A circular hole of radius $$\frac{a}{2}$$ is cut out of a circular disc of radius $$a$$ as shown in figure. The centroid of the remaining circular portion with respect to point O will be:

Let us consider Centroid 1 as

$$C_1=\left(x_1,\ y_1\right)$$

where $$C_1=\left(a,\ 0\right)$$

and Centroid 2 as

$$C_2=\left(x_2,\ y_2\right)$$

where 

$$C_2=\left(\frac{3a}{2},\ 0\right)$$

Area of bigger circle is $$A_1=\ \pi\ R^2$$

Area of the cut part is 

$$A_2=\ \ \frac{\pi R^2}{4}$$

Now formula is $$X\ =\ \frac{\left(A_1x_1+A_2x_2\right)}{A_1+A_2}$$

$$X\ =\ \frac{\left(\pi\ R^2\left(a\right)-\frac{\pi R^2}{4}\left(\frac{3a}{2}\right)\right)}{\pi\ R^2-\frac{\pi R^2}{4}}$$

$$X\ =\ \frac{\frac{5\pi R^2}{8}}{\frac{3\pi R^2}{4}}$$

$$X\ =\frac{5}{6}$$