When Yes/No questions ask whether a certain algebraic expression is true, as you see here where the question asks about an algebraic inequality, it is generally most efficient to try to make the given statements look like the question. If you can effectively "mirror" the question with the statements, you can prove sufficiency.

For example, with statement 1, you can take the given information:

And try to make it look like the question, which has a positive 

 on one side of the inequality and the  and  terms on the other. To accomplish that, add  to each side:

And then subtract  from each side:

This statement then directly answers the question, so the answer must be "yes." Statement 1 is therefore sufficient.

You can work on Statement 2 the same way. If you take the given statement, , and subtract  from each side, you'll again have mirrored the question:

This also proves that the answer is "yes" and is therefore sufficient information. The answer, then, is "EACH statement ALONE is sufficient to answer the question asked".

This Yes/No Data Sufficiency problem rewards those who play devil's advocate. Each statement screams  could be ! (which would give the answer "yes,  but each allows for an alternative value of  (each of which would give the answer "no" to the question "is ).

With statement 1, it is important to recognize that  allows for two answers:  and . Remember: although on a day to day basis you think about positive numbers much more often than you think about negatives, you must always consider the possibility of negative values on the GMAT! Since each of these values provides a different answer to the main question (7 is greater than 6 but -7 is less than 6), statement 1 is not sufficient, so you can eliminate answer choices "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked" 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 also allows for different values of . While you might be tempted to just divide both sides by , note that you cannot do so unless you know for certain that . Dividing by 0 is not allowed, so if  could be 0 you have to treat this problem like a quadratic and solve that way. If you do so, you will subtract  from both sides of the given equation , leaving:

Then you can factor out the common  term:

And then solve:  or . Since each gives a different answer to the overall question, this statement is also insufficient and you can eliminate choice "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked".

When you combine the statements, you know from statement 1 that 
 must be either -7 or 7 and from statement 2 that   must be 0 or 7. Since 
 is the only possible solution consistent with both statements, you have a concrete value for  and can definitively answer the question: "yes, ." This is sufficient information, so the correct answer is " Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient". 

This problem rewards students who remember to “Play Devil’s Advocate” when picking numbers. Remember to examine your assumptions around the numbers you have picked and test the boundaries of the problem. Statement (1) gives that . If , then 

Since 

, this gives an answer of “Yes” to the question posed.

 However, consider:  could equal 0, in which case 

This would mean that , giving you an answer of “No” to the question posed.

 Because you can get both a “Yes” and a “No”, statement (1) is insufficient, eliminating "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) states that  is an integer. This means that  could be equal to -2 , in which case . This would give you an answer of “Yes” to the question posed. However, since could also be 0, using the work you did for statement (1) shows that you can also get a “no” for the information given. Since statement (2) allows for multiple answers to the question posed, you must conclude that statement (2) is insufficient and eliminate "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked".

 Taking both statements together, you should recognize that the two statements together are still insufficient based on the work you have already done. The numbers chosen to test statement (2) also were within the boundaries established by statement (1). Since  and  fit both statements and give an answer of “Yes” and “No” respectively, you can conclude that statements (1) and (2) are not sufficient even when taken together, so you must eliminate "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient" and choose answer choice "Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed".

This is a “yes/no” question that asks you to determine whether circular region X has a larger area than does circular region Y. Because statement 2 is much less wordy, it may be easier to evaluate, so you should start there.

Statement 2: This statement gives you the relationship between the radius of circle X and that of circle Y. Circle X has the larger radius. Since the formula for the area of a circle, , is a constant, it is true that a larger radius definitely means a larger area of the circle.

 If you need to prove this to yourself, remember that you can quickly do this by picking numbers.If circles X and Y had radii of 4 and 3 respectively, they would have areas of  and . A quick look at the equation for the area of a circle should confirm that the relationship between these two will remain constant as long as the radius of circle X is larger than that of circle Y, so statement 2 is sufficient. Eliminate "Statement (1) ALONE is sufficient, but statement (2) 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 1: This statement compares the circumference of circles X and Y with that of circle Z. The statement can be rewritten as “circumference X > circumference Z” and “circumference Y > circumference Z.” However, this does not allow for a comparison of circles x and y; it simply indicates that each are larger than circle Z.

