Cracks in materials induce highly localized displacement gradients in the vicinity of the crack tip, posing significant challenges for numerical simulation. Traditional approaches typically rely on local mesh refinement or enrichment techniques. The former substantially increases the number of degrees of freedom, resulting in computational inefficiency. The latter alleviates the computational burden but generally depends on explicit asymptotic solutions, which are unavailable in most situations and motivates further improvement and development. In this study, we propose a novel Rational-Adaptive Tip Element (RATE) for finite element method (FEM) and boundary element method (BEM) as a natural extension of enrichment techniques, introducing a rational enrichment scheme implicitly defined through weight parameters. The enrichment functions are constructed using rational polynomials formed by the weighted normalization of the partition of unity polynomial bases. A scaled mapping is employed to accurately capture the singular behavior near the crack tip, enabling precise approximation of tip fields without requiring explicit asymptotic expressions. This formulation facilitates indirect fitting of a wide range of crack tip field behaviors, thereby extending its applicability to problems with complex or unknown asymptotics. The effectiveness and robustness of the RATE are demonstrated through benchmark problems. It is observed that the RATE yields highly accurate results even on coarse meshes. Furthermore, it is shown that the method significantly reduces the degrees of freedom for accurate modeling of crack tip asymptotics, offering an efficient numerical tool for analyzing cracked structures.