In a 30°-60°-90° triangle, the hypotenuse is twice the length of the short leg. Here, it is convenient to assign segment BH the length of 1.
BH is the short leg of ∆BCH, so BC will have length 2.
BC is the short leg of ∆BAC, so BA will have length 4.
Now, we have
... (AH + HB) : HB = 4 : 1
So the ratio we want is ...
... AH : HB = 3 : 1
