If you have two equations, you can solve for two variables.
This rule is a cornerstone of algebra. It’s how we solve for values when we’re given a relationship between two unknowns:
If I can buy 2 kumquats and 3 rutabagas for $16, and 3 kumquats and 1 rutabaga for $9, how much does 1 kumquat cost?
We set up two equations:
2k + 4r = 16
3k + r = 9
Then we can use either substitution or elimination to solve. (Try it out yourself; answer* below).
On the GMAT, you’ll be using the “2 equations à 2 variables” rule to solve for a lot of word problems like the one above, especially in Problem Solving. Be careful, though! On the GMAT this rule doesn’t always apply, especially in Data Sufficiency. Here are some sneaky exceptions to the rule…
2 Equations aren’t always 2 equations
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.
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:
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:
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.
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”
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.”
On the GMAT, you may see a 3 to 5 ratio expressed in a variety of ways:
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), 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:
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)
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:
“Many a true word is said in jest.”—I don’t know, but I heard it from my mother.
Once 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.
This 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.)
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):
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.
Raise 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?
Have you got an answer? Pick one anyway. Pretend it’s the real test: you can’t keep going till you pick an answer.
Many a true word is said in jest.—I don’t know, but I heard it from my mother.
It’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.
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.
My 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.