### Archives For Data Sufficiency

Have you heard of the C-Trap? I’m not going to tell you what it is yet. Try this problem from GMATPrep® first and see whether you can avoid it

* “In a certain year, the difference between Mary’s and Jim’s annual salaries was twice the difference between Mary’s and Kate’s annual salaries. If Mary’s annual salary was the highest of the 3 people, what was the average (arithmetic mean) annual salary of the 3 people that year?

“(1) Jim’s annual salary was \$30,000 that year.

“(2) Kate’s annual salary was \$40,000 that year.”

I’m going to do something I normally never do at this point in an article: I’m going to tell you the correct answer. I’m not going to type the letter, though, so that your eye won’t inadvertently catch it while you’re still working on the problem. The correct answer is the second of the five data sufficiency answer choices.

How did you do? Did you pick that one? Or did you pick the trap answer, the third one?

Here’s where the C-Trap gets its name: on some questions, using the two statements together will be sufficient to answer the question. The trap is that using just one statement alone will also get you there—so you can’t pick answer (C), which says that neither statement alone works.

In the trickiest C-Traps, the two statements look almost the same (as they do in this problem), and the first one doesn’t work. You’re predisposed, then, to assume that the second statement, which seemingly supplies the “same” kind of information, also won’t work. Therefore, you don’t vet the second statement thoroughly enough before dismissing it—and you’ve just fallen into the trap.

How can you dig yourself out? First of all, just because two statements look similar, don’t assume that they either both work or both don’t. The test writers are really good at setting traps, so assume nothing.

Second, imagine that you’re teaching your 10-year-old niece how to do algebra. She’s never done this before but she’s pretty bright. She understands your explanation of what variables are and how they work. She knows that, if you give her an equation with 3 variables, and then give her values for 2 of those variables, she’ll be able to solve for the third one. What answer is she going to pick on the above problem?

Hmm. She’d pick (C) also, since that gives her values for two of the three variables in the equation that she can write from the question stem.

It’s obvious, in fact, that using the two statements together will allow you to find all three salaries, in which case you can average them. In the test-prep world, this is what’s known as a Too Good To Be True answer. If your 10-year-old niece, who just learned algebra, could get to the same answer, then chances are you’re falling into a trap. Stop, take a deep breath, and scrutinize those statements individually!

Here’s how to solve the problem.

Take a quick glance; what have you got? DS. Story problem: understand the story before writing.

The question asks for the average of the three salaries. What do you actually need to know in order to find an average? Right, the sum. So can you find the sum of the three salaries?

Jot that on your scrap paper: M + J + K = ?

Step 2: Reflect Organize

The first sentence provides an equation, so translate it. (Note that the second sentence says Mary’s salary is the highest.)

The positive difference between Mary’s and Jim’s salaries has to be MJ, since M is larger. Likewise, the positive difference between Mary’s and Kate’s salaries has to be MK, since M is larger.

Here’s the translated formula:

MJ = 2(MK)

Step 3: Work

By itself, that doesn’t look very helpful, but anytime DS gives you a formula that isn’t simplified, simplify it. Multiply out the right-hand side and also get “like” variables together:

MJ = 2(MK)

MJ = 2M – 2K

- J = M – 2K

Notice two things: first, negatives are annoying. Second, this formula (so far) doesn’t look anything like the question: M + J + K = ?

Is there any way to remedy those two things?

Move the –J over: 0 = M – 2K + J.

Notice that 2K is never going to fit the question, which has only K. Move that away from the others: 2K = M + J.

Interesting. The right-hand side now matches part of the question. In fact, you could substitute:

M + J + K = ?

2K = M + J

Therefore, the question becomes 2K + K = ?

If you know what K is—only K!— then you can solve. (Note: we call this process Rephrasing. Use the information given in the question stem to rephrase the question in a more simplified form.)

“(1) Jim’s annual salary was \$30,000 that year.”

J = 30,000. If you plug that into M + J + K = ?, it isn’t sufficient. If you plug that into 2K = M + J, you get 2K = M + 30,000, which still isn’t sufficient. Knowing only J doesn’t get you very far. This statement is not sufficient; eliminate answers (A) and (D).

“(2) Kate’s annual salary was \$40,000 that year.”

Bingo! If you know Kate’s salary, then you know the sum of all three. This statement is sufficient to answer the question.