Remember that you can often prove a statement insufficient by Playing Devil's Advocate and picking numbers even if the wording seems to make it abstract. Notice that if circle Z had a radius of  and a circumference of , that as long as X and Y had larger radii, they would fulfill the constraints of the statement. Both could have the same radius (for instance ) and have the same circumference, or one could be bigger than the other. Since there is no way of knowing, this information is not sufficient. Eliminate "EACH statement ALONE is sufficient to answer the question asked".

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

From the question stem, the average of the current portfolio is 50 photos (). Statement (1) indicates that each of the two new subjects has less than 117 photos. At most, each of the two new subjects would contain 116 photos. This would bring the new average to a maximum 56 photos (). While this is the maximum, there could always be fewer photos. This allows for both greater than 55 photos on average and fewer than 55 photos on average. Since this statement yields both "yes" and "no" as a response, statement (1) is not sufficient. Statement (2 tells us that each new subject has more than 71 photos. At least, each of the two new subjects would contain 72 photos. This means that the new average must be at least 52 photos (). Of course, this also allows for both greater than 55 and fewer than 55 photos on average, so statement (2) is not sufficient. Taking the two statements together, we know that the new average is between 52 and 56; however this is still not sufficient to say whether the average is greater than 55 or not. The two statements together are not sufficient and the answer is accordingly E. Tip: A quicker way to evaluate the statements is to maintain the average of 50 first, and then see whether there are enough additional photos to push the average to 55. For statement (1), subtract 50 photos from the maximum possible of 116. That 50 subtracted is enough to maintain the average; so at this point, we have an average of 50 with an additional 66 photos per subject. Since we have 2 new subjects, we have 132 additional photos (66 * 2) over 22 subjects to try and push the average over 55. If yields a number larger than 5, then the total average will be greater than 55. This technique requires much easier calculating than dividing 1232 by 22. Repeat the same procedure for statement (2).

A crucial strategy for Data Sufficiency problems is making sure to leverage all of the information given in the question stem. Here the fact that hops, yeast, and malt expenditures total $100,000 allows you to begin - before you ever get to the statements - with the equation:

H + Y + M = $100,000

When you then combine this with statement 1, you should see that the information remains insufficient. Statement 1 can be expressed as Y = 1.2M, allowing you to streamline the given equation to:

H + 1.2M + M = $100,000

But since you still have two variables you cannot solve for either. Since statement 1 is insufficient you can eliminate choices "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".

When you combine the given information with statement 2, however, note that you have:

Given: H + Y + M = $100,000

Statement 2: Y + M = H

This then lets you substitute for the expression Y + M in the initial equation, giving you:

H + H = $100,000

Since 2H = $100,000, that means that H = $50,000. Since the question asks you for H, you have sufficient information. Choice "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked" is correct.

As you assess this question, first note the given information in the question stem: the average for the 3 months was $75. Since the question requires you to separate the three months' bills to determine which is highest, it is a good idea to just multiply this average by the number of terms (3) right away to get the total amount for the 3 months: $225 total.

Statement 1 tells you that the lowest bill was received in July. This means that the July bill must be lower than $75, however, the highest bill could still be for June or August. Since this is not specific it is not sufficient. Eliminate choices "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 tells you that the total for July and August was $160. This gives you the exact amount for June. $225 – 160 = $65 for June. But remember - at this point you are only assessing statement 2 alone, so be careful not to include the information from statement 1 at this point. You do not know if July or August is the highest bill. This statement alone is not sufficient. 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 know that the June bill is $65 and that the July bill is less than this amount. Clearly August is the highest bill since both June and July are below the average. Because the statements together (but not alone) are sufficient, the correct answer is "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient".

For all “what is the value” data sufficiency questions, you are looking for consistency in the value you are asked to find. Remember - that means that you don’t necessarily have to solve for the value in question, so often you can leave these questions one step short if you recognize that an equation has only one solution.

