Data Sufficiency: Basic Principles

Data Sufficiency is a question type you’ve never seen before. This chapter will show you how to use basic POE techniques to make this format your new favorite kind of math.

Almost half of the 31 math questions on the GMAT will be Data Sufficiency questions. We’re about to show you how to use POE to make this strange question type easy.

WHAT IS DATA SUFFICIENCY?

If you’ve never heard of Data Sufficiency, that’s because this question type is unique to the GMAT, and these questions definitely require some getting used to. If you have already taken a GMAT practice exam, or the actual GMAT, you may have spent several minutes just trying to understand the directions for Data Sufficiency questions.

However, Data Sufficiency questions really just test the same math concepts as Problem Solving questions, but with a twist—a strange question format.

Here’s what a Data Sufficiency question looks like on the GMAT:

GMAT上的31個數學問題中，幾乎有一半是數據充足性問題。 我們將向您展示如何使用POE簡化這種奇怪的問題類型。

What is the value of y ?

(1) y is an even integer such that –1.5 < y < 1.5

(2) Integer y is not prime

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Each Data Sufficiency question consists of a question followed by two statements. There are also five possible answer choices, as shown. The answers are the same for every Data Sufficiency question, so once you learn what each means, you won’t need to spend time rereading them. You’ll just be able to think about them as answers (A), (B), (C), (D), and (E), which is how we’ll refer to them.

Notice that there are two words that the answer choices keep repeating—alone and sufficient. So, it looks like we’re supposed to evaluate the statements on their own—at least at first. Moreover, our task is evidently to determine whether we have sufficient information to answer the question.

That’s how Data Sufficiency differs from Problem Solving. In Problem Solving questions, you are asked to give a numerical answer to the question. In fact, the inclusion of five numerical answer choices tells you that you can assume that the question can be solved. For Data Sufficiency questions, however, you’re not being asked to solve the question but to decide WHETHER the question can be solved. It may, in fact, turn out that the statements do not provide sufficient information to answer the question.

y的值是多少？

（1）y是一個偶數，使得–1.5 <y <1.5

（2）整數y不是素數

Here’s How to Crack It

The first answer choice—(A)—indicates that we should first look at Statement (1) by itself to see if it is sufficient to answer the question.

In fact, the best way to work Data Sufficiency problems is to look at one statement at a time. So, ignore Statement (2). Here, we’ve replaced Statement (2) with question marks to indicate that we are looking only at the first statement—almost as though we had covered up the second statement.

What is the value of y ?

(1) y is an even integer such that –1.5 < y < 1.5.

(2) ????

Now, we’re ready to evaluate Statement (1) alone. There are three integers between –1.5 and 1.5: –1, 0, and 1. Of those, as you may recall from Chapter 7, only 0 is even. So, Statement (1) does provide sufficient information to answer the question.

We’re not ready to choose the first answer—(A)—yet, however, because the second part of the answer choice states that Statement (2) alone is not sufficient. Now, forget that you have ever seen Statement (1).

What is the value of y ?

(1) ????

(2) Integer y is not prime.

The second statement tells us only that y is not prime. So, possible values for y include 1, 4, 6, 8, etc. Do we know the value of y? No way. So, Statement (2) is not sufficient. Because (1) is sufficient and (2) is not, the answer to this question is

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Or, in other words, the correct answer is (A).

y的值是多少？

（1）y是一個偶數整數，使得–1.5 <y <1.5。

（2）????

y的值是多少？

（1）????

（2）整數y不是素數。

DATA SUFFICIENCY: GETTING STARTED

Now that you’ve seen and worked a Data Sufficiency question, it’s time to learn how to make this weird question type your own. The first step is to understand what each of the answer choices means.

By making small changes to the example you’ve just seen, we can provide examples of each of the answer choices. Next to each example, you’ll find a graphic that provides a quick and dirty way to understand and remember each answer choice. Here’s the example for (A) again:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

What is the value of y ?

(1) y is an even integer such that

–1.5 < y < 1.5.

y的值是多少？

（1）y是一個偶數，使得

–1.5 <y <1.5。 (2) Integer y is not prime.

Now, let’s make some changes to the statements, to get an example of (B).

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

What is the value of y ?

(1) Integer y is not prime.

（2）整數y不是質數。

y的值是多少？

（1）整數y不是質數。 (2) y is an even integer such that

–1.5 < y < 1.5.

