GMAT Strategy / Data Sufficiency Strategy

If x is a positive integer less than 30, is x odd? (1) When x is divided by 3, the remainder is 2. (2) When x is divided by 5, the remainder is 2.

While it's possible to solve this question with a conceptual understanding, it is much easier to put some numbers to work for you. When you do employ numbers, remember that your goal is to play devil's advocate. Your goal is to determine whether x is an odd number, so you will likely start with an odd number that satisfies statement (1). 5 works here, as 5 divided by 3, as 5 3 = 1 remainder 2. So x could be odd. Now that you've found an odd value of x—the answer yes to the overall question—your goal should change. You want to find an even value, because that would show that the statement is not sufficient. If you try everything you can think of and cannot find an even value of x, then you can conclude that it is sufficient. You want to play devil's advocate to ensure that either x must be odd, or conclude that the statement is not sufficient. With that in mind, you might try 8: 8 divided by 3 provides a remainder of 2 ( 8 3 = 2 remainder 2). So now you have an even potential x—and the answer no to conclude that statement (1) is not sufficient. The same process works for statement (2). 7 is an odd number that does the same, so x could still be odd, providing a yes answer. But 12 is an even number that satisfies statement (2), so you can get the answer no, and the answer is thus still maybe. Statement (2) is not sufficient. Taken together, the statements provide a bit more information, as now you know that x provides a remainder of 2 when divided by 3 and when divided by 5. You might recognize 17 as such a number, noting that 15 is the least common multiple of 3 and 5, so 17 will divide out that 15 and leave 2 remaining. Here's where you really need to play devil's advocate: If you chart out the values that work with each statement and look for matches between them, you may well conclude that 17 is the only such value less than 30: Remainder 2 when divided by 3 5 8 11 14 17 (MATCH; x could be odd) 20 23 26 29 32 (MATCH; x could be even—but not even and less than 30) Remainder 2 when divided by 5 7 12 17 22 27 32 37 26 67 LESSON But still play devil's advocate. Is there any even number that could fit the bill? There is, but it's very hard to find unless you remember what happens when you divide a smaller number by a larger number. 2 also works. When 2 is divided by 3, the quotient is 0 and the remainder is 2. When 2 is divided by 5, the quotient is 0, and the remainder is 5. 2 is the even counterpart, and although it may not be as readily clear as 17, if you force yourself to play devil's advocate and consider the entire range of numbers available to you, you will often find that "catch" upon which correct answers often depend. The correct answer to this problem is E, but the authors of the question are betting that you will forget to consider 2 and therefore fall into the trap of selecting C.

Just Do It

You should treat many Data Sufficiency problems as problem-solving questions and just do the math until you can see the answer. • Quadratics, inequalities, and many word problems are examples of question types in which you must solve to prove sufficiency.

Remember the 12TEN pneumonic - this stands for the data sufficiency options which are

1 alone is sufficient, but not 2 2 alone is sufficient, but not 1 Together, statements 1 and 2 are sufficient Either statement is sufficient Neither statement is sufficient

For Yes or No questions, two common traps exist. What are these?

1. A statement allows for multiple values, but all values provide the same answer to the Yes or No question. 2. The statement only allows you to obtain the answer of no - but since always no is a consistent answer, that means that the statement is sufficient (even though the answer is not yes). Remember: you are not trying to prove that the answer is yes, only that it is either yes or no, and not both.

When determining data sufficiency questions, you need to follow this process:

1. Look at statement 1 alone and make your decision 2. Look at statement 2 alone and make your decision 3. Look at both statements together and make your decision

What is the value? questions... What are the two biggest pitfalls that people face with these types of questions?

1. People assume that values must be integers and/or positive. With that assumption, a statement appears to be sufficient when there are actually multiple values possible. 2. People miss restrictions in the problem that do guarantee that the numbers involved are, for instance, integers so they think there are multiple possibilities, when indeed there that statement proves one exact value.

Don't carry information

Always be sure to assess each statement and the, independent of the other, with only the question and that one statement

Manipulate Algebraically

Any time you are given algebra that can be manipulated, look to do that strategically. Often you will be able to make the statement mirror the question or vice versa. • Since algebra "tells the truth," this is often the safest and most efficient way to show sufficiency. Don't waste time with number picking or conceptual thinking if you can prove something algebraically. • Don't forget that it is often more important to manipulate the question than the statements.

Choosing between answer choices C&E

Choices between C and E are particularly hard on the GMAT. Why? Because there are no more hints to leverage from the other statement. Your best hope in choosing between answer choices C and E on hard problems is construct thinking. Be wary of ever picking answer choice E when it just seems like the information is not sufficient, but you have not proven that is not sufficient. And if it seems to be answer choice C, you should be playing devil's advocate by trying to prove answer choice E.

