Archives For Algebra

We need to know a lot of different facts, rules, formulas, and techniques for the quant portion of the test, but there are 4 math strategies that can be used over and over again, across any type of math—algebra, geometry, word problems, and so on.

Do you know what they are?

Try this GMATPrep® problem and then we’ll talk about the first of the four strategies.

* ” If mv < pv < 0, is v > 0?

“(1) m < p

“(2) m < 0”

All set?

How did you do the problem? Most quant questions have more than one possible approach and this one is no exception—but I want to use this problem to talk about a particular technique called Testing Cases.

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. When we have these kinds of questions, we can use theory to solve—but that can get very confusing very quickly. Instead, try testing real numbers to “prove” the theory to yourself.

(Note: I chose a particularly tough question for this exercise; testing cases can also be useful and fast on easier questions!)

This problem gives one inequality:

“mv < pv < 0″

The test writers are hoping that you’ll say, “Oh, let’s just divide by v to get rid of it, so the equation is really m < p < 0.” But that’s a trap! Why?

When you divide an inequality by a negative, you have to flip the signs. But you don’t know whether v is positive or negative, so you don’t know whether to flip the signs! Never divide an inequality by a variable if you don’t know the sign of the variable.

The question itself contains a clue (two, actually!) pointing to this trap. The given inequality asks about “< 0” and the question also asks whether v > 0? Less than zero and greater than zero are code for “I’m testing you on positives and negatives.”
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On the GMAT, you may see a 3 to 5 ratio expressed in a variety of ways:

3:5
3 to 5
x/y = 3/5
5x = 3y (Yes, that’s the same as the other 3. Think about it.)

In the real world, we encounter ratios in drink recipes more often than anywhere else (3 parts vodka, 5 parts cranberry),don't drink and derive perhaps explaining why–after drinks that strong–we forget how to handle them.

Keep in mind: ratios express a “part to part” relationship, whereas fractions and percentages express a “part to whole” relationship. So the fraction of the above drink is 3/8 vodka (or 37.5% of the whole). Either way, hold off on mixing that drink until after this post.

I like to set up ratios using a “ratio box.” The box is a variant on the “Unknown Multiplier” technique from page 65 of our FDPs book, but it’s a nice way to visually manage ratios without resorting to algebra.

Let’s take the beginning of a typical ratio question:

“The ratio of men to women in a class is 3:2…”

Instead of doing anything fancy with variables, I just set up a tracking chart:

Men Women Total
Ratio 3 2 5

From this point alone, I have sufficient information to answer a bunch of questions.

-What fraction of the students are men? (3/5)

-What percent of the students are women? (40%)

-What is the probability of choosing a man? (3/5)

-etc.

However, I have nowhere near enough information to answer anything about the REAL numbers of students in this class. Suppose the GMAT were to add a little more information:

“The ratio of men to women in a class is 3:2. If there are 35 students in the class…”

Now we can calculate almost everything about the real numbers of people. First, make a bigger box with 3 lines. The unfilled box looks like this:

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“Many a true word is said in jest.”—I don’t know, but I heard it from my mother.

math languageOnce upon a time in America, when I was a boy, my father, an engineer, said to me, “You can make numbers do anything you want them to do.”  This was the beginning of my cynicism.  But never mind that.  My father was fluent in four languages: English, German, French, and Algebra.  My father was also a very honest man.  His comment relied on the fact that most people can’t read Algebra—he just let people fool themselves.  Teaching GMAT classes, I combat the fact that many people can’t read Algebra.  Like my father, the GMAT exploits that weakness and lets—nay, encourages—people to fool themselves.  Thus, for many, preparing for the quantitative portion of the GMAT is akin to studying a foreign language.  (I know that even many native speakers feel that preparing for the verbal portion of the GMAT is also akin to studying a foreign language.  But that’s a different topic.)  In any case, you want to make your Algebra as fluent as your French. . .yes, for most of you, that was one of those jokes.

