document.write( "Question 616847: In a class, 25 students have been on a plane, 20 on a train, 27 on a boat, 9 on a plane and train, 12 on a train and boat, 14 on a plane and boat, 3 on all three, and 1 on none of them How many students are in the class? \n" ); document.write( "
Algebra.Com's Answer #387992 by EdwinParker(16)\"\" \"About 
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In a class, 25 students have been on a plane, 20 on a train, 27 on a boat, 9 on a plane and train, 12 on a train and boat, 14 on a plane and boat, 3 on all three, and 1 on none of them How many students are in the class?\r
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document.write( "Let P = the set of all students who have been on a plane, regardless\r\n" );
document.write( "of whether they have been on a train or a boat.\r\n" );
document.write( "Let T = the set of all students who have been on a train, regardless\r\n" );
document.write( "of whether they have been on a plane or a boat.\r\n" );
document.write( "Let B = the set of all students who have been on a boat, regardless\r\n" );
document.write( "of whether they have been on a plane or a train.\r\n" );
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document.write( "There are 8 regions which I have labeled with arbitrary small\r\n" );
document.write( "letters. [I skipped little \"b\" because I used capital \"B\" for the\r\n" );
document.write( "set of all students who have been on a boat.]\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( ">>...3 on all three,...<<
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document.write( "Those 3 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 3 for f\r\n" );
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document.write( "The most EXcluding clue we have is this one:\r\n" );
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\n" ); document.write( ">>...and 1 on none of them...<<
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document.write( "That 1 is outside all three circles, and the only\r\n" );
document.write( "region outside all three circles is region i, so we \r\n" );
document.write( "put 1 in region i:\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( ">>...9 on a plane and train,...<<
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document.write( "3 of those 9 are already accounted for in the middle region,\r\n" );
document.write( "so that leaves 9-3 or 6 to go in the other region common\r\n" );
document.write( "to circles P and T, which is region c, so we put 6 in that region:\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( ">>...12 on a train and boat,...<<
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document.write( "3 of those 12 are already accounted for in the middle region,\r\n" );
document.write( "so that leaves 12-3 or 9 to go in the other region common\r\n" );
document.write( "to circles T and B, which is region g, so we put 9 in that region:\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( ">>...14 on a plane and boat,...<<
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document.write( "3 of those 14 are already accounted for in the middle region,\r\n" );
document.write( "so that leaves 14-3 or 11 to go in the other region common\r\n" );
document.write( "to circles P and B, which is region e, so we put 11 in that region:\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( ">>...25 students have been on a plane,...<<
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document.write( "We have numbers in 3 of the regions of circle P, so \r\n" );
document.write( "6+3+11 or 20 of those 25 are already accounted\r\n" );
document.write( "for so that leaves 25-20 or 5 to go in the remaining\r\n" );
document.write( "region of circle P, region a, so we put 5 in that region:\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( ">>...20 on a train,...<<
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document.write( "We have numbers in 3 of the regions of circle T, so \r\n" );
document.write( "6+3+9 or 18 of those 20 are already accounted\r\n" );
document.write( "for so that leaves 20-18 or 2 to go in the remaining\r\n" );
document.write( "region of circle T, region d, so we put 2 in that region:\r\n" );
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document.write( "Now we look at this:\r\n" );
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\n" ); document.write( ">>...27 on a boat,...<<
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document.write( "We have numbers in 3 of the regions of circle B, so \r\n" );
document.write( "11+3+9 or 23 of those 27 are already accounted\r\n" );
document.write( "for so that leaves 27-23 or 4 to go in the remaining\r\n" );
document.write( "region of circle B, region h, so we put 4 in that region:\r\n" );
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document.write( "We are now ready to look at the question, for now\r\n" );
document.write( "we can answer anything we are asked because we have\r\n" );
document.write( "the complete Venn diagram:\r\n" );
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\n" ); document.write( ">>...How many students are in the class?...<<
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document.write( "Just add up all the 8 numbers in the regions:\r\n" );
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document.write( "5+6+2+11+3+9+4+1 = 41\r\n" );
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document.write( "Answer: There are 41 students in the class.\r\n" );
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document.write( "Edwin
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