Archives For gmat quant

gmat-quant-tipsMemorize 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.

Screen Shot 2014-05-29 at 2.40.57 PM

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.

Step 1: Glance Read Jot

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:

Screen Shot 2014-05-29 at 2.31.24 PM

Let’s see.

Screen Shot 2014-05-29 at 2.32.25 PM

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:

Screen Shot 2014-05-29 at 2.33.22 PM

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.

Screen Shot 2014-05-29 at 2.34.28 PM

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

The numerator: Screen Shot 2014-05-29 at 2.35.25 PM

The denominator: Screen Shot 2014-05-29 at 2.36.02 PM

They’re almost identical! Both of the Screen Shot 2014-05-29 at 2.37.06 PMterms cancel out, as do the Screen Shot 2014-05-29 at 2.37.41 PMterms, leaving me with:

Screen Shot 2014-05-29 at 2.38.29 PM

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:

Screen Shot 2014-05-29 at 2.39.05 PM

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

Screen Shot 2014-05-29 at 2.39.54 PM

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.

 

 

geometry-math-tipsThe 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®

Screen Shot 2014-05-07 at 1.38.36 PM

In the figure shown, the measure of slopePRS is how many degrees greater than the measure of slopePQR?

(1) The measure ofslope QPR is 30 degrees.

(2) The sum of the measures of slope PQR and slopePRQ 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 slope 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 slopePRS and slopePRQ 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 slopePRS and slopePQR. Since slopePRS is in the question, start by labeling slopePRS as x. Since slopePRS and slope PRQ sum to 180 degrees, you can also label slopePRQ as (180 – x) and slope 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 slope 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, slopeQPR, using your newfound companions, x and y. slopeQPR 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.

Screen Shot 2014-05-07 at 1.44.12 PM

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 slopeQPR is 30 degrees. Using your xy expression from the newly labeled diagram as the value of slope 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 slopeQPR 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.

 

gmat-math-strategies-01A 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.

data-sufficiencySome 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:

Screen Shot 2014-04-10 at 12.37.53 PM

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):

Screen Shot 2014-04-10 at 12.38.22 PM

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:

Screen Shot 2014-04-10 at 12.38.58 PM

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:

Screen Shot 2014-04-10 at 12.39.24 PM

(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!).

Screen Shot 2014-04-10 at 12.39.59 PM

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.

Screen Shot 2014-04-10 at 12.40.21 PM

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!

 

gmat-quantStop! Before you dive in and start calculating on a math problem, reflect for a moment. How can you set up the work to minimize the number of annoying calculations?

Try the below Percent problem from the free question set that comes with your GMATPrep® software. The problem itself isn’t super hard but the calculations can become time-consuming. If you find the problem easy, don’t dismiss it. Instead, ask yourself: how can you get to the answer with an absolute minimum of annoying calculations?

 

District

Number of Votes

Percent of Votes for Candidate P

Percent of Votes for Candidate Q

1

800

60

40

2

1,000

50

50

3

1,500

50

50

4

1,800

40

60

5

1,200

30

70

 

* ” The table above shows the results of a recent school board election in which the candidate with the higher total number of votes from the five districts was declared the winner. Which district had the greatest number of votes for the winner?

“(A) 1

“(B) 2

“(C) 3

“(D) 4

“(E) 5”

 

Ugh. We have to figure out what they’re talking about in the first place!

The first sentence of the problem describes the table. It shows 5 different districts with a number of votes, a percentage of votes for one candidate and a percentage of votes for a different candidate.

Hmm. So there were two candidates, P and Q, and the one who won the election received the most votes overall. The problem doesn’t say who that was. I could calculate that from the given data, but I’m not going to do so now! I’m only going to do that if I have to.

Let’s see. The problem then asks which district had the greatest number of votes for the winner. Ugh. I am going to have to figure out whether P or Q won. Let your annoyance guide you: is there a way to tell who won without actually calculating all the votes?

Continue Reading…

GMAT-geometryA couple of months ago, we talked about what to do when a geometry problem pops up on the screen. Do you remember the basic steps? Try to implement them on the below GMATPrep® problem from the free tests.

* ”In the xy-plane, what is the y-intercept of line L?

“(1) The slope of line L is 3 times its y-intercept
“(2) The x-intercept of line L is – 1/3”

My title (3 Steps to Better Geometry) is doing double-duty. First, here’s the general 3-step process for any quant problem, geometry included:

Screen Shot 2014-02-05 at 12.13.43 PM

All geometry problems also have three standard strategies that fit into that process.

First, pick up your pen and start drawing! If they give you a diagram, redraw it on your scrap paper. If they don’t (as in the above problem), draw yourself a diagram anyway. This is part of your Glance-Read-Jot step.