If you don’t rephrase up front, and instead go through all of the work of plugging in the values for statements (1) and (2), then you may still discover the correct answer. You’ll take longer, though. You may also fall into the trap of assuming that statement (2) won’t work because it looks so very similar to statement (1) and that one didn’t work.

Key Takeaways: Data Sufficiency

(1) Don’t just write down the information in the question stem, shrug, and go straight to the statements. Push yourself to try to rephrase the question before you go to the statements.

(2) Use standard math steps and your test-taker savvy to help you know how to simplify. It’s standard algebra to try to get “like” variables together in equations. A negative sticking out in front of an equation is ugly, so that was clue #2. Finally, you’re ultimately trying to match the information in the question (M + J + K = ?), so try to rearrange your rephrased equation to match the question as much as possible. Then see whether you can substitute in to make that question simpler!

(3) Keep an eye out for Too Good to Be True answers. If an answer seems pretty obvious, then there’s a good chance you’re falling into a trap!

* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

Some Data Sufficiency questions present you with scenarios: stories that could play out in various complicated ways, depending on the statements. How do you get through these with a minimum of time and fuss?

Try the below problem. (Copyright: me! I was inspired by an OG problem; I’ll tell you which one at the end.)

* “During a week-long sale at a car dealership, the most number of cars sold on  any one day was 12. If at least 2 cars were sold each day, was the average daily number of cars sold during that week more than 6?

“(1) During that week, the second smallest number of cars sold on any one day was 4.

“(2) During that week, the median number of cars sold was 10.”

First, do you see why I described this as a “scenario” problem? All these different days… and some number of cars sold each day… and then they (I!) toss in average and median… and to top it all off, the problem asks for a range (more than 6). Sigh.

Okay, what do we do with this thing?

Because it’s Data Sufficiency, start by establishing the givens. Because it’s a scenario, Draw It Out.

Let’s see. The “highest” day was 12, but it doesn’t say which day of the week that was. So how can you draw this out?

Neither statement provides information about a specific day of the week, either. Rather, they provide information about the least number of sales and the median number of sales.

The use of median is interesting. How do you normally organize numbers when you’re dealing with median?

Bingo! Try organizing the number of sales from smallest to largest. Draw out 7 slots (one for each day) and add the information given in the question stem:

Now, what about that question? It asks not for the average, but whether the average number of daily sales for the week is more than 6. Does that give you any ideas for an approach to take?

Because it’s a yes/no question, you want to try to “prove” both yes and no for each statement. If you can show that a statement will give you both a yes and a no, then you know that statement is not sufficient. Try this out with statement 1

(1) During that week, the least number of cars sold on any one day was 4.

Draw out a version of the scenario that includes statement (1):

Can you find a way to make the average less than 6? Keep the first day at 2 and make the other days as small as possible:

The sum of the numbers is 34. The average is 34 / 7 = a little smaller than 5.

Can you also make the average greater than 6? Try making all the numbers as big as possible:

(Note: if you’re not sure whether the smallest day could be 4—the wording is a little weird—err on the cautious side and make it 3.)

You may be able to eyeball that and tell it will be greater than 6. If not, calculate: the sum is 67, so the average is just under 10.

Statement (1) is not sufficient because the average might be greater than or less than 6. Cross off answers (A) and (D).

Now, move to statement (2):

(2) During that week, the median number of cars sold was 10.

Again, draw out the scenario (using only the second statement this time!).

Can you make the average less than 6? Test the smallest numbers you can. The three lowest days could each be 2. Then, the next three days could each be 10.

The sum is 6 + 30 + 12 = 48. The average is 48 / 7 = just under 7, but bigger than 6. The numbers cannot be made any smaller—you have to have a minimum of 2 a day. Once you hit the median of 10 in the middle slot, you have to have something greater than or equal to the median for the remaining slots to the right.

The smallest possible average is still bigger than 6, so this statement is sufficient to answer the question. The correct answer is (B).

Oh, and the OG question is DS #121 from OG13. If you think you’ve got the concept, test yourself on the OG problem.

Key Takeaway: Draw Out Scenarios

(1) Sometimes, these scenarios are so elaborate that people are paralyzed. Pretend your boss just asked you to figure this out. What would you do? You’d just start drawing out possibilities till you figured it out.

(2) On Yes/No DS questions, try to get a Yes answer and a No answer. As soon as you do that, you can label the statement Not Sufficient and move on.

