For a uniform rectangular sheet shown in the figure, the ratio of...

We will use Parallel Axes Theorem here,

Distance between $O{O}^{'}$ is,

$O{O}^{'} = \dfrac{1}{2}\sqrt{{a}^{2} + {b}^{2}} = \sqrt{\dfrac{{a}^{2}}{4} + \dfrac{{b}^{2}}{4}}$

So, according to parallel axis theorem we have,

$\dfrac{{I}_{O}}{I{O}^{'}} = \dfrac{\dfrac{M}{12}\left({a}^{2} + {b}^{2}\right)}{\dfrac{M}{12}\left({a}^{2} + {b}^{2}\right) + M\left(\dfrac{{a}^{2}}{4} + \dfrac{{b}^{2}}{4}\right)} = \dfrac{\dfrac{M}{12}\left({a}^{2} + {b}^{2}\right)}{\dfrac{M}{3}\left({a}^{2} + {b}^{2}\right)} = \dfrac{1}{4}$