Lattice-structural, magnetic, and magnetoelectric (ME) properties have been investigated for single crystals of prototypical polar ferrimagnet ${\mathrm{Ga}}_{2\ensuremath{-}x}{\mathrm{Fe}}_{x}{\mathrm{O}}_{3}$ $(0.8\ensuremath{\leqslant}x\ensuremath{\leqslant}1.4)$ as melt-grown by a floating-zone (FZ) method. Magnetization measurements show that the saturated magnetization as well as the ferrimagnetic phase transition temperature $({T}_{C})$ increases with as increase of $\mathrm{Fe}$ content $x$, while the coercive force decreases. A neutron powder diffraction study indicates fairly low ordering of $\mathrm{Ga}$ and $\mathrm{Fe}$ arrangement at cation sites, which is likely related to the lower ${T}_{C}$ in the FZ crystals than in the corresponding flux-grown crystals. Coefficients of linear and quadratic ME effects have been obtained with measurements of change in electric polarization induced by sweeping a magnetic field. Electric polarization was largely modulated in a magnetic field applied parallel to the direction of spontaneous magnetization, but not in a field parallel to that of the spontaneous polarization. A simple model to explain the sharp contrast is presented.