Statement 1 alone is not enough to prove  rational or irrational. Examples:

If , then

If  , then 

In both cases,  is irrational, but in only one case,  is rational.

Assume Statement 2 alone.  is rational, so, by closure of the rational numbers under multiplication,

 is rational. The rationals are closed under addition, so the sum

is rational.

If  is rational, then, since the product of two rational numbers is rational,  is rational. If Statement 1 alone is assumed, then, since  is irrational,  must be irrational.

Assume Statement 2 alone, and note that

In other words,  is the square root of . Since both rational and irrational numbers have irrational square roots,  being irrational does not prove or disprove that  is rational.

Statement 1 alone provides insufficient information.  is a number with a rational cube root, , and a rational square root, .  is a number with a rational cube root, , but an irrational square root.

Now assume Statement 2 alone.

In other words, is the square of . The rational numbers are closed under multiplication, so if  is rational,   is rational.

Statement 1 alone is not enough to determine whether  is rational or not;  and  both have rational cubes, but only  is rational. By a similar argument, Statement 2 alone is insufficient.

Assume both statements are true. , the quotient of two rational numbers, which must itself be rational.

We examine two examples of situations in which both statements hold.

Example 1: 

Then 

32 is not a perfect square, so  is not an integer.

Example 2: 

Then , making   an integer.

In both cases, both statements hold, but in only one,  is an integer. This makes the two statements together insufficient.

When we are faced with a radical in the denominator of a fraction, the first step is to multiply the top and bottom of the fraction by the numerator:

We can then reduce the fraction to:

A negative integer to an even power is positive:

Example: 

A negative integer to an odd power is negative:

Example: 

A positive integer to an odd power is positive:

Example: 

So, as seen in the first two statements, knowing only that base  is negative is insufficient to detemine the sign of ; as seen in the last two statements, knowing only that exponent  is odd is also insufficient. But by the middle statement, knowing both facts tells us  is negative.

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

This is a question that may initially seem to require more information to solve than it actually does, requiring you to leverage assets in order to “move up” the data sufficiency ladder. For this problem, be wary of "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient". Multiples tend to follow set patterns, so if you can find one of those patterns, you should be able to use small sets of smaller numbers to prove a rule, meaning that you need less information than you think you do.

From the information given, you know that T contains only multiples of 14. Based on your knowledge of factors and multiples, you should recognize that you can break up your numbers into factors to make them more manageable. If you are looking for multiples of 21, you should recognize that each multiple of 21 must also be a multiple of 3 and 7, since those are the prime factors of 21. Similarly, every multiple of 14 will also be a multiple of 2 and 7. Leverage what you know here: since every multiple of 14 is already a multiple of 7, that means that you are looking for how many multiples of 14 are also multiples of 3 so that you can satisfy the factors of 21 (3 and 7).

Statement (1) may at first seem insufficient since so little information is given. If there are 30 integers in the set, you should ask yourself: does the number of multiples of 3 depend more on where the set starts or how many items are in the set? You can come to a conclusion by using your printing press. Multiples of 14 start with 0 and continue:

0, 14, 28, 42, 56, 70, 84, 98… etc.

Notice that every third number (0, 42, 84, etc.) is a multiple of 3. Based on this, you should recognize that, as long as the number of terms in the set is divisible by 3, it doesn’t matter where you start. If you need to prove this to yourself, you can take 3 sets:

Set 1: 0, 14, 28 Set 2: 14, 28, 42 Set 3: 28, 42, 56

You can see that, although you start at different points in the pattern, because each set has three consecutive terms, you are guaranteed to have a single multiple of 3 in each set (remember that 0 is a multiple of all numbers). If you extrapolate from here, you should see that a set of 30 numbers would have ten times as many multiples of 3, or 10 multiples of 3 (and therefore of 21).

