Assume Statement 1 alone. The median of five numbers (an odd number) is the number in the middle when they are arranged in ascending order. By Statement 1, this number is , so the question is answered.

Statement 2 alone, however, gives that the mean is . It is possible that the mean and the median can be one and the same or two different numbers.

Case 1: 

The mean is 

making this consistent with Statement 2.

The median is the middle element, .

Case 2: 

again, making this consistent with Statement 2.

The median is the middle element, .

In a data set of 19 elements, 19 being odd, the median is the element that occurs in the  position when they are arranged in ascending (or descending) order. Neither statement alone tells us what that middle element is. But put together, they tell us what the ninth and eleventh elements would be when arranged in ascending (or descending) order. Since both are 72, the element between them - the tenth element, and, thus, the median - must be 72. 

The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

From the initial information, we can determine that  of the balls are green.

The first statement alone tells you that half of the blue marbles and half of the red marbles are large, so half of the green marbles must be large as well. But this alone does not tell you how many marbles there are total.

The second statement alone tells you how many large marbles are red or green, but you do not have any way of figuring out how many small marbles or total marbles are. 

But if you put the statements together, you know the following:

Half of the red marbles and half of the green marbles comprise the thirty-six large marbles:

There are 120 marbles.

The answer is that both statements together are sufficent to answer the question, but neither statement alone is sufficient.

Both  and  satisfy the conditions of Statement 1; the first fraction is in lowest terms, the second is not.

Both  and  satisfy the conditions of Statement 2; the first fraction is in lowest terms, the second is not.

Neither statement alone is enough to answer the question. Both together, however, are enough to prove the fraction is not in lowest terms; a fraction which satisfies both statements has both its numerator and denominator divisible by 2.

The two statements together are insufficient.  and  each satisfy the conditions of both statements, but the first fraction is in lowest terms and the second is not.

The two statements together provide insufficient information.

  and   are two examples of fractions that satisfy both conditions; note that the first is represented in decimal form by a terminating decimal, and the second, by a repeating decimal.

From the question, we know that only one of the fractions in the answer choices is larger than . The correct answer must therefore be the largest of the answer choices. This observation is important, because it means we can eliminate some of the answer choices without performing any calculations. For example, is smaller than , so eliminate from the possibilities. is smaller than both and , so eliminate it as well. Similarly, is smaller than , so eliminate too.

This leaves us with only two answer choices: and

First consider . We can rewrite this as , which is clearly smaller than . Eliminate .

By process of elimination, we've thus shown that is the correct answer choice.

Statement 1 alone does not prove is or is not an integer.

For example, if , then

.

If , then

.

is an integer in only one scenario, but Statement 1 is true in both.

Now assume Statement 2 alone, and let . Then is an integer by Statement 2, and

Suppose is an integer. Then , the difference of integers, is itself an integer. But we know that this is equal to , which is not an integer, so, by contradiction, is not an integer.

A fraction in lowest terms can be expressed as a terminating decimal if and only if its denominator has no prime factors other than 2 or 5; the numerator is irrelevant. Statement 1 is unhelpful since it only gives information about the numerator. If Statement 2 is assumed - that is, if we know the denominator has 7 as a prime factor - we know that the decimal representation of  is repeating.

Assume Statement 1 alone. A necessary and sufficient condition for a lowest-terms fraction to be equivalent to a terminating decimal is that its only prime factors are 2 and 5. Statement 1 says this about both fractions; the sum of terminating decimals is itself an integer or a terminating decimal.

Statement 2 alone provides insuffiicient information. Examine these examples:

Both fit the conditions of the main premise and Statement 2, but in only one case is the sum equal to a terminating decimal.