In this GRE Math Tutorial, we show how to best study for the GRE Exam.
GRE Math Tutorial [Video Transcript]
Jim: Hi there and welcome to Grockit.com. This is the GRE video, beginning of the video series on the GRE Quantitative tests, the math side of the regular GRE exam. My name is name is Jim Jacobson. Did I say that already? I’ve been doing test prep for about ten years now, helping people get better scores, and showing them tips, tricks, strategies. And so that’s what I’m gonna be doing with you today. Hopefully, no. Not hopefully. With practice–and there will be practice, there is homework for you to do after this lesson–with practice, you can incorporate these strategies and this information into your test-taking approach and get a higher score. You’re having a better opportunity to show what you really know and what you’re capable of to graduate schools, so that when you apply they’ll look at your score and say thumbs up.
Alright, so let’s get started. So today, we’re gonna talk about the intro to the quantitative strategy. It’s kind of an intro to the quantitative section overall. As well as talk about number properties, one of the key elements to…kind of basis of the quantitative section. We do get to bigger topics later; you can’t avoid doing algebra and stuff like that. But we do need to talk about number properties first because that underlies all of these things. So in case you didn’t believe me, so like I said today, intro to the quantitative strategies and number properties. Then we start getting into word problems, which can be anything from geometry, to algebra, to statistics, to whatever. Equations and coordinate geometry, the types of things that you are likely to see most the often. We gradually move into things that you’ll see in smaller percentages on the test, but you really are still expected to know. So we’ll cover averages and percentages, rates, inequalities, absolute value. Two lessons on geometry because for most people, it’s actually been a while since they’ve done any geometry. If that’s not you, I guess you don’t have to listen to my videos. I mean, it’s up you. We may still be able to give you useful information. But for a lot of people, geometry was like sophomore year in high school. So if you haven’t done this stuff in a while, it may take a bit to shake the rust off, which is why we’re covering it in more detail.
Also, standardized tests love the geometry stuff, because it’s very discrete, with an ETE set of information. And they can come up with a lot of creative problems using that same small set of rules and formulas that you can be expected to know on test day. And then we’ll cover the data analysis and statistics and probability. Data analysis, of course, is not escapable. Statistics and probability are higher end questions, so it is possible you wouldn’t see one. But I’m very certain that you’re interested in a good score; why would you be watching this if you weren’t interested in a good score? We need to cover that topic. So without further ado, let’s get into those quantitative strategies.
So four question types on the test, and some of these are newer than others: multiple choice, select one answer. Since the dawn of time, mankind has been answering multiple choice questions. And these are like those ones that you’ve seen before. Multiple choice where you have to select one or more answer are a little bit different. You may not have done one like this before. In this case, one answer may be correct, but it may also be more than one and you have to select, they may ask you to select three or they may ask you to select all that apply. So be aware that it won’t just be a one…choose one of five, that there may be a more complicated approach that’s required.
Numeric entry is new, requires you to actually to enter a number. That’s the name: numeric entry. And these ones can ask you for more complicated decimals, things like that, because you do have to make use of a calculator in the exam. Finally qualitative comparisons, and we’ll show you what these look like. These are comparing two quantities, hence the name, and you’re trying to decide which one is greater. They will usually not just give you a number like which is greater, two or three. There will give you something algebraic or some element of geometry, some problem that you have to solve in order to solve that one. And there’re two ways these questions can appear. One is that they appear absolutely as an individual question. The other is that it will appear as one of a series of questions on data interpretation where you’re given a body of information and then asked to answer several questions on it. It’s rather like a math version of reading comprehension in that way, that you’ll have a set of stuff…I’ll show you what that looks like too. Let’s take a look at some samples, because that’s what I said we were going to do.
