\({l_{Total}} = \frac{1}{2}M{R^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....\left( i \right)\)

Mass of circular hole (removed)

\( = \frac{M}{4}\left( {As\,M = \pi {R^2}t\therefore M \propto {R^2}} \right)\)

\(M.I.\) of removed hole about its own ax is 

\( = \frac{1}{2}\left( {\frac{M}{4}} \right){\left( {\frac{R}{2}} \right)^2} = \frac{1}{{32}}M{R^2}\)

\(M.I.\) of removed hole about \(O'\)

\(\begin{array}{l}
{I_{removed\,hole}} = {I_{cm}} + m{x^2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{M{R^2}}}{{32}} + \frac{M}{4}{\left( {\frac{R}{2}} \right)^2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{M{R^2}}}{{32}} + \frac{{M{R^2}}}{{16}} = \frac{{3M{R^2}}}{{32}}
\end{array}\)

\(M.I.\) of complete disc can also be written as

\(\begin{array}{l}
{I_{Total}} = {I_{removed\,hole}} + {I_{re\min ing\,disc}}\\
{I_{Total}} = \frac{{3M{R^2}}}{{32}} + {I_{remaining\,disc}}\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)
\end{array}\)

Form eq. \((i)\) and \((ii)\),

\(\begin{array}{l}
\frac{1}{2}M{R^2} = \frac{{3M{R^2}}}{{32}} + {I_{remaining\,disc}}\\
 \Rightarrow {I_{remaining\,disc}}\\
\,\,\,\,\,\,\,\,\, = \frac{{M{R^2}}}{2} + \frac{{3M{R^2}}}{{32}} = \left( {\frac{{13}}{{32}}} \right)M{R^2}
\end{array}\)