GMAT Quantitative Reasoning - GMAT Quantitative Reasoning

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

r, you can leave this one step short. Since you can solve for the arclength, you can determine that Statement (1) is sufficient and eliminate (B), (C), and (E).

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

q the same way you solved for r in statement (1), you should realize that statement (2) is also sufficient. However, if that isn't immediately apparent, you can also do the math.

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

(−8,6)

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

(6,6), you must conclude that statement (1) is insufficient. 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) 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.

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 .

At that point you should see that you have sufficient information with statement 1 alone, as adding 1 to any number makes that number bigger - it just moves it one place to the right on the number line. So statement 1 is sufficient without the need to pair it with statement 2 (which pretty clearly should be insufficient on its own).

The other way to rearrange the algebra in statement 1 is to break the fractional addition apart from the beginning, making look like:

Which is relatively convenient, as the right-hand fraction just nets to 1 (anything divided by itself = 1). Then with the left-hand fraction, you can flip the bottom fraction and multiply, yielding:

, which is . Add the left and right terms and you have, again, which is one greater than , again proving statement one to be sufficient.

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

And then to:

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".