I know that some of you disagreed with the above and feel that the problem is an inability to understand math.  But that’s not true, at least on the level necessary to succeed on the GMAT.  If you really didn’t have enough synapses, they wouldn’t let you out without a keeper—because you couldn’t tip, or comparison shop, or count your change.  It’s a literacy problem.  Think about the math units in the course.  Truthfully, the first one is often a death march.  By the end, as country folk say, I often feel like I’m whipping dead horses.  On the other hand, the lesson concerning probability and combinations, putatively a more advanced topic, usually goes really well.  Why?  Because folks can read the words and understand their meaning.  Conversely, folks just stare at the algebraic symbols as if they were hieroglyphics.  The problem is that putting a Rosetta Stone in the book bag would make it weigh too much. . .kidding.  But if you can’t read the hieroglyphics, the mummy will get you—just like in the movies.

It really is a literacy issue and should be approached in that fashion.  You still don’t believe me?  You want specific examples?  I got examples, a pro and a con.  On the affirmative side, I once worked one on one with a man who came to me because his math was in shreds.  Because he couldn’t read what the symbols were saying.  Partly because his mother had once said, “Your sister is the one that’s good at math.”  As far as the GMAT is concerned, she was wrong, and so was your mother, if she said that.  Anyway, one day I gave him a high level Data Sufficiency word problem concerning average daily balances on a credit card.  He looked at it for about 30 seconds, and he didn’t write anything on his scrap paper.  Then he turned to me and said the answer was blah blah.  And he was right.  I looked at him and said, “How did you do that?  You’re not that good.”  (Yes, this is also an example of how mean I am to private students.)  But—and here’s the real punch line—he said, “It was about debt; I understood what the words meant.”  And there you go.  As a by the way, he worked very hard, became competent although not brilliant quantitatively, scored 710—97%V, 72%Q*—and went to Kellogg.

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gmat median & meanThis is the second of a series of posts that offer alternate ways to solve certain GMAT problems (check out the first here: DS Value Problems). Just like last time, if you like the method, steal it! And if you don’t, I promise not to lose any sleep. There’s a lot of ways to solve most questions on the GMAT and the best way will always be the way that works best for you. So without further ado, let’s check out a GMATPrep question and see how fast you can solve:

Last month 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than $165,000.

II. At least one of the homes was sold for more than $130,000 and less than $150,000.

III. At least one of the homes was sold for less than $130,000.

 

(A) I only

(B) II only

(C) III only

(D) I and II

(E) I and III

First things first, if you answered this question using algebra, you’re in great company. Another one of our instructors, Stacey Koprince, has a great write up on the algebra in this question, and it’s definitely worth a read-through right here. But a lot of questions on the GMAT, including this one, can be solved by thinking of extremely simple scenarios, rather than the algebra that determines all of them.

The first thing I noticed on this question is that this is one of those awful questions where there’s a whole lot of wiggle room with the information that they give you. What was the cheapest house? What was the cost of the third most expensive house? Were any of the houses all the same price? If the second cheapest house is half as expensive as the most expensive, how does that affect the cost of the other houses? It’s easy to get lost when you start to think about how little you know in this scenario.

But before I jump around and start picking values out of thin air, the most important part of this problem are the (few) things that MUST be true. In this case, there are two: the 15 house prices have a mean of $150,000 and a median of $130,000. And on my paper, I would write out a few slots to represent the house prices like this: (note- I wouldn’t write out all 15 slots. Just the first few, the last few, and, since this is a median problem, one in the middle.)

 

____

____

____

____

____

1

2

7

14

15

 

Again, there are two things that they tell me here, but I want to start with the most restrictive element in this problem. There are lots of different ways to get a mean of $150,000, but in order to get a median of $130,000, I would need at least one house to cost EXACTLY $130,000. So I add that to my chart (ignoring the $ sign and extra zeroes):

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gmat 750+A few months ago, I wrote a couple of articles targeted toward those students looking for a super-high score (one for quant, one for verbal). I challenged students to answer those questions in much less time than we typically average on test questions.

Well, I’m back with another one in the series. This problem is a bit different though: it’s from our Challenge Problem archive, a question bank consisting of what we call 800+ level problems. (Some might qualify as 750+ but most are harder than anything you’ll ever see on the real test.)