Use Conceptual Understanding

Conceptual thinking is the best approach for most Arithmetic problems. Because problems about ratios, the number line, percents, etc. are more about the concept than actual calculations, doing math and or algebra can often be avoided. • If you are not sure conceptually, then you should prove sufficiency by doing some math or picking numbers. • Testmakers are good at finding exceptions to concepts that you think you understand well, so be careful when solving Data Sufficiency problems on a purely conceptual basis.

Example of Yes/No drill trap: Is x greater than 5? (1) 2x-15=17(x-15)+171

Here you know that you have a linear equation that will simplify to one value of x. Once you have that value, you will know whether or not it is greater than 5. You don't need to do the work, as you can see that simply taking each step will always produce one exact value and therefore one distinct answer.

Think like the testmaker

In Yes or No Data Sufficiency questions, if a statement is sufficient, it almost always gives an affirmative yes answer. However, in certain types of questions, testmakers will cleverly insert statements that give a definitive no answer. This question is a classic example: You are desperately trying to figure out how to determine that k is prime when you should be trying to show that it is not prime. Remember also the equilateral triangle problem from the previous section. Testmakers know when you will forget to disprove things, and they cleverly create problems that exploit this weakness.

Construct Thinking: A street vendor sells only apples and pineapples, and all apples weigh 6.5 ounces, while all pineapples weigh 13 ounces. If she sells twice as many apples as pineapples, how many apples does she sell? (1) She sells 8 more apples than pineapples. (2) She sells an equal amount, by weight, of apples and pineapples.

In problem #14, the word problem format begs you to set up equations. The given information states that, of the fruit that she sold, A = 2P. Statement (1) should again be pretty straightforward: If she sells 8 more apples than pineapples, then A = 8 + P. Using both equations together, we can plug in 2P for A to get 2P = 8 + P, and solve for P. Statement (2) also seems to offer the same type of information—a second equation to pair with the given information that A = 2P. But wait! Your senses should be heightened for the counter-intuitive statement (2) now that statement (1) has proven to be a bit too easy. Statement (2), as an equation, is that 6.5A = 13P. Divide by 6.5, and you'll find that A = 2P—the same equation that we already have! Statement (2) offers no new information and is therefore not sufficient (making answer choice A correct here). Having been on guard for a sneaky other statement after a straightforward first statement, you should be looking for that clever restatement of already-known information and avoid this trap. NOTE: For ease of teaching, the recent examples used the construct "Statement (1) is obvious; (2) is counterintuitive." There's nothing magic about statement (1), and the GMAT could well feature the statements of any of these questions in opposite order, baiting you into handling the easier statement (2) first and then being caught unaware on statement (1). You will see examples of this in the homework to follow; just know that when one statement is a little too easy, there's a high likelihood that the other has some sneaky difficulty built into it that you should anticipat

Learning to Play the Game: A recipe for mixed nuts includes only whole peanuts and cashews and calls for a strict peanut:cashew ratio of 7:3. How many peanuts are in a bag? (1) The packaging facility guarantees that each bag will contain no fewer than 95 and no more than 105 nuts.

In the first example you have to leverage the fact you are dealing with integers and that there are 10 total parts in the ratio. Because you need a whole-number multiplier, and because the number of total nuts is between 95 and 105, then there must be a total of 100 nuts and thus 70 peanuts. People get this wrong because they do not leverage all the mathematical assets given in the problem.

Learning to Play the Game: A recipe for mixed nuts includes only whole peanuts and cashews and calls for a ratio of 7 ounces of peanuts for every 3 ounces of cashews. How many ounces of peanuts are in a bag? (2) The packaging facility guarantees that each bag will contain no less than 30 ounces and no more than 33 ounces of nuts.

In the second example, particularly after doing the first one, you may think that it is a similar situation. There are still 10 total parts, and the total amount is greater than or equal to 30 but less than 33. It has to be 30 total ounces, right? Not in this case, because you can have a fractional multiplier with weight. You do not need to have integer ounces and the multiplier is not known from that information. People get this wrong because they overvalue information and do not play devil's advocate.

Play Devil's Advocate and Pick Numbers

Number picking is an important strategy, but it is one that should only be used when necessary, as it can be time-consuming and make you error-prone. • If you have to number pick, make sure that you pick with the purpose of playing devil's advocate and finding the exception. Smart number picking is key! • Certain questions types and scenarios lend themselves to number picking. Quotient/remainder problems and scenario-driven min/max word problems are great examples of problems in which you should number pick.