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A while back, we talked about the 4 GMAT math strategies that everyone needs to master. Today, I’ve got some additional practice for you with regard to one of those strategies: Testing Cases.
Try this GMATPrep® problem:
* ” If xy + z = x(y + z), which of the following must be true?
“(A) x = 0 and z = 0
“(B) x = 1 and y = 1
“(C) y = 1 and z = 0
“(D) x = 1 or y = 0
“(E) x = 1 or z = 0
How did it go?
This question is called a “theory” question: there are just variables, no real numbers, and the answer depends on some characteristic of a category of numbers, not a specific number or set of numbers. Problem solving theory questions also usually ask what must or could be true (or what must not be true). When we have these kinds of questions, we can use theory to solve—but that can get very confusing very quickly. Testing real numbers to “prove” the theory to yourself will make the work easier.
The question stem contains a given equation:
xy + z = x(y + z)
Whenever the problem gives you a complicated equation, make your life easier: try to simplify the equation before you do any more work.
xy + z = x(y + z)
xy + z = xy + xz
z = xz
Very interesting! The y term subtracts completely out of the equation. What is the significance of that piece of info?
Nothing absolutely has to be true about the variable y. Glance at your answers. You can cross off (B), (C), and (D) right now!
Next, notice something. I stopped at z = xz. I didn’t divide both sides by z. Why?
In general, never divide by a variable unless you know that the variable does not equal zero. Dividing by zero is an “illegal” move in algebra—and it will cause you to lose a possible solution to the equation, increasing your chances of answering the problem incorrectly.
The best way to finish off this problem is to test possible cases. Notice a couple of things about the answers. First, they give you very specific possibilities to test; you don’t even have to come up with your own numbers to try. Second, answer (A) says that both pieces must be true (“and”) while answer (E) says “or.” Keep that in mind while working through the rest of the problem.
z = xz
Let’s see. z = 0 would make this equation true, so that is one possibility. This shows up in both remaining answers.
If x = 0, then the right-hand side would become 0. In that case, z would also have to be 0 in order for the equation to be true. That matches answer (A).
If x = 1, then it doesn’t matter what z is; the equation will still be true. That matches answer (E).
Wait a second—what’s going on? Both answers can’t be correct.
Be careful about how you test cases. The question asks what MUST be true. Go back to the starting point that worked for both answers: z = 0.
It’s true that, for example, 0 = (3)(0).
Does z always have to equal 0? Can you come up with a case where z does not equal 0 but the equation is still true?
Try 2 = (1)(2). In this case, z = 2 and x = 1, and the equation is true. Here’s the key to the “and” vs. “or” language. If z = 0, then the equation is always 0 = 0, but if not, then x must be 1; in that case, the equation is z = z. In other words, either x = 1 OR z = 0.
The correct answer is (E).
The above reasoning also proves why answer (A) could be true but doesn’t always have to be true. If both variables are 0, then the equation works, but other combinations are also possible, such as z = 2 and x = 1.
Key Takeaways: Test Cases on Theory Problems
(1) If you didn’t simplify the original equation, and so didn’t know that y didn’t matter, then you still could’ve tested real numbers to narrow down the answers, but it would’ve taken longer. Whenever possible, simplify the given information to make your work easier.
(2) Must Be True problems are usually theory problems. Test some real numbers to help yourself understand the theory and knock out answers. Where possible, use the answer choices to help you decide what to test.
(3) Be careful about how you test those cases! On a must be true question, some or all of the wrong answers could be true some of the time; you’ll need to figure out how to test the cases in such a way that you figure out what must be true all the time, not just what could be true.
* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.
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.
Recently, we talked about how to create Official Guide (OG) problem sets in order to practice for the test. I have one more component to add: track your work and analyze your results to help you prioritize your studies.
In the first half of this article, we talked about making problem sets from the roughly 1,500 problems that can be found in the three main OG books. These problems are generally regarded as the gold standard for GMAT study, but how do you keep track of your progress across so many different problems?
