Davis: Hey everybody, this is GRE Bites. My name is Davis, and I’m an educator with over ten years of experience.
Orion: And I’m Orion, the founder of StellarGRE.
Davis: We’re here to bring you your weekly bite-sized episode on GRE prep and grad school admissions. Check out our top-rated GRE self-study program at stellargre.com. And don’t forget, you can use the code “BITES” for 10% off any membership.
Okay, so we’ve talked about different sections, and one is the quant section, which often gives people a lot of anxiety. We’ve talked about different tools. I’m interested in today specifically to pick your brain about when you have the type of quant question that gives you variables. And rather than getting an analytic solution, where you’re solving the equations for a general solution, it’s much simpler to just plug in a few different numbers and hone in on one of the multiple-choice answers quickly. In that kind of situation, what are the go-to numbers when you’re just plug-and-play, and you want to get…?
Orion: It’s a great question. Let’s start with the first thing that you have alluded to. But I want to make sure that everybody understands what you’re talking about, because some people might not. So, there’s a lot of algebra on the GRE. And in the stellar system, we try to forego 99% of all algebra.
Davis: Why is that?
Orion: Because algebra, in most cases, is abstract. It’s kind of infinite, because an algebraic variable can represent, in many cases, an infinite number of values, which doesn’t give students great odds. If you try to solve questions algebraically, you kind of have to derive the one correct value out of the infinity of incorrect values in order to get that point. And the only way you can do that is by understanding in depth the equations that are in play in the problem, and then solving it algebraically to find that specific value.
And the other reason why I don’t like the algebraic solutions, in general, is that I’ve yet to encounter a person who is more accurate and more efficient with algebra than with arithmetic. Even the biggest math hotshots that I have tutored in my time, and I’ve tutored several of them who knew way more about math than me. Some of them were committed to doing it the hard way. They still couldn’t beat me on the GRE quant section on some level, because over a long enough timeline, over four hours, over hundreds of questions, they’re just slightly more likely to make a careless mistake on the level of abstraction using algebra than the level of concreteness using arithmetic.
Davis: And this points out something that has been a source of frustration for me, but it’s understandable in the context of the GRE test, of standardized tests in general, which is the test is not after your innate or intimate knowledge of the math, no, at the core of a question, or it’s after your ability to recognize a problem, solve it quickly and efficiently under time, in the scope of the larger test.
Orion: Yeah, it’s not really a math test. Like if you get to a certain level, it’s like Neo at the end of the Matrix; you can see the code, you can kind of see through it. Math is the means by which they are attempting to assess an aptitude, which is a general ability, simply, but you know, they have to do it in something. They can’t just scan you for general ability; what does that look like?
So, on some level, they use words, and that’s the verbal section; on some level, they use quantitative concepts and numbers, that’s the other section. So, but those are just like a means to a greater end. Now, how successful the GRE is, in assessing that general ability, is another question. And in many respects, we know that there are some significant failings in that assessment, but it’s not a completely terrible tool.
So, back to the question at hand, which is, we understand the GRE is not testing a person’s ability to understand and recognize the math at the core of a question. Because, at the end of the day, it’s just whether you got the question right or wrong, and how much time it took you. It’s not even about that. It’s like you can do the question right and hit the wrong button. Sometimes what I tell students is the GRE is a button-pressing task. And the only thing that matters, these days, is you push the button that gets the credit.
And so even if you’re a math whiz, there are certain strategies that can help you be much more efficient. Sure, use less mental resources and therefore be less prone to hitting the wrong button. Keep your field clear so that you’re just honing in on the right choice with the mental capacity to still just click the right button and move on to the next question.
So, one of these is, instead of solving it algebraically and understanding the math at the core, you can solve it arithmetically, with arithmetic, to hone in on a multiple-choice answer, or even the multiple answers too. So that’s absolutely correct.
So, rather than do the algebra, we’re going to do arithmetic instead. And we can do that by plugging in our own values. And plugging in is like the staple technique of most test prep systems. So, I don’t think I invented it; I don’t know if anybody invented it. But it’s the bread and butter. It’s the most useful quantitative technique on standardized tests because you have a calculator.
And it’s the most useful technique because about a fifth to a quarter of the questions on any given set will be amenable to this technique. We know that they’re going to be amenable at a glance; in the solar system, we call it structural diagnosis, where you try to figure out the way that a specific question is being presented. And the easiest way to figure that out is to look at the answer choices. And frankly, if there are variables, letters, for the most part, in the answer choices, you know that you can plug in, funny and it works because remember, an algebraic variable represents every value in a specific domain of numbers. And so if that variable represents every value, then any value that we choose will work, because any is always contained in every. So we don’t have to really worry about it so much. So by plugging in concrete values, we can transform abstract algebra into concrete arithmetic, and then just generally add, subtract, multiply, and divide our way to an answer.
Davis: I’m imagining it serves a lot more than to pick really simple numbers that may make the arithmetic really simple and show us a direction to choose the next number if we’re going to plug in more than one. We don’t always do this, right? So, what are those numbers?
Orion: You’re absolutely correct. So, if we’re only plugging in once, or for our first round of plugging in, as indicated, we want to use the nicest, easiest numbers that there are, because the test is hard enough. Why are we going to go out of our way to make it harder on ourselves? So, the nicest, easiest number that there is, is the number two. I love the number two.
Davis: So, real quick here, why avoid zero or one? Is it because they’re weird.
Orion: Okay, there we go. Zero and one are very important numbers for us. But they’re important because they do such unusual, unique, and unexpected things in arithmetic. And so, they’re unlike really any other numbers that exist.
