document.write( "Question 99531: A group of workers was put on a job. From the second day onwards one
\n" ); document.write( "worker was withdrawn each day. The work was finished when the last
\n" ); document.write( "worker was withdrawn. If no worker was withdrawn at any stage, the
\n" ); document.write( "group would have finished the job in two-thirds time. How many workers
\n" ); document.write( "were there in the group? \n" ); document.write( "
Algebra.Com's Answer #72420 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
A group of workers was put on a job. From the second day onwards one
\n" ); document.write( "worker was withdrawn each day. The work was finished when the last
\n" ); document.write( "worker was withdrawn. If no worker was withdrawn at any stage, the
\n" ); document.write( "group would have finished the job in two-thirds time. How many workers
\n" ); document.write( "were there in the group?
\n" ); document.write( "
\r\n" );
document.write( "Let n be the number of workers.\r\n" );
document.write( "

\n" ); document.write( ">>...The work was finished when the last worker was withdrawn...<<
\n" ); document.write( "
\r\n" );
document.write( "Since the nth worker was withdrawn on the nth day, the number of days\r\n" );
document.write( "required to finish the job was n days. \r\n" );
document.write( "\r\n" );
document.write( "We will assume that all n workers worked at the same rate. Let the \r\n" );
document.write( "fraction of a job per day that each worker could do be r.  Suppose\r\n" );
document.write( "they finished in n days.\r\n" );
document.write( "\r\n" );
document.write( "The 1st worker only worked 1 day before he was withdrawn, so his \r\n" );
document.write( "fraction of the job was r.\r\n" );
document.write( "\r\n" );
document.write( "The 2nd worker worked 2 days before he was withdrawn, so his \r\n" );
document.write( "fraction of the job was 2r.\r\n" );
document.write( "\r\n" );
document.write( "The 3rd worker worked 3 days before he was withdrawn, so his \r\n" );
document.write( "fraction of the job was 3r.\r\n" );
document.write( "\r\n" );
document.write( "...(etc.)...\r\n" );
document.write( "\r\n" );
document.write( "The nth worker worked n days before he was withdrawn, so his \r\n" );
document.write( "fraction of the job was nr.\r\n" );
document.write( "\r\n" );
document.write( "If we add the fractions of the job each did it should add to \r\n" );
document.write( "one job\r\n" );
document.write( "\r\n" );
document.write( "r + 2r + 3r + ... + nr = 1\r\n" );
document.write( "

\n" ); document.write( ">>...If no worker was withdrawn at any stage, the group would have
\n" ); document.write( "finished the job in two-thirds time...<<
\n" ); document.write( "
\r\n" );
document.write( "If all n workers had stayed on the job throughout, it would \r\n" );
document.write( "have taken only \"2%2F3\"n days instead of the n days.\r\n" );
document.write( "\r\n" );
document.write( "Then each worker would have worked for \"2%2F3\"n days, and \r\n" );
document.write( "since the fraction of the job each worker would have \r\n" );
document.write( "completed per day would have been r, in \"2%2F3\"n days each\r\n" );
document.write( "worker would have done this fraction of the job ---> \"2%2F3\"nr.  \r\n" );
document.write( "\r\n" );
document.write( "Since there were n workers, the sum of all the fractions \r\n" );
document.write( "of the job they would have done would have been n times the \r\n" );
document.write( "fraction each worker would have done. So we multiply \"2%2F3\"nr \r\n" );
document.write( "by n and get \"2%2F3\"nēr.\r\n" );
document.write( "\r\n" );
document.write( "Since the n workers would have completed one full job, we set \r\n" );
document.write( "that = 1 job:\r\n" );
document.write( "\r\n" );
document.write( "\"2%2F3\"nēr = 1\r\n" );
document.write( "\r\n" );
document.write( "So we have this system of equations:\r\n" );
document.write( "\r\n" );
document.write( "r + 2r + 3r + ... + nr = 1\r\n" );
document.write( "\"2%2F3\"nēr = 1\r\n" );
document.write( "\r\n" );
document.write( "The left side of the first equation is the sum of an \r\n" );
