Would that be the same as taking the Area of $A$ which would be a random point inside the circle divided by the Area of $S$ inside the circle $\frac{(\pi(r/2)^2)}{(\pi r^2)}$?
I am just confused as to how to to find this.
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$\begingroup$As the comments point out, the question doesn't make much sense as written.
If the book means "circumference", then you're right, just compare the areas of the two circles. Note that your expression can be simplified to just $\frac14$. This is the same question: A point in a circle is selected at random. Calculate probability that point is closer to centre than circumference
If the book really means "a given radius", then the region of points closer to the center than to the rest of the radius would be the half-circle opposite from the given radius, and the answer would be $\frac12$.
Since the back of the book says $\frac14$, it seems that it meant to say "circumference".
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