You can attend the first session of any of our online or in-person GRE courses absolutely free. Ready to take the plunge? Check out our upcoming courses here.
Think of an absolute value as a simple machine that looks like this: ||. You put a value into it, and the machine answers a single question for you: how far away from zero was the value that you put in?
The basic operation of the machine is simple. Take any number, put it into the machine, and find out how far from zero that number is. The absolute value of 12, |12|, is equal to 12. The absolute value of -10, |-10|, is equal to 10. That’s because -10 is 10 units away from zero.
It starts to get complicated when the GRE asks you to put things into the machine that are more complex than simple numbers. Imagine that somebody else is operating the machine. She puts values in, but she doesn’t tell you what those values are. All you can see is the answer that the machine gives when it receives those values. Read more
You can attend the first session of any of our online or in-person GRE courses absolutely free. Crazy, right? Check out our upcoming courses here.
Imagine a world where every conversation went like this:
Student: When is our final project due?
Professor: Three days after the first Wednesday after your rough draft is due.
Professor: The rough draft is due 15 days after the date 6 days before May 14.
Solving a GRE math word problem is a little bit like having this kind of conversation. That’s why word problems can be so infuriating. The problem isn’t lying to you. It’s just telling you the truth in a really annoying, backwards way. (Reading Comprehension problems do that too—it’s not just a Quant thing.)
In the conversation above, how would you work out the due date of the final project? Personally, I’d start by getting out my calendar. I’d start at May 14, then count 6 days backwards. Then, I’d count 15 days forwards, put a star on the calendar, and mark it ‘rough draft.’ Then I’d find the first Wednesday after that date, and finally, I’d count three days forward from there. That would give me my answer. Read more
Did you know that you can attend the first session of any of our online or in-person GRE courses absolutely free? We’re not kidding! Check out our upcoming courses here.
You know what I love about GRE Discrete Quant problems? Specifically, multiple-choice Discrete Quant? The answer choices. Think about it: out of the infinite number of numbers in the universe, the GRE has already narrowed it down to just five possibilities. They’ve done almost all of the work for you. And that makes Discrete Quant a huge opportunity for People Who Hate Math. Read more
Did you know that you can attend the first session of any of our online or in-person GRE courses absolutely free? We’re not kidding! Check out our upcoming courses here.
You know what’s really frustrating? Making a ridiculous math mistake on a GRE Quant problem, totally by accident, and never noticing it. Add a three-second sanity check to your GRE Quant routine, and you’ll be more likely to catch small mistakes before they turn into huge disasters. Read more
You may already know the basic rules of exponents for the GRE. These rules tell you what to do if you want to multiply or divide two exponential numbers, or raise an exponent to another power. Once you’ve memorized them, exponent problems become exponentially easier (I’m so sorry). But there are two types of exponent problems that many students find intimidating, because the basic rules just don’t seem useful. In this article, we’ll go over those two problem types, how to recognize them, and what to do if you see one. Read more
GRE high-scorers might not be smarter than everyone else, but they do think about the test differently. One key difference is in how high-scorers do algebra. They make far fewer algebraic mistakes, because, either consciously or subconsciously, they use mathematical rules to check their work as they simplify. Here’s how to develop that habit yourself. Read more
Imagine that you asked a friend of yours what she got on the Quant section of the GRE. Instead of answering you directly, she said “let’s just say that 4 times my score is a multiple of 44, and 3 times my score is a multiple of 45.”
Could you tell what score she got? If not… you may need to work on your GRE translation skills! Read more
Some of you may have already read an excellent post discussing how you should study for the GRE, differentiating the application of skill as opposed to the application of knowledge. (Hint: you need both, but many people struggle to progress past pure knowledge!) If you have not read that post, you can find it here.
Today (or whenever you may be reading this) I would like to “riff” on that concept inside the quantitative section. Many, many students that I work with want to treat the GRE quantitative section as a math test: there’s an equation I should use, and a number I should solve for.
