The speed of the child when running with the sidewalk is

     = 12 ft per second = u + v    (1)

where u is the sidewalk's rate and v is the child's rate on a still sidewalk.



The speed of the child when running against the sidewalk is

     =  4 ft per second = u - v    (2)

where u is the child's rate on a still sidewalk and v is the sidewalk rate.



Now you have two equations (1) and (2) for 2 unknowns, u and v.
  
To solve the system, add the equations (1) and (2). You will get

    2u = 12 + 4 = 16  ====>  u = 16/2 = 8 ft/s  is the sidewalk' speed.


Next substitute this found value of u into either equation (1) or (2).

You will find then  ( from (1) )  v = 12 - 8 = 4 ft/s as the rate of the child.


Answer.  Sidewalk's rate is  8 ft/s;  child's rate is 4 ft/s.