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.

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".