document.write( "Question 708306: A man and his wife have three children, John, Ben and Mary.
\n" ); document.write( "The difference between their parents' ages is the same as between John and Ben
\n" ); document.write( "and between Ben and Mary.
\n" ); document.write( "The ages of John and Ben, multiplied together, equals the age of the father.
\n" ); document.write( "The ages of Ben and Mary multiplied together equals the age of the mother.
\n" ); document.write( "The combined ages of the family amounts to 90 years.
\n" ); document.write( "What is the age of each person? \n" ); document.write( "
Algebra.Com's Answer #436089 by Edwin Parker(36)\"\" \"About 
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A man and his wife have three children, John, Ben and Mary.
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document.write( "Let d be the difference between the father's and the mother's ages,\r\n" );
document.write( "which is also the difference between John's and Ben's ages, and \r\n" );
document.write( "also the difference between Ben's and and Mary's ages.\r\n" );
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document.write( "We cannot tell which parent is older.\r\n" );
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document.write( "We can only tell that Bill is the middle child, but we cannot tell\r\n" );
document.write( "which of John and Mary is the oldest and youngest.\r\n" );
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document.write( "Let Bill's age be x\r\n" );
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document.write( "Then the younger child's age is x-d\r\n" );
document.write( "And the older child's age is x+d\r\n" );
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\n" ); document.write( "The ages of John and Ben, multiplied together, equals the age of the father.
\n" ); document.write( "The ages of Ben and Mary multiplied together equals the age of the mother.
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document.write( "The product of Bill's age and the older child's age = the older parent's age\r\n" );
document.write( "The product of Bill's age and the younger child's age = the younger parent's age\r\n" );
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document.write( "x(x-d) = younger parent's age\r\n" );
document.write( "x(x+d) = older parent's age\r\n" );
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document.write( "The younger parent's age + d = the older parent's age.\r\n" );
document.write( "                    \r\n" );
document.write( "                  x(x-d) + d = x(x+d)\r\n" );
document.write( "                     x²-dx+d = x²+dx\r\n" );
document.write( "                           d = 2dx\r\n" );
document.write( "                           0 = 2dx - d\r\n" );
document.write( "                           0 = d(2x-1)\r\n" );
document.write( "                      d=0, or 2x-1 = 0\r\n" );
document.write( "                                2x = 1  \r\n" );
document.write( "                                 x = \"1%2F2\"\r\n" );
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document.write( "Certainly Bill is not half a year old (a 6 month-old baby), so d=0\r\n" );
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document.write( "That means all three children are the same age, and the two parents are\r\n" );
document.write( "the same age. \r\n" );
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document.write( "John, Bill, and Mary are the same age, x.  They are either triplets or\r\n" );
document.write( "adopted children.\r\n" );
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document.write( "x(x-d) = younger parent's age\r\n" );
document.write( "x(x+d) = older parent's age\r\n" );
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document.write( "And since d=0, they are both x² years old.\r\n" );
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document.write( "So each child is x years old and each parent is x² years old.\r\n" );
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\n" ); document.write( "The combined ages of the family amounts to 90 years.
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document.write( "Since the sum of their ages is 90,\r\n" );
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document.write( "x + x + x + x² + x² = 90\r\n" );
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document.write( "2x² + 3x - 90 = 0\r\n" );
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document.write( "That factors as\r\n" );
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document.write( "(x - 6)(2x + 15) = 0\r\n" );
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document.write( "x - 6 = 0;   2x + 15 = 0\r\n" );
document.write( "    x = 6         2x = -15\r\n" );
document.write( "                   x = -15/2, which we ignore.\r\n" );
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document.write( "So the children are 6 years old each, \r\n" );
document.write( "and the parents are 6² or 36 years old each. \r\n" );
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document.write( "Edwin
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