The best tool out there (okay, I’m biased) is our GMAT Navigator program, though you can also build your own tracking tool in Excel, if you prefer. I’ll talk about how to get the most out of Navigator, but I’ll also address what to include if you decide to build your own Excel tracker.
(Note: GMAT Navigator used to be called OG Archer. If you used OG Archer in the past, Navigator brings you all of that same functionality—it just has a new name.)
What is GMAT Navigator?
Navigator contains entries for every one of the problems in the OG13, Quant Supplement, and Verbal Supplement books. In fact, you can even look up problems from OG12. You can time yourself while you answer the question, input your answer, review written and video solutions, get statistics based on your performance, and more.
Everyone can access a free version of Navigator. Students in our courses or guided-self study programs have access to the full version of the program, which includes explanations for hundreds of the problems.
How Does Navigator Work?
First, have your OG books handy. The one thing the program does not contain is the full text of problems. (Copyright rules prevent this, unfortunately.)
When you sign on to Navigator, you’ll be presented with a quick tutorial showing you what’s included in the program and how to use it. Take about 10 minutes to browse through the instructions and get oriented.
When you reach the main page, your first task is to decide whether you want to be in Browse mode or Practice mode.
Practice mode is the default mode; you’ll spend most of your time in this mode. You’ll see an entry for the problem along with various tools (more on this below).
Browse mode will immediately show you the correct answer and the explanation. You might use this mode after finishing a set of questions, when you want to browse through the answers. Don’t reveal the answers and explanations before you’ve tried the problem yourself!
Here’s what you can do in Practice mode:
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!
I’m reviving an old article I first wrote five years ago (time flies!) because the topic is so important. I hope that no one ever again experiences a significant score drop on the real test (or even a practice one!), but the reality is that this does happen. The big question: now what?
If this happens to you, the most important thing to do next is figure out why this happened. If you can figure out why, then you may be able to do something to prevent a score drop from happening again.
1: Official Test Conditions
Did you take your practice tests under official test conditions? Did you:
- Do the essay and IR sections?
- Take only two 8-minute breaks (the first between IR and quant, the second between quant and verbal)?
- Complete the test in one sitting (e.g., you didn’t do the verbal section later that evening or the next day)?
- Pause the test, look at books or notes, eat and drink during the test, or do anything else that wouldn’t be allowed on test day
If you did not take your practice tests under official testing conditions, then your practice scores were likely inflated—possibly just a little or possibly a lot, depending upon how far you were from official test conditions. If your practice test scores were inflated, then the bad news is that your scoring level wasn’t as good as you thought it was. In other words, your official test didn’t represent as much of a drop as you first thought (and, possibly, the official test didn’t represent any drop at all).
While this is not great news, it is crucial to know, because it tells you what the problem is. You need to figure out in which areas you’re falling short and do what you need to do (math, grammar, time management, problem-solving skills) in order to improve. (And don’t forget to take tests under official conditions in future, so that you get a true picture of your current scoring level.)
Mismanaged timing might be the most common cause of big score drops on the test. If your scores keep jumping up and down on practice tests and you’re not sure why, your timing may be the culprit.
Timing is so crucial because of certain consequences that can kill your score. You tend to make more careless mistakes when you’re rushing. You may get multiple questions wrong in a row. You may run out of time entirely before the section is over. All of these things will have a negative impact on your score.
There are two major categories for mismanaged timing: too slow and too fast. These two categories lead to three common scenarios:
- Run out of time before the section is over
- Finish the section with lots of time left
- Finish just on time—but a review of the timing patterns shows that you spent too long on some, then rushed on others to catch up. (We call this “up and down timing.”)
The vast majority of students who mismanage time badly enough to experience a big score drop will do so by going too slowly at some point on the test and, consequently, either running out of time with questions left or being forced to move too quickly at other points, thereby increasing the error rate.