And sometimes, yes, that’s exactly what the test wants you to do. But there are other questions. Questions that don’t feel quite so … “math-y”. If you’ve taken a practice test, you probably know what I’m talking about, even if you can’t put your finger on an exact definition. You saw some questions that didn’t have an equation, or questions that had an equation but no definitive “x = 243” final answer. If you had a gut reaction of “This doesn’t feel like math?!?” to these questions, congratulations! You are well on your way to a more nuanced understanding of what the GRE quantitative section wants from you!
This is what I mean in the title “Calculation versus Principle”. Some GRE quant questions are best approached through the application of various math principles; running calculations on these questions is often too time-consuming.
(As an aside, when I use the term “calculation” I am not referring to questions you would plug into a calculator. Any questions that require mathematic manipulations to find a definitive numerical result are calculation questions.)
If I were teaching a class, this is about the point where I would get tired of talking. I’m tired of talking, let’s see an example!
Ah, yes, a lovely quant comparison question. What follows is a transcription of a hypothetical test-taker’s calculation approach. Feel free to skim the next two paragraphs; the purpose here is NOT for you to know the calculation approach, but instead to compare this approach to a principle-based approach.
***Begin hypothetical calculation test-taker.***
“I need to compare the area of a triangle to the area of a square. Well that’s easy! Area = ½ b*h , and Area = side*side. Ok, what’s the …. Uh-oh. What am I supposed to do with this? They haven’t given me numbers. No wait, when they don’t give me numbers, I’m allowed to choose numbers that fit the problem. Ok, a triangle and a square have the same perimeter. Let’s make the perimeter 12, so I can easily make a 3-sided and 4-sided figure. Ok, square with sides = 3, area is 3*3 = 9. All right, quantity B is 9. Let’s get quantity A.”
“What triangle should I make? Right triangles are easy, could I make a right triangle? Hey, a 3-4-5 right triangle has a perimeter of 12! Ok, so it’s ½ b*h, and that’s ½ (3)(4) so the triangle has an area of 6 – that’s definitely less than 9. But the problem didn’t tell me it was a right triangle; am I allowed to assume that? No, I should probably try another triangle. Well, I could make an equilateral triangle – 4-4-4. What would the area of this triangle equal? The base is 4, but what’s the height? Ok, I’ll have to draw the height. Ah, I have a 30-60-90 triangle inside here, and the 60 side is going to be . This will have an area of ½ (4)(3.7) – that will be 2*3.7, which is 7.4. Still less than 9. Ok, the answer is B.”
***End hypothetical calculation test-taker.***
Well, this person is correct. The answer is B, quantity B is always larger. But wow, that was a lot of work, and in all honesty, I tried to make this hypothetical test-taker an extremely accomplished GRE quant test-taker. The immediate jump to number testing, the recognition that we need to actively try to find the maximum area triangle to correctly compare that to the square area, the immediate recognition of easy right triangles and the immediate ability to calculate the area of the equilateral triangle, the quick estimation of … these are all possible, but to do them all in the same problem, and do them correctly? I would prefer an easier way.
So let’s see what happens when we apply a general principle to this problem.
***Begin hypothetical principle approach test-taker***
I’m comparing the area of a square to the area of a triangle. The perimeters have to be the same. Ok, I know that all else being equal, if I want to maximize the area of a shape, I want it to be symmetrical. A square has more area than a rectangle with the same perimeter. What’s the most symmetrical shape? A circle. So the closer my shape gets to a circle – the more sides I put in it – the more I’m maximizing my area. Ok, the square has more sides, and therefore the larger area. B.
***End hypothetical principle approach test-taker***
Hopefully you agree that the principle-based approach is far simpler, just as accurate, and requires much less time.
So now comes the fun part – how do we learn the principles, and how do we know when to apply them?
