New experimental results on the ratio R-dp of the quasi-elastic charge-exchange yield at the outgoing proton angle theta(p,Lab) = 0 degrees for the nd -> p(nn) reaction to the elastic np -> pn charge-exchange yield are presented. The measurements were carried out at the Nuclotron of the Veksler and Baldin Laboratory of High Energies of the JINR (Dubna) at the neutron beam kinetic energies of 0.55, 0.8, 1.0, 1.2, 1.4, 1.8 and 2.0 GeV. The intense neutron beam with small momentum spread was produced by break-up of deuterons which were accelerated and extracted to the experimental hall. In both reactions mentioned above the outgoing protons with the momenta p(p) approximately equal to the neutron beam momentum p(n), beam were detected in the directions close to the direction of incident neutrons, i.e. in the vicinity of the scattering angle theta(p,Lab) = 0 degrees. Measured in the same data taking runs, the angular distributions of the charge-exchange reaction products were corrected for the well-known instrumental effects and averaged in the vicinity of the incident neutron beam direction. These corrected angular distributions for every of nd. p( nn) and np -> pn charge-exchange processes were proportional to the differential cross-sections of the corresponding reactions. The data were accumulated by the Delta-Sigma set-up magnetic spectrometer with two sets of multiwire proportional chambers located upstream and downstream of the momentum analyzing magnet. Inelastic processes were considerably reduced by the additional detectors surrounding the hydrogen and deuterium targets. The time-of-flight system was applied to identify the detected particles. The new R-dp data are compared with the existing ones, which were obtained below 1 GeV, and with the calculations which were made using the phenomenological NN amplitude sets.