Question 729601
10 men can do a piece of work in 24 days. After working 6 days, a few men were added to finish the work in 21 days from the begining. Find the added men.
<pre>
>>...10 men can do a piece of work in 24 days...<<  

So the 10 men's combined working rate is {{{1_job/24_days}}} or {{{1/24}}}{{{job/day}}}.  

Also it would take 1 man 240 days (ten times as long as it would take 10
men.  So 1 man's working rate is {{{1_job/240_days}}} or {{{1/240}}}{{{job/day}}}. 
</pre>
>>...After working 6 days,...<<
<pre>
In those 6 days the 10 men at the rate of {{{1/24}}}{{{job/day}}} have done
{6/24}}}ths or {{{1/4}}}th of job and so there is still {{{3/4}}}ths of the job
left to be done.  
</pre>
>>...A few men were added to finish the work in 21 days from the begining...<<
<pre>
Let the number of men added be N.  So for the remaining 21-6 or 15 days,
we have 10+N men working at the combined rate of (10+N)·{{{1_job/240_days}}}
or {{{(10+N)job/240_days}}} or {{{(10+N)/240}}}{{{job/day}}} doing {{{3/4}}}ths of the job.
 
So 15 days times their {{{(10+N)/240}}}{{{job/day}}} rate should equal {{{3/4}}}ths of the job.

                               {{{15(10+N)/240}}}{{{""=""}}}{{{3/4}}}

Cross multiply:

                                 60(10+N) = 3·240
                                 60 + 60N = 720
                                      60N = 660
                                        N = 11 

So they added 11 men.

Edwin</pre>