In this paper we introduce reasoning procedures for $\mathcal{ALCQ}^{+}_{F}$, a fuzzy description logic with extended qualified quantification. The language allows for the definition of fuzzy quantifiers of the absolute and relative kind by means of piecewise linear functions on ℕ and ℚ∩[0,1] respectively. In order to reason about instances, the semantics of quantified expressions is defined based on recently developed measures of the cardinality of fuzzy sets. A procedure is described to calculate the fuzzy satisfiability of a fuzzy assertion, which is a very important reasoning task. The procedure considers several different cases and provides direct solutions for the most frequent types of fuzzy assertions.