document.write( "Question 1131586: In a classroom, two-third of the students are wearing blue jeans and three-fourth of those students are wearing T-shirts. How many students are wearing jeans and a T-shirt? show work. \n" ); document.write( "
Algebra.Com's Answer #748846 by Edwin McCravy(20059)\"\" \"About 
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document.write( "THE ABOVE SOLUTION IS WRONG!  It's 5/12ths of the students\r\n" );
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document.write( "Let J = the set of students wearing jeans\r\n" );
document.write( "Let T = the set of students wearing T-shirts\r\n" );
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document.write( "Let x = the number wearing jeans but not T-shirts = N(J∩T')\r\n" );
document.write( "Let y = the number wearing both jeans and a T-shirt = N(J∩T)\r\n" );
document.write( "Let z = the number wearing T-shirts but not jeans = N(J'∩T)\r\n" );
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document.write( "N(J) = N(J∩T') + N(J∩T) = x+y\r\n" );
document.write( "N(T) = N(J∩T) + N(J'∩T) = y+z\r\n" );
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document.write( "This is the Venn diagram:\r\n" );
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document.write( "So \"system%28x%2By=expr%282%2F3%29n%2Cy%2Bz=expr%283%2F4%29n%29\"\r\n" );
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document.write( "After clearing of fractions:\r\n" );
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document.write( "\"system%283%28x%2By%29=2n%2C4%28y%2Bz%29=4n%29\"\r\n" );
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document.write( "This says that n must be divisible by both 3 and 4, so n must be\r\n" );
document.write( "divisible by 12, so n = 12k\r\n" );
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document.write( "Since two-third of the students are wearing blue jeans,\r\n" );
document.write( "2/3 of 12k = 8k are wearing blue jeans.\r\n" );
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document.write( "Since three-fourths of the students are wearing T-shirts,\r\n" );
document.write( "3/4 of 12k = 9k are wearing T-shirts.\r\n" );
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document.write( "N(J or T) = N(J) + N(T) - N(J and T)\r\n" );
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document.write( "       12k = 8k + 9k - N(J and T)\r\n" );
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document.write( "       12k = 17k - N(J and T)\r\n" );
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document.write( "N(X and Y) = 5k\r\n" );
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document.write( "So 5k students are wearing both jeans and T-shirts.\r\n" );
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document.write( "So there are many solutions depending upon the choice of k:\r\n" );
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document.write( "Since there are 8k students in set J, the blue circle, and 5k students\r\n" );
document.write( "in the overlapping part of the two circles, there must be 8k-5k or 3k\r\n" );
document.write( "in the left part of the blue circle.\r\n" );
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document.write( "Similarly, since there are 9k students in set T, the red circle, and 5k\r\n" );
document.write( "students in the overlapping part of the two circles, there must be 9k-5k \r\n" );
document.write( "or 4k in the right part of the blue circle.\r\n" );
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document.write( "Since there are 5k students wearing both, out of 3k+5k+4k=12k students,\r\n" );
document.write( "the answer is always 5/12 of the students. \r\n" );
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document.write( "The smallest possible solution is when k=0, no students at all in the\r\n" );
document.write( "classroom, 0 wearing blue jeans, and 0 wearing T-shirts. That is a\r\n" );
document.write( "possibility because 0 is both 2/3rds as well as 3/4ths of 0.  Also 5/12ths\r\n" );
document.write( "of zero is zero. It's probably not the solution your teacher wanted, but\r\n" );
document.write( "mathematically it fits.\r\n" );
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document.write( "The next smallest possible solution is when k=1, \r\n" );
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document.write( "In that case, there are 12 students in the classroom, 8 wearing blue jeans\r\n" );
document.write( "and 9 wearing T-shirts. That is a possibility because 8 is 2/3rds of 12 and\r\n" );
document.write( "9 is 3/4ths of 12. The answer is 5 wearing both, which is 5/12ths of 12. \r\n" );
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document.write( "The next smallest possible solution is when k=2\r\n" );
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document.write( "In that case, there are 24 students in the classroom, 16 wearing blue jeans\r\n" );
document.write( "and 18 wearing T-shirts. That is a possibility because 16 is 2/3rds of 24 \r\n" );
document.write( "and 18 is 3/4ths of 24. The answer is 10 wearing both, which is 5/12ths of 24. \r\n" );
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document.write( "The next smallest possible solution is when k=3\r\n" );
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document.write( "In that case, there are 36 students in the classroom, 24 wearing blue jeans\r\n" );
document.write( "and 27 wearing T-shirts. That is a possibility because 24 is 2/3rds of 36 \r\n" );
document.write( "and 27 is 3/4ths of 36. The answer is 15 wearing both, which is 5/12ths of 36. \r\n" );
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document.write( "We can let k be any integer and get a different solution.  The solution\r\n" );
document.write( "in every case will be 5k, a multiple of 5.\r\n" );
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document.write( "Edwin
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