Learning the Principles
There is no easy answer to this, but I can provide some guidelines. Look through your GRE study sources. If they look anything like mine (which are, of course the Manhattan Strategy Guides), there are certain concepts that are in boldface. Compare the following options, all of which at least partly appear in bold in my strategy guides:
1) “Sides correspond to their opposite angles…. The longest side is opposite the largest angle, and the smallest side is opposite the smallest angle.”
2) “The internal angles of a triangle must add up to 180 degrees.”
3) “Rate x Time = Distance”
4) “For some grouping problems, you may want to think about the most or least evenly distributed arrangements of the items.”
Items 1 and 4 are what I would call principle statements. They give relationships or strategies, but don’t readily lend themselves to equations. Items 2 and 3 are calculation statements. They either state clearly defined numerical quantities (and therefore easily lend themselves to equation creation, a la “a+b+c = 180”) or literally state an equation.
Look through your study materials. The more the content seems to address relationships or ideas that don’t correspond to exact numbers or exact equations, the more you should consider applying these ideas as large principles.
There is one particular area that I feel deserves special mention: number properties. GRE questions that revolve around positive vs. negative, even vs. odd, prime vs. composite numbers are more often than not principle based. There are broad principles that define specific relationships across these types of numbers. Similarly, the GRE often asks questions that either revolve around or take advantage of what I call “trick” numbers: -1, 0, and 1; and proper fractions, either positive or negative. These numbers have special properties; learning these properties, as opposed to needing to do exact calculations, can save you much heartache on the test.
Applying the Principles
When should we apply the principles? This question relies on you closely reviewing your work. Whenever a question asks for a relationship between items without providing solid numbers, perhaps you could apply a broad principle. Whenever a question seems to rely less on solving for a specific quantity, and more on identifying what kind of quantity will result – “which of the following must be odd” – perhaps you could apply a general principle. And finally, if a question permits trick numbers, there may be a principle you could apply.
As you review your work, ask yourself the following question: “Is there a way I could have answered this question without doing any actual math?” If the answer is yes, you have found a principle question.
Good luck, and happy studying!!
Have you ever gotten a GRE question wrong because you thought you were supposed to take a square root and get two different numbers but the answer key said only the positive root counted? Alternatively, have you ever gotten one wrong because you took the square root and wrote down just the positive root but the answer key said that, this time, both the positive and the negative root counted? What’s going on here?
There are a couple of rules we need to keep straight in terms of how standardized tests (including the GRE) deal with square roots. The Official Guide does detail these rules, but enough students have found the explanation confusing – and have complained to us about it – that we decided to write an article to clear everything up.
Doesn’t the OG say that we’re only supposed to take the positive root?
Sometimes this is true – but not always. This is where the confusion arises. Here’s a quote from the OG 2nd edition, page 212:
“All positive numbers have two square roots, one positive and one negative.”
Hmm. Okay, so that makes it seem like we always should take two roots, not just the positive one. Later in the same paragraph, though, the book says:
“The symbol √n is used to denote the nonnegative square root of the nonnegative number n.”
Translation: when there’s a square root symbol given with an actual number underneath it – not a variable – then we should take only the positive root. This is confusing because, although they’re not talking about variables, they use the letter n in the example. In this instance, even though they use the letter n, they define n as a “nonnegative number” – that is, they have already removed the possibility that n could be negative, so n is not really a variable.
If I ask you for the value of √9, then the answer is 3, but not -3. That leads us to our first rule.
Rule #1: √9 = 3 only, not -3
If the problem gives you an actual number below that square root symbol, then take only the positive root.
Note that there are no variables in that rule. Let’s insert one: √9 = x. What is x? In this case, x = 3, because whenever we take the square root of an actual number, we take only the positive root; the rule doesn’t change.
Okay, what if I change the problem to this: √x = 3. Now what is x? In this case, x = 9, but not -9. How do we know? Try plugging the actual number back into the problem. √9 does equal 3. What does √-9 equal? Nothing – we’re not allowed to have negative signs underneath square root signs, so √-9 doesn’t work.
