A thin uniform circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity $\omega$. Another disc of the same dimensions but of mass $M/4$ is placed gently on the first disc coaxially. The angular velocity of the system now becomes

Angular momentum about the common axis is conserved. For the first disc, $$I_1=\frac{1}{2}MR^2$$ For the second disc, $$I_2=\frac{1}{2}\cdot \frac{M}{4}R^2=\frac{1}{8}MR^2$$ Initially only the first disc rotates, so $$L_i=I_1\omega$$ After they rotate together, $$L_f=(I_1+I_2)\omega'$$ Using conservation, $$I_1\omega=(I_1+I_2)\omega'$$ $$\omega'=\frac{I_1}{I_1+I_2}\omega$$ $$\omega'=\frac{\frac{1}{2}}{\frac{1}{2}+\frac{1}{8}}\omega=\frac{\frac{1}{2}}{\frac{5}{8}}\omega=\frac{4\omega}{5}$$ This matches option $\text{D}$.