So, we’ll get to that in a second. But two is the best one to use. Because it’s small, it’s even, we use it all the time. It’s a relatively low magnitude number; calculations can go fast and accurately. So, if you can use small positive integers, 2, 3, 4, etc., if you have multiple variables. Now, if you have variables in the answer choices, but it’s a choose one problem, which they call a problem-solving, so there are five options, only one of them is correct, then a lot of times you can get away with just plugging in once. You plug in numbers, like two or three for the variables into the question, you solve for that, and then you realize that the answer choices are really just that target number in disguise. You can plug the numbers into the answer choices to transform them into values. And then, you choose the one that matches up with the number that you’re looking for. More or less, right?
Davis: Yeah, no, that makes sense to me. And if anyone wants to understand this more in-depth, it’s best to get your hands on some practice questions, and sign up with GRE to understand how this works in principle, but if you already know what we’re talking about with StellarGRE.
Orion: Yeah, I mean, the great thing about Stellar is that it’s a very, very, very regimented system. It’s like there is a protocol for every question on the entire test. And if you recognize what kind of question you’re dealing with, and you just activate the protocol, and you kind of do what you’ve rehearsed, it’s going to make you feel, you’re going to see all the code behind something like that. I mean, a lot of the test is about realizing how the test is attempting to trick you, misguide you.
Orion: And if you can, just, if those attempts become invisible to you, through your training and your rehearsal, so much the better, right?
Davis: Okay, so back to plugging it. So you plugged in the first two, for example, two, three, or four? If it’s a single answer, you match it up, boom, you got your answer. You’re done.
Orion: Yeah, but sometimes there are questions with variables, the answer choices that are “choose many.” These are multiple answers, which are kind of “choose all that apply,” or in my system, quantitative comparisons. Technically, you’re only choosing one, A, B, C, or D for quantitative comparison, but I’d like to think of quantitative comparisons like old-timey scales. So the pans can go up and down. This one could be heavier, this one could be lighter, or they could be the same; it could be more than one thing. So in my system, quantitative comparison questions are considered multiple answers. Anyway, for these questions, it’s imperative, always, always, always, to plug in twice.
Davis: Why is that?
Orion: Well, if we plug in once and we get an answer, we haven’t really proved that it has to be that value. That has to be correct. We’ve more or less demonstrated that it could be that it could be this, it could be true that A is the correct answer, or it could be true that quantity B is larger, but we haven’t really conclusively demonstrated that it is true. So if possible, we want to stress the system, we have a tentative hypothesis. But if we keep plugging in nice, easy numbers, 3, 4, 5, we’re unlikely to get a different result. And so, they’re too similar in nature.
So for the second round of plugging in, it’s really important to try to stress the system as much as possible. Because if we stress the system, and it breaks, great, we know what the answer is, the answer is D on a quantitative comparison question. And if we stress the system as hard as we can, and we still get a consistent result, that’s actually still not proof, but much more convincing evidence that we hit upon the correct solution. So not every number is going to stress the system; we want to use numbers that are weirder than other numbers because they’re more likely to provoke something that is unexpected or different. Does that make sense?
Davis: Give me the weirdest number you’re going to use.
Orion: Yeah, so we have a list of five I’ve rank-ordered them. And one is more than 213, et cetera. So we can just kind of go down this list in a mechanical fashion when we’re plugging in for our second round. And you’ve already mentioned two of them.
So the weirdest number of all is zero. That is a freaky number, dude, it’s the valueless value, it kind of doesn’t really make sense, breaks math in a lot of cases and a lot, a lot of the identity properties dividing by zero, a calculator will blow up. Anything times zero is itself; there’s a lot of weird exponents and arithmetic things involving exceptions that involve zero, right, so if you can use zero always use zero because it kind of destroys math, and it’s very easy to use. Number two is number one. Also, because of its identity properties, that’s weird and annoying for number one to be number two, but then after zero and one, it’s numbers between zero and one, which sometimes people call fractions or decimals.
But as a math nerd, I’m here to say that that’s technically not accurate, like the number two is a fraction, two over one, the number two is a decimal, it’s 2.0. And those aren’t weird numbers; I decided that two is actually the nicest number that there is. So numbers between zero and one are often expressed as fractions or decimals. But it’s not that expression that makes them weird. It’s the fact they live between zero and one on the number line. It’s like a haunted neighborhood. They’re good numbers, but that just makes them do weird, spooky stuff, right?
Mostly with exponents and radicals and things like that, they move in the opposite direction of all other positive numbers. Number four are negatives, mostly because people forget about them. We like to use positives because we deal with positive quantities in real life. And we’re more fluent with them in our calculations. And number five is what I call pushing the extremes, which means either really, really big or really, really small, depending on the context of the problem. And this one is number five, because even if you can’t use the other four, you can always, always, always push the extremes, because big and small are relative terms that are dictated by the question itself.
Davis: So there’s, even if it’s like numbers between zero and one, well, a really, really big number could be 0.99, and a really, really small number could be 0.01. So you can always find something that’s big or small in a given domain.
Orion: That’s right. So those are the five again: 01 numbers between zero and one, negatives, and pushing the extremes. You can go down that list in a rank-ordered fashion, somewhat algorithmically, at least in the beginning, until you get a bit more advanced. There are some advanced considerations, but they don’t get you through like 97% of the time for that quarter of the quantitative questions.
Davis: You can use plug and play.
Orion: Yeah, you can’t plug in on questions that don’t have variables. But think about it. This one technique is applicable to up to a quarter of the problems.
Davis: Yes, that’s a lot of bang for your buck.
Thanks, everybody, for tuning in. We’ll be back next week with another bite-sized episode of GRE Bites. If you have a topic you’d like discussed on a future episode, let us know at email@example.com. And if you’re ready to take your prep to the next level, check out our top-rated GRE self-study program at stellargre.com. You can use the code “BITES” for 10% off all memberships there. Talk to you soon.