Algebra

$\cfrac{1}{1 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{2}}}}.$

Workng from the bottom-up

4 + 1/2  =  9/2

4 + 1/ (9/2)  =   4 + 2/9  =  (38/9)

1 + 1/(38/9)  =    1 + 9/38  = 47/38

1 / (47/38)  =   

38 / 47