### Archives For Quant

Memorize what? I’m not going to tell you yet. Try this problem from the GMATPrep® free practice tests first and see whether you can spot the most efficient solution.

All right, have you got an answer? How satisfied are you with your solution? If you did get an answer but you don’t feel as though you found an elegant solution, take some time to review the problem yourself before you keep reading.

Take a quick glance; what have you got? PS. A given equation, xy = 1. A seriously ugly-looking equation. Some fairly “nice” numbers in the answers. Hmm, maybe you should work backwards from the answers?

Jot the given info on the scrap paper.

Step 2: Reflect Organize

Oh, wait. Working backwards isn’t going to work—the answers don’t stand for just a simple variable.

Okay, what’s plan B? Does anything else jump out from the question stem?

Hey, those ugly exponents…there is one way in which they’re kind of nice. They’re both one of the three common special products. In general, when you see a special product, try rewriting the problem usually the other form of the special product.

Step 3: Work

Here’s the original expression again:

Let’s see.

Interesting. I like that for two reasons. First of all, a couple of those terms incorporate xy and the question stem told me that xy = 1, so maybe I’m heading in the right direction. Here’s what I’ve got now:

And that takes me to the second reason I like this: the two sets of exponents look awfully similar now, and they gave me a fraction to start. In general, we’re supposed to try to simplify fractions, and we do that by dividing stuff out.

How else can I write this to try to divide the similar stuff out? Wait, I’ve got it:

The numerator:

The denominator:

They’re almost identical! Both of the terms cancel out, as do the terms, leaving me with:

I like that a lot better than the crazy thing they started me with. Okay, how do I deal with this last step?

First, be really careful. Fractions + negative exponents = messy. In order to get rid of the negative exponent, take the reciprocal of the base:

Next, dividing by 1/2 is the same as multiplying by 2:

That multiplies to 16, so the correct answer is (D).

Key Takeaways: Special Products

(1) Your math skills have to be solid. If you don’t know how to manipulate exponents or how to simplify fractions, you’re going to get this problem wrong. If you struggle to remember any of the rules, start building and drilling flash cards. If you know the rules but make careless mistakes as you work, start writing down every step and pausing to think about where you’re going before you go there. Don’t just run through everything without thinking!

(2) You need to memorize the special products and you also need to know when and how to use them. The test writers LOVE to use special products to create a seemingly impossible question with a very elegant solution. Whenever you spot any form of a special product, write the problem down using both the original form and the other form. If you’re not sure which one will lead to the answer, try the other form first, the one they didn’t give you; this is more likely to lead to the correct answer (though not always).

(3) You may not see your way to the end after just the first step. That’s okay. Look for clues that indicate that you may be on the right track, such as xy being part of the other form. If you take a few steps and come up with something totally crazy or ridiculously hard, go back to the beginning and try the other path. Often, though, you’ll find the problem simplifying itself as you get several steps in.

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

With GMAT INTERACT™ coming June 16th, we’d like to take you behind-the-scenes to explore some fun facts about GMAT INTERACT and the creation process that has made all of this possible. Here are a few fun facts we’d like to share.

1. GMAT INTERACT was years in the making.

It took over 6,000 hours of development to bring GMAT INTERACT to life. An expert team of Manhattan Prep designers, coders, developers, and instructors worked for over three years on the design and development of the platform to create a user experience that is unlike anything else in test prep.

2. This is the first GMAT learning platform that is truly interactive.

GMAT INTERACT is a comprehensive on demand, self-paced program that features 35+ lessons that are interactive, funny, and completely directed by you. No two people see the same thing. Designed around the student-teacher connection, an expert Manhattan Prep instructor will guide you through each section of the GMAT, asking you questions and prompting you to think about the content presented. What’s more: every response you give tailors the lesson you’ll receive.

Manhattan GMAT is known for our incredible instructors (just check out our Beat The GMAT Verified Reviews). Not only are our teachers top scoring GMAT experts, they’re also fun and engaging—and we’ve put them front and center in GMAT INTERACT. And, we may have also thrown in a sock puppet or two…

To give you a taste of the fun you can expect, here are some facts about GMAT INTERACT:

o Number of times you get to see Tommy dance: 3
o Number of times Whitney Garner laughs on camera: uncountable
o Most takes we needed for a clip: 16
o Number of times we cursed on camera and had to toss the clip: 11
o Number of dolphin drawings used: 1
o Number of dinosaur cat robots destroyed in production: 1
o Number of bubbles used in the Evil Grammar Lab: 521
o Number of cavemen used in production: 1

4. With GMAT INTERACT, you don’t get 1 Manhattan Prep instructor – you get 11!

When we say that GMAT INTERACT is comprehensive – we mean it! We put eleven of our most accomplished instructors in front of the camera, take-after-take, and are delivering them to your computer and mobile devices wherever you are. Not just a video, our instructors will engage with you based on the responses and answers you input.

5. You don’t have to wait until June 16th to try GMAT INTERACT!

While the full version of GMAT INTERACT won’t be available until June 16th for purchase, you can try a FREE GMAT INTERACT Geometry Lesson right now, for free. So what are you waiting for? Jump in and have some fun! Test prep doesn’t have to be boring ever again!