In this question, you are asked for the total number of executives, given that each employee was given a bonus ($75 for non-executives and $125 for executives) and that there were 100 non-executives. Notice that the wording of the problem means that there can only be two categories – people who are executives and people who are not. That means that there aren’t other groups that aren’t accounted for.

Statement (1) gives that there are 120 total employees. Since you know that there are 100 non-executives, that means that you can set up the simple equation

100 + E = 120, where E is the number of executives.

Because this is a one-variable equation, statement (1) is obviously 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 the company gave a total of $10,000 in bonuses, you can set up and solve another single-variable equation, since you know that

Total Bonus Amount = (100)(75) + (E)(125)

Since you know that there are 100 non-executives and that they gave a total of $10,000 in bonuses, you can plug in to the equation to get:

10,000 = (100)(75) + (E)(125)

Since you have a single-variable linear equation, you know that if you solve you will get only one value for E. You already know, then, that statement (2) is sufficient and can leave the equation unsolved. Therefore, eliminate "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked" and choose "EACH statement ALONE is sufficient to answer the question asked" – both statements are independently sufficient.

Any time you are set up questions for a problem on the GMAT, remember that you should pick meaningful variables that are clearly tied to the problem you are working with. In this case, instead of “x” and “y”, you can pick “S” for the shorter (18-inch) chain and “L” for the longer (28-inch) chain.

Your question can then be written as “What is the value of L?” You know from the information given in the prompt that S and L have only one value each since there is no such thing as a “volume” discount for buying in bulk, so if you can solve for a single, consistent answer for L, you know that a statement must be sufficient.

Statement (1) gives that the total price for buying one 18-inch and one 28-inch chain is $68. If you translate this into an equation, this gives you

S + L = 68.

Since this is a two-variable equation, it is impossible to solve for L without additional information since there are multiple possible combinations for values of S and L. For example, S = 12 and L = 56 would satisfy the equation, but so would S = 1 and L = 67. Statement (1) is therefore insufficient – 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 the price of two 18-inch necklaces and one 28-inch necklace is $98. If you translate this into an equation, this gives you

2S + L = 98

Again, since you are left with a two-variable equation and no additional information, it is impossible to solve for S or L. Eliminate "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked".

Taken together, the two statements give you a system of linear equations:

S + L = 68

2S + L = 98

You may be tempted to go ahead and solve this, but remember that sufficiency requires consistency, not an answer. So ask yourself: what condition needs to be met in order for a system of equations to have only one answer? The two variables must be in different ratios in the two different equations. So in this case, S and L are in a 1:1 ratio in the first equation and a 2:1 ratio in the second. Because these are different, you can conclude that system will yield an answer and that the statements are sufficient together and pick (C).

If this wasn’t immediately obvious, you could always solve the system of equations by negating the first equation and adding:

-S – L = - 68

+2S + L = 98

Which yields S = 30. Substituting back into the first equation then gives you that L = 38.

The correct answer is "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient".

For data sufficiency questions that give multiple algebraic expressions, it is usually easier and more accurate to perform some algebraic manipulation to “make your math look like theirs” than it is to pick numbers. Remember, algebra doesn’t lie, and leveraging your assets can make the question stem easier to evaluate.

The question stem gives that and asks for the value of 

Notice the similarities between the two fractions – it is possible to leverage this information to simplify the question stem.

If you take the first equation and multiply both sides by , you get:

You can then substitute  in for  in the question stem to get

What is the value of ? – a much simpler question to deal with.

Statement (1) is then automatically recognizable as sufficient – it gives a value for . There is no room for any other value for the fraction, so this 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  have no common factors greater than 1. This means that both  and  are either primes or have no prime factors in common. However, it gives no other information. You can quickly prove this insufficient by picking numbers:

It is possible that  and  are 3 and 5 respectively and that 

However, it is also possible that  and  are 7 and 9 and that 

Statement (2) is therefore insufficient, eliminating "EACH statement ALONE is sufficient to answer the question asked" and leaving you with answer choice "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".