Gamma and fast neutron shielding performances of some borate-based glasses, recently synthesized by other researchers, in the form 39H$_3$BO$_3$ + 30PbO+20MO+10Bi$_2$O$_3$+1Dy$_2$O$_3$ (with M=Ca, Sr, Ba, Na$_2$ and K$_2$) have been investigated. The gamma shielding ability of the glasses with doped heavy metals was studied for photons with 20, 30, 40, 60, 100, 200, 284, and 511 keV energies by exploiting the mass attenuation coefficients, half-value layers, mean free paths and effective atomic numbers. It was concluded that the glass with SrO has abetter shielding performance at 20 and 30 keV. For photons with energies above 30 keV, up to 511 keV, the glass having BaO is superior in terms of gamma shielding ability. Furthermore, the glass containing SrO was shown to have the best performance against fast neutrons among the glasses under investigation.