(3) After a while, you might have to go back to your boss and say, “Sorry, I can’t figure this out.” (Translation: you might have to give up and guess.) There isn’t a fantastic way to guess on this one, though I probably wouldn’t guess (E). The statements don’t look obviously helpful at first glance… which means probably at least one of them is!

In honor of the final season of Breaking Bad, we decided to put together our ultimate Breaking Bad GMAT quiz. Those of you who fall in the overlapping section of the “Breaking Bad Fan” “GMAT student” Venn diagram should test your skills below… yo!

## 1. Data Sufficiency

Does x+4 = Walter White?

(1) x+4 is the danger
(2) x+4 is the one who knocks

A. Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient
B. Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D. EACH statement ALONE is sufficient
E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed

## 2. Discrete Quant

The front portion of Walter White’s Roof is a 7 ‘ by 15’ rectangle. If the diameter of a pizza is 22”, what is the approximate area of the shaded region of this diagram?

A. 13,600 inches sq.
B. 14,740 inches sq.
C. 15,120 inches sq.
D. 15,500 inches sq.
E. 16,640 inches sq.

## 3. Critical Reasoning

Today, Walter White will cook 100 pounds of methamphetamine.

This argument is flawed primarily because:

A. Cooking methamphetamine presents a moral dilemma for Walter White.
B. Walter White has to prioritize the needs of his wife and children and be a better father.
C. Walter has already paid for his cancer treatment and no longer needs to cook methamphetamine.
D. There is a fly in the laboratory.
E. He was told not to cook that day and is obeying his instructions.

## 4. Critical Reasoning

Hank’s collection of rocks includes over 400 different items. Hank’s rock collection is clearly the most impressive in New Mexico.

This argument is flawed primarily because:

A. Rock collections are not judged by the total number of rocks but by the rarity of each item included.
B. Rock collections are not impressive to anyone.
C. Hank’s rock collection is a metaphor and therefore cannot be judged against other rock collections.
D. Hank’s wife stole most of the rocks and it is therefore ineligible for any superlatives.
E. They aren’t rocks, they are minerals.

## 5. Discrete Quant

Walter Junior eats 3 eggs for breakfast every morning. Given that Walter Junior never misses breakfast, how many eggs does Walter Junior consume in March?

A. 60
B. 74
C. 82
D. 93
E. 107

Data sufficiency question are a strange animal that exists only in GMAT land.  The newness of this question type creates high levels of anxiety because we don’t know how to react when we see something new (How do you think you would react if you were standing face to face with a unicorn?).   Once we get over this newness, data sufficiency questions all follow a specific morphology, and in my opinion actually contain less diversity than problem solving questions.  There is always either a yes/no question (is ab even?) or value question (how many boys are in the class?), followed by two statements, and the five answer choice are always the same and in the same order. (If you are completely unfamiliar with data sufficiency questions take a look at an example here)

Because of this very confined structure, there are actually cases where the structure of question and statements can give you information regardless of the specifics of the problem. There are at least four instances where a specific form of the statement(s) will allow you to eliminate several responses without evaluating the full content of the problem.

1) A value statement for a yes/no question

If a statement provides a value for the sole variable in the question, it is definitely sufficient to answer any yes or no question.

For example:

One of the hardest parts about becoming an instructor with Manhattan GMAT was relearning how to solve GMAT questions. That sounds absurd, considering I had already scored a 780 on the GMAT when I applied to become an instructor, but it’s true. During the interview process, I went through online and in-person classroom simulations with 99th percentile instructors playing students, testing my ability to explain a question using algebra instead of plugging numbers or using a rate chart instead of adding rates. Over the years, I’ve found that many of our instructors felt the same way: overwhelmed by how hard it is to go along with someone else’s preferred method without skipping a beat. Ultimately, I realized that teaching the GMAT is a hundred times harder than taking the GMAT because every question has several valid ways of being solved.

Which leads to the problem of what solution is the BEST solution. Any student who has worked with me over the years has heard me say the following- I don’t care what method you use to solve a problem. But I do care that you get great at that method. It’s the reason why the Official Guide has an explanation for each quant problem and Manhattan has an OG Companion with different explanations, along with online video explanations that will sometimes differ from either of those methods. With so many different ways of solving a question, it’s important to not get bogged down finding the best way to solve a problem, but instead focus on finding the fastest way from start to submit.

