### FAST Math for the GMAT (Part 2 of 5)

*Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We’re not kidding! **Check out our upcoming courses here**.*

Welcome to part 2 of our Fast Math series! **In Part 1**, I acquainted you with the fact that I’m a lazy math person: I don’t want to do any more than I have to in order to answer the question. And this series shows you how! ☺

**Principle #2: Learn shortcuts for when you do have to do the math.**

You already saw the first example of this in Principle #1:

*Shortcut #1:* When multiplying a string of numbers, pair off the 5’s and 2’s and multiply them first.

Let’s say that that problem hadn’t had a 20 in it. If we had to multiply 5 and 81…how would you do that? Read more

### Here’s How to Avoid Calculations on GMAT Quant Problem Solving

Last time, we talked about how to avoid annoying calculations on Data Sufficiency. It’s not so surprising that you can do this on DS, since you don’t “really” have to solve all the way on this question type.

But you can avoid annoying calculations on Problem Solving, too! Try this problem from the GMATPrep® free exams to learn how. Read more

### GMAT Data Sufficiency: Ratio Stories – Part 2

Recently, we took a look at a story problem dealing with ratios, and I finished up by giving you a second problem to test your skills. How did you do?

If you haven’t already, try the GMATPrep® problem below and then we’ll talk about it. Give yourself about 2 minutes. Go! Read more

### GMAT Data Sufficiency Ratio Stories — Part 1

How are you with story problems? Most math concepts can be presented in story form on the test and the GMAT test writers do like to get wordy with us. You’ve got a double task: you have to translate the words into math and then you still have to do the math! How can we get through these as efficiently as possible?

Try the GMATPrep® problem below and then we’ll talk about it. Give yourself about 2 minutes. Go!

### The Importance of Getting to No on the GMAT — Part 2

Last time, we talked about how crucial it is to develop the instinct to go for the “No” when taking the GMAT. If you haven’t read the first installment, do so right now, then come back here to learn more.

I left you with this GMATPrep® problem from the free exams.

“*If 0 <

r< 1 <s< 2, which of the following must be less than 1? “I.“II.

rs“III.

s–r“(A) I only

“(B) II only

“(C) III only

“(D) I and II

“(E) I and III”

Let’s talk about it now!

### Geometry – Who Said Geometry? It’s Algebra!

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 (*x* – *y*)*. *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 *x* – *y*?”

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 *x* – *y* expression from the newly labeled diagram as the value of QPR, you can jot down the equation *x* – *y* = 30. Mission accomplished! The statement is sufficient to answer the question “what is the value of *x* – *y*?”

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 13^{th} DS 79 and Quant Supplement 2^{nd} editions PS 157, 162 and DS 60, 114 and 123.

### The 4 GMAT Math Strategies Everyone Must Master: Testing Cases Redux

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.

### Tackling Multi-Shape Geometry on the GMAT

What do you do when you realize a geometry problem has just popped up on the screen? Try this GMATPrep© problem from the free practice test and then we’ll talk about what to do!

In the figure above, the radius of the circle with center *O* is 1 and *BC* = 1. What is the area of triangular region *ABC*?

What’s your first step? Let’s use this problem as an opportunity to practice the Quant Process.

At a glance, you can see that the problem provides a diagram. Draw! Make it big enough that you can add labels as you calculate new pieces of information (and, of course, jot down any information given in the problem).

Finally, write down any formulas you’ll need, as well as whatever the problem asks you to find. Your scrap paper might look something like this:

Read more

### GMAT Challenge Problem Showdown: December 16, 2013

We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!

Here is this week’s problem:

A set of

nidentical triangles with anglex° and two sides of length 1 is assembled to make a parallelogram (ifnis even) or a trapezoid (ifnis odd), as shown. Is the perimeter of the parallelogram or trapezoid less than 10?

### GMAT Challenge Problem Showdown: October 21, 2013

We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!

Here is this week’s problem:

A sheet of paper

ABDEis a 12-by-18-inch rectangle, as shown in Figure 1. The sheet is then folded along the segmentCFso that pointsAandDcoincide after the paper is folded, as shown in Figure 2 (The shaded area represents a portion of the back side of the paper, not visible in Figure 1). What is the area, in square inches, of the shaded triangle shown?