### GMAT Data Sufficiency Strategy: Test Cases

If you’re going to do a great job on Data Sufficiency, then you’ve got to know how to Test Cases. This strategy will help you on countless DS problems.

Try this GMATPrep® problem from the free exams. Give yourself about 2 minutes. Go!

* “On the number line, if the number *k* is to the left of the number *t*, is the product *kt* to the right of *t*?

“(1) *t* < 0

“(2) *k* < 1”

If visualizing things helps you wrap your brain around the math (it certainly helps me), sketch out a number line:

*k* is somewhere to the left of *t*, but the two actual values could be anything. Both could be positive or both negative, or *k* could be negative and *t* positive. One of the two could even be zero.

The question asks whether *kt* is to the right of *t*. That is, is the product *kt* greater than *t* by itself?

There are a million possibilities for the values of *k* and* t*, so this question is what we call a theory question: are there certain characteristics of various numbers that would produce a consistent answer? Common characteristics tested on theory problems include positive, negative, zero, simple fractions, odds, evens, primes—basically, number properties.

“(1) *t* < 0

This problem appears to be testing positive and negative, since the statement specifies that one of the values must be negative. Test some real numbers, always making sure that *t* is negative.

Case #1:

t |
k |
Valid case? |
Is kt > t? |

-1 | -2 | Valid: t < 0 and k < t |
2 > -1? Yes. |

Testing Cases involves three consistent steps:

First, choose numbers to test in the problem

Second, make sure that you have selected a valid case. All of the givens must be true using your selected numbers.

Third, answer the question.

In this case, the answer is Yes. Now, your next strategy comes into play: try to prove the statement *insufficient*.

How? Ask yourself what numbers you could try that would give you the opposite answer. The first time, you got a Yes. Can you get a No?

Case #2:

t |
k |
Valid case? |
Is kt > t? |

-1 | 2 | Invalid! k is not less than t! |

Careful: this is where you might make a mistake. In trying to find the opposite case, you might try a mix of numbers that is invalid. Always make sure that you have a valid case before you actually try to answer the question. Discard case 2.

Case #3:

t |
k |
Valid case? |
Is kt > t? |

-1 | -5 | Valid: t < 0 and k < t |
5 > -1? Yes. |

Hmm. We got another Yes answer. What does this mean? If you can’t come up with the opposite answer, see if you can understand why. According to this statement, *t* is always negative. Since *k* must be smaller than *t*, *k* will also always be negative.

The product *kt*, then, will be the product of two negative numbers, which is always positive. As a result, *kt* must always be larger than *t*, since *kt* is positive and *t *is negative.

Okay, statement (1) is sufficient. Cross off answers BCE and check out statement (2):

“(2) *k* < 1”

You know the drill. Test cases again!

Case #1:

k |
t |
Valid case? |
Is kt > t? |

0 | 1 | Valid: k < 1 and k < t |
0 > 1? No. |

You’ve got a No answer. Try to find a Yes.

Case #2:

k |
t |
Valid case? |
Is kt > t? |

-1 | 1 | Valid: k < 1 and k < t |
-1 > 1? No. |

Hmm. I got another No. What needs to happen to make *kt* > *t*? Remember what happened when you were testing statement (1): try making them both negative!

In fact, when you’re testing statement (2), see whether any of the cases you already tested for statement (1) are still valid for statement (2). If so, you can save yourself some work. Ideally, the below would be your path for statement (2), not what I first showed above:

“(2) *k* < 1”

Case #1:

k |
t |
Valid case? |
Is kt > t? |

-2 | -1 | Valid: k < 1 and k < t |
Same case, still Yes. |

All you have to do is make sure that the case is valid. If so, you’ve already done the math, so you know that the answer is the same (in this case, Yes).

Now, try to find your opposite answer: can you get a No?

Case #2: Try something I couldn’t try before. *k* could be positive or even 0…

k |
t |
Valid case? |
Is kt > t? |

0 | 1 | Valid: k < 1 and k < t |
0 > 1? No. |

A Yes and a No add up to an insufficient answer. Eliminate answer (D).

