From the condition, you have these two equations


      x +     y +    z =    500    (1)

  748*x + 712*y + 45*z = 300000    (2)


where x is the number of men, y is the number of women and z is the number of children.


From equation (1), express z = 500-x-y  and substitute it into equation (2). You will get


  748*x + 712*y + 45*(500-x-y) = 300000.


Simplify it


  703x  + 667y  = 300000 - 45*500 = 277500.


Now we should find integer solution/solutions to the last equation.


The simplest way is to express y = , put this formula into Excel, run for x= 1, 2, 3, 4, . . . and search for integer y


In this way I got the integer solution  x = 149,   y = 259.


Then z = 500-x-y = 500-149-259 = 92.


ANSWER.  149 men,  259 women and 92 children.