As you can see from this example, (B) is pretty much the flip side of (A). In this case, the first statement provides no help in determining the value of y, but the second statement tells us that y = 0.

A few more changes produce an example of (C).

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

What is the value of y ?

(1) y is an even integer.

（2）y是一個偶數，使得

–1.5 <y <1.5。

y的值是多少？

（1）y是偶數整數。 Now that you’ve seen and worked a Data Sufficiency question, it’s time to learn how to make this weird question type your own. The first step is to understand what each of the answer choices means.

By making small changes to the example you’ve just seen, we can provide examples of each of the answer choices. Next to each example, you’ll find a graphic that provides a quick and dirty way to understand and remember each answer choice. Here’s the example for (A) again:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

What is the value of y ?

(1) y is an even integer such that

–1.5 < y < 1.5.

(2) Integer y is not prime.

Now, let’s make some changes to the statements, to get an example of (B).

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

What is the value of y ?

(1) Integer y is not prime.

(2) y is an even integer such that

–1.5 < y < 1.5.

As you can see from this example, (B) is pretty much the flip side of (A). In this case, the first statement provides no help in determining the value of y, but the second statement tells us that y = 0.

A few more changes produce an example of (C).

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

What is the value of y ?

(1) y is an even integer.

(2) –1.5 < y < 1.5

The first statement tells us that y is even, but there are a lot of even integers. The second statement gives us a range of values for y, but, by itself, we don’t even know that y is an integer from the second statement. So, neither statement is sufficient on its own. But, when we put them together, we know that y = 0.

Now, let’s get an example of (D).

EACH statement ALONE is sufficient.

What is the value of y ?

(1) y is an even integer such that

–1.5 < y < 1.5.

y的值是多少？

（1）y是一個偶數，使得

–1.5 <y <1.5。

（2）整數y不是素數。

y的值是多少？

（1）整數y不是素數。

（2）y是一個偶數，使得

–1.5 <y <1.5。

y的值是多少？

（1）y是偶數整數。

（2）–1.5 <y <1.5

y的值是多少？

（1）y是一個偶數，使得

–1.5 <y <1.5。 (2) For any integer a ≠ 0, ay = 0.

As pointed out in previous examples, the information in Statement (1) allows us to conclude that y = 0. The information in the second statement also tells us that y is 0, because the only way for the product of ay to equal 0 is if either a or y is 0. Since a can’t be 0, y must be 0. Note how the statements independently allow us to arrive at the conclusion that y = 0 for (D).

Finally, let’s look at an example of (E).

Statements (1) and (2) TOGETHER are not sufficient.

What is the value of y ?

(1) y is an even integer.

（2）對於任何a≠0的整數，ay = 0。

y的值是多少？

（1）y是偶數整數。 (2) Integer y is not prime.

For this example, there’s no way to determine the value of y. The first statement doesn’t work because y could be any even integer. The second statement also doesn’t help because y can be any integer that isn’t prime. Even when we combine the statements, we don’t know the value of y because any even integer except 2 fits the conditions. So, (E) is the no way, no how answer.

Below, you’ll find the full graphic for all of the answers. You may find it helpful to keep the graphic handy until you are completely comfortable with what each answer choice means.

（2）整數y不是素數。 DATA SUFFICIENCY: BASIC POE STRATEGY

One of the reasons the test-writers decided to include Data Sufficiency questions on the GMAT is that when this format was first dreamed up they thought these questions would be immune to Process of Elimination (POE). Were they ever wrong! If anything, it’s even easier to apply POE to Data Sufficiency questions. Let’s see why.

First, however, let’s restate one of the most important strategies for working any Data Sufficiency question: Evaluate the statements one at a time before you think about combining them. Many people mistakenly pick (C)—you need both statements together—when it would have been possible to answer the question with only the information in the first statement or the second statement. Generally, people make this mistake when they read both statements right after reading the question stem. In fact, this mistake is the most common mistake that test-takers make when working Data Sufficiency questions.

To avoid this common mistake, read the question stem and only the first statement. Ignore the second statement. Pretend it isn’t there. You may even go as far as covering Statement (2) with your finger if you find the temptation to read both statements too overpowering. Once you have evaluated Statement (1), forget it. Ignore it. It doesn’t exist anymore. Cover it up if you need to and read and evaluate Statement (2).

