Using the quantum molecular dynamics model, we study the role of mass asymmetry of colliding nuclei on the fragmentation at the balance energy and on its mass dependence. The study is done by keeping the total mass of the system fixed as 40, 80, 160, and 240 and by varying the mass asymmetry of the ($eta$ = $frac{A_{T}-A_{P}}{A_{T}+A_{P}}$; where $A_{T}$ and $A_{P}$ are the masses of the target and projectile, respectively) reaction from 0.1 to 0.7. Our results clearly indicate a sizeable effect of the mass asymmetry on the multiplicity of various fragments. The mass asymmetry dependence of various fragments is found to increase with increase in total system mass (except for heavy mass fragments). Similar to symmetric reactions, a power law system mass dependence of various fragment multiplicities is also found to exit for large asymmetries.