document.write( "Question 774783: Mary is twice as old as Ann was when Mary was half as old as Ann will be when she is three times as old as Mary was when she was three times as old as Ann. \n" ); document.write( "
Algebra.Com's Answer #472493 by Edwin McCravy(20054)\"\" \"About 
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document.write( "Mary is now M years old\r\n" );
document.write( "Ann is now A years old\r\n" );
document.write( "x = the first mentioned number of years in the past\r\n" );
document.write( "y = the mentioned number of years in the future\r\n" );
document.write( "z = the second mentioned number of years in the past\r\n" );
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document.write( "Mary is twice as old as Ann was (x years ago)\r\n" );
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document.write( "M = 2(A-x)\r\n" );
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document.write( "(x years ago) was when Mary was half as old as Ann will be (y years from now)\r\n" );
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document.write( "M-x = \"1%2F2\"(A+y)\r\n" );
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document.write( "(y years from now is) when she (Ann) is three times as old as Mary was \r\n" );
document.write( "(z years ago) \r\n" );
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document.write( "A+y = 3(M-z)\r\n" );
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document.write( "when she was three times as old as Ann (was z years ago)\r\n" );
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document.write( "M-z = 3(A-z)\r\n" );
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document.write( "M = 2(A-x)\r\n" );
document.write( "M-x = \"1%2F2\"(A+y)\r\n" );
document.write( "A+y = 3(M-z)\r\n" );
document.write( "M-z=3(A-z)\r\n" );
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document.write( "That's 4 equations in 5 unknowns, so there probably are\r\n" );
document.write( "more than one answer\r\n" );
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document.write( "  M - 2A          +  x = 0\r\n" );
document.write( " 2M -  A - y      - 2x = 0\r\n" );
document.write( "-3M +  A + y + 3z      = 0\r\n" );
document.write( "  M - 3A     + 2z      = 0\r\n" );
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document.write( "Solving that by matrices gives:\r\n" );
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document.write( "(M,A,y,z) = (6x,10x,12x,4x)\r\n" );
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document.write( "The youngest they could be is when x=1 year ago, y=12 years from now,\r\n" );
document.write( "and z=4 years ago:  Mary is now 6, Ann is now 10.\r\n" );
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document.write( "The next youngest they could be is when x=2 year ago, y=24 years from now,\r\n" );
document.write( "and z=8 years ago:  Mary is now 12, Ann is now 20.\r\n" );
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document.write( "The next youngest they could be is when x=3 year ago, y=36 years from now,\r\n" );
document.write( "and z=12 years ago:  Mary is now 18, Ann is now 30.\r\n" );
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document.write( "etc. etc.\r\n" );
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document.write( "Edwin
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