Just as an aside, if the test did want us to take the negative root of some positive number under a square root sign, they’d give us this: -√9. First, we’d take the square root of 9 to get 3 and then that negative sign would still be hanging out there. Voilà! We have -3.
What else does the OG say?
Here’s the second source of confusion on this topic in the OG. On the same page of the book (212), right after the quotes that I gave up above, we have a table showing various rules and examples, and these rules seem to support the idea that we should always take the positive root and only the positive root. Note something very important though: the table is introduced with the text “where a > 0 and b > 0.” In other words, everything in the table is only true when we already know that the numbers are positive! In that case, of course we only want to take the positive values!
What if we don’t already know that the numbers in question are positive? That brings us to our second and third rules.
Rule #2: x2 = 9 means x = 3, x = -3
How are things different in this example? We no longer have a square root sign – here, we’re dealing with an exponent. If we square the number 3, we get 9. If we square the number -3, we also get 9. Therefore, both numbers are possible values for x, because both make the equation true.
Mathematically, we would say that x = 3 or x = -3. If you’re doing a Quantitative Comparison problem, think of it this way: either one is a possible value for x, so both have to be considered possible values when comparing Quantity A to Quantity B.
Rule #3: √(x)2 = 3 means x = 3, x = -3
Okay, we’re back to our square root sign, but we also have an exponent this time! Now what? Do we take only the positive root, because we have a square root sign? Or do we take both positive and negative roots, because we have an exponent?
First, solve for the value of x: square both sides of √(x)2 = 3 to get x2 = 9. Take the square root to get x = 3, x = -3 (as in our rule #2).
If you’re not sure that rule #2 (take both roots) should apply, try plugging the two numbers into the given equation, √x2 = 3, and see whether they make the equation true. If we plug 3 into the equation √x2 = 3, we get: √(3)2 = 3. Is this true? Yes: √(3)2 = √9 and that does indeed equal 3.
Now, try plugging -3 into the equation: √(-3)2= 3. We have a negative under the square root sign, but we also have parentheses with an exponent. Follow the order of operations: square the number first to get √9. No more negative number under the exponent! Finishing off the problem, we get √9 and once again that does equal 3, so -3 is also a possible value for x. The variable x could equal 3 or -3.
How am I going to remember all that?
Notice something: the first example has either a real number or a plain variable (no exponent) under the square root sign. In both circumstances, we solve only for the positive value of the root, not the negative one.
The second and third examples both include an exponent. Our second rule doesn’t include any square root symbol at all – if we have only exponents, no roots at all, then we can have both positive and negative roots. Our third rule does have a square root symbol, but it also has an exponent. In cases like this, we have to check the math just as we did in the above example. First, we solve for both solutions and then we plug both back into the original equation. Any answer that “works,” or gives us a “true” equation, is a valid possible solution.
Takeaways for Square Roots:
(1) If there is an actual number shown under a square root sign, then take only the positive root.
(2) If, on the other hand, there are variables and exponents involved, be careful. If you have only exponents and no square root sign, then take both roots. If you have both an exponent and a square root sign, you’ll have to do the math to see, but there’s still a good chance that both the positive and negative roots will be valid.
(3) If you’re not sure whether to include the negative root, try plugging it back into the original to see whether it produces a “true” answer (such as √(-3)2 = 3) or an “invalid” situation (such as √-9, which doesn’t equal any real number).
* The text excerpted above from The Official Guide to the GRE 2nd Edition is copyright ETS. The short excerpts are quoted under fair-use statutes for scholarly or journalistic work; use of these excerpts does not imply endorsement of this article by ETS.
I know that for people who’ve been away from math for a while, the GRE requires a lot of refreshment on topics and skills. Even for those of us who’ve been around math all along, there may be topics we haven’t seen since high school on the exam.
