3 numbers are given in increasing order.

I. The second number is given.

is determined

II. The arithmetic mean of the first and third numbers is given.

is given, and therefore is given.

We are given a system of equations by the prompt:

The arithemetic mean of the first two is 5 less than the arithmetic mean of all three.

The sum of the first two numbers is equal to the arithmetic mean of the last two.

Combining these two, we get:

Thus, the second statement doesn't actually provide us with any information (we are still left with 2 equations and 3 variables, which cannot be solved for any particular number).

On the other hand, if y is determined, then relates and .  Similarly,  relates and .  Since these give different equations, we could use them to solve for both and .

So the first gives sufficient data while the second does not.

For statement (1), there are different combinations that satisfy the condition. For example, the five numbers can be  or the five numbers can be . Therefore, we cannot determine how many of the numbers are greater than  by knowing the first statement.

For statement (2), even though we know three of them, the two unknown numbers can both be greater than , or one smaller and one greater. Thus statement (2) is not sufficient.

Putting the two statements together, we know that the sum of the two unknown numbers is

Since none of them is greater than 100, both of them have to be greater than 50. Therefore when we combine the two statements, we know that there are two numbers that are greater than 50.

The median gives information about the center of a set of numbers, but is insufficient for calculating the mean. Additionally, the mode merely indicates which number is the most repeated value. Therefore, more information is required to calculate the mean.

Statement (1) provides the value of y in terms of x, which is not enough to determine the value of x in the list we are given.

Statement (2) gives the arithmetic mean of the list. We can the write the following equation:

However, we cannot find the value of x using the information in Statement (2) only.

Using the information in Statement (1), we can replace y by x-4 in the previous equation:

We need both statements to calculate the value of x.

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

The mean class grade is the sum of all average class grades divided by the number of grades:

Using Statement (1):

Therefore Statement (1) is sufficient to calculate the arithmetic mean for these grades.

Using statement (2):

Therefore 

Therefore Statement (2) is sufficient to calculate the arithmetic mean for these grades.

Each Statement ALONE IS SUFFICIENT to answer the question

To find the mean, we find the sum of all of the values, and divide by how many there are:

To find the mean we rearrange the values in ascending numerical order and select the middle value:

The product, then, is 

The arithmetic mean of  and  is equal to .

Statement 1 alone gives us this value directly. 

From Statement 2 alone, the value can be determined by dividing both sides by 4:

The median of three or five numbers (both odd numbers) is the number in the middle when they are arranged in ascending order. Statement 1 alone does not answer the question of which one is the median; both of the following orderings are consistent with that statement:

 - median 

 - median .

For similar reasons, Statement 2 alone does not answer the question.

Now, assume both statements to be true. From Statement 1, exactly one of  and  is less than , and the other is greater. From Statement 2, exactly one of  and  is less than , and the other is greater. Therefore, two of the five values are less than  and two are greater, making  the middle element, or median.

Assume Statement 1 alone. The sequences

and

are both arithmetic, each term being the previous term plus the same number (in the first case, this common difference is 40; in the second, it is 20). The fourth term is 120 in both cases, The second and third terms of the first sequence have arithmetic mean ; the second and third terms of the second sequence have arithmetic mean .  Therefore, the mean of those two terms cannot be determined for certain. A similar argument holds for Statement 2 alone being insufficient.

Now assume both statements. Let  be the common difference of the sequences mentioned in Statement 2. By Statement 2, 0 is the first term, so the sequence will be

By the first statement, the fourth term is 120, so

The second terms is and the third term is , and their arithmetic mean is .

An arithmetic sequence is one in which each term is formed by adding the same number to its preceding term - the common difference.

Let  be the first term, and  be the common difference. The first five terms are

The arithmetic mean of two numbers is half the sum of the numbers. The arithmetic mean of the first and third terms is

,

which is the second term. Statement 2 alone gives this number as 100.

Now assume Statement 1 alone. Consider these two sequences, both of which can be seen to be arithmetic with fifth term 130:

The arithmetic mean of the first and third terms differ, as can be seen by looking at the second terms; in the first sequence, it is 127, and in the second, it is 100. That makes Statement 2 inconclusive.