Second, identify the “wanted” element and mark this element on your diagram. You’ll do this as part of the Glance-Read-Jot step, but do it last so that it leads you into the Reflect-Organize stage. Where am I trying to go? How can I get there?

Third, start Working! Infer from the given information. Geometry on the GMAT can be a bit like the proofs that we learned to do in high school. You’re given a couple of pieces of info to start and you have to figure out the 4 or 5 steps that will get you over to the answer, or what you’re trying to “prove.”

Let’s dive into this problem. They’re talking about a coordinate plane, so you know the first step: draw a coordinate plane on your scrap paper. The question indicates that there’s a line L, but you don’t know anything else about it, so you can’t actually draw it. You do know, though, that they want to know the y-intercept. What does that mean?

They want to know where line L crosses the y-axis. What are the possibilities?

Infinite, really. The line could slant up or down or it could be horizontal. In any of those cases, it could cross anywhere. In fact, the line could even be vertical, in which case it would either be right on the y-axis or it wouldn’t cross the y-axis at all. Hmm.
Continue Reading…

Math-strategies-gmatLast time, we talked about the first 2 of 4 quant strategies that everyone must master: Test Cases and Choose Smart Numbers.

Today, we’re going to cover the 3rd and 4th strategies. First up, we have Work Backwards. Let’s try a problem first: open up your Official Guide, 13th edition (OG13), and try problem solving #15 on page 192. (Give yourself about 2 minutes.)

I found this one by popping open my copy of OG13 and looking for a certain characteristic that meant I knew I could use the Work Backwards technique. Can you figure out how I knew, with just a quick glance, that this problem qualified for the Work Backwards strategy? (I’ll tell you at the end of the solution.)

For copyright reasons, I can’t reproduce the entire problem, but here’s a summary: John spends 1/2 his money on fruits and vegetables, 1/3 on meat, and 1/10 on treats from the bakery. He also spends $6 on candy. By the time he’s done, he’s spent all his money. The problem asks how much money he started out with in the first place.

Here are the answer choices:

“(A) $60

“(B) $80

“(C) $90

“(D) $120

“(E) $180”

Work Backwards literally means to start with the answers and do all of the math in the reverse order described in the problem. You’re essentially plugging the answers into the problem to see which one works. This strategy is very closely tied to the first two we discussed last time—except, in this instance, you’re not picking your own numbers. Instead, you’re using the numbers given in the answers.

In general, when using this technique, start with answer (B) or (D), your choice. If one looks like an easier number, start there. If (C) looks a lot easier than (B) or (D), start with (C) instead.

This time, the numbers are all equally “hard,” so start with answer (B). Here’s what you’re going to do:

(B) $80

 

F + V (1/2)

M (1/3)

B (1/10)

C $6

Add?

(B) $80

$40

…?

$6

Set up a table to calculate each piece. If John starts with $80, then he spends $40 on fruits and vegetables. He spends… wait a second! $80 doesn’t go into 1/3 in a way that would give a dollar-and-cents amount. It would be $26.66666 repeating forever. This can’t be the right answer!

Interesting. Cross off answer (B), and glance at the other answers. They’re all divisible by 3, so we can’t cross off any others for this same reason.

Try answer (D) next.

 

F + V (1/2)

M (1/3)

B (1/10)

C $6

Add to?

(B) $80

$40

…?

$6

?

(D) $120

$60

$40

$12

$6

$118

 

In order for (D) to be the correct answer, the individual calculations would have to add back up to $120, but they don’t. They add up to $118.

Okay, so (D) isn’t the correct answer either. Now what? Think about what you know so far. Answer (D) didn’t work, but the calculations also fell short—$118 wasn’t large enough to reach the starting point. As a result, try a smaller starting point next.

 

F + V (1/2)

M (1/3)

B (1/10)

C $6

Add?

(B) $80

$40

…?

$6

?

(D) $120

$60

$40

$12

$6

$118

(C) $90

$45

$30

$9

$6

$90

 

It’s a match! The correct answer is (C).

Now, why would you want to do the problem this way, instead of the “straightforward,” normal math way? The textbook math solution on this one involves finding common denominators for three fractions—somewhat annoying but not horribly so. If you dislike manipulating fractions, or know that you’re more likely to make mistakes with that kind of math, then you may prefer to work backwards.

Note, though, that the above problem is a lower-numbered problem. On harder problems, this Work Backwards technique can become far easier than the textbook math. Try PS #203 in OG13. I would far rather Work Backwards on this problem than do the textbook math!

So, have you figured out how to tell, at a glance, that a problem might qualify for this strategy?

