document.write( "Question 889580: Ages of three persons are now in the proportion of 2:3:4 and in five years from now, the proportion will be 5:7:8. What is the present age of the youngest person?? \n" ); document.write( "
Algebra.Com's Answer #538310 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The 8 should have been a 9.  Then there would have been a solution.\r\n" );
document.write( "But with an 8 there, there is no solution.  I'll show you first,\r\n" );
document.write( "why with the 8 there, there is no solution, and also if we change the\r\n" );
document.write( "8 to 9 there will be a solution.\r\n" );
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document.write( "First let's try for a solution with the 8 there.\r\n" );
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document.write( "Since 2, 3, 4, have no common factor, the only ages\r\n" );
document.write( "they could be are multiples of these, or\r\n" );
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document.write( "2x, 3x, 4x,  where n is a positive integer\r\n" );
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document.write( "Similarly, \r\n" );
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document.write( "Since 5, 7, and 8 have no common factor, the only ages\r\n" );
document.write( "they could be in 5 years are multiples of these, or\r\n" );
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document.write( "5y, 7y, 8y, where y is a positive integer.\r\n" );
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document.write( "So in 5 years we must have \r\n" );
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document.write( "2x+5=5y, 3x+5=7y, 4x+5=8y or\r\n" );
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document.write( "For there to be a solution to the problem these 3 equations must\r\n" );
document.write( "have a common integer solution, however they do not!\r\n" );
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document.write( "The first two equations have a common integer solution (x,y) = (10,5)\r\n" );
document.write( "However the first and third do not! Their only common solution is \r\n" );
document.write( "(3.75,2.5), which not only is not (10,5), it is not an integer \r\n" );
document.write( "solution. Also the second and third do not either! Their only common\r\n" );
document.write( "solution is (1.25,1.25).\r\n" );
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document.write( "Since the first two have a common integer solution, the culprit has to\r\n" );
document.write( "be the 3rd equation!\r\n" );
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document.write( "If we change the 8 to 9, there is a solution.  So let's start all over \r\n" );
document.write( "and pretend your problem is:\r\n" );
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\n" ); document.write( "Ages of three persons are now in the proportion of 2:3:4 and in five years from now, the proportion will be 5:7:9. What is the present age of the youngest person??
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\r\n" );
document.write( "Since 2, 3, 4, have no common factor, the only ages\r\n" );
document.write( "they could be are multiples of these, or\r\n" );
document.write( "\r\n" );
document.write( "2x, 3x, 4x,  where n is a positive integer\r\n" );
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document.write( "Similarly, \r\n" );
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document.write( "Since 5, 7, and 9 have no common factor, the only ages\r\n" );
document.write( "they could be in 5 years are multiples of these, or\r\n" );
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document.write( "5y, 7y, 9y, where y is a positive integer.\r\n" );
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document.write( "So in 5 years we must have \r\n" );
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document.write( "2x+5=5y, 3x+5=7y, 4x+5=9y or\r\n" );
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document.write( "For there to be a solution to the problem the 3 equations must\r\n" );
document.write( "have a common integer solution.\r\n" );
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document.write( "The first two equations have a common integer solution (x,y) = (10,5)\r\n" );
document.write( "The first and second equations NOW have a common integer solution (x,y) = (10,5)\r\n" );
document.write( "The second and third equations NOW have a common integer solution (x,y) = (10,5)\r\n" );
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document.write( "So x=10 and y=5\r\n" );
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document.write( "So their ages now are 2x, 3x, and 4x or 20, 30, and 40\r\n" );
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document.write( "So the present age of the youngest is 20.\r\n" );
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document.write( "Now let's check:\r\n" );
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document.write( "Certainly 20,30 and 40 are in the ratio 2:3:4,\r\n" );
document.write( "since dividing them by 10 gives that ratio.\r\n" );
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document.write( "In 5 years they will be 25, 35, and 45.\r\n" );
document.write( "This also checks with 5y, 7y, 9y.  And\r\n" );
document.write( "certainly 25,35 and 45 are in the ratio 5:7:9,\r\n" );
document.write( "since dividing them by 5 gives that ratio.\r\n" );
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document.write( "So apparently your 8 was a typo and should have been 9.\r\n" );
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document.write( "Edwin
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