Memorize this and pick up 2 or 3 GMAT quant questions on the test!
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 uglylooking 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.
The 4 Math Strategies Everyone Must Master, part 2
Last 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 3^{rd} and 4^{th} strategies. First up, we have Work Backwards. Let’s try a problem first: open up your Official Guide, 13^{th} 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 dollarandcents 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 lowernumbered 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 x – y, 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.
“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?
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.
Five Strategies for Conquering 700 Level Quant Questions
Let me start off by saying that hard work and mastering each question topic is the best way to conquer the GMAT. There is no UpUpDownDownLeftRightLeftRight B, A, Start cheat code that can replace months of intense studying. That said, getting a 700+ score on the GMAT sometimes means having a few tricks up your sleeves. Here’s a few strategies that I’ve found to be helpful with gaining a few extra points at the very top of the GMAT curve:
1) Know your PEMDAS and your SADMEP
In other words, you have to know your parenthesis, exponents/roots, multiplication, division, addition, and subtraction, backwards and forwards. For as many students as I have worked with, I have yet to come across a student who can barely work through a multiplication table, yet still manages to consistently finish the quant portion of the GMAT. Even though you only need to answer 37 quantitative questions, this will entail hundreds of math calculations calculations that far too many of us have left to the machines (I for one welcome our new calculator overlords). If the average straightforward calculation takes five seconds and a student sees two hundred of these calculations over an average test, that’s sixteen minutes and forty seconds of just doing simple arithmetic. And if it takes you twice as long to do each of those calculations, that’s going to take, umm, well, it’s…. it’s going to take a lot longer.
Announcing the New Advanced GMAT Quant Strategy Guide
Exciting news “ our Advanced Quant Strategy Guide is finally ready for prime time! We’re also launching a Foundations of Verbal book; click on the link to read about that one.
Who should use this book? Great question. Are you already at the 70^{th}plus percentile (minimum) on quant and you’re looking to push yourself well into the 90s? This book is for you. In addition, please note that this book assumes that you have already worked through our five regular Strategy Guides (or the equivalent material from another company).
To give you an idea of what to expect, excerpts from the new Advanced Quant guide are below. The main point I want to make is that this book covers both advanced concepts / mathematical material, and advanced problem solving processes. Both are critical for a 90thplus percentile testtaker.
Okay, without further ado, here’s excerpt #1, an introduction to a methodical solving style inspired by mathematician George Polya. Read more