, making  a right triangle. Since , both segments being radii,  is also a 45-45-90 triangle.

From Statement 1 alone, it can be determined by way of the 45-45-90 Theorem that .

From Statement 2 alone, since the area of a right triangle is half the product of its legs, 

Since  and ,

From either statement alone, the radius of the circle can be calculated. From there, the circumference can be calculated, and the length of the arc can be found by multiplying the circumference by 

The two statements together are not enough unless you know the size of the minute hand; without this information, you cannot tell the angular position of the minute hand, so, even if you know the angle the hands are making, you do not know the position of the hour hand either.

The first event happens numerous times over the course of twelve hours, so the first statement is not enough to deduce the time; all the second statement tells you is that it is forty minutes after an hour (12:40, 1:40, etc.)

Suppose we put the two statements together. It is  from one number to the next, so the 8:00 position is the  position. If the hour hand makes a  with the minute hand, then the hour hand is either at  or . Since forty minutes is two-thirds of an hour, however, the hour hand must be two-thirds of the way from one number to the next.

Case 1: If the hour hand is at , then it is at the  position - in other words, two-thirds of the way from the 5 to the 6. This is consistent with our conditions.

Case 2: If the hour hand is at , then it is at the  position - in other words, one-third of the way from the 10 to the 11. This is inconsistent with our conditions. 

Therefore, only the first case is possible, and if we are given both statements, we know it is 5:40.

Since there are twelve numbers on the clock, the angular measure from one number to the next is ; this means   represents two and a half number positions.

Suppose we know both statements. Since the minute hand is on the 6, the hour hand is either midway between the 3 and the 4, or midway between the 8 and the 9. Both scenarios are possible, as they correspond to 3:30 and 8:30, respectively, so the question is not answered even if we know both statements.

From Statement 1, the measure of the arc can be determined by doubling the measure of the intercepting angle, which is . From Statement 2, the measure of the arc can be calculated by subtracting from  the degree measure of the corresponding major arc, which is . 

According to Greg's erroneous information, the number of people who voted was

.

213 voted for Douglas, meaning that his sector will have degree measure

Based on the new information, 

 people voted.

213 voted for Douglas, meaning that his sector will have degree measure

The reduction in degree measure will be .

The number of people who voted:

176 people voted for Starr, so the sector representing Starr will have measure

Let  be the radius of Circle B. Then Circle A has radius  and, subsequently, area . Since the area of Circle B is , the area of Circle A is 36 times that of Circle B.

The given sector of Circle A has the same area as Circle B, so the sector is one thirty-sixth of the circle. That makes the angle measure of the sector

The radius of Circle A and the length of a side of the square are the same - we will call each . The area of the circle is ; that of the square is . Therefore, a sector of the circle with area  will be  of the circle, which is a sector of measure

The number of people who voted:

245 people voted for Starr, so the sector representing Starr will have measure