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