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).

This question asks for the length of the arc XYZ given that arc XYZ is a semicircle. Because you know that to find the length of an arc you need the length of the diameter, you should recognize that you will either need to be directly given that value or leverage your assets in order to find the diameter.

Before you even begin work on this one, recognize that for a more difficult problem, C is much too easy an answer. If you know the diameter of a semicircle (which statements 1 and 2 together hand you on a silver platter), you can easily find the arclength of that semicircle. Don’t take the bait on C – or at least recognize that you should try to leverage your assets as much as possible before concluding that either statement is insufficient.

Recognize also that statements 1 and 2 each give you the same kind of information – one segment of the diameter and the side of a smaller right triangle within the larger right triangle. So if one statement is sufficient, so is the other.

Statement (1) gives you that . Since you know that the smaller right triangle has a second leg of 4 you can set up the Pythagorean theorem and solve for length XY. Since the numbers are relatively small and easy to work with, it doesn't hurt to go ahead and solve. However, if the numbers were messy to work with, you should remember that you could just take this as a "known number" and go from there.

If you do the math, 

,or . Since you know that you'll be dealing with the Pythagorean theorem for other parts of this problem and will need the value 

, it doesn't hurt to just leave this as-is and move on.

Now, it is tempting to say that without information about r it is impossible to continue. However, take a look at the two possible Pythagorean set ups that remain. Can you leverage your assets to solve for r and therefore the arc length?

, which becomes

.
Notice that because you have the expression "" in both equations, you can substitute in to get:

, which can be simplified to

.

This is a single-variable linear equation, so you will have only one value for r. And since you know that the only piece of information you need to solve for the arclength is 

Turning your attention to Statement (2), notice that the information given is identical in value to the information given in statement (1). If you automatically recognize that you can solve for 

You can find using the formula

, which simplifies to 

.

And, just as with statement (1), you can then set up two equations:

 and

 which can be rewritten as

And, similar to in statement (1), you can substitute in for  to get:

And just as with Statement (1), you can eliminate the quadratic by subtracting the squared value from both sides, leaving you with a linear, single-variable equation:

As with statement (1), you should recognize that because you can find r and q, you will be able to find the arclength and that statement (2) is sufficient. The correct answer is "EACH statement ALONE is sufficient to answer the question asked".

 Notice that this is a problem where the answer that is handed to you on a silver platter, "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient", is not correct. For harder problems, if you are tempted to pick "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient" with little work, try to "move up" the data sufficiency ladder by leveraging your assets, especially in geometry.

This problem is a classic example of the "Why Are You Here" strategy. Clearly statement 2 is not sufficient on its own, so why was it written?

In statement 1, the "obvious" answer for x is that x = 2013 and (x + 1) would then equal 2014. Which looks pretty sufficient. But there's one additional, not as obvious possibility: x = -2014 and (x + 1) = -2013. Since negative-times-negative is positive, that would give the same result. So statement 1 looks pretty sufficient but it is not. Statement 2 provides that little clue by emphatically stating that x is odd. That should get you thinking "how could x not be odd?" and of course that would be if x were -2014 and x + 1 were -2013. With both statements together, that negative-negative possibility is off the table, so the correct answer is "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient".

In this problem you should notice two critical elements in the question stem: 1) there’s an entire equation given to you (A = 2B), and 2) the question is asking about a combination (B – C) and not an individual variable. Whenever that is the case, you should see if you can solve directly for the combination, which generally requires less information (so you can get a “more sufficient” answer) then it would take to solve for each variable individually.

When you assess statement (1), you can set up the equation A – 4 = 2C. If you then combine the two known equations at this point, you have:

A = 2B

A – 4 = 2C

If you then plug in 2B for A in the second equation, you have:

2B – 4 = 2C

You can then add 4 to and subtract 2C from each side to get the B and C terms together (to match the question “What is B – C?”) and you have:

2B – 2C = 4 Divide both sides by 2 and you’ve solved for exactly what they asked:

B – C = 2

Therefore, statement 1 is sufficient.

