This question is a perfect example of when you need to use algebraic manipulation to make the question look like the statements. Generally speaking, students try to change the statements to match the question or somehow leverage the statements with the question as given, but often the best (and only) approach is to manipulate the question.

You should take the given question isand use your algebra toolkit to simplify it. While your first instinct might be to multiply both sides by 2z to eliminate most of the denominators, you don't know the sign of z so you cannot make this manipulation. However you can take the terms on the right side of this inequality with 2z in the denominator and move them to the left (addition and subtraction is always allowed). With this manipulation the question becomes:

Now combine all the terms on the left with the common denominator 2z to get:

is which after canceling the x's and y's in the numerator is the same as:

is  which after canceling the z's in the numerator and the denominator is the same as:

is  or rewriting it one last time:

is 

With that simplification of the question stem, statement 1 is all of a sudden very useful: it matches the question stem exactly and is thus sufficent! Yes:. Statement 2 is clearly not sufficient as it only tells you that x is positive and y is negative or vice versa. This does not allow you to determine whether  as you don't know the actual values of x and y. As a result, the correct answer is "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".

For this data sufficiency question, it helps to pick numbers to illustrate the logical constraints put forth in the statements and to play Devil’s Advocate – remember that sometimes non-intuitive combinations of numbers (that still fall within the constraints of the problem) can help you see how a seemingly-sufficient answer choice is in fact insufficient.

Statement (1) requires some logic and number picking to determine sufficiency. If you know that every factor of  must be a factor of , then  must be a factor of . To prove this, try picking numbers for  and  based on your restrictions. If  and you know that every factor of  must also be a factor of , you should recognize that  must be divisible by 

1,2,3,4,6,8,12,1,2,3,4,6,8,12, and 24. This means that any value of  must be a multiple of 24 greater than or equal to 24 since  must be a positive integer. So in this case,  could be 24, 48, 72, 96, etc. Since any of these divided by 24 must yield an integer, you can conclude that Statement (1) is sufficient, eliminating "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) is very similar to (1) on the surface. However, do not go straight to assuming that statement (2) is sufficient – pick numbers, do the math, and think through the logic!

If every factor of , then you can conclude that  is divisible by . What does this mean for divisibility?

If , it has factors of  1,2,3,4,6,8,12,1,2,3,4,6,8,12, and 24. That means that, when ,  must be a multiple of 24. If you pick numbers, you see that, if 

, which is an integer.

However, there is no limit on what  can be, only that it must be a multiple of 24 if . Thus,  could equal 48, which would yield

, which is not an integer.

Since you got both a “yes” and a “no”, you can conclude that statement (2) is insufficient, eliminating "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked" and leaving you with the correct answer, "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".

When you assess statement 1 in this problem, consider the possibilities for the sum of any three terms (at this point you don't know how many terms) in a set to equal 21:

Case 1: there are exactly three terms, and the sum is 21 (for example 6 + 7 + 8 = 21). Since the average is the sum divided by the number of terms, the average will be 7.

Case 2: there are more than three terms. Here one case that works is if they're all 7, so whenever you pick three terms and add them together, you're adding 7 + 7 + 7 = 21. Can any other cases work? As soon as there is some diversity to the numbers (e.g. 6, 7, 8, 6, 7, 8) you cannot guarantee that the sum of any three of them will be 21. If you were to sum three consecutive numbers in that proposed set (6 + 7 + 8) that work, but as soon as you pick a repeat value (6 + 6 + __ or 8 + 8 + __) you cannot get to a sum of 21. So if there are more than three terms, all terms must be 7. And then the average will have to be 7 as well.

So with statement 1 alone, you know that the average must be 7, so statement 1 is sufficient.

Statement 2 is there to make you think you need to know the number of terms to go along with the information from statement 1. But as proven above you do not need that. Clearly statement 2 alone is insufficient (Average = Sum of Terms / Number of Terms, and statement 2 only gives you the number of terms...the sum could be anything), so the correct answer must be "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".

Neither statement alone is sufficient, as in each case the x and y terms can appear in any part of the number line (both large positive numbers, both large negative numbers - each of which would give a "yes" answer - or one of each, in which case you'd have a negative product and a "no answer). So the problem really "gets started" when you take the statements together to assess "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient" vs. "Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed".

In doing so, the common mistake that most students make is to pick numbers to reason out a solution rather than apply conceptual understanding and algebraic manipulation. Number picking has its place on some problems (some can only be solved using numbers to show patterns, etc.) but most can be solved quickly and efficiently with algebra. Using your understanding of combining inequalities, it is possible to isolate x and y and learn more about them individually. First let’s eliminate x and isolate y:

Step 1: Rewrite the inequalities to line up variables

y – x < 2

2y – x > 8

Step 2: Multiply top inequality by -1 to get the signs pointing the same way and then combine to eliminate x:

-y + x > -2

2y – x > 8

y > 6

Repeat step 2 to eliminate y by multiplying the top inequality by -2 to get the signs pointing the same way and then combine:

-2y + 2x > -4

2y - x > 8

x > 4

If y > 6 and x > 4 then you know that the product of xy must be greater than 24 and the answer to the question is "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient".

This problem features a classic example of "algebraic mirroring," a technique via which you use algebraic manipulation to either make one of the statements look like the question or make the question look like one of the statements.

With statement 1, your clue that you should use algebraic mirroring is that number 16 that appears as a coefficient in statement 1 and as a constant in the question. Can you isolate 16 on the right hand side of the equation in statement 1 to make it look like the question? If you divide both sides by  you can:

becomes

And from here you can use a bit of "reverse engineering" with the process of adding fractions. If you were to add two fractions, you'd find a common denominator and then sum the numerators; by that same logic, since you have a "common denominator" (one single denominator) of , you can break apart the numerator:

This means that you can rephrase the equation as:

And then you can factor out the common terms in each fraction:

This directly mirrors the question, so you know that the answer is "yes" and statement 1 is sufficient.

