Ordering fuzzy quantities is a challenging problem in fuzzy sets theory that has attracted the interest of many researchers. Despite the multiple indices introduced for this purpose and due to the fact that fuzzy quantities do not have a natural order, there is still a chance to provide a new approach for ranking this type of quantities from the acceptability and foundation point of view. This paper aims at developing a new approach to ranking fuzzy numbers (FNs), fuzzy rank acceptability analysis (FRAA), which not only implements a ranking of the FNs, but also provides a degree of confidence for all ranks. Additionally, the FRAA can be efficiently implemented by using different fuzzy preference relations including both transitive and intransitive ones. Properties of FRAA ranking, their dependence on the fuzzy preference relations, and correspondence with the basic axioms for ranking FNs are analyzed. Finally, a comparison of the FRAA ranks to ranks from other methods is analyzed along with a discussion of the advantages of FRAA ranking.