A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?
$\textbf{(A)}\ 4/9 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 5/6 \qquad \textbf{(D)}\ 3/2 \qquad \textbf{(E)}\ 9/4$
Solution 1
Let $r_1$ and $r_2$ be the radii of circles $A$ and$B$, respectively. 
It is well known that in a circle with radius $r$, a subtended arc opposite an angle of $\theta$ degrees has length $\frac{\theta}{360} \cdot 2\pi r$. 
Using that here, the arc of circle A has length $\frac{45}{360}\cdot2\pi{r_1}=\frac{r_1\pi}{4}$. The arc of circle B has length $\frac{30}{360} \cdot 2\pi{r_2}=\frac{r_2\pi}{6}$. We know that they are equal, so $\frac{r_1\pi}{4}=\frac{r_2\pi}{6}$, so we multiply through and simplify to get $\frac{r_1}{r_2}=\frac{2}{3}$. As all circles are similar to one another, the ratio of the areas is just the square of the ratios of the radii, so our answer is $\boxed{\textbf{(A) } 4/9}$.

Solution 2
The arc of circle $A$ is $\frac{45}{30}=\frac{3}{2}$ that of circle $B$.
The circumference of circle $A$ is $\frac{2}{3}$ that of circle $B$ ($B$ is the larger circle).
The radius of circle $A$ is $\frac{2}{3}$ that of circle $B$.
The area of circle $A$ is ${\left(\frac{2}{3}\right)}^2=\boxed{\textbf{(A) } 4/9}$ that of circle $B$.

Solution 3
We can use a very cheap way to find the answer based on the choices. Firstly, if a higher angle measure equals a lower angle measure, that means that the ratio of areas must be lower, cutting D and E. Then we realize that the ratio has to be with square numbers, as there are no square roots. Therefore, it is $(A)$
-dragoon

Solution 4
Written by a 6th grader, please don't mind*
Set Circle A's arc to 45 degrees. In a full circle, there is 360 degrees, so 360/45 = 8. If we set the arc to be pi degrees, then in total there will be 8 pi in the circumference of this circle, yielding 4 as the radius and 16pi as the area. 
Since the arc of 30 degrees in Circle B is also pi, and there are 12 30 degree arcs in a circle, there will be 12pi in the circumference of Circle B, yielding 6 as the radius and 36pi as the area. 
Now, we put 16pi/36pi as our ratio, yielding 16/36=4/9. Therefore, it is $(A)$4/9. 
~MathKatana