So with that said, over the next few months, I’d like to share a few methods that I personally use when solving a few different types of GMAT questions. Some of these methods might click for you, and I hope you practice them. Some of them won’t and I hope you stick with a method that works better for you. So without further ado- let’s take a look at a fairly straightforward GMATPrep problem and think about how you would attack this question:

A sum of \$200,000 from a certain estate was divided among a spouse and three children. How much of the estate did the youngest child receive?

(1)  The spouse received 1/2 of the sum from the estate, and the oldest child received 1/4 of the remainder.

(2)  Each of the two younger children received \$12,500 more than the oldest child and \$62,500 less than the spouse.

The first two things that I notice about this problem is that it is a word problem, giving us a real-world scenario, and a value Data Sufficiency question, asking us to find a single value for the amount that the youngest child received. And if I wanted to set this up algebraically, I could assign variables (s = spouse, x, y, z = oldest, middle, youngest child), write out several equations (s + x + y + z = 200,000. (1) s = 1/2*200,000; x = 1/4 * (1/2*200,000); y + z = 75,000. (2) y = z; z = x + 12,500; z = s âˆ’ 62,500), and eventually solve for z using Statement 2: the correct answer is (B). Different students at different levels of comfort with Data Sufficiency will be able to stop at different points after realizing that there either will or will not be a single variable in the equation that they’ve set up.

The other week, we discussed the overall process for Data Sufficiency. This week, we’re going to test out the process using a GMATPrep question “ and take a look at a couple of very common DS traps.

Set your timer for 2 minutes. and GO!

*  A bookstore that sells used books sells each of its paperback books for a certain price and each of its hardcover books for a certain price. If Joe, Maria, and Paul all bought books in this store, how much did Maria pay for 1 paperback book and 1 hardcover book?

(1) Joe bought 2 paperback books and 3 hardcover books for \$12.50.

(2) Paul bought 4 paperback books and 6 hardcover books for \$25.00.

Note that I haven’t listed the answer choices for you. Because DS answers are always the same, we should memorize them. If you don’t have them memorized yet, look back at the How DS Works article linked in the first paragraph.

All right, let’s tackle this problem.

Step 1: Read the Question Stem

The first sentence tells us that each paperback book sells for the same price and each hardcover book also sells for the same price (but possibly a different price than the paperback books).

The question asks how much Maria paid for 1 of each type of book. Is this a value or a yes/no question?

They’re asking for a specific amount; this is a value question. We’ve also got lots of words; we’re going to have to translate.

Let’s just put it right out there: data sufficiency is bizarre. If you’re just starting out, you’re probably thinking, What is this thing? Even if you’ve been studying for a while, unless you really like math, you probably feel a little uncomfortable whenever a DS question pops up on the screen.

Why? Because we all realize that we could completely mess up a DS question and still get to one of the 5 answer choices, clueless that we’ve messed up. It’s not like Problem Solving, where at least I know when I mess up because my answer isn’t in the answer choices! (Actually, a lot of the time, we still get an answer that’s in the answer choices even on PS but we persist in feeling that PS is more straightforward because the answers are real.)

## What is DS?

The GMAT really isn’t a math test. These tests are actually trying to test us on our executive reasoning skills “ that is, how well we make decisions and prioritize when faced with too many things to do in too short a length of time.

Data Sufficiency questions test our ability to (quickly) analyze a collective set of data and figure out which pieces are needed to do the job. Imagine your boss dumping a bunch of stuff on you and saying, Hey, our client wants to know whether they should raise the price on this product. Can you answer that question from this data? If so, which pieces do we need to prove the case?

We do, of course, have to do some math “ and sometimes that math is quite annoying. We usually don’t, however, have to do as much as is necessary on the more normal quant questions (PS).

In an earlier post, we tackled a medium-level GMATPrep weighted average question; click here to read that article before reading this one. This week, we’re trying a harder GMATPrep  weighted average question in order to test whether you learned the concept as well as you thought you did. : )

As we discussed earlier, every weighted average problem I’ve seen (so far!) on GMATPrep is a Data Sufficiency question. This doesn’t mean that they’ll never give us a Problem Solving weighted average problem, but it does seem to be the case that the test-writers are more concerned with whether we understand how weighted averages work than with whether we can actually do the calculations. Last week, we focused on understanding how weighted averages work via writing some equations. We’ll try to apply that understanding to our harder problem this week, along with a more efficient solution method.