A simple four-wave-mixing (FWM)-based method for measuring the ratio between the third- and the fourth-order dispersion coefficients $({\beta }_3∕{\beta }_4)$ in optical fibers is reported. The FWM interaction involves a low power laser and a low power amplified spontaneous emission noise source. The method is applied in several dispersion-shifted and non-zero-dispersion-shifted fibers with lengths varying from 0.03 to $25\mathrm{km}$ , and we have obtained an error of less than 3% in measuring ${\beta }_3∕{\beta }_4$ . In highly nonlinear fibers the error has presented a strong dependency with longitudinal variations of the zero-dispersion wavelength $\left({\lambda }_0\right)$ ; an error less than 20% could be obtained in most of the tested fibers. Our method also allows for a fast estimation of the distribution of fluctuations of ${\lambda }_0$ along the fiber.