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In this work we will introduce the concept of ratio-covariety, as a nonemptyfamily $\mathscr{R}$ of numerical semigroups verifying certain properties. Thisconcept will allow us to:  \begin{enumerate}  \item Describe an algorithmic process to compute $\mathscr{R}.$  \item Prove the existence of the smallest element of $\sR$ that contains aset of positive integers.  \item Talk about the smallest ratio-covariety that contains a finite set ofnumerical semigroups.  \end{enumerate}  In addition, in this paper we will apply the previous results to the study ofthe ratio-covariety $\mathscr{R}(F,m)=\{S\mid S \mbox{ is a numerical semigroupwith Fro-}  \mbox {benius number }F \mbox{ and multiplicity }m\}.$

The ratio-covariety of numerical semigroups with fixed multiplicity and Frobenius number