We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that $\displaystyle m \angle ACB = 30 ^{\circ }$. $\displaystyle \angle ACB$ is an inscribed angle that intercepts the arc $\displaystyle \widehat{AB}$; therefore, the arc - and the central angle $\displaystyle \angle AOB$ that intercepts it - has twice its measure, or $\displaystyle 2 \times 30^{\circ }= 60 ^{\circ }$. From angle addition, this can be subtracted from $\displaystyle 180^{\circ }$ to yield the measure of central angle $\displaystyle \angle BOD$ of the shaded sector, which is $\displaystyle 120^{\circ }$. That makes that sector $\displaystyle \frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\displaystyle \frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\displaystyle \frac{2}{3} : \frac{1}{3}$, or $\displaystyle 2:1$.

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle $\displaystyle \angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $\displaystyle 2 \pi$ to yield radius $\displaystyle 28 \pi \div 2 \pi = 14$, and that can be substituted for $\displaystyle r$ in the formula $\displaystyle A = \pi r^{2}$ to find the area: $\displaystyle A = \pi \cdot 14^{2} = 196 \pi$.

However, it provides no clue that might yield $\displaystyle m \angle BOD$.

From Statement 2 alone, we can find $\displaystyle m \angle BOD$. $\displaystyle \angle BCD$, an inscribed angle, intercepts an arc twice its measure - this arc is $\displaystyle \widehat{BAD}$, which has measure $\displaystyle 240^{\circ }$. $\displaystyle \widehat{B D}$, the corresponding minor arc, will have measure $\displaystyle 360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$. This gives us $\displaystyle m \angle BOD$, but no clue that yields the area.

Now assume both statements are true. The area is $\displaystyle 196 \pi$ and the shaded sector is $\displaystyle \frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\displaystyle \frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$.

Assume Statement 1 alone. No clues are given about the measure of $\displaystyle \angle BOD$, so that of $\displaystyle \angle AOB$, and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\displaystyle \widehat{BD }$ , or, subsequently, $\displaystyle \widehat{AB}$, is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined. 

Now assume both statements are true. Let $\displaystyle r$ be the radius of the circle and $\displaystyle N ^{\circ }$ be the measure of $\displaystyle \angle BOD$. Then:

$\displaystyle \frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and 

$\displaystyle \frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$\displaystyle r ^{2}N = 17,280$

and 

$\displaystyle r N= 1,440$

From these two statements:

$\displaystyle \frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

$\displaystyle r = 12$; the second statement can be solved for $\displaystyle N$:

$\displaystyle 12 N= 1,440$

$\displaystyle N = 120$.

$\displaystyle m \angle BOD = 120^{\circ }$, so $\displaystyle m \angle AOB = 60^{\circ }$.

Since $\displaystyle r = 12$, the circle has area $\displaystyle A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$. Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

$\displaystyle \frac{1}{6} \times 144 \pi = 24\pi$.

The area of a $\displaystyle 60^{\circ}$ sector of radius $\displaystyle r$ is 

$\displaystyle A= \frac{60}{360} \pi r^{2} = \frac{1}{6} \pi r^{2}$

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the $\displaystyle 60^{\circ}$ arc is 

$\displaystyle L= \frac{60}{360} \cdot 2 \pi r = \frac{\pi r}{3}$

Given that length, you can find the radius:

$\displaystyle 8 \pi = \frac{\pi r}{3}$

$\displaystyle 24=r$

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

The time alone is insufficient without the length of the hand. The second statement does not give us that information, only the relationship between the lengths of the two hands, which is useless without the length of the minute hand.

The answer is that both statements together are insufficient to answer the question.

The arc with the greater degree measure is the one which is the greater part of its circle.

If both arcs have the same length, then the one that is the greater part of its circle must be the one on the smaller circle; Statement 1 alone tells us both have the same length, but not which circle is smaller.

If $\displaystyle OA > PC$ , then $\displaystyle \odot O$ has the greater radius and, sqbsequently, the greater circumference; it is the larger circle. But we know nothing about the measures of the arcs, so Statement 2 alone is insufficient.

If we know both statements, however, we know that, since the arcs have the same length, and $\displaystyle \odot O$ is the larger circle - with greater circumference - $\displaystyle \widehat{AB}$ must be take up the lesser portion of its circle, and have the lesser degree measure of the two.

$\displaystyle \angle AOB$ and $\displaystyle \angle CPD$ are the central angles that intercept $\displaystyle \widehat{AB}$ and $\displaystyle \widehat{CD}$, respectively. The measure of an arc is equal to that of its central angle, so, is we are given Statement 2, that $\displaystyle m \angle AOB = m \angle CPD$, we know that $\displaystyle m \widehat{AB} = m \widehat{CD}$. The arcs are the same portion of their respective circles. The larger of  $\displaystyle \odot O$ and $\displaystyle \odot P$ determines which arc is longer; this is given in Statement 1, since, if $\displaystyle OA = 2 \cdot PC$, then $\displaystyle \odot O$ has the greater radius and circumference.

Both statements together are sufficient to show that $\displaystyle \widehat{AB}$ is the longer of the two, but neither alone is suffcient. From Statement 1, the relative sizes of the circles are known, but not the degree measures of the arcs; it is possible for an arc on a larger circle to have length less than, equal to, or greater than the arc on the smaller circle. From Statement 2 alone, the degree measures of the arcs can be proved equal, but not thier lengths.

From Statement 1 alone, $\displaystyle m \angle AOB = 60 ^{\circ }$, so $\displaystyle \widehat{AB}$ can be determined to be a $\displaystyle 60 ^{\circ }$ arc. But no method is given to find the length of the arc.

From Statement 2 alone, neither $\displaystyle m \angle AOB$ nor radius $\displaystyle O A$ can be determined, as the area of a triangle alone cannot be used to determine any angle or side.

From the two statements together, $\displaystyle m \angle AOB = 60 ^{\circ }$, and the common sidelength of the equilateral triangle can be determined from the formula

$\displaystyle A = \frac{s^{2} \sqrt{3}}{4}$

This sidelength $\displaystyle s$ is the radius of the circle. Once $\displaystyle s$ is calculated, the circumference can be calculated, and the arc length will be $\displaystyle \frac{60}{360} = \frac{1}{6}$ of this.

If either or both Statement 1 and Statement 2 are known, then then only thing about $\displaystyle \widehat{AB}$ that can be determined is that it is an arc of measure $\displaystyle 110 ^{\circ }$. Without knowing any of the linear measures of the circle, such as the radius or the circumference, it is impossible to determine the length of $\displaystyle \widehat{AB}$.

Statement 1 only establishes that $\displaystyle \widehat{AB}$ is one-third of the circle. Without other information such as the radius, the circumference, or the length of an arc, it is impossible to determine the length of the chord. Statement 2 alone is also insufficient to give the length of the chord, for similar reasons.

The two statements together, however, establish that $\displaystyle 40 \pi$ is the length of the major arc of a $\displaystyle 120^{\circ }$ central angle, and therefore, two-thirds the circumference. The circumference can therefore be calculated to be $\displaystyle 40 \pi \div \frac{2}{3} = 60 \pi$, and minor arc $\displaystyle \widehat{AB}$ is one third of this, or $\displaystyle 20 \pi$.