The GMAT quant section has many faces – there are a number of content areas, and it is best to try to master as many of them as you can before test day. It is important, however, that you not compartmentalize too much. In many of the harder questions in fact, two or more topics often show up together. You can easily find quadratics in a consecutive integer question, coordinate geometry in a probability question, number properties in a function question, for example. One common intersection of two topics that I find surprises many students is that of geometry and algebra. Many people expect a geometry question to be about marking up diagrams with values or tick marks to show equality and/or applying properties and formulas to calculate or solve. While these are no doubt important skill sets in geometry, don’t forget to pull out one of the most important skills from your  GMAT tool bag – the almighty variable! x’s and y’s have a welcomed home in many a geometry question, though you might find that you are the one who has to take the initiative to put them there!

Take a look at this data sufficiency question from GMATPrep®

In the figure shown, the measure of PRS is how many degrees greater than the measure of PQR?

(1) The measure of QPR is 30 degrees.

(2) The sum of the measures of  PQR and PRQ is 150 degrees.

How did you do? Don’t feel bad if you’re a little lost on this one. This is a difficult question, though you’ll see that with the right moves it is quite doable. At the end of this discussion, you’ll even see how you could put up a good guess on this one.

As is so often the case in a data sufficiency question, the right moves here start with the stem – in rephrasing the question. Unfortunately the stem doesn’t appear to provide us with a lot of given information. As indicated in the picture, you have a 90 degree angle at  PQR and that seems like all that you are given, but it’s not! There are some other inherent RELATIONSHIPS, ones that are implied by the picture. For example PRS and PRQ sum to 180 degrees. The problem, however, is how do you CAPTURE THOSE RELATIONSHIPS? The answer is simple – you capture those relationships the way you always capture relationships in math when the relationship is between two unknown quantities – you use variables!

But where should you put the variables and how many variables should you use?  This last question is one that you’ll likely find yourself pondering  a number of times on the GMAT. Some believe the answer to be a matter of taste. My thoughts are always use as few variables as possible. If you can capture all of the relationships that you want to capture with one variable, great. If you need two variables, so be it. The use of three or more variables would be rather uncommon in a geometry question, though you could easily see that in a word problem.  Keep one thing in mind when assigning variables: the more variables you use, usually the more equations you will need to write in order to solve.

As for the first question above about where to place the variables, you can take a closer look in this question at what they are asking and use that as a guide. They ask for the   (degree) difference between PRS and PQR. Since PRS is in the question, start by labeling PRS as x. Since PRS and  PRQ sum to 180 degrees, you can also label PRQ as (180 – x) and  RPS as (180 – x – 90) or (90 – x).

Can you continue to label the other angles in triangle PRQ in terms of x or is it now time to place a second variable, y? Since you still have two other unknown quantities in that triangle, it’s in fact time for that y. The logical place of where to put it is on  PQR since that is also part of the actual question. The temptation is to stop there – DON’T! Continue to label the final angle of the triangle, QPR, using your newfound companions, x and y. QPR can be labeled as [180 – y – (180 – x)] or (xy). Now all of the angles in the triangle are labeled and you are poised and ready to craft an algebraic equation/expression to capture any other relationships that might come your way.

Before you rush off to the statements, however, there is one last step. Formulate what the question is really asking in terms of x and y. The question rephrases to “What is the value of xy?”

Now you can finally head to the statements. Oh the joy of a fully dissected data sufficiency stem – 90% of the work has already been done!

Statement (1) tells you that the measure of QPR is 30 degrees. Using your xy expression from the newly labeled diagram as the value of  QPR, you can jot down the equation xy = 30. Mission accomplished! The statement is sufficient to answer the question “what is the value of xy?”

Statement (2) indirectly provides the same information as statement (1). If the two other angles of triangle PQR sum to 150 degrees, then QPR is 30 degrees, so the statement is sufficient as well. If you somehow missed this inference and instead directly pulled from the diagram y + (180 – x) and set that equal to 150, you’d come to the same conclusion. Either way the algebra saves the day!

The answer to the question is D, EACH statement ALONE is sufficient to answer the question asked.

NOTE here that from a strategic guessing point of view, noticing that statements (1) and (2) essentially provide the same information allows you to eliminate answer choices A, B and C: A and B because how could it be one and not the other if they are the same, and C because there is nothing gained by combining them if they provide exactly the same information.

The takeaways from this question are as follows:

(1) When a geometry question has you staring at the diagram, uncertain of how to proceed in marking things up or capturing relationships that you know exist – use variables! Those variables will help you move through the relationships just as actual values would.

(2) In data sufficiency geometry questions, when possible represent the question in algebraic form so the target becomes clear and so that the rules of algebra are there to help you assess sufficiency.

(3) Once you have assigned a variable, continue to label as much of the diagram in terms of that variable. If you need a second variable to fully label the diagram, use it. If you can get away with just one variable and still accomplish the mission, do so.

