It is known that for coprime integers p > q > 0, the lens space L(p^2,pq-1) bounds a rational ball, B_{p,q}, arising as the 2-fold branched cover of a (smooth) surface in B^4 bounding the associated 2-bridge knot or link. Lekilli and Maydanskiy give handle decompositions for each B_{p,q}. Whereas, Yamada gives an alternative definition of rational balls, A_{m,n}, bounding L(p^2,pq-1) by their handlebody decompositions alone. We show that these two families coincide - answering a question of Kadokami and Yamada. To that end, we show that each A_{m,n} admits a Stein filling of the universally tight contact structure on L(p^2,pq-1) investigated by Lisca. Furthermore, we construct boundary diffeomorphisms between these families. Using the carving process, pioneered by Akbulut, we show that these boundary maps can be extended to diffeomorphisms between the spaces B_{p,q} and A_{m,n}.
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