Statement 2, however, is not sufficient. When you take your initial equation (A = 2B) and combine with the equation that statement 2 tells you (A = 8 + C), note that you cannot get B and C together with the same coefficient. When you substitute 2B for A, you get:

2B = 8 + C

But this doesn’t allow you to get directly to B – C, so this statement is not sufficient. Accordingly, the correct answer is "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked".

As you get started on this problem, remember that with problems asking for a ratio between two unknowns that you often need much less information than you do for problems that ask for exact values. Students who do well on this problem will leverage this information and manipulate the statements in order to get as much information from them as possible.

Statement (1) gives a weighted average of the items sold. While this may seem to not give enough information, remember that weighted averages are essentially another way of expressing the ratio of the “weight” of two categories in an average – and because this “weight” is determined by the ratio of the number of items in each category, you can use this to solve for the ratio of cookies sold to brownies sold. If you recognize this, you can go ahead and determine that Statement (1) is sufficient. However, if y you don’t immediately recognize this, you can go ahead and solve for the ratio.

One easy way to illustrate this is with the Mapping Strategy, which can be set up as below, where Categories 1 and 2 are Cookies sold and Brownies sold, respectively.

Category 1 ---------Distance 1 ----------- Average -------Distance 2 ------ Category 2

Inserting what you know and finding the distance between each gives you

Cookies ------ 0.12 ------- 1.42 -------0.08 ------- 1.5

The ratio of the distances is therefore 12:8, which simplifies to 3:2. The ratio of the distances is always the inverse of the ratio between the amounts, so the ratio of the number of cookies sold to brownies sold is 2:3. Statement (1) 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 the total value of items sold was $14.20. This strikes many students immediately as insufficient, leading them to pick "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked". However, don’t forget to leverage your assets! Because you can only have whole numbers, there are only so many combinations of brownies and cookies sold that could give you $14.20 – it pays to experiment to see if there is in fact only one solution. While you should never “bring down” information from a previous statement, you can “borrow” from another to give you a place to start experimenting. Notice that the total amount, $14.20 is ten times the average price given in statement (1). This means that you could conclude that there could be a total of ten items sold. Using this information, you can set up two equations:

1.3C + 1.5B = 14.20

And

C + B = 10

Notice that you have a system of equations. Remember that if you solve a (non-dependent) linear system, you will get only one value each for C and B, meaning that you will have a consistent ratio. So it is possible to solve for one ratio. You don’t need to solve since you already know that there is a ratio that works with this number based on your work on Statement (1) – 2:3. The question is whether it is possible to have others.

What if, for example, there were 11 items? One way to “test the limits” of this is to ask if it would be possible to have 11 of any combination of items. If, for example, there were 11 chocolate chip cookies sold (the less expensive item), you would get:

(11)(1.3) = $15.40.

Because this is greater than the given price, $14.20, you should recognize that it is impossible to have a total of $14.20 with eleven items sold.

Similarly, you can test 9 or fewer items by seeing if it is possible to get to a total of $14.20 with 9 of the more expensive brownies. That would give you:

(9)($1.5) = $13.50

This means that there is no way to reach a total of $14.20 by selling 9 items. You can therefore conclude that the ratio from 10 items is the only possible one since you can’t sell a fraction of a brownie or cookie. Therefore, 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".

This question asks for the specific value of  give that , , and  are distinct positive integers and that . The problem also state that  .

Since square roots can be difficult to conceptualize in data sufficiency questions, it helps to simplify the question by first squaring both sides to get:

Since you know that  is positive (and therefore not 0), you can divide both sides by  to get:

. 

While this may have already been apparent because of the definitions of squares and square roots (for  algebraic manipulation confirms your initial assumption.