Do you need to be able to answer a question like this in order to score 750+? Absolutely not. (In fact, after my colleague Ron Purewal submitted this question, I tested it out on several of my fellow instructors, all of whom have scored 760+ on the test. Not everyone answered correctly.) Mostly, I’m offering this to stretch your brains, drive you a little crazy, and make one important point (see my second takeaway at the end).

If, however, quant is your strength and you’re hoping to score 51 in that section”you can certainly score 51 without getting this one right, but if you do get this one right in 2 minutes, then you know you’re ready for the quant section.

One more tidbit before we dive in. I chose this question because it is SO very hard. As of right now (as I’m typing this), 254 people have tried this problem and 44 have answered it correctly.

Do a little math here. What percentage of people answered the question correctly?

17%. Random guess position is 20%. Wow.

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gmat pumpRaise your hand if you love rate and work questions. They’re awesome, right? They tend to be fairly long, and the set-up is pretty complex, plus we get to build a table before we dive into the equations!

Oh, wait no those are all reasons why we can’t stand these problems.

Give yourself approximately 2 minutes to try the below GMATPrep problem. When you’re done, take a look at it again and ask yourself, Is there a better way to do this thing?

* Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours; pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?

(A) 1/3

(B) 1/2

(C) 2/3

(D) 5/6

(E) 1

Have you got an answer? Pick one anyway. Pretend it’s the real test: you can’t keep going till you pick an answer.

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Many a true word is said in jest.—I don’t know, but I heard it from my mother.

gmat timingIt’s a funny thing—folks get good at doing OG problems at their desks.  Then they take a practice CAT, with the clock on the monitor running down, like sands in the hourglass.  Suddenly they are seized by amphetamine psychosis.  Like NFL rookies, the big adjustment is to the speed of the game.  When you’re taking the test, if you can’t do it* in two to three minutes, you can’t do it.*  However, timing problems are an effect, not a cause.  People have timing problems because their math foundation sucks.  People have timing problems because they don’t get a good rephrasing.  People have timing problems because they don’t compare SC choices vertically.  People have timing problems because they don’t have the discipline to guess.  And so on.  All of these problems are fixable.  Like most GMAT issues, timing problems are the result of either a poor foundation or bad behavior.

Take foundation work. . .please—that’s a joke from your grandparents’ day.  When I say 7 times 13, you say 91.  Think of it as a rap.  When you see .625, you say 5/8.  Woot.  All seriousness aside, people waste 30 seconds a question in the quant because they don’t know their times tables or squares or the fractional decimal percentage equivalencies.  Or their algebra isn’t smooth and silky.  Think about how much time that uses up during the section.  How do you fix that?  How do you get to Carnegie Hall?  Practice, practice, practice.  That’s a New York joke—LA classes hate it.  You have to want it enough to do the work that you need to do.  That amount varies, person to person.

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Remember Your Units!

Andrea Pawliczek —  January 14, 2013 — 1 Comment

Did you ever have one of those anal teachers in high school math or science who would take off points if you did not include the correct units? So an answer of 7 would only receive partial credit when the answer was 7 inches.  Although this practice likely seemed frustrating at the time, I hope to provide some method behind this madness “ or specifically how awareness of units can help you on the GMAT.

remember gmat unitsMy appreciation of units first began during college. I was a chemistry major in college, and as part of my major I had to take physics.  The topics in physics never came naturally for me so I was always looking for little tricks that would lead me towards a correct answer.  One trick I found that was surprisingly effective was to just combine the numbers in the way such that the answer was in the appropriate units.  For example if the question asked for an acceleration (the rate at which speed is changing or the second derivative of distance for the calculus-inclined), I knew that acceleration is always in the form of units of distance / units of time^2 (e.g. meters/ seconds^2).  So unless I combined the numbers in a way that resulted in these units as the answer “ for example by dividing a speed in meters per second by a time in seconds “ I knew I had done something wrong.

Since units are not required on the GMAT, I find many students exclude them entirely from their note taking and calculations.  But keeping track of units, while it may cost a little time, can help lead you towards right answers and prevent you from doing illogical algebra.
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