Let u be the child's own speed in ft/s, and

let v be the moving sidewalk speed.


The child's effective velocity running with he sidewalk's motion is 

u + v = 120/12 = 10  ft/s


The child's effective velocity running against he sidewalk's motion is 

u - v = 32/8 = 4  ft/s


Again

u + v = 10
u - v =  4
-------------------------Add the equations to exclude v

2u = 10 + 4 = 14   ====>  u = 7 ft/s.


Then v = 10-u = 10-7 = 3 ft/s.


Answer.  The child own speed is 7 ft/s.  The running sidewalk speed  is 3 ft/s.