$C^{\ast }$-algebra extensions associated to continued fraction expansions of rational numbers

We solve the isomorphism problem for essential unital $C^{\ast }$-algebra extensions of the form $0\to \mathcal{K}\oplus \mathcal{K}\to E\stackrel{\pi }{\to }{M}_n\otimes C(\mathbb{T})\to 0$. We then relate these to analogs of the Effros Shen AF algebras for rational numbers. This involves $C^{\ast }$-algebras constructed from categories of paths built from certain nonsimple continued fraction expansions.