document.write( "Question 530267: 1) 2 men and 7 boys can do a piece of work in 4 days. The same work is done by 4 men and 4 boys in 3 days. How long would it take 1 man and 1 boy to do the same? \n" ); document.write( "
Algebra.Com's Answer #349988 by Edwin McCravy(20059) $\"\"$ $\"About$
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2 men and 7 boys can do a piece of work in 4 days. The same work is done by 4 men and 4 boys in 3 days. How long would it take 1 man and 1 boy to do the same?
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document.write( "Let x = the number of days it would take a man to do 1 job working alone\r\n" );
document.write( "Let y = the number of days it would take a boy to do 1 job working alone\r\n" );
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document.write( "Make this chart:\r\n" );
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document.write( "                jobs   days    jobs/day\r\n" );
document.write( "1 man                         \r\n" );
document.write( "1 boy            \r\n" );
document.write( "2 men            \r\n" );
document.write( "7 boys           \r\n" );
document.write( "2 men & 7 boys    \r\n" );
document.write( "4 men             \r\n" );
document.write( "4 boys             \r\n" );
document.write( "4 men & 4 boys    \r\n" );
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document.write( "Since x = the number of days it would take 1 man to do 1 job working alone\r\n" );
document.write( "and y = the number of days it would take 1 boy to do 1 job working alone\r\n" );
document.write( "we fill this much in:\r\n" );
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document.write( "                jobs   days    rate in jobs/day\r\n" );
document.write( "1 man             1     x                         \r\n" );
document.write( "1 boy             1     y\r\n" );
document.write( "2 men            \r\n" );
document.write( "7 boys           \r\n" );
document.write( "2 men & 7 boys    \r\n" );
document.write( "4 men             \r\n" );
document.write( "4 boys             \r\n" );
document.write( "4 men & 4 boys    \r\n" );
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document.write( "Now we fill in their rates in jobs/day by dividing\r\n" );
document.write( "jobs by days:\r\n" );
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document.write( "                jobs   days    rate in jobs/day\r\n" );
document.write( "1 man             1     x            1/x                         \r\n" );
document.write( "1 boy             1     y            1/y \r\n" );
document.write( "2 men            \r\n" );
document.write( "7 boys           \r\n" );
document.write( "2 men & 7 boys    \r\n" );
document.write( "4 men             \r\n" );
document.write( "4 boys             \r\n" );
document.write( "4 men & 4 boys    \r\n" );
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document.write( "The rate for 2 men is 2 times the rate for 1 man\r\n" );
document.write( "The rate for 4 men is 4 times the rate for 1 man\r\n" );
document.write( "The rate for 7 boys is 7 times the rate for 1 boy\r\n" );
document.write( "The rate for 4 boys is 4 times the rate for 1 boy\r\n" );
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document.write( "So fill those rates in:\r\n" );
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document.write( "                jobs   days    rate in jobs/day\r\n" );
document.write( "1 man             1     x            1/x                         \r\n" );
document.write( "1 boy             1     y            1/y \r\n" );
document.write( "2 men                                2/x            \r\n" );
document.write( "7 boys                               7/y\r\n" );
document.write( "2 men & 7 boys    \r\n" );
document.write( "4 men                                4/x \r\n" );
document.write( "4 boys                               4/y\r\n" );
document.write( "4 men & 4 boys    \r\n" );
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document.write( "Since in every case above we are talking about completing 1 job,\r\n" );
document.write( "we put 1's all the way down the \"number of jobs\" list.\r\n" );
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document.write( "                jobs   days    rate in jobs/day\r\n" );
document.write( "1 man             1     x            1/x                         \r\n" );
document.write( "1 boy             1     y            1/y \r\n" );
document.write( "2 men             1                  2/x            \r\n" );
document.write( "7 boys            1                  7/y\r\n" );
document.write( "2 men & 7 boys    1\r\n" );
document.write( "4 men             1                  4/x \r\n" );
document.write( "4 boys            1                  4/y\r\n" );
document.write( "4 men & 4 boys    1\r\n" );
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document.write( ">>...2 men and 7 boys can do a piece of work in 4 days...<<\r\n" );
