document.write( "Question 544951: Working together, a job takes Tom and Joe 8 days, Tom and Dave 13 1/3 days, and Joe and Dave 6 2/3 days. How long does it take each person to do the job alone? \n" ); document.write( "
Algebra.Com's Answer #355368 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
Working together, a job takes Tom and Joe 8 days, Tom and Dave 13 1/3 days, and Joe and Dave 6 2/3 days. How long does it take each person to do the job alone?
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document.write( "Let x = the number of days Tom requires working alone.\r\n" );
document.write( "Let y = the number of days Joe requires working alone.\r\n" );
document.write( "Let z = the number of days Dave requires working alone.\r\n" );
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document.write( "Make this chart and fill in the unknown times x, y, and z\r\n" );
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document.write( "                               Number of       Number of       Rate in\r\n" );
document.write( "                               jobs done     days required     jobs/day \r\n" );
document.write( "------------------------------------------------------------------------\r\n" );
document.write( "Tom & Joe working together   |            |                 |           |                           \r\n" );
document.write( "Tom & Dave working together  |            |                 |           |\r\n" );
document.write( "Joe & Dave working together  |            |                 |           |\r\n" );
document.write( "Tom working alone            |            |        x        |           |  \r\n" );
document.write( "Joe working alone            |            |        y        |           | \r\n" );
document.write( "Dave working alone           |            |        z        |           | \r\n" );
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document.write( "In all 6 cases exactly 1 whole job is done.  So we fill in 1 for all the number of\r\n" );
document.write( "jobs done:\r\n" );
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document.write( "                               Number of       Number of       Rate in\r\n" );
document.write( "                               jobs done     days required     jobs/day \r\n" );
document.write( "------------------------------------------------------------------------\r\n" );
document.write( "Tom & Joe working together   |     1      |                 |           |                           \r\n" );
document.write( "Tom & Dave working together  |     1      |                 |           |\r\n" );
document.write( "Joe & Dave working together  |     1      |                 |           |\r\n" );
document.write( "Tom working alone            |     1      |        x        |           |  \r\n" );
document.write( "Joe working alone            |     1      |        y        |           | \r\n" );
document.write( "Dave working alone           |     1      |        z        |           | \r\n" );
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document.write( "Two of the given times are in mixed fractions, so let's make them into improper fractions:\r\n" );
document.write( "Number of days required by Tom and Dave working together = \"13%261%2F3\" = \"40%2F3\" \r\n" );
document.write( "Number of days required by Joe and Dave working together = \"6%262%2F3\" = \"20%2F3\" \r\n" );
document.write( "So we fill in the given number of days required in the first three cases:\r\n" );
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document.write( "                               Number of       Number of       Rate in\r\n" );
document.write( "                               jobs done     days required     jobs/day \r\n" );
document.write( "------------------------------------------------------------------------ \r\n" );
document.write( "Tom & Joe working together   |     1      |        8        |           |                           \r\n" );
document.write( "Tom & Dave working together  |     1      |      40/3       |           |\r\n" );
document.write( "Joe & Dave working together  |     1      |      20/3       |           |\r\n" );
document.write( "Tom working alone            |     1      |        x        |           |  \r\n" );
document.write( "Joe working alone            |     1      |        y        |           | \r\n" );
document.write( "Dave working alone           |     1      |        z        |           | \r\n" );
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document.write( "Next we fill in the rates in jobs/day by dividing jobs by days:\r\n" );