What happens when you evaluate the statements one at a time? Something magical, that’s what! POE comes roaring back. Consider the following partial example:

What is the value of x ?

(1) x + 7 = 12

x的值是多少？

（1）x + 7 = 12

We don’t even have Statement (2), but we can still do a lot with this partial question. (Don’t worry. There won’t be any partial questions on the real test!) First, you want to see if the statement is sufficient to answer the question. In this case, you could subtract 7 from both sides of the equation to discover that x = 5. We’ll take this as an opportunity to remind you, however, that you don’t really need to solve the equation—you just need to know that you can solve the equation. After all, to pick an answer to the problem, you just need to know if you have sufficient information.

Since Statement (1) is sufficient in this case, which answer choices can be eliminated? From the chart, you can see that there are only two answer choices—(A) and (D)—that have Statement (1) circled to indicate that, for that answer choice, Statement (1) is sufficient. So, you no longer need to worry about (B), (C), or (E). They’ve been eliminated! If the first statement is sufficient, the answer to the problem must be (A) or (D)! You’re down to 50/50 just based on looking at the first statement! Now, check out this example:

What is the value of x ?

(1) x is an integer.

Now, what are the possible answers? If you said (B), (C), or (E), you are well on your way to getting this Data Sufficiency stuff under control. If you said something else, take a look at the steps outlined previously. In this case, the first statement is insufficient to determine the value of x. So, you want the answer choices that have 1 crossed off, and that is (B), (C), or (E).

x的值是多少？

（1）x是整數。 In the following drill, each question is followed by only one statement. Based on the first statement, decide if you are down to AD or BCE. The answers can be found in Part VI.

1. What is the value of x ?

(1) y = 4

(2) ????

2. Is y an integer?

(1) 2y is an integer.

(2) ????

3. A certain room contains 12 children. How many more boys than girls are there?

(1) There are three girls in the room.

(2) ????

4. What number is x percent of 20 ?

(1) 10 percent of x is 5.

(2) ????

1. x的值是多少？

（1）y = 4

（2）????

2. y是整數嗎？

（1）2y是整數。

（2）????

3.某個房間可容納12個孩子。 男孩比女孩多多少？

（1）房間裡有三個女孩。

（2）????

4. 20的百分之x是多少？

（1）x的10％為5。

（2）????

From AD or BCE to the Answer

Every time you start a Data Sufficiency question, you should read the question and only the first statement. If the first statement is sufficient, your possible answers are (A) or (D). If the first statement is insufficient, your possible answers are (B), (C), or (E). So, you can always get rid of either two or three answer choices just by evaluating the first statement. The AD/BCE split is so important that you’ll want to write down AD or BCE on your noteboard as you work every Data Sufficiency question.

But what happens next? How do you get to the answer? Let’s take a look.

If x + y = 3, what is the value of xy ?

(1) x and y are integers.

(2) x and y are positive.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Here’s How to Crack It

As always, ignore Statement (2) and look only at Statement (1). If x and y are integers and x+y = 3, do we know what they are? Not really—x could be 1 and y could be 2 (in which case, xy would be 2). But x could also be 0 (yes, 0 is an integer) and y could be 3 (in which case, xy would be 0). Because Statement (1) alone is not sufficient, we are down to BCE, a one in three shot.

Now, ignore Statement (1) and look at Statement (2). By itself, this statement doesn’t begin to give us values for x and y—x could be 1 and y could be 2, but x could just as easily be 1.4 and y could be 1.6. Because there is still more than one possible value for xy, cross off (B).

We’re down to (C) or (E). Now it’s finally time to look at both statements at the same time. See how late in the process we combine the statements? Get into the habit of physically crossing off (B) before you think about combining the statements. That’s how you can avoid making the most common GMAT Data Sufficiency mistake of putting the statements together too early.

Because we know from the first statement that x and y are integers, and from the second statement that they must be positive, do we now know specific values for x and y?

Well, we do know that there are only two positive integers that add up to 3: 2 and 1. (Remember, zero is an integer but it is neither positive nor negative.)

Do we know if x = 1 and y = 2, or vice versa? Not really, but frankly, it doesn’t matter in this case. The question is asking us the value of xy.

Because neither statement by itself is sufficient, but both statements together are sufficient, the answer is (C).