Statement (1) gives that . Your given information then becomes

Since  and  are distinct positive integers and , the only combination is , and . (While 1 and 8, remember that  and so the two cannot be equal.) Since this means you oly get one possible value for , state (1) 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 you that the average of , , and  is . Because 

 this means that the sum of , , and  must be 14, or: 

This statement is tricky. It may seem like you cannot allocate the 14 between , , and  in only one way. Many students therefore quickly assume that statement (2) is insufficient and pick "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked". Leverage what your assets, however, and you will quickly see that there is only one where  is for , , and  It just takes a careful consideration of what you know.

In order to get a sum of 14, an even number, you must either have:

Odd + Odd + Even
or
Even + Even + Even

If you had Odd + Odd + Even, it would be impossible for  since if  and  were odd, it would be impossible for  to be even, since the product of two odd numbers will always be odd. Similarly, it would be impossible for  and  to be odd if  were even since the product of an even and odd number must be even. Thus, you know that all three numbers have to be even.

And since the only three positive, even numbers that add together to 14 with no repeats are 2, 4, and 8, you know that 

Statement (2) is also sufficient, so the answer is "EACH statement ALONE is sufficient to answer the question asked".

This is yes/no data sufficiency question asks whether line M run through point (6,6)

You are given that Line M is tangent to a circle centered on point (3,4)

However, you are not given any information as to the size of the circle or where M is tangent to that circle.

Statement (1) gives you that Line M runs through point 

It DOES NOT say that this is the point of tangency, an important distinction. Because of this, you have no indication as to the size of the circle. In addition, because it takes two points to make a line, you also have no indication as to orientation of the line. Since this means that you have no evidence whether or not line M goes through point 

Statement (2) states that Line M is tangent to the circle at point (3,6)

If you are trying to rush through this problem, you may automatically assume that, like statement (1), this statement does not give enough information about the line and must be insufficient and must be paired with statement (1) and pick (C). However, before you jump to this stage remember that you only get to use both statements when neither statement is sufficient alone. So take a close look at statement (2) and put the work in to leverage your assets before you discard it.

The definition of tangent is very important. A line that is tangent to a circle touches that circle at only one point and is perpendicular to the circle's radius at that point. Given that the center of the circle is at (3,4), knowing that the point of tangency is at (3,6) means that a radius that the tangent line must be perpendicular to is part of the vertical line x=3. 

For the tangent line to be perpendicular to a vertical line, it must be horizontal. That means M runs along the line y=6

Since this encompasses all values of x as long as y=6, this means that line M does pass through (6,6).

Statement (2) is sufficient alone.

Remember that, for geometry problems especially, there can be multiple ways to sufficiency within data sufficiency problems, so just because one statement doesn't work doesn't mean that a very similar statement will also not work. Put the work in to prove the statements sufficient or not by leveraging your assets!

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

This question asks for a consistent, specific answer for the remainder of the quotient . 

Remember that the most important thing here is consistency. If you can prove that a certain statement could yield more than one answer, you know that the statement is insufficient.

Statement (1) gives that which can be rewritten as 

Since the only restrictions you have on  and  are that they must both be positive integers, you can pick numbers for  to try to force different results for the remainder when 

 is divided by . Because  or , must be a whole number (since remainders are always integers), you should recognize that  must be a multiple of 5 in order produce a whole number.

If , 

This means that 

Conversely, if  then , which would yield 

Since you got different remainders for different starting values, you can conclude that statement (1) is not sufficient, 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) should be clearly insufficient. If you know nothing else about  and , that still leaves a huge number of possible numbers. However, the obvious insufficiency should lead you to ask yourself: why is that statement there?

If you take the two statements together, its usefulness becomes more apparent: it limits the number of combinations of  and . For , . Since there are 3 total digits between  and this is one possible set of numbers. For ,  

However, since there are 5 total digits between the two numbers, you should recognize that this set (and any sets for larger values of  is invalid. Since only one set works, you can conclude that, taken together, the two statements are sufficient.

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