document.write( "Question 39237: The ages of Old and Young total 48. Old is twice as old as Young was
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Algebra.Com's Answer #24665 by AnlytcPhil(1807) $\"\"$ $\"About$
You can put this solution on YOUR website!
The ages of Old and Young total 48. Old is twice as old as Young was\r\n" );
document.write( "when Old was half as old as Young will be when Young is three times as\r\n" );
document.write( "old as Old was when Old was three times as old as Young.\r\n" );
document.write( "How old is Old?\r\n" );
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document.write( "Let x = Old's present age\r\n" );
document.write( "Let y = Young's present age\r\n" );
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document.write( "I will insert some parentheses to help clarify:\r\n" );
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document.write( "1. The (present) ages of Old (x years old) and Young (y years old) total 48.\r\n" );
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document.write( "2. Old (now) is twice as old as Young was (z years ago)\r\n" );
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document.write( "3. (z years ago was) when Old was half as old as Young will be (w years from\r\n" );
document.write( "now)\r\n" );
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document.write( "4. (w years from now will be) when Young is three times as old as Old was (v\r\n" );
document.write( "years ago)\r\n" );
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document.write( "5. (v years ago was) when Old was three times as old as Young (was v years ago).\r\n" );
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document.write( "Now for each of those five sentences we'll set up five equations in five\r\n" );
document.write( "unknowns, x, y, z, w, and v:\r\n" );
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document.write( "1. >>...The (present) ages of Old (x years old) and Young (y years old) total\r\n" );
document.write( "48...<<\r\n" );
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document.write( "x + y = 48\r\n" );
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document.write( "2. >>...Old is (now) twice as old as Young was (z years ago)...<<\r\n" );
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document.write( "Old is now x, and z years ago, Young was y - z, so\r\n" );
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document.write( "x = 2(y - z)\r\n" );
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document.write( "3. >>...(z years ago was) when Old was half as old as Young will be (w years\r\n" );
document.write( "from now)...<<\r\n" );
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document.write( "z years ago, Old was x - z\r\n" );
document.write( "w years from now, Young will be y + w, so\r\n" );
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document.write( "x - z = 1/2(y + w)\r\n" );
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document.write( "4. >>...(w years from now will be) when Young is three times as old as Old was\r\n" );
document.write( "(v years ago)...<<\r\n" );
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document.write( "w years from now, Young will be y + w\r\n" );
document.write( "v years ago, Old was x - v, so\r\n" );
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document.write( "y + w = 3(x - v)\r\n" );
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document.write( "5.  >>...(v years ago was) when Old was three times as old as Young (was v years\r\n" );
document.write( "ago)...<<\r\n" );
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document.write( "v years ago, Old was x - v\r\n" );
document.write( "v years ago, Young was y - v, so\r\n" );
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document.write( "x - v = 3(y - v)\r\n" );
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document.write( "So we have this system of 5 equations and 5 unknowns:\r\n" );
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document.write( "x + y = 48\r\n" );
document.write( "x = 2(y - z)\r\n" );
document.write( "x - z = 1/2(y + w)\r\n" );
document.write( "y + w = 3(x - v)\r\n" );
document.write( "x - v = 3(y - v)\r\n" );
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document.write( "We can simplify these to give this system\r\n" );
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document.write( "  x  +  y + 0z + 0w + 0v = 48\r\n" );
document.write( "  x  - 2y + 2z + 0w + 0v =  0\r\n" );
document.write( " 2x  -  y - 2z -  w + 0v =  0\r\n" );
document.write( "-3x  +  y + 0z +  w + 3v =  0\r\n" );
document.write( "  x  - 3y + 0z + 0w + 2v =  0 \r\n" );
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document.write( "The best way to solve this system is with a TI-83, finding the\r\n" );
document.write( "\"rref\" (row-reduced echelon form) of this augmented 5×6 matrix:\r\n" );
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document.write( "[ 1   1   0   0   0 | 48]\r\n" );
document.write( "[ 1  -2   2   0   0 |  0]\r\n" );
document.write( "[ 2  -1  -2  -1   0 |  0]\r\n" );
document.write( "[-3   1   0   1   3 |  0]\r\n" );
document.write( "[ 1  -3   0   0   2 |  0]\r\n" );
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document.write( "which is \r\n" );
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document.write( "[ 1   0   0   0   0 | 30]\r\n" );
document.write( "[ 0   1   0   0   0 | 18]\r\n" );
document.write( "[ 0   0   1   0   0 |  3]\r\n" );
document.write( "[ 0   0   0   1   0 | 36]\r\n" );
document.write( "[ 0   0   0   0   1 | 12]\r\n" );
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document.write( "So x is 30 and y is 18.  z = 3, w = 36, v = 12\r\n" );
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document.write( "You were asked \"How old is Old?\", so the answer is 30.\r\n" );
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document.write( "---------------------------------------------------------\r\n" );
document.write( "Now let's check:\r\n" );
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document.write( "1. The (present) ages of Old (30 years old) and Young (18 years old) total 48.\r\n" );
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document.write( "2. Old (now 30) is twice as old as Young was (3 years ago, when Young was 15)\r\n" );
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document.write( "3. (3 years ago was) when Old was half as old as Young will be (36 years from\r\n" );
document.write( "now)\r\n" );
document.write( "[3 years ago Old was 27 and that is half as old as Young will be 36 years from\r\n" );
document.write( "now when he will be 54]\r\n" );
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document.write( "4. (36 years from now will be) when Young is three times as old as Old was (12\r\n" );
document.write( "years ago)\r\n" );
document.write( "[36 years from now Young will be 54. 12 years ago Old was 18, and 3 times 18 is\r\n" );
document.write( "54]\r\n" );
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document.write( "5. (12 years ago was) when Old was three times as old as Young (was 12 years ago)\r\n" );
document.write( "[12 years ago, Old was 18 and Young was 6, and 18 is 3 times 6]\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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