Alternatively, you could manipulate the question to look like the statement. If you take

 and multiply each side by  the question then asks:

Is 

And the fractions factor to:

Is ?

To which statement 1 loudly proclaims "yes." Again, statement 1 is sufficient.

Statement 2 is not sufficient, as it allows for both "yes" and "no" answers. When you equate  and  to make the question look like:

The answer is "yes" if  and "no" if the value is anything else, so statement 2 is not sufficient and the correct answer is "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".  

This exponent-based problem involves an important lesson in Data Sufficiency strategy: if the problem asks for the value formed by a combination of variables (such as here), there is usually a way to solve specifically for that combination without having to solve for the variables individually!

Here you can do that given statement 1. If you employ the first guiding principle of exponents, "find common bases," you can factor the 3, 9, and 27 all into base 3s so that all your bases are common. That means that:

Which then means that you can employ the rule for taking one exponent to another (in other words, ) and rephrase this as:

Then you can combine the terms on the left using the rule that when you multiply two exponents with the same base, you add the exponents. Therefore:

And here you can set the exponents equal using another law of exponents, meaning that , and making Statement 1 sufficient.

Statement 2 is not sufficient, as merely knowing that  alone does not allow you to find a specific value for . Therefore the correct answer is "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".

This yes/no data sufficiency question asks is 

Statement 1 states that . 

This statement may not appear sufficient at first, since it cannot be easily solved to give a value for  or for . However, it's always important to remember that sometimes things that would be insufficient to solve a "what is the value" question may be perfectly suitable for a "yes/no" question.

In this case, look to simplify the question stem using what you know from statement (1). Since statement (1) matches what's on the right hand side of the equation in the question stem, you can substitute it in to get:

Is 

A squared number cannot be negative, so the answer must always be yes. This statement is consistent so it is 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 that  and 

Again, this statement may not appear to be sufficient. It does not give specific values for  or . However, if you “Just Do It” or if you plug in the numbers then you will see that it is sufficient.

Conceptually it looks like this: so long as  is greater than 3,  will be greater than . Whatever number  is can only take away from the - it cannot add to it. In fact, statement (2) gives more information than strictly necessary;  would have been sufficient. The correct answer is "EACH statement ALONE is sufficient to answer the question asked".

If this wasn't apparent, you could also pick numbers. Remember as you plug in numbers, however, that you want to test the limits of the problem to try to force statement (2) to be inconsistent. One way to do this is to pick extreme numbers for  and . 

If  and , the statement becomes

Is  Since  is definitely bigger than , the answer is yes.

But what if  were bigger than 

If  and ,  the problem becomes

Is . Since  is obviously much bigger than any negative number, the answer is also yes. Statement (2) is sufficient. Eliminate "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked" and choose answer choice "EACH statement ALONE is sufficient to answer the question asked".

There are essentially two ways to manipulate the algebra for statement 1 to get it to look a bit more like the question stem.

One way is to multiply top and bottom by 

 means that in the top set of fractions   yields simply . And for the bottom, the  terms will cancel, leaving just 

. So the new fraction is:  And here's where one more step working with fractions really pays off. You can express that as  which leaves you with .

This question asks for a specific number value for x given that x is a two-digit integer.

 Statement (1) states that the product of the two digits is 14. To figure out potential digits that would fit this statement, consider that factor-pairs of 14: 1 and 14, and 2 and 7. Notice that 1 and 14 isn't an option since you're looking for single-digit numbers to make up a two-digit number. Therefore, the two digits must be 2 and 7. One possibility for the value of x is therefore 27.

However, you must remember to Play Devil's Advocate. What happens if you switch the positions of the digits? 72 still fulfills the constraints of the statement and is also therefore a possible value of x. You must therefore conclude that Statement (1) is not sufficient and eliminate "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked" and "EACH statement ALONE is sufficient to answer the question asked".

Statement (2) gives that x is divisible by 9. In other words, x is a multiple of 9. However, there are many two-digit multiples of 9, such as 18, 27, 36, etc. This is clearly not sufficient since the statement can yield multiple values for x. Eliminate choice "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked".

When you take the statements together, you should recognize that the two, separate values you found in statement (1) still work here. Both 27 and 72 are multiples of 9, and both have digits that multiply to 14. Because the two statements together still give you two separate values, you must conclude that the two statements are not sufficient. The correct answer is " Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed".

In beginning this problem, it is a good idea to take inventory of everything you know. For one, you can sum the "known" values: 3 + 8 + 17 = 28. Secondly, you can use the given equation  to put the entire list in terms of : if , then the sum of the values:

Can be expressed as 

And then you can also note that for the average of 5 terms to be less than 8, the sum of those terms must be less than 40. So this question is also asking:

Is ?

Which simplifies to:

Is 

Is 

As you progress to the statements, statement 1 may look tempting: If , wouldn't the least possible value of  be 3?

No - that's only true if you know that  is an integer, which is not necessarily the case.  could be something like 2.5, which would give the answer "yes" to the question. Or  could of course be anything else greater than 2 (like 10 or 100), which would give the answer "no." So statement 1 is not sufficient. And there is an important lesson here: you cannot assume that a value is an integer unless you're either told so or given information that proves it so.

Statement 2, however, is sufficient. When you factor down  that means that . This means that  must be less than 3, guaranteeing that the answer to the question is "yes." Therefore the correct answer is "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked".