region (g))\r<BR>\n" );
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document.write( "Question (3). How many student study French if and only if study Yoruba<BR>\n" );
document.write( "Answer: 5 students (combine regions (e) and (f))<BR>\n" );
document.write( "I interpret this question to be asking \"how many students are studying Yoruba and French at the same time?\". This includes the 3 people taking all three languages.<BR>\n" );
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