The correct answer is (A).

Guess what? The technique can also work on some Problem Solving problems. Try it out on the following GMATPrep problem, then join me next week to discuss the answer:

* “For which of the following functions *f* is *f*(*x*) = *f*(1 – *x*) for all *x*?

“(A) *f*(*x*) = 1 – *x*

“(B) *f*(*x*) = 1 – *x*^{2}

“(C) *f*(*x*) = *x*^{2} – (1 – *x*)^{2}

“(D) *f*(*x*) = *x*^{2}(1 – *x*)^{2}

“(E)

### Key Takeaways: Test Cases on Data Sufficiency

(1) When DS asks you a “theory” question, test cases. Theory questions allow multiple possible scenarios, or cases. Your goal is to see whether the given information provides a consistent answer.

(2) Specifically, try to disprove the statement: if you can find one Yes and one No answer, then you’re done with that statement. You know it’s insufficient. If you keep trying different kinds of numbers but getting the same answer, see whether you can think through the theory to prove to yourself that the statement really does always work. (If you can’t, but the numbers you try keep giving you one consistent answer, just go ahead and assume that the statement is sufficient. If you’ve made a mistake, you can learn from it later.)

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

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### Avoiding the C-Trap in 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.

Read more

### Save Time and Eliminate Frustration on DS: Draw It Out!

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!

### How Would You Do On A Breaking Bad GMAT?

*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

Answers are after the jump…

### Giveaway Data Sufficiency Statements

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:

### Another Way to Solve Data Sufficiency Value Problems, The Goldilocks Method, & Finding a Number That’s Just Right

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.

### GMATPrep Data Sufficiency: Maria’s Books

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.

### How Data Sufficiency Works

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).

### Breaking Down GMATPrep Weighted Average Problems #2

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.

Let’s start with a sample problem. Set your timer for 2 minutes. and GO!

* A contractor combined

xtons of gravel mixture that contained 10 percent gravel G, by weight, withytons of a mixture that contained 2 percent gravel G, by weight, to produceztons of a mixture that was 5 percent gravel G, by weight. What is the value ofx?(1)

y= 10(2)

z= 16

There are two kinds of gravel: 10% gravel and 2% gravel. These are our two sub-groups. When the two are combined (in some unknown “ for now! “ amounts), we get a 3^{rd} kind:5% gravel. The number of tons of 10% gravel (x) and the number of tons of 2% gravel (y) will add up to the number of tons of 5% gravel (z), or *x* + *y* = *z*. We need to find the number of tons of 10% gravel used in the mixture.

The problem this week throws in a new wrinkle: we’re not just trying to calculate a ratio this time. We have to have enough info to calculate the actual amount of 10% gravel used. Last week, we never had to worry about the actual number of employees. We’ll have to keep that in mind to see how things might change.

### An Olympics-Inspired GMAT Data Sufficiency Question

In honor of Gabby Douglas’ gold medal win, as well as the U.S. women’s gymnastics team’s all-around gold medal win, here is an Olympics-inspired Data Sufficiency problem.

A particular gymnastics tournament awards a gold, a silver, and a bronze medal in each of four events: Floor, Beam, Bars, and Vault. A platinum Best All-Around medal is awarded to the competitor who gains the most points from winning the other medals: 3 points for gold, 2 points for silver, 1 point for bronze. If McKenzie won the Best All-Around medal, and no one can win more than one medal in any of the four events, did she win at least one gold medal?

- All of the gold, silver, and bronze medals were won by fewer than six competitors, including McKenzie
- Another competitor in the tournament has 8 points.
(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) Statements (1) and (2) TOGETHER are NOT sufficient.

Choose your answer before proceeding!

First, you may ask, How could someone win the All-Around without winning a single gold medal?

Easy “ just imagine that McKenzie won ALL of the silver medals (8 points), and that no one else won more than one medal (the other medals are won by 8 separate people), so each person who has a gold has just 3 points, and each person who has a bronze has just 1 point.