So here’s a sample question of a multiple choice with one answer. So we have if C + A = 20, and B – C = 8, then A + B = what? And there’re several different strategies that you can adopt, and we’ll talk about those later. But in this case, well in any case of a question like this, only one one answer is correct. You can use process of elimination or plugging things back into the question, but only one of these is right. If you’re able to eliminate for example two of the three by some other means, you get your odds of guessing down to 50%. A 50% chance of getting it right even if you don’t which of these two it is. And this is a problem from the Grockit vaults, so you may see the problem again, in your practice. Okay, so when you have those have those single answer questions, where you only have to select one answer, one right answer, we have a couple of strategies that we want to give you. One is plugging numbers in. So of course, your number one strategy, your number one best way of solving any problem is to just know how to do it. Know the answer. So if you know how to do something algebraically, it’s almost always going to be faster than either of these strategies that we’re giving you here. Not always, but usually. And so if you can do it the “right” way, that’s great. If you can’t do that, getting the right answer is still the way to go. So we have a couple of other strategies that can help you.
First one, plugging things in. So if you have word problems with variables, and then numbers in the answer choices, in particular, that’s a good candidate for taking for taking things and plugging them back into the original problem. I’ll show you how that works. So we’ll cover algebra in a couple of sessions. But plugging things in, we would just try an answer; we wanna start with either B or D. And I’ll show you why. Let’s start with answer choice B here. No, let’s start with D. No, we’ll start with B. Okay, anyway it doesn’t matter; you won’t always know which one is good. We’ll start with B. So you just plug this in. So one-third of one plus…and we take this value in answer choice B, plug it in for X. And we set it as equal even though we know that it may well not be. What we’re doing here is trying to zero in the right answer. So we figure out, well, one-third of one and two-thirds. Three, four, five. So this is five-thirds. So one-thirds times five-thirds…well one-third of five-thirds is not three. It’s actually five-ninths, which is a little bit less, or excuse me, a little bit more than one-half. So it’s not even close to three.
So the reason why you start with B or D is because if you can tell, and you won’t always be able to, but if you can tell whether the answer that you have is too big or too small, you can go in either direction. If you know for a fact that your answer was too big, and you started with B, A has to be the right answer. So that’s really great. If it’s not, then you know you have to go in the other direction. You make D your second one. If D is the right answer, you’re in good shape. If not, if you need to go lower, it has to be C, and you if you have to go higher, it has to be E. So it’s pretty great. There will be some problems where because of the way the problem is worded, you won’t always be able to tell whether your number is too high or too low. And in most cases, you’ll probably have to try all the answer choices until you get the right one. That’s the price you pay. And that’s why I said that knowing the answer is still the best approach. But if you can’t, plugging things in is pretty great. Just to show you the rest, we tried D here. So one-third of one plus three equals three, so one-third of four equals three.
No, actually it does not equal three, it equals three-fourths. Oh, excuse me, four-thirds. Four-thirds is one and one-third, which is still less than half of the number that we’re after, which is three. So we have to go still higher. It has to be E. But just for the sake of illustration, I will show you how that would work. So one-third of one plus eight, using this value in here, equals three. Well sure enough, this ends up being one-third of nine. One-third of nine is three. So that’s how that works. It can get you a lot of points on test day. So just practice this strategy. Whenever you’re feeling a little bit lost and you have numbers in the answers choices, try plugging them back into the original question. I mean, it can’t hurt, right? It may get you more points.
So the other strategy here is using numbers. You don’t have to do every problem algebraically. Sometimes you can actually just ‘magic-up’ some numbers that work for the problem. And sometimes there’s really good ones to choose where it really is easier to do it this way. So an example here, when you have a percentage problem in particular, or variables in the answer choices and in the original problem, Picking numbers for them can make a big difference. With percentages, your buddy, your best friend ever, your BFF number is 100, okay? Because the first increase or decrease on 100 if you have to do a 50% increase on 100, then your number is 150. That’s why you’ll start with 100, because the first increase is done for you basically. So our example here, Jessica’s sales increased by 20% from 1997 to ’98, and by 15% from 1998 to 1999. By what total percent did her sales increase from 1997 to 1999? Now the trick here, of course, is that her sales went up, and then they went up on top of that. So the amount of increase in the sales from the 2 time periods, we can’t just add 20 and 15, because the 15% increase is done after that first 20% increase has happened, it’s not on the original number. So we can’t just add these 2 together and say 35. This is the one answer we know it isn’t.
And in fact, if we wanted to magic our way, logic our way the rest of the way, we could say well if it increased 20%, and then increased 15% on top of that, it can’t be this and it can’t be this, because it went 20% in the first year. Since both of these were positive increases and this one was the simple sum, it has to this one. Now let’s just say it wasn’t that obvious. Let’s say we just had to actually Pick a number. We Pick 100. So in 1997, she’s at 100, and 1998 she has a 20% increase, she’s now at 120. See how easy that is? You start with 100 and whatever the percent increase is, you just add that on to 100, and that’s the number. If it’s negative, you subtract it. The trick of course, well it’s not really a trick, but the problem is we actually have to a little bit of math because then from 1998 to 1999, she went up 15%. Now I’m lazy about math, to tell you the truth. I mean, yes you have a calculator. Sometimes it’s faster to do things in your head though than to actually sit there like punch, punch, punch, punch, punch, punch, punch, punching a bunch numbers in. And in this case, it’s easier to say well since it’s a 15% increase, 10% of 120, you can just move the decimal point over 1. That’s how I figure out the tip in a restaurant. I know, go me.
So you move the decimal point over one. 10% of 120 is 12. 15% will be another half of 12, which is…so 15% of 120 is 12. The additional 5%…sorry, 10% is 12. The additional 5% is 6, for a total of 18. We add 18 onto 120, and we’re at 138 in 1999. 138 is a 38% increase over 100. Again, the problem could be much more complicated than that, and 100 could still be the right answer because it makes the math that much easier. So with one-answer strategies, you know, you need to be aware of the easy answers especially in difficult sections. As you’ve been getting a bunch right, the questions get harder. So the further along you’ve gone, the more likely it is that that easy looking thing is probably not the right one. Also beware of not answering the questions asked. So the easy answer in the one we just looked at was that 35%. Or 20 or 15, which were also just numbers right from the problem. Also beware of not answering the questions asked, that could be a problem too. So the figure to the left shows two concentric circles. The nine-inch diameter of this guy is equal to the nine-inch radius of this guy. And so as you may remember, and as we will cover when we get to that geometry, the area of a circle is Pi times the radius squared.
So to find out the area of the donut without the whole, we subtract the whole circle, the area of this whole thing, minus the area of the donut hole that will equal the area of the donut itself. So it’s actually gonna be the area of one circle minus the area of the other circle. We subtract the little one from the big one. The big one is going to be Pi times its radius squared. So Pi times nine squared. This one is going to have a radius of 4.5. I’ll put that in blue so you can read it. The radius is gonna be 4.5 here because the diameter is 9. So Pi times 4.5 squared. Anyway, we go the rest of the way through this one. 4.5…notice all these are in fractions. So we probably want that as nine-halves.
So let’s put that down here. So we have 81 Pi minus 81-fourths Pi. And if we convert 81 into fourths, so here’s where you could use your calculator. You can either subtract it and then convert it into fourth, or convert it into fourths and then subtract it. We’ll get into the geometry later, and so I don’t really want to get into this in too much detail now. But you end up with 81 Pi minus 20 and a quarter Pi. You end up with 60 and three-quarters Pi, which isn’t in your answer choices, but 243 over 4 Pi, which is what this is, is in the answer choices. Note that we have the nine-halves, which is what the radius was, but that’s only one of the circles. 243 over 4 is almost the right answer but it’s missing Pi. So you can see how they’ll give you answer choices that are suspiciously like what you need. So if you’re careless, if you forget to include Pi in a circle question for example, they may get you with one of the answer choices without Pi. So be careful, don’t be that person.
So this is the one that I alluded to earlier, the type of question that’s new on the GRE, where it’s multiple choice where one or more answers will actually be correct. And they’ll specify whether you have to select a certain number or whether it’s open-ended. So for example, here’s one from our Grockit database: which of the 3 following integers have a product that is less than -85? So it will actually be three of the answer choices, three of the six in this case. So selecting only two or more than two will not be sufficient. There is only one correct answer, even though we need to select three of the numbers. They won’t always specify the number; it may be ‘select all that apply’ as you may have done in surveys. So pay very close attention to the format that…because otherwise you’ll get it wrong, and you don’t want to do that. So your strategy with the one or more answers, definitely understand how many answer choices you have been asked to fill in. Write it down if you need to, keep it straight, chant it to yourself. No, don’t chant it to yourself. Stroke your chin the number of times equal to the number of answers you have to select. If it’s open ended, you just keep stroking your chin the whole time. I’m joking of course. Realistically though, it could be…because it’s the type of careless mistake that someone might make under pressure, they are counting on some people making it. So don’t be that person.
Statistically, significant number of people have to make a mistake on all sorts of problems on the test in order for it to be useful to college admission. So don’t be the one making that mistake. So make sure you do that right. When you’re eliminating, find the upper and the lower bounds. So some types of questions will give you a range of numbers, in particular inequalities and absolute values. You can use that to your advantage. You’ll be able to tell sometimes that certain answer choices are just not going to be possible. So if you can tell from the problem that the biggest answer can be eight, then the answer that says…the answer choice that’s nine or ten is automatically wrong without you even having to test it. So look for little tricks like that. And then again, recognising that overall predictability of math. The rules are pretty set. They’re not inventing a new…the style doesn’t change for how we compute the area of a circle. The equation of a line also doesn’t change. So getting comfortable with the patterns that develop in sequences and sets and formulas can really make your life easier.
If you’re the sort of person who gets anxious about math, doing a lot of practice can help alleviate that anxiety, especially if you do targeted practice at your actual level of ability, as opposed to shooting too high and getting frustrated. So getting used to the predictability of math can really make your life better, or at least your GRE testing life better. So numeric entry was the other one that I mentioned, the kind where you actually have to enter in the exact number. And you can have two different formats in this case. Both of these are from the Grockit database. So the first one, an insect colony doubles in population every ten years. So it doubles every ten. Its population is reduced by 20% every 7 years. After 50 years, the population of this colony is how many times it’s initial population, rounded to the nearest integer? So note that we’ve been given some specific instructions about how to round, and then we have to enter in a particular answer. So even if the real answer is 4.7, we need to round that to 5. If the real answer is 10.3, we need to round that to 10, nearest integer.
This is to be contrasted with the kind that we have down here. Charlie and Lucy are painting an office. Working alone, Charlie can paint the room in six hours and Lucy can finish the job in four hours. How long will it take them to finish, in hours, if they work together at their respective individual rates? Express your answer as an improper fraction. So in this particular case, it needs to be an improper fraction which would mean the numerator is larger than the denominator. So there’ll be a big number up here and a small number down here. What you can’t do is put a decimal in one these two slots. You can’t put three-fourths or four-thirds up in the numerator and leave this blank. You going to have to put the four at the top and the three at the bottom. So that’s what those’ll look like, and they can appear in a variety of question types. So with numeric entry questions, absolutely…and make sure you’re answering the right question, because unlike multiple choice answer choice ones, there’s no kind of guidance there. With multiple choice, if your answer is way different from the multiple answer choices, you know you did it wrong. Since you’re just entering in a number, you don’t have that guidance from the answers. So you really need to be really sure you’re answering what you were asked.
The format that the answer is in matters, whether they ask you for a decimal, an integer, or a fraction. Definitely round your answer to the nearest figure that you’re asked. Remember that with decimals it’s tenths, hundredths, thousandths, ten thousandths. And then powers of ten to the right of the decimal point, even though it’s units and then tens and then hundreds on the left. Rounding, you do not round until the end. So if the question asks you to round your answer, you do not round until the final answer. You leave all of the other answers, messy decimals and fractions and stuff, you leave those unrounded until the end. On multiple choice ones, you can round along the way if the answer choices are far apart. Because the answer choices are far apart and whichever one you’re closest to is likely to be the right answer. On numeric entry ones, do not round until the end, if even then. Sometimes they will still insist that you put a decimal in.
And then finally, be familiar with how you’re asked to do this, you have to put in each of the blanks separately. And this is particularly important because their little calculator that they have has a transfer button on it. So you can take a number, like a whole decimal that appears in your calculator, and transfer it to a blank. But that’s really only gonna work if you put it in the right blank. And if you have one like the one we had before where there’s two blanks, that transfer thing is going to be trouble. So just be warned, practice it in a place that gives you online tools similar to the one on the test just make sure you get it right.
So then quantitative comparisons is the fourth of our major question types. You may have seen this in previous versions of the GRE if you’ve taken it before. Basically, you have two quantities and you are asked to compare them. You’ll get column A and column B. The answers are the same every time. The right answer is not the same every time. It’s not like it’s not like it’s C every time. But the answer choices that A is, the quantity in column A is always choice A. B is greater is always choice C. The fact that they are the same is C and if you can’t tell from the information given is always D. So you’ll have this memorised if you’ve done this right, because you’ll have done enough practice problems that you’re like, yeah. You’re saying the quantitative answer choices in your sleep, or quantitative comparison answer choices anyway. So this one has some common information in the middle and then that applies to both columns. And we have to do a little figuring before that. Sometimes there will be no common information and you just have to compute something that’s in column A and then compute something that’s in column B and figure out which is greater.
So we’ll cover more information on that later. But just know that it can vary a little bit in format. So your strategy overall, like I said, the answer choices are always the same. You don’t need to do all the math, you just need to do enough that you can actually compare the two columns. And this is an important distinction to make. Sometimes you will actually have to do all the math. But if you already know that column A is bigger, don’t finish the computation the rest of the way. That’s time that you can devote to getting another question right elsewhere on the exam. So that could really make your life easier. So just recognize when it’s time to stop. And for shapes, note if the figures are drawn to scale. If they are drawn to scale, you can use that to your advantage by eyeballing some of the figures, and try and figure out which one is plausibly bigger. Now they’re usually not gonna make something super obvious visually. But you can at least determine whether your answer choices are plausible based on the information you’re given. If the figures aren’t drawn to scale, just pretend that you know the figure is a liar in some way because it might be. So if it’s not drawn to scale, don’t use it for anything other than telling that it’s…you can only use the information that you’re definitely given. We’ll talk about that more later.
And then if you’re unsure, test with the things that break given statements. So negatives, positives, fractions, zero, and two, if it’s a question about primes, are all things that can kind of be exceptions to the rules. And if you’re trying to determine whether column A or column B is always greater than the other one, testing the expectations helps you make sure. There could be cases where because of these exceptions, the answer choice will be D, that it cannot be determined. So sample data interpretation is that final, it’s not really a question type, it’s more of a question format. Because you’re asked the same types of questions where you have multiple choice, answer one, like this. Or it will be numeric entry which we’ve already talked about. In this case though, you’re given the information on the graph. And you’ll note that there’s all sorts of stuff; we have different bars for males and females and different departments and these are the percents of the total numbers here. And then we’re given additional information in the problems themselves. So there’s a lot to comprehend and it’s in your best interest to get a sense of what’s on there first before you start tackling questions. So your strategy is of course, number one, get a broad feel for the graph and chart, try to look at the axis. What is it actually saying? What are the trends, if any?
It’s kind of like reading comprehension, I think I mentioned that already, in that you have to do a little bit of an initial time investment here where you have to kind of understand what’s going on before you can tackle those questions. So titles, units, labels, captions, all that stuff, they could test you on anything they give you. You can eyeball graphs; they are drawn to scale. So you can tell whether the answer is just totally implausible sometimes, and then you can just eliminate it. Or you can tell whether the answer that you got is wrong, and then it tells you to either start over or guess and move on depending on you’re doing on time. And then finally, all the information you need in front of you is going to be in front of you. The previous ones about doctors working in different specialties, you don’t have to be a doctor, in fact, if you’re studying for the GRE. You probably aren’t one. But if you don’t need to know anything about the topic, everything you need will be given to you.
And so what you’re expected to do is a variety of things. Sometimes you’ll be asked to just interpret what’s on the graph. What were division three’s sales in the first quarter? You have to correctly identify division three out of multiple divisions, presumably, and correctly identify the third quarter ones. People can mess this up; it’s easy to make a careless mistake, especially when you have a lot of information that’s close together on a graph, like a lot of bars that are tiny and small and next to each other. Sometimes you’ll be asked to manipulate the information on the graph. So you may be asked to do something about division three’s sales in the third quarter. Like compare it, or is half of division three’s sales in the third quarter less than a quarter of division four’s is in the fourth? You know we’ll have to do something with it. Sometimes there will be approximation and rounding. We’ll look for words like approximately to be your clues. This is actually true of the quantitative test in general. Just know that sometimes you aren’t going to be dealing with the exact numbers.
And finally, like reading comprehension, I’ve made this connection a couple of times now, sometimes the answer comes from more than one place within the information that you’re given. So just like the answer to a reading comp might be in more than one place in the passage, like part of the answer might be in paragraph one, part of it might be in paragraph four, with these guys, part of it might be in the graph and part of it might in the chart or a table that came with it. So just be prepared to not have everything in one place that you’ll have make connections, and transfer information from one part of your data set to another.
So now we’re getting into arithmetic, those building blocks upon which algebra, geometry, and other fun stuff will be built in the course of this video series. Factors, multiples, and divisors. First off some basic terminology we need to cover. And integer is a whole number. So positive and negative whole numbers. So numbers like…actually we’ll bring that up there. Examples include negative three, two, zero. Yeah, so the markings on the number line on either side of zero and including zero. Not examples would be things that are not whole numbers. Negative 3.5, two and one-fourth, the square root of two, things like that. Not-integers. So guys like this, not like this. A prime number is a divisible but only by one and itself. So that’s the dictionary definition. So they’re always positive, they’re always whole natural numbers. So they look like integers, but of course they’re never negative. So they have two factors, okay? One and itself. So a prime number, for example, would be three. The only two numbers, the only two different numbers that you can multiply together to get three are one and three, one times three equals three. That’s how you define a prime number.
So a number like six which you could do one and six, or two and three is not a prime because there’s four factors. Factors are the numbers that you can multiply together to get a bigger number. A number is always its own factor as is one, but primes are the ones that have only those two. So the important questions: are there any even prime numbers? Are there any numbers that are even that you can only divide by themselves and one? I’ll just answer that question for you, yes. And standardized tests love the number two. Remember that you can only multiply one times two to get two, in terms of two different factors. So definitely an even prime number. Is one prime? Now it you’ve been following along, our definition of prime is the number has to have two factors, two different factors. How many numbers can you multiply together to get one? Only one, which is one. One times one is one. So one is in its own special case; it’s called an identity. Or you can just call it not prime like the rest of all the other numbers. But basically, one has only one factor, which is itself. Is one prime? No. Is zero prime? No. Zero is prime for the opposite reason, since you can multiply any numbers together with zero to get zero. Zero is also not prime. You need have two and only two distinct factors for zero. Whereas zero could be zero and one or zero and eleventy billion.
Are there any negative prime numbers? No. Remember part of definition is that they’re positive. Even a number like negative three, you can multiply one times negative three equals negative three. Or negative one times three equals three. So if you want to even introduce the idea of being able to multiply other things together to give you negative three, it still has more than two factors. So no, there are no negative prime numbers, okay. So here’s an example of a number properties problem from the Grockit vaults: Which of the following integers cannot possibly be the sum of two prime numbers? Select all that apply. So this is a select one or more answer choice question. And so there’s going to be at least one, and it could be all of them, in theory. So we already established from the previous [inaudible 00:31:18] that a negative number can’t be prime. And the only way you can get a negative number by adding things together is if you’re adding together negative numbers at least once. So like you could have four plus negative seven equals negative three. But in that case, this can’t be prime and this isn’t prime. We can’t even do one plus…or two plus negative five. Because even though five is prime, negative five is not. So, it can’t be this one. This one cannot be the sum of two prime numbers. So that’s one of our correct answers.
What about three? What numbers can we add together to get three? We can add one plus two, but remember, one is not prime. We can add three plus zero, but remember zero is not prime. So this actually is another one of our options. So we can add one point five and one point five, but those are also not prime because primes don’t have decimals or fractions. So three is also a number that cannot be the sum of two prime numbers, and therefore it’s one of our answers. What about eight? Well eight, it could be three plus five. Three and five are both prime numbers. It may help you when you have a prime question if you’re not as comfortable with primes to memorize the first few prime numbers and then write them down. So the first few prime numbers, two, three, five seven, not nine because nine could also be three times three. If it were only one times nine it would be prime. But since three times three is another way to get nine, it’s not prime. However 11 is prime, 13 is prime, 15 isn’t. All the rest of these are odd except for the first one. But 13 is, 15 isn’t, 17 is and so is 19. Anyway, you could…there’s more of them. In fact nobody even knows how many there are. In theory, they’re infinite. But knowing these first few, especially the ones up to 10 or 11 really make your life easier. So eight could be three plus five, two numbers on our list.
D, can we get two numbers on our list to be nine? Well seven plus two, that’s nine. Seven plus two equals nine, three plus five equals eight. So since it can be the sum of two prime numbers, it can’t be the right answer. We need the ones that cannot possibly…we could do it so that there’s two not prime numbers, one and seven. But it still could be this other one, sorry, like five and four. That would be a another one. Anyway, not C, not D. 18, well we could do 5 and 13. Two more prime numbers. And since we were able to come up with a way for two primes, it’s also not the right answer, leaving us only with A and B.
Some other number properties to talk about. Negative times negative is always positive. Just remember it. It always is. Positive times positive is also always positive. It’s only when you have a negative times a positive that you end up with a negative number. So memorize these if you have any spare brain space. These are absolutely mission critical to getting so many questions right on the task, on the quantitative side anyway. So negative times negative, positive times positive be can be positive. Only negative times positive gives you negative. So putting that into practice, so which of the three–and we actually did this. We had this as an example of a choose one or more answer choice when we were doing our sample problems–which three of the following integers have a product that is less than negative 85? So remember, less than negative 85 means more negative. Like if you have negative $85, that’s more money than negative $185. They are both not good amounts of money to have. But negative 85 is greater than something that’s more negative. Anyway, so we’ve got a big negative number. How can we get a big negative number? And we have to do it through three numbers. So knowing what we just learned about positives and negatives, to get a negative number at the end out of three numbers, we either have to have positive times positive times negative, or we need to have negative times negative times negative.
Anything else will give us a positive number. If we do plus, plus, plus, that’s positive. Also if we do minus, minus, plus two negative numbers, these two, negative times negative is positive–and we could turn it into a smiley face, I guess–turns into a positive number. So it has to be one of these two. Now there’s only two positive numbers in our list. So let’s try this out first. This would be seven times two, and it will give me a negative number. We’re trying to make it as big as possible, we are trying to make the number part of it big because it has to be more negative than negative 85. So our biggest one is negative six. So 7 times negative 6 is negative 42, times 2 equals negative 84. Just a bit shy. Negative 84 is actually greater than negative 85, because it’s less negative. Anyway we need something like negative 86. So it’s not positive, positive, negative. The only other option is negative, negative, negative. We can get rid of these two answer choices just by doing that. And now again we still want as big a number as possible, so we choose all the three biggest ones, six, five and four. Negative six, negative five, negative four. Negative 5 times negative 4 becomes positive 20, but positive 20 times negative 6 becomes negative 120, which fits the bill of being less than negative 85. So it is these three.
Evens and odds; another set of things that really you do need to memorize. I mean, I’m sorry to tell you that, nobody likes to hear that they have to this in addition to learning words and grammar and stuff. But your ability to prepare for this test is one of the things that graduate schools are testing. So commit to memory the following: even times even is always even. Even times odd is also always even. It’s only odd times odd that always gives you a odd number. Whenever you’re not sure about this rules, for whatever reason, just pick numbers like two and three and test them out. So even times even would be two times two equals four. Yeah, that stayed even. Even times odd, two times three equals six. Stays even. Whereas three times three equals nine. It stayed odd. When you do even plus or minus even, so let’s say four plus or minus two, four plus two is six, four minus two is two. So even plus or minus even stays even. Even plus or minus odd. So let’s do four plus or minus three. So it’s either seven or one, both odd. So this one is odd. Odd plus or minus odd. Again, let’s do five plus or minus three, we get eight or two. So odd plus or minus odd is even. Again, whichever is easier, I guess. Either memorizing these rules or being very comfortable in just reinventing them, re-engineering them, rederiving them on the spot if you have a problem that involves these things.
So, but I think they are worth having…being very comfortable with no matter how you do it. Properties of zero. Is zero positive or negative? Neither. If you look at a number line, pretend that’s a line. One’s here, negative one’s here. All the negative numbers start at zero and move that way. All the positive numbers start at zero and go that way. But neither of those includes zero. Zero is neither positive or negative. It is neutral in all disputes of the cosmic struggle between positivity and negativity. Is it odd or even? Now this one it does actually have a value, because it is one of the intervals on the number line, so like here we would have negative two and here we have positive two. So this is an even number, an odd number. This must be an even number because we alternate between evens and odds on the number line, even, odd, even, odd. So because two and negative two are even, and then you have the odds before it, zero is considered an even number. Zero times any number is gonna stay zero, whereas zero plus any number will stay that number. So zero times X gives you zero, zero plus X gives you X.
Any number divided by zero is undefined. If you think about it, you can put an infinite number of zeroes into something. Because you’re putting nothing in; you can do that as much as you want and it doesn’t change the value. So if you have a savings account, with $100 in it, you can make $0 deposits all day, for the rest of your life, and you’ll still have $100. But it’s undefined how many times you could really do that. So maybe that doesn’t help you remember it. But just dividing by zero is naughty. Maybe it helps if I just write that. Dividing by zero is naughty. The GRE will not do that; it will go out of its way to avoid it. I’ll show you what I mean. Alright, so the order of operation, PEMDAS. If you can’t remember P-E-M-D-A-S, there’s a mnemonic device: Please Excuse My Dear Aunt Sally. You can come up with something funnier or naughtier if you want, as long as you remember it on test day and actually while you’re doing your homework. This represents the order that you should do things. If you don’t do operations, a complex thing like this guy here, if you don’t do it in this order, you will get a different number and that number will be wrong. So the GRE is expecting you to have this internalized, memorized on test day.
You do parenthesis, anything that’s in parenthesis you do first. Then you do exponents, then you do multiplication and division, in that order from left to right. Technically they can be done in any order, whatever that property is, communicative or something like that. But if you want to be consistent and make sure that you don’t miss things, from left to right in that order. Same thing with addition and subtraction. You can do them individually but it is best to do it from left to right just to keep yourself straight. So, do it in this order, otherwise bad things happen. So we would do the stuff in the parenthesis here first. So this just becomes five X quantity squared. Then we would do the exponents. 5X quantity squared becomes 25X squared. And then we would have 2 times 25X squared plus 6X minus 3, all over 3. Then we’d go from left to right through the multiplication. This then becomes 50X squared. And then we have like this factoring thing that we might have to deal with. Anyway, that’s how that works. Memorize that. Sorry to keep telling…and not everything that I’m going to be telling you is memorize stuff. But there are some basics that are considered fundamental to the rest of the exam.
Fractions, something that you may not have to deal with very often in your real life but totally show up on the GRE. So the main thing here is knowing that when you add or subtract fractions, the guys on the left of my divider down to the middle of the room, these guys need a common denominator. It is naughty just adding across. So three-fourths plus five-sixths does not equal eight-tenths. If you do that you will get every question wrong because that’s not how it works. You need a common denominator, some multiple of both four and six. When in doubt, you can just multiply four times six and turn this whole thing into twenty-fourths. But if you can choose a smaller number, that’s usually better. In this case 6 and 4 both have 12 as a multiple. So this is really nine-twelfths, because we’ll multiply the numerator and the denominator by three to turn twelfths. And on this one, we’ll multiply numerator and denominator by the same number two to get twelfths. So this is ten-twelfths. So that equals nineteen-twelfths. Sometimes it will be further reducible at that point but you can’t predict that. Nineteen-twelfths is very different from eight-tenths, and that’s how wrong you’ll be. So don’t do that.
Subtraction works the same way. So this will be, using same numbers, this will be nine-twelfths minus ten-twelfths, which would equal negative one-twelfth. Again you can’t just go across. This is to be contrasted with multiplication and division which do not require a common denominator. Multiplication, you do multiply across. You can just treat this as three times five over four times six. So that’s 15 over 24. It really is that simple. You just multiply across the numerator and denominator. Division on the other hand, you don’t necessarily do that. I mean you do do that. In fact you do it even more than you would think. If you don’t remember how to do divisions of fractions, you’ll take the second of the two that you’re dividing, you invert it, you flip it the other way around, and you multiply the two together. So three-fourths divided by five-sixths is the same thing as three-fourths times six-fifths, really. And then you just multiply across. 3 times 6 equals 18, 4 times 5 is 20. That’s how you do those.
Decimals, the main thing is keeping track of your decimal places. When in doubt, add dummies in. