document.write( "Question 1019477: This quarter, a survey of 110 students at De Anza College
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Algebra.Com's Answer #635857 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We draw this Venn diagram.  \r\n" );
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document.write( "The capital letters  M, E, and H represent the \r\n" );
document.write( "three subjects Math, English and History.  The \r\n" );
document.write( "small letters a through j represent the number \r\n" );
document.write( "of elements (students) in each of the 8 regions.\r\n" );
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document.write( "In these problems the clues are usually given \r\n" );
document.write( "in the reverse order that you need them to\r\n" );
document.write( "place the proper numbers in the 8 regions of\r\n" );
document.write( "the Venn diagram.\r\n" );
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document.write( "Here are the clues:\r\n" );
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\n" ); document.write( "Clue #1. 110 students,
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\n" ); document.write( "Clue #5. 25 take English and math,
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document.write( "We begin with clue #8 and go backward:\r\n" );
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\n" ); document.write( "Clue #8. 15 take all three subjects.
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document.write( "This is the most inclusive of all the clues, so\r\n" );
document.write( "the 15 are placed in the region that is common\r\n" );
document.write( "to all three circles, the region with \"e=?\".\r\n" );
document.write( "So we put \"e=15\" in the middle region.\r\n" );
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\n" ); document.write( "Clue #7. 20 take math and history,
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document.write( "This is the pair of regions that are common\r\n" );
document.write( "to circle M and circle H, which are regions\r\n" );
document.write( "with e=15 and d=?. So e+d=20, and since e=15,\r\n" );
document.write( "d=20-15 or 5.  So we put d=5 in that region.\r\n" );
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\n" ); document.write( "Clue #6. 20 take English and history,
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document.write( "This is the pair of regions that are common\r\n" );
document.write( "to circle E and circle H, which are regions\r\n" );
document.write( "with e=15 and f=?. So e+f=20, and since e=15,\r\n" );
document.write( "f=20-15 or 5.  So we put f=5 in that region.\r\n" );
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\n" ); document.write( "Clue #5. 25 take English and math,
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document.write( "This is the pair of regions that are common\r\n" );
document.write( "to circle E and circle M, which are regions\r\n" );
document.write( "with e=15 and b=?. So e+b=25, and since e=15,\r\n" );
document.write( "b=25-15 or 10.  So we put b=10 in that region.\r\n" );
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\n" ); document.write( "Clue #4. 40 take history,
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document.write( "This is the 4 regions of circle H, which are regions\r\n" );
document.write( "with e=15,d=5,f=5,g=?. So e+d+f+g=40, and since e=15,\r\n" );
document.write( "d=5,f=5, the g=40-15-5-5=15 So we put g=15 in that region.\r\n" );
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\n" ); document.write( "Clue #3. 50 take English,
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document.write( "This is the 4 regions of circle E, which are regions\r\n" );
document.write( "with e=15,b=10,f=5,c=?. So e+b+f+c=50, and since e=15,\r\n" );
document.write( "b=10,f=5, then c=50-15-10-5=20 So we put c=20 in that region.\r\n" );
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\n" ); document.write( "Clue #2. 60 take math,
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document.write( "This is the 4 regions of circle M, which are regions\r\n" );
document.write( "with e=15,b=10,d=5,a=?. So e+b+d+a=50, and since e=15,\r\n" );
document.write( "b=10,d=5, then a=60-15-10-5=30 So we put a=30 in that region.\r\n" );
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\n" ); document.write( "Clue #1. 110 students,
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document.write( "This is the total of all the students in all the 8 regions.\r\n" );
document.write( "So this is a+b+c+d+e+f+g+h=110.  We have the numbers in all\r\n" );
document.write( "the regions except the region with h=?,  which is outside \r\n" );
document.write( "all the three circles yet inside the big rectangle.\r\n" );
document.write( "So h=110-30-10-20-5-15-5-15 = 10.  So we put h=10 in that \r\n" );
document.write( "last region.\r\n" );
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document.write( "Now our Venn diagram is complete.\r\n" );
document.write( "We can answer any question asked:\r\n" );
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\n" ); document.write( "a. The number of students taking math
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document.write( "These are the students is the region with a=30.\r\n" );
document.write( "Answer: 30\r\n" );
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\n" ); document.write( "b. The number of students taking English
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document.write( "These are the students in the region with a=30,\r\n" );
document.write( "b=10 and c=20. \r\n" );
document.write( "Answer: 30+10+20 = 60\r\n" );
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\n" ); document.write( "c. The number of students taking none of
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document.write( "These students are the ones in the region outside \r\n" );
document.write( "all three circles, that is, h=10.\r\n" );
document.write( "Answer: 10.\r\n" );
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document.write( "Edwn
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