In a circle with center $O$, central angle $AOB$ has a measure of $\dfrac{7\pi}{4}$. The area of the sector formed by central angle $AOB$ is what fraction of the area of the circle?

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Don't overthink this one. An entire circle is $360 \degree$ or $2\pi \text{ radians}$. Therefore, the area of the sector to the area of the entire circle is:

$$\frac{\frac{7\pi}{4}}{2 \pi}$$ $$= \frac{7 \pi}{4} \cdot \frac{1}{2\pi}$$ $$= \boxed{\frac{7}{8}}$$

Here's a picture of the situation:

Note that the fraction $\dfrac{7}{8}$ is the ratio for most corresponding properties of the sector to the entire circle (arc length, sector area, central angle, etc.).