Let w be the uniform width of the stone border.


Then the exterior dimensions of the larger rectangle are (10+2w) meters by (20+2w) meters, 
and the area of the larger rectangle is the product  (10+2w)*(20+2w) square meters.


Hence, the area of the border itself is  (10+2w)*(20+2w) - 10*20,  and it is equal exactly to  64 square meters.


It gives you an equation for w:

(10+2w)*(20+2w) - 10*20 = 64,

40w + 20w + 4w^2 = 64,

4w^2 + 60w - 64 = 0,

w^2 + 15w - 16 = 0,

(w+16)*(w-1) = 0  ====>   the only positive root  w= 1 gives you the answer to the problem's question.