Statement (1) is therefore sufficient; eliminate "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked", "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient", and "Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed". Statement 2 gives only a starting point for the Set and no end point. Thus, the Set could have 1 multiple of 21 or an infinite number of multiples of 21. There is no way of telling. Therefore (2) is insufficient, eliminating answer choice "EACH statement ALONE is sufficient to answer the question asked".

Notice that if you hadn’t done the work to leverage statement (1) you may have concluded that you needed to know the starting point of the set in order to come to a conclusion and might have chosen "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient". Remember that, especially for harder questions like this one, to be leery of choosing any “easy” answer – generally the correct answer is going to require you to put in some work to make your answer choices work.

Choice "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked" is correct.

This problem heavily rewards those who "play the game" of Data Sufficiency effectively. While the two statements should look just about identical, those who Play Devil's Advocate and/or ask "Why Are You Here?" can spot the ever-important difference and avoid the trap answer.

Your inclination on both statements should be to use algebraic mirroring to factor coefficients and arrive at the expression j + k on the left hand side. For statement 1 that's:

m(j + k) = 2m

And for statement 2 that's:

5(j + k) = 10

Note that in statement 2, you can simply divide both sides by 5 and arrive at j + k = 2, making statement 2 sufficient.

Most people try to do the same thing on statement 1, dividing both sides by m. But you cannot do that! Why? Because m could equal 0, and you cannot divide by 0. You can demonstrate that by setting m equal to 0, in which case statement 1 would be:

0j + 0k = 0(2), in which case j and k could be absolutely anything.

So statement 1 is insufficient and the correct answer is "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked". And the lesson: you can avoid that trap (note that most examinees choose "EACH statement ALONE is sufficient to answer the question asked.") by:

Playing Devil's Advocate - when a statement seems a little too easy, ask yourself whether negative numbers, fractions, zero, or any other "edge cases" might give a different answer.

Asking "Why Are You Here?" - when one statement is extremely easy (as statement 2 is here), that's a signal that the more-nuanced statement likely has some difficulty to it, and that the easy statement might provide a clue. The difference between the statements here is statement 1 uses a variable where statement 2 uses the coefficient 5. Why would that distinction matter? Because if you can't rule out 0 as a value of a variable, you can't divide by it.

This Data Sufficiency problem requires you to pick numbers carefully and play Devil’s Advocate. Notice that the question stem tells you that x is a positive number less than 20. However, the question says nothing about any other number in the problem – you will probably need to leverage this distinction.

Statement (1) gives that x is the sum of two consecutive integers. It should quickly be clear that you can pick numbers to prove statement (1) is insufficient.

If the numbers are 1 and 2, x would be 3, but if the consecutive numbers are 2 and 3, x would be 5. Because you get two different values for x using the same information, you can conclude that Statement 1 is insufficient, eliminating (A) and (D).

Statement (2) gives that x is the sum of five consecutive numbers. Intuitively, that might lead you to say that x is the sum of the first 5 numbers, or

x = 1 + 2 + 3 + 4 + 5 = 15.

Since 2 + 3 + 4 + 5 + 6 = 20, which is not a valid value for x, it may seem that statement (2) is sufficient. However, play Devil’s Advocate! Nothing says that the consecutive numbers have to be positive. If the set instead starts with 0, you get

x = 0 + 1 + 2 + 3 + 4 = 10. You can therefore conclude that Statement (2) is insufficient, eliminating (B).

Taking statements (1) and (2) puts a restriction on what x can be. Since the sum of two consecutive numbers will always be an Even + an Odd, x must always be odd. You might automatically then conclude that x must equal 15 (since that is 1 + 2 + 3 + 4 + 5 and 7 + 8), but remember to once again Play Devil’s Advocate. If you can make another odd answer, then it’s possible to have another answer. You can do this by starting with a negative number. Remember, only x has to be positive – the numbers that make it up don’t have to be!

What about x = -1 + 0 + 1 + 2 + 3 = 5 and x = 2 + 3 = 5? Since you can construct another value for x, you must conclude that (1) and (2) together are not sufficient. Therefore, the answer is (E).