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document.write( "1÷8 = \"1%2F8\",  1÷\"40%2F3\" = 1×\"3%2F40\" = \"3%2F40\",  1÷\"20%2F3\" = 1×\"3%2F20\" = \"3%2F20\"\r\n" );
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document.write( "                               Number of       Number of       Rate in\r\n" );
document.write( "                               jobs done     days required     jobs/day \r\n" );
document.write( "------------------------------------------------------------------------ \r\n" );
document.write( "Tom & Joe working together   |     1      |        8        |    1/8    |                           \r\n" );
document.write( "Tom & Dave working together  |     1      |      40/3       |   3/40    |\r\n" );
document.write( "Joe & Dave working together  |     1      |      20/3       |   3/20    |\r\n" );
document.write( "Tom working alone            |     1      |        x        |    1/x    |  \r\n" );
document.write( "Joe working alone            |     1      |        y        |    1/y    | \r\n" );
document.write( "Dave working alone           |     1      |        z        |    1/z    | \r\n" );
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document.write( "The three equations come from adding the rates to get the combined rates:\r\n" );
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document.write( "                \"%28matrix%284%2C1%2C%22Tom%27s%22%2Crate%2Cworking%2Calone%29%29\" + \"%28matrix%284%2C1%2C%22Joe%27s%22%2Crate%2Cworking%2Calone%29%29\" = \"%28matrix%286%2C1%2C%22Tom%22%2Cand%2C%22Joe%27s%22%2Crate%2Cworking%2Ctogether%29%29\"\r\n" );
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document.write( "                \"%28matrix%284%2C1%2C%22Tom%27s%22%2Crate%2Cworking%2Calone%29%29\" + \"%28matrix%284%2C1%2C%22Dave%27s%22%2Crate%2Cworking%2Calone%29%29\" = \"%28matrix%286%2C1%2C%22Tom%22%2Cand%2C%22Dave%27s%22%2Crate%2Cworking%2Ctogether%29%29\"\r\n" );
document.write( "               \r\n" );
document.write( "                \"%28matrix%284%2C1%2C%22Joe%27s%22%2Crate%2Cworking%2Calone%29%29\" + \"%28matrix%284%2C1%2C%22Dave%27s%22%2Crate%2Cworking%2Calone%29%29\" = \"%28matrix%286%2C1%2C%22Joe%22%2Cand%2C%22Dave%27s%22%2Crate%2Cworking%2Ctogether%29%29\"\r\n" );
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document.write( "So we have this system of three equations in three unknowns:\r\n" );
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document.write( "                \"system%281%2Fx%2B1%2Fy=1%2F8%2C1%2Fx%2B1%2Fz=3%2F40%2C+1%2Fy%2B1%2Fz=3%2F20%29\"\r\n" );
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document.write( "Important: DO NOT clear of fractions! That's because doing so would introduce\r\n" );
document.write( "terms in two variables multiplied together, that is, terms in xy, xz, and yz,\r\n" );
document.write( "which would complicate matters further.\r\n" );
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document.write( "Instead let u = \"1%2Fx\", v = \"1%2Fy\", and w = \r\n" );
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document.write( "Then we have\r\n" );
document.write( "                 ì u + v     = \"1%2F8\"\r\n" );
document.write( "                 í u     + w = \"3%2F40\"\r\n" );
document.write( "                 î     v + w = \"3%2F20\"\r\n" );
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document.write( "NOW we MAY clear of fractions because we won't get multiplied variables.\r\n" );
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document.write( "                 ì 8u +  8v       = 1\r\n" );
document.write( "                 í40u       + 40w = 3\r\n" );
document.write( "                 î      20v + 20w = 3\r\n" );
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document.write( "Solve that system by elimination  (I'm sure you know how).\r\n" );
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document.write( "You get u = \"1%2F40\",  v = \"1%2F10\", and w = \"1%2F20\"\r\n" );
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document.write( "Then since u = \"1%2Fx\", then \"1%2F40\" = \"1%2Fx\", and x = 40, so Tom\r\n" );
document.write( "working alone takes 40 days to do the job.\r\n" );
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document.write( "Since v = \"1%2Fy\", then \"1%2F10\" = \"1%2Fy\", and y = 10, so Joe\r\n" );
document.write( "working alone takes 10 days to do the job.\r\n" );
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document.write( "Since w = \"1%2Fz\", then \"1%2F20\" = \"1%2Fz\", and z = 20, so Joe\r\n" );
document.write( "working alone takes 20 days to do the job.\r\n" );
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document.write( "Edwin
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