Most GMAT test-takers know that they need to develop clear strategies when it comes to different types of word problems, and most of those involve either muscling your way through the problem with some kind of practical approach (picking numbers, visualizing, back-solving, logical reasoning) or writing out algebraic equations and solving.  There are of course pluses and minuses to all of the approaches and those need to be weighed by each person on an individual basis. What few realize, however, is that geometry questions can also demonstrate that level of complexity and thus can often also be solved with the tools of algebra. When actual values are few and far between, don’t hesitate to pull out an “x” (and possibly also a “y”) and see what kind of equations/expressions you can cook up.

For more practice in “algebrating” a geometry question, please see OG 13th DS 79 and Quant Supplement 2nd editions PS 157, 162 and DS 60, 114 and 123.

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 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.

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!

So you’ve been told over and over that guessing is an important part of the GMAT. But knowing you’re supposed to guess and knowing when you’re supposed to guess are two very different things. Here are a few guidelines for how to decide when to guess.

But first, know that there are two kinds of guesses: random guesses and educated guesses. Both have their place on the GMAT. Random guesses are best for the questions that are so tough, that you don’t even know where to get started. Educated guesses, on the other hand, are useful when you’ve made at least some progress, but aren’t going to get all the way to an answer in time.

Here are a few different scenarios that should end in a guess.

Scenario 1: I’ve read the question twice, and I have no idea what it’s asking.

This one is pretty straightforward. Don’t worry about whether the question is objectively easy or difficult. If it’s too hard for you, it’s not worth doing. In fact, it’s so not worth doing that it’s not even worth your time narrowing down answer choices to make an educated guess. In fact, if it’s that difficult, it may even be better for you to get it wrong!

To make the most of your random guesses, you should use the same answer choice every time. The difference is slight, but it does up your odds of getting some of these random guess right.

Scenario 2: I had a plan, but I hit a wall.

Often, when this happens, you haven’t yet spent 2 minutes on the problem. So why guess? Maybe now you have a better plan for how to get to the answer. I know this is hard to hear, but don’t do it! To stay on pace for the entire section, you have to stay disciplined and that means that you only have one chance to get each question right.

The good news is that no 1 question you get wrong will kill your score. But, 1 question can really hurt your score if you spend too long on it! Once you realize that your plan didn’t work, it’s time to make an educated guess. You’ve already spent more than a minute on this question (hopefully not more than 2!), and you probably have some sense of which answers are more likely to be right. Take another 15 seconds (no more!) and make your best educated guess.

Scenario 3: I got an answer, but it doesn’t match any of the answer choices.

This is another painful one, but it’s an almost identical situation to Scenario 2. It means you either made a calculation error somewhere along the way, or you set the problem up incorrectly to begin with. In an untimed setting, both of these problems would have the same solution: go back over your work and find the mistake. On the GMAT, however, that process is too time-consuming. Plus, even once you find your mistake, you still have to redo all the work!

Once again, though it might hurt, it’s still in your best interest to let the question go. If you can narrow down the answer choices, great (though don’t spend longer than 15 or 20 seconds doing so). If not, don’t worry about it. Just make a random guess and vow to be more careful on the next one (and all the rest after that!).

Scenario 4: I checked my pacing chart and I’m more than 2 minutes behind.

Pacing problems are best dealt with early. If you’re more than 2 minutes behind, don’t wait until another 5 questions have passed and you realize you’re 5 minutes behind. At this point, you want to find a question in the next 5 that you can guess randomly on. The quicker you can identify a good candidate to skip, the more time you can make up.

This is another scenario where random guessing is best. Educated guessing takes time, and we’re trying to save as much time as possible. Look for questions that take a long time to read, or that deal with topics you’re not as strong in, but most importantly, just make the decision and pick up the time.

Wrap Up

Remember, this test is not like high school exams; it’s not designed to have every question answered. This test is about consistency on questions you know how to do. Knowing when to get out of a question is one of the most fundamental parts of a good score. The better you are at limiting time spent on really difficult questions, the more time you have to answer questions you know how to do.

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We are happy to announce that the latest version of our free GMAT app, Pocket GMAT Flashcards, is now available for download via the App store! New updates include:

• Back-end and usability fixes
• Content overhaul
• Updated for iOS7
• Shiny new icon

Containing over 350 GMAT quant flash cards, Pocket GMAT uses an adaptive algorithm developed by Manhattan Prep instructors to help you target cards you most need help with. Allowing you to strengthen your GMAT quantitative skills anywhere and at any time, the Pocket GMAT app is an indispensable tool for iPhone users.

The app also now works better on iOS6 devices and we have fixed issues with scrolling and swiping, so overall navigation is smoother. We’ve also fixed content errata and made the images look better.

Manhattan Prep has teamed up with Learningpod to make Pocket GMAT free for everyone! In addition to the adaptive algorithm, there is also a sequential practice mode that lets you flip through the cards however you want. You also have the ability to enter a Target Date to keep you on pace and track your progress. The flash cards are organized into “KeyRings” by topic and include algebra, number properties, word problems, geometry, fractions, decimals, and percents.

We hope the new updates improve your studying experience, and if you’re as excited as we are about the revisions, please let us know in the review section of the App store. We use your feedback to make our study tools the best they can possibly be!