It has to do with the form of the answer choices. First, they need to be numeric. Second, the numbers should be what we consider “easy” numbers. These could be integers similar to the ones we saw in the above two problems. They could also be smaller “easy” fractions, such as 1/2, 1/3, 3/2, and so on.

Further, the question should ask about a single variable or unknown. If it asks for x, or for the amount of money that John had to start, then Work Backwards may be a great solution technique. If, on the other hand, the problem asks for xy, or some other combination of unknowns, then the technique may not work as well.

(Drumroll, please) We’re now up to our fourth, and final, Quant Strategy that Everyone Must Master. Any guesses as to what it is? Try this GMATPrep© problem.

 

geometry

“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?

Screen Shot 2013-12-29 at 3.26.36 PM

If the radius is 1, then the bottom line (the hypotenuse) of the triangle is 2. If you drop a line from point B to that bottom line, or base, you’ll have a height and can calculate the area of the triangle, since A = (1/2)bh.

You don’t know what that height is, yet, but you do know that it’s smaller than the length of BC. If BC were the height of the triangle, then the area would be A = (1/2)(2)(1) = 1. Because the height is smaller than BC, the area has to be smaller than 1. Eliminate answers (C), (D), and (E).

Now, decide whether you want to go through the effort of figuring out that height, so that you can calculate the precise area, or whether you’re fine with guessing between 2 answer choices. (Remember, unless you’re going for a top score on quant, you only have to answer about 60% of the questions correctly, so a 50/50 guess with about 30 seconds’ worth of work may be your best strategic move at this point on the test!)

The technique we just used to narrow down the answers is one I’m sure you’ve used before: Estimation. Everybody already knows to estimate when the problem asks you for an approximate answer. When else can (and should) you estimate?

Glance at the answers. Notice anything? They can be divided into 3 “categories” of numbers: less than 1, 1, and greater than 1.

Whenever you have a division like this (greater or less than 1, positive or negative, really big vs. really small), then you can estimate to get rid of some answers. In many cases, you can get rid of 3 and sometimes even all 4 wrong answers. Given the annoyingly complicated math that sometimes needs to take place in order to get to the final answer, your best decision just might be to narrow down to 2 answers quickly and then guess.

Want to know how to get to the actual answer for this problem, which is (B)? Take a look at the full solution here.

The 4 Quant Strategies Everyone Must Master

Here’s a summary of our four strategies.

(1) Test Cases.

-      Especially useful on Data Sufficiency with variables / unknowns. Pick numbers that fit the constraints given and test the statement. That will give you a particular answer, either a value (on Value DS) or a yes or no (on Yes/No DS). Then test another case, choosing numbers that differ from the first set in a mathematically appropriate way (e.g., positive vs. negative, odd vs. even, integer vs. fraction). If you get an “always” answer (you keep getting the same value or you get always yes or always no), then the statement is sufficient. If you find a different answer (a different value, or a yes plus a no), then that statement is not sufficient.

-      Also useful on “theory” Problem Solving questions, particularly ones that ask what must be true or could be true. Test the answers using your own real numbers and cross off any answers that don’t work with the given constraints. Keep testing, using different sets of numbers, till you have only one answer left (or you think you’ve spent too much time).

(2) Choose Smart Numbers.

-      Used on Problem Solving questions that don’t require you to find something that must or could be true. In this case, you need to select just one set of numbers to work through the math in the problem, then pick the one answer that works.

-      Look for variable expressions (no equals or inequalities signs) in the answer choices. Will also work with fraction or percent answers.

(3) Work Backwards.

-      Used on Problem Solving questions with numerical answers. Most useful when the answers are “easy”—small integers, easy fractions, and so on—and the problem asks for a single variable. Instead of selecting your own numbers to try in the problem, use the given answer choices.

-      Start with answer (B) or (D). If a choice doesn’t work, cross it off but examine the math to see whether you should try a larger or smaller choice next.

(4) Estimate.

-      You’re likely already doing this whenever the problem actually asks you to find an approximate answer, but look for more opportunities to save yourself time and mental energy. When the answers are numerical and either very far apart or split across a “divide” (e.g., greater or less than 0, greater or less than 1), you can often estimate to get rid of 2 or 3 answers, sometimes even all 4 wrong answers.

The biggest takeaway here is very simple: these strategies are just as valid as any textbook math strategies you know, and they also require just as much practice as those textbook strategies. Make these techniques a part of your practice: master how and when to use them, and you will be well on your way to mastering the Quant portion of the GMAT!

Read The 4 Math Strategies Everyone Must Master, Part 1.

challenge problem
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:

Which of the following is the largest?

(A) 227.3
(B) 318.2
(C) 511.1
(D) 79.1
(E) 115.1

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