（1）x和y是整數。

（2）x和y為正。

Here’s a handy flowchart that shows you what to do for any Data Sufficiency problem. You should keep the flowchart next to you and consult it as you first start practicing Data Sufficiency questions. After you have done 10 or 20 questions, you’ll probably find that you have learned the basic POE process well enough that you don’t need the chart anymore. However, if you ever find yourself having trouble with Data Sufficiency, pull out the chart again and do some more problems, using it as a guide. YES/NO DATA SUFFICIENCY: THE BASICS

If you were going to provide the answer to most Data Sufficiency questions, your response would be a number. However, as many as half of all the Data Sufficiency questions that you will see on your test will ask a yes-or-no question instead.

Leave it to GMAC to come up with a way to give you five different answer choices on a yes-or-no question. Let’s look at an example.

Did candidate x receive more than half of the 30,000 votes cast in the general election?

(1) Candidate y received 12,000 of the votes cast.

(2) Candidate x received 18,000 of the votes cast.

Here’s How to Crack It

When all is said and done, the answer to this question is either yes or no. Start by ignoring Statement (2) and evaluating Statement (1). Does Statement (1) alone answer the question? If you were in a hurry, you might think so. Many people assume that there are only two candidates in the election. They reason that if candidate y got 12,000 votes, then candidate x must have received 18,000 votes. However, there’s no reason to assume that there are only two candidates. So, Statement (1) is insufficient. Write down BCE. Does Statement (2) alone answer the question? Yes, it’s pretty clear that candidate x received more than half of the votes. So, the correct answer is (B).

（1）候選人獲得12,000張選票。

（2）候選人x獲得18,000張選票。

That didn’t seem so bad, did it? Yet, you may have heard that yes/no Data Sufficiency questions have a reputation for being hard. Let’s change our example to see why.

Did candidate x receive more than half of the 30,000 votes cast in the general election?

(1) Candidate y received 12,000 of the votes cast.

(2) Candidate x received 13,000 of the votes cast.

Here’s How to Crack It

As always, start by ignoring Statement (2) so that you can properly evaluate Statement (1) alone. As in our previous example, Statement (1) is insufficient, so be sure to write down BCE. Statement (2) seems pretty straightforward. Candidate x received fewer than half of the votes cast. At this point many people say, “Since the guy clearly got fewer than half the votes, this statement doesn’t answer the question, either.” But those people are wrong!

（1）候選人獲得12,000張選票。

（2）候選人x獲得13,000張選票。

Just Say No

Broken down to its basics, the question we were asked was, “Did he get more than half of the vote—yes or no?

Statement (2) does answer the question. The answer is, “No, he didn’t.” So, the answer is the same as that of the first example. The answer is (B).

On a yes/no Data Sufficiency problem, if the statement answers the question in either the affirmative or the negative, it is sufficient.

Yes/No/Maybe

Yes/no questions really should be called yes/no/maybe questions. Even if that’s not their “official” name, it’s still worthwhile to think about them in that fashion.

Let’s look at one last example to see why.

Did candidate x receive more than half of the 30,000 votes cast in the general election?

(1) Candidate y received 12,000 of the votes cast.

(2) Candidate x received at least 13,000 of the votes cast.

Here’s How to Crack It

（1）候選人獲得12,000張選票。

（2）候選人x至少獲得13,000張選票。

MORE ON DATA SUFFICIENCY

Although Data Sufficiency problems test the same material covered by regular Problem Solving questions, some readers find it distracting to learn the more complicated subtleties of this new question type at the same time that they are learning (or relearning) math concepts. That’s why we’ve put our main chapter on Data Sufficiency at the end of our math review.

However, you will find Data Sufficiency problems sprinkled throughout the math drills—and you should feel free at any time to dip into Chapter 16, where you’ll find everything in one place, including more advanced strategy, several more drills, and some great techniques to handle the most complicated yes/no questions.

Summary

“Data Sufficiency” means just that, sufficiency. These questions are asking you if the data presented is enough to solve the problem.

Every Data Sufficiency problem consists of a question followed by two statements. You must decide whether the question can be answered based on the information in the two statements.

The best strategy for Data Sufficiency problems is to look at one statement at a time. Cover up the other statement with your hand, so that you can completely focus on one statement at a time.

AD or BCE: These are always your options when you first start eliminating. Memorize them.

“數據充分性”就意味著充分性。 這些問題問您所提供的數據是否足以解決問題。