Most of getting good at GRE math is practicing your skills, learning to recognize clues and patterns on the exam, and knowing what material is being tested and how it is tested. One key step is knowing the definition of math words, because those definitions often come with important restrictions.
For example, when a question starts by specifying that x is an integer, that restriction will probably be a key to the problem. There is an infinite amount of numbers that are not integers, including fractions and radicals. It’s also important to remember that integers don’t have to be positive – there are negative integers, and zero is in integer as well.
My suggestion is that you clarify the definitions, but not simply memorize them. Let’s say that I realize knowing the definition of “integer” is important, so I decide to make a flashcard that says “integer” on one side and “a member of the set of whole numbers” on the other.
Great. That’s true, and if the test were going to ask me to define the word “integer”, that would be a great thing to know. But remember: for the most part, the quant section of the GRE is a skill test, not a knowledge test. It tests your ability to notice patterns and details, perform math tasks, plan an efficient road to a solution, and reason with numbers. So the definition of “integer” that I want to know is something that will help me.
I am not the biggest fan of flashcards for the quant portion of the GRE, but if I were going to make one for “integer”, I’d want to make sure the back of the card included:
• My own definition in my own words,
• Key trouble issues to watch out for, and
• How the concept tends to show up on the exam.
As I did additional problems, I might add information to the back of the card, so that eventually it would look something like this:
• not decimals or fractions
• Includes zero and negatives!
• When they say “non-negative integer”, think “positive OR zero”
• When they say “number,” think about fractions
• When the exponent is a positive integer, the value usually gets bigger. UNLESS that positive integer is one – value stays the same.
The key is that your definition should include all the things that tripped you up, written in your own language, and written in a way that tells you what to do, not what not to do. (Notice my card doesn’t say anything like, “don’t test only positive numbers”, because generally it’s much harder for us to remember directions given in the negative.) It’s less of a definition and more of a collection of key points that help you clarify how this topic is applied on the exam. In this way, you become a better issue-spotter and avoid common mistakes.
Thinking of definitions in this way can help you to realize their importance while also learning them in a way that’s directly applicable to the exam. The next paragraph is a big, long list of terms for which you might find a definition card useful. All these terms are covered in ETS’s math review for the GRE. You certainly don’t need to make definition cards for each of these words, but if you think it would help you, go for it!
You might find it helpful to make definition cards for the following terms: integer, even, odd, positive, negative, divisible, factor, multiple, greatest common factor, least common multiple, remainder, prime number, prime factor, composite number, zero, one, rational number, reciprocal, square root, terminating decimal, real number, less than, greater than, absolute value, ratio, proportion, percent, percent increase, percent decrease, domain, compound interest, slope, y-intercept, reflection, symmetric, x-intercept, parallel, perpendicular, line of symmetry, parabola, vertex, circle, stretched, shrunk, shifted, line segment, congruent, midpoint, bisect, perpendicular bisector, opposite angles, verticle angles, right angle, acute, obtuse, polygon, triangle, quadrilateral, pentagon, hexagon, octagon, regular polygon, perimeter, area, equilateral triangle, right triangle, hypotenuse, legs, square, rectangle, parallelogram, trapezoid, chord, circumference, radius, diameter, arc, measure of an arc, length of an arc, sector, tangent, point of tangency, inscribed, circumscribed, rectangular solid, face, cube, volume, surface area, circular cylinder, lateral surface, axis, right circular cylinder, frequency, count, frequency distribution, relative frequency, relative frequency distribution, univariate, bivariate, central tendency, mean, median, mode, weighted mean, quartiles, percentiles, dispersion, range, outliers, interquartile range, standard deviation, sample standard deviation, population standard deviation, standardization, finite set, infinite set, nonempty set, empty set, subset, list, intersection, union, disjoint, mutually exclusive, universal set, factorial, probability, permutation, combination, and normal distribution.