document.write( "Question 710238: There are 100 seats and $100 to spend. $5.00 per seat for men, $2.00 per seat for women, $.10 per seat for children. How many of each to fill 100 seats? \n" ); document.write( "
Algebra.Com's Answer #436961 by Edwin McCravy(20060) $\"\"$ $\"About$
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document.write( "     m + w + c = 100\r\n" );
document.write( "5m + 2w + .10c = 100\r\n" );
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document.write( "The number of children is a multiple of 10, so \r\n" );
document.write( "it will come out to a whole number of dollars.\r\n" );
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document.write( "Let c = 10k\r\n" );
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document.write( "    m +  w + 10k = 100\r\n" );
document.write( "   5m + 2w +   k = 100\r\n" );
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document.write( "Since these both = 100, set them equal:\r\n" );
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document.write( "  m + w + 10k = 5m + 2w + k\r\n" );
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document.write( "      9k - 4m = w\r\n" );
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document.write( "Substitute 9k - 4m for w in the first equation\r\n" );
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document.write( "       m +  w + 10k = 100\r\n" );
document.write( "m + (9k - 4m) + 10k = 100\r\n" );
document.write( "  m + 9k - 4m + 10k = 100\r\n" );
document.write( "          -3m + 19k = 100\r\n" );
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document.write( "the smallest coefficient in absolute value is 3\r\n" );
document.write( "So we write 19 and 100 in terms of their nearest\r\n" );
document.write( "multiple of 3. So we write 19 as 18+1 and 100 as 99+1\r\n" );
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document.write( "      -3m + (18+1)k = 99+1\r\n" );
document.write( "      -3m + 18k + k = 99 + 1 \r\n" );
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document.write( "Divide through by 3\r\n" );
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document.write( "       -m + 6k +  = 33 + \r\n" );
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document.write( "Isolate the fractions:\r\n" );
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document.write( "        -  = 33 + m - 6k\r\n" );
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document.write( "The right side is an integer, and so is the left side,\r\n" );
document.write( "so set both sides = A, an integer:\r\n" );
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document.write( " -  = A;     33 + m - 6k = A\r\n" );
document.write( " k - 1 = 3A\r\n" );
document.write( "     k = 3A+1\r\n" );
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document.write( "Substitute 3A+1 for k in     33 + m - 6k = A\r\n" );
document.write( "                        33 + m - 6(3A+1) = A\r\n" );
document.write( "                        33 + m - 18A - 6 = A \r\n" );
document.write( "                                       m = 19A - 27\r\n" );
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document.write( "Substitute 3A+1 for k  and 19A-27 for m in            9k - 4m = w \r\n" );
document.write( "                                          9(3A+1) - 4(19A-27) = w\r\n" );
document.write( "                                          27A + 9 - 76A + 108 = w\r\n" );
document.write( "                                                    117 - 49A = w\r\n" );
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document.write( "So we have  m = 19A-27 men, 117-49A = w women, and c = 10k = 10(3A+1) \r\n" );
document.write( "= 30A+10 children  \r\n" );
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document.write( "We can't have a negative number of men, women or children\r\n" );
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document.write( "        m ≧ 0 \r\n" );
document.write( " 19A - 27 ≧ 0\r\n" );
document.write( "      19A ≧ 27\r\n" );
document.write( "        A ≧ \r\n" );
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document.write( "        w ≧ 0 \r\n" );
document.write( "117 - 49A ≧ 0\r\n" );
document.write( "     -49A ≧ -117\r\n" );
document.write( "        A ≦ \r\n" );
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document.write( "Thus  ≦ A ≦ \r\n" );
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document.write( "Therefore A = 2 since that is the only integer that\r\n" );
document.write( "satisfies that inequality.\r\n" );
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document.write( " m = 19A - 27 = 19(2) - 27 = 38 - 27 = 11 men\r\n" );
document.write( " w = 117 - 49A = 117 - 49(2) = 117 - 98 = 19 women\r\n" );
document.write( " c = 30A + 10 = 30(2) + 10 = 60 + 10 = 70 children\r\n" );
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document.write( "Answer:  11 men, 19 women, and 70 children.\r\n" );
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document.write( "Edwin
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