The three-particle impulse approximation with truncation (TIAT) has been applied to the investigation of d(p, 2p)n and d(p, pn)p quasi-free knock-out reactions (QFR) in the nonrelativistic scattering region (E<160 MeV). It is stated that the truncation procedure leads to the approximate restoration of the three-particle unitary relationships which are violated explicitly within the usual formulation of the single-collision mechanism. It is shown that the TIAT explains sufficiently well all the available experimental data for the value of the truncation parameter used by the authors in their earlier papers for the investigation of other QFR.