>>...10 men can do a piece of work in 24 days...<<  

So the 10 men's combined working rate is  or .  

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  or . 

In those 6 days the 10 men at the rate of  have done
{6/24}}}ths or th of job and so there is still ths of the job
left to be done.  

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)·
or  or  doing ths of the job.
 
So 15 days times their  rate should equal ths of the job.

                               

Cross multiply:

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

So they added 11 men.

Edwin