### Instruction: If an arc of $45^{\circ}$ on circle $A$ has the same length as an arc of $30^{\circ}$ on circle $B$, then what is the ratio of the area of circle $A$ to the area of circle $B$? Express your answer as a common fraction.### Response: To compare the areas of the circles, I need to find their radii.
I can use the formula for the length of an arc, which is $s = r\theta$, where $s$ is the arc length, $r$ is the radius, and $\theta$ is the central angle in radians.
Since the arc lengths are equal, I can equate the expressions for $s$ for both circles and solve for the ratio of the radii.
I have $r_A \cdot \frac{45}{180} \cdot \pi = r_B \cdot \frac{30}{180} \cdot \pi$.
Simplifying, I get $\frac{r_A}{r_B} = \frac{2}{3}$.
The ratio of the areas of the circles is the square of the ratio of their radii, by the formula for the area of a circle, $A = \pi r^2$.
So the ratio of the areas is $\frac{r_A^2}{r_B^2} = \frac{4}{9}$.