The arithmetic \({\mathbf {R}}^{\boldsymbol {\sharp }}\) is obtained by postulating the Peano axioms on the basis of the relevant logic R. \({\mathbf {R}}^{\boldsymbol {\sharp }}\) is a remarkable arithmetic, not least in that it has finite models. In this paper we examine the options for extending \({\mathbf {R}}^{\boldsymbol {\sharp }}\) from natural numbers to rational numbers, as this is the essential next step towards providing a relevant basis for mathematics and for applications. Relevant rational number theory is problematic in that the most obvious approaches lead to non-conservative extensions of \({\mathbf {R}}^{\boldsymbol {\sharp }}\) . We consider three ways in which relevant theories of rational arithmetic can be formulated, and note in particular how these fare in the finite models of \({\mathbf {R}}^{\boldsymbol {\sharp }}\) .