For larger solid sphere about diameter $Y$-axis,

$$I_{\text {whole }}=\frac{2}{5} M(2 R)^2=\frac{8}{5} M R^2$$

Density of sphere is uniform

$$\begin{aligned} \& \Rightarrow \frac{M}{V_{\text {whole }}}=\frac{M_{\text {smaller }}}{V_{\text {smaller }}} \Rightarrow \frac{M}{\frac{4}{3} \pi(2 R)^3}=\frac{M^{\prime}}{\frac{4}{3} \pi R^3} \\ \& \Rightarrow M^{\prime}=\frac{M}{8} \end{aligned}$$

Using parallel axis theorem for smaller sphere,

$$\begin{aligned} \& I^{\prime}=I_{\mathrm{cm}}+M^{\prime} R^2=\frac{2}{5} \frac{M R^2}{8}+\frac{M R^2}{8}=\frac{7}{40} M R^2 \\ \& \therefore \text { Ratio }=\frac{I_{\text {smaller }}}{I_{\text {remaining }}}=\frac{I^{\prime}}{I_{\text {whole }}-I^{\prime}}=\frac{\frac{7}{40} M R^2}{\left(\frac{8}{5}-\frac{7}{40}\right) M R^2}=\frac{7}{64-7}=\frac{7}{57} \end{aligned}$$