document.write( "Question 288071: In an interview of 50 math majors,
\n" ); document.write( "12 liked calculus and geometry
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Algebra.Com's Answer #208788 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let A = the set of all students who liked algebra, regardless\r\n" );
document.write( "of whether they liked the other two subjects.\r\n" );
document.write( "Let G = the set of all students who liked geometry, regardless\r\n" );
document.write( "of whether they liked the other two subjects.\r\n" );
document.write( "Let C = the set of all students who liked calculus, regardless\r\n" );
document.write( "of whether they liked the other two subjects.\r\n" );
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document.write( "The most inclusive clue we have is this one:\r\n" );
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\n" ); document.write( "4 liked calculus, algebra, and geometry
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document.write( "Those 4 are in all three circles, and the only\r\n" );
document.write( "region that is common to all three circles is\r\n" );
document.write( "the middle region f, so we put 4 for f\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( "12 liked calculus and geometry
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document.write( "4 of these 12 are in the middle region,\r\n" );
document.write( "so the other 12-4 or 8 are in j, so\r\n" );
document.write( "we put 8 in region j:\r\n" );
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document.write( "Next we look at this clue:\r\n" );
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\n" ); document.write( "18 liked calculus but not algebra
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document.write( "So 18 are in circle C but not in circle A.\r\n" );
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document.write( "8 of these 18 we just got through finding so\r\n" );
document.write( "the other 18-8 or 10 must be in region k,\r\n" );
document.write( "since they do not like algebra.  So we put\r\n" );
document.write( "10 in region k:\r\n" );
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document.write( "Next we look at:\r\n" );
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\n" ); document.write( "25 liked calculus
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document.write( "Now we can complete the circle C, \r\n" );
document.write( "because we have accounted for the\r\n" );
document.write( "numbers in three of its four regions\r\n" );
document.write( "of its regions, that's 4+8+10, or 22,\r\n" );
document.write( "so the other 25-22 or 3 are in region h.\r\n" );
document.write( "So we put 3 in region h:\r\n" );
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document.write( "Next we look at this clue:\r\n" );
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\n" ); document.write( "10 liked algebra but neither calculus nor geometry
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document.write( "These 10 are all in region d because they are the\r\n" );
document.write( "only ones in the algebra circle who are not also\r\n" );
document.write( "in either of the other two circles.  So we can\r\n" );
document.write( "put 10 in region d\r\n" );
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document.write( "Next clue:\r\n" );
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\n" ); document.write( "2 liked geometry and algebra but not calculus.
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document.write( "These 2 are in region e because that region is\r\n" );
document.write( "part of the geometry and algebra circles but\r\n" );
document.write( "not part of the calculus circle,  So we put\r\n" );
document.write( "2 in region e:\r\n" );
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document.write( "Next clue:\r\n" );
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\n" ); document.write( "15 liked geometry
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document.write( "So there are 15 students in the geometry cirle,\r\n" );
document.write( "and we have accounted for 2+4+8 or 14 of the 15\r\n" );
document.write( "in the geometry circle, so that leaves 15-14 or\r\n" );
document.write( "just 1 student in region i. So we put 1 student\r\n" );
document.write( "in region i:\r\n" );
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document.write( "Finally we look at,\r\n" );
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\n" ); document.write( "In an interview of 50 math majors,
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\r\n" );
document.write( "So of these 50 math majors, we have accounted for\r\n" );
document.write( "10+2+1+3+4+8+10=38 of them, so the remaining 50-38\r\n" );
document.write( "or 12 math majors are in the region outside all\r\n" );
document.write( "three circles, which is m.  So we put 12 in region\r\n" );
document.write( "m.\r\n" );
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document.write( "So 12 math majors didn't like any of\r\n" );
document.write( "the three math subjects, so those 12\r\n" );
document.write( "had better change their major to history\r\n" );
document.write( "or English or a foreign language, don't\r\n" );
document.write( "you think!!\r\n" );
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document.write( "Anyway you only have to answer this:\r\n" );
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document.write( "Of those surveyed, how many liked calculus and algebra?\r\n" );
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document.write( "Those are the students in the two region with 3 \r\n" );
document.write( "and 4 in them for they are in both the calculus and\r\n" );
document.write( "the algebra circles. So the only thing you have to\r\n" );
document.write( "answer is 3+4 or 7.\r\n" );
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document.write( "Yes, I know, we could have answered that long before we \r\n" );
document.write( "finished all the regions.  However I went through the \r\n" );
document.write( "whole Venn diagram anyway, because in other Venn diagram \r\n" );
document.write( "problems you will likely have to complete the whole Venn \r\n" );
document.write( "diagram to answer the question asked.\r\n" );
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document.write( "Answer: 7\r\n" );
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document.write( "Edwin
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