document.write( ">>...The same work is done by 4 men and 4 boys in 3 days...<<\r\n" );
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document.write( "So we fill in 4 days and 3 days for those cases:\r\n" );
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document.write( "                jobs   days    rate in jobs/day\r\n" );
document.write( "1 man             1     x            1/x                         \r\n" );
document.write( "1 boy             1     y            1/y \r\n" );
document.write( "2 men             1                  2/x            \r\n" );
document.write( "7 boys            1                  7/y\r\n" );
document.write( "2 men & 7 boys    1     4\r\n" );
document.write( "4 men             1                  4/x \r\n" );
document.write( "4 boys            1                  4/y\r\n" );
document.write( "4 men & 4 boys    1     3 \r\n" );
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document.write( "Now we get their rates by dividing jobs by days:\r\n" );
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document.write( "                jobs   days    rate in jobs/day\r\n" );
document.write( "1 man             1     x            1/x                         \r\n" );
document.write( "1 boy             1     y            1/y \r\n" );
document.write( "2 men             1                  2/x            \r\n" );
document.write( "7 boys            1                  7/y\r\n" );
document.write( "2 men & 7 boys    1     4            1/4\r\n" );
document.write( "4 men             1                  4/x \r\n" );
document.write( "4 boys            1                  4/y\r\n" );
document.write( "4 men & 4 boys    1     3            1/3\r\n" );
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document.write( "Now we make our equations from:\r\n" );
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document.write( " +  = \r\n" );
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document.write( "and\r\n" );
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document.write( " +  = \r\n" );
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document.write( "2/x + 7/y = 1/4\r\n" );
document.write( "4/x + 4/y = 1/3\r\n" );
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document.write( "DO NOT CLEAR OF FRACTIONS! Do by elimination:\r\n" );
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document.write( "Multiply the first equation by -2 to eliminate the terms in x:\r\n" );
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document.write( "-4/x - 14/y = -1/2\r\n" );
document.write( " 4/x +  4/y =  1/3\r\n" );
document.write( "------------------\r\n" );
document.write( "      -10/y = -1/6\r\n" );
document.write( "         -y = -60\r\n" );
document.write( "          y = 60\r\n" );
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document.write( "2/x +  7/y = 1/4\r\n" );
document.write( "2/x + 7/60 = 1/4\r\n" );
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document.write( "Multiply through by LCD of 60x\r\n" );
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document.write( "120 + 7x = 15x\r\n" );
document.write( "     -8x = -120 \r\n" );
document.write( "       x = 15\r\n" );
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document.write( "So it takes 1 man 15 days to do one job,\r\n" );
document.write( "and it takes 1 boy 60 days to do one job. \r\n" );
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document.write( "Now we have a new problem:\r\n" );
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document.write( "Let z = the number of hours required for 1 man and 1 boy\r\n" );
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document.write( "Make this new chart\r\n" );
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document.write( "                jobs   days    rate in jobs/day\r\n" );
document.write( "1 man             1     15            1/15                         \r\n" );
document.write( "1 boy             1     60            1/60 \r\n" );
document.write( "1 man & 1 boy     1      z            1/z\r\n" );
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document.write( "We get our equation from:\r\n" );
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document.write( " +  = \r\n" );
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document.write( "1/15 + 1/60 = 1/z\r\n" );
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document.write( "Multiply through by 60z\r\n" );
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document.write( "4z + z = 60\r\n" );
document.write( "    5z = 60\r\n" );
document.write( "     z = 12\r\n" );
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document.write( " So it would take 1 man and 1 boy 12 hours to do the job.\r\n" );
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document.write( "Edwin
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