document.write( "arithmetic series, so we use the formula\r\n" );
document.write( "\r\n" );
document.write( "Sn = n(a1+an)/2\r\n" );
document.write( "\r\n" );
document.write( "where a1 = r and an = nr\r\n" );
document.write( "\r\n" );
document.write( "r + 2r + 3r + ... + nr = n(r+nr)/2\r\n" );
document.write( "\r\n" );
document.write( "Since the left side equals 1, so does the right side:\r\n" );
document.write( "\r\n" );
document.write( "n(r+nr)/2 = 1\r\n" );
document.write( "\r\n" );
document.write( "Multiply both sides by 2 to clear the fraction\r\n" );
document.write( "\r\n" );
document.write( "n(r+nr) = 2\r\n" );
document.write( "\r\n" );
document.write( "nr+nēr = 2\r\n" );
document.write( "\r\n" );
document.write( "Now we work on the second equation\r\n" );
document.write( " \r\n" );
document.write( "\"2%2F3\"nēr = 1\r\n" );
document.write( "\r\n" );
document.write( "Multiply both sides by 3 to clear the fraction\r\n" );
document.write( "\r\n" );
document.write( "2nēr = 3\r\n" );
document.write( "\r\n" );
document.write( "So now our simplified system of equations is\r\n" );
document.write( "\r\n" );
document.write( "nr + nēr = 2\r\n" );
document.write( "2nēr = 3\r\n" );
document.write( "\r\n" );
document.write( "Solve the second equation for r:\r\n" );
document.write( "\r\n" );
document.write( "r = \"3%2F%282n%5E2%29\"\r\n" );
document.write( "                 \r\n" );
document.write( "Substitute in the first equation:\r\n" );
document.write( "\r\n" );
document.write( "nr + nēr = 2\r\n" );
document.write( "\r\n" );
document.write( "n(\"3%2F%282n%5E2%29\") + nē\"3%2F%282n%5E2%29\" = 2\r\n" );
document.write( "\r\n" );
document.write( "which simplifies to\r\n" );
document.write( "\r\n" );
document.write( "\"3%2F%282n%29\" + \"3%2F2\" = 2\r\n" );
document.write( "\r\n" );
document.write( "Multiply through by LCD = 2n to clear of fractions:\r\n" );
document.write( "\r\n" );
document.write( "(2n)\"3%2F%282n%29\" + (2n)\"3%2F2\" = (2n)2\r\n" );
document.write( "\r\n" );
document.write( "3 + 3n = 4n\r\n" );
document.write( "\r\n" );
document.write( "Subtract 3n from both sides\r\n" );
document.write( "\r\n" );
document.write( "     3 = 4n - 3n\r\n" );
document.write( "     3 = n\r\n" );
document.write( "\r\n" );
document.write( "So there were three workers, and it took\r\n" );
document.write( "three days to finish the job.\r\n" );
document.write( "\r\n" );
document.write( "-----------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "Checking:  If all 3 workers had stayed on the job,\r\n" );
document.write( "the job would have been finished in 2 days instead\r\n" );
document.write( "of 3, so in one day, all three workers would have \r\n" );
document.write( "done 1/2 the job and each worker would have done \r\n" );
document.write( "1/3 of 1/2 of the job, or 1/6 of the job.  \r\n" );
document.write( "\r\n" );
document.write( "So the first day there were 3 workers on the job,\r\n" );
document.write( "each doing 1/6 of the job. 3 times 1/6 is 1/2, so\r\n" );
document.write( "half the job got finished on the first day only.\r\n" );
document.write( "\r\n" );
document.write( "The second day there were 2 workers on the job,\r\n" );
document.write( "each doing 1/6 of the job, 2 times 1/6 is 1/3, so\r\n" );
document.write( "1/3 of the job got finished on the second day only.\r\n" );
document.write( "\r\n" );
document.write( "The third day there was only 1 worker on the job, and\r\n" );
document.write( "he did 1/6 of the job, so 1/6 of the job got finished\r\n" );
document.write( "at the end of the second day.\r\n" );
document.write( "\r\n" );
document.write( "If we add together \r\n" );
document.write( "1. the 1/2 of the job that got done on the first day,\r\n" );
document.write( "2. the 1/3 of the job that got done on the second day,\r\n" );
document.write( "and\r\n" );
document.write( "3. the 1/6 of the job that got done on the third day,\r\n" );
document.write( "\r\n" );
document.write( "we get 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 6/6 = 1 job.\r\n" );
document.write( "\r\n" );
document.write( "So 3 workers is correct.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );