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2014-10-10T08:46:22+05:30

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Let there be N men in the fort.  Let the amount of provisions required for each man for each day be P.  Let us assume that this does not change when new men come in.
 
   Provisions for N men per day is P N.  
   Provisions for N men for 50 days = 50 P N 

   Provisions consumed in 10 days by N men = 10 P N
   Remaining provisions after 10 days = 50 P N - 10 PN = 40 P N   ---- equation 1
   
   Total number of men after 10 days = N + 500
   Provisions required for these men for 35 days = 35 * P * (N + 500)   --- equation 2
 
     So provisions in equation 1 and equation 2 are equal.
 
        40 P N = 35 * P ( N + 500)
         8 N = 7 (N+500)

          N = 3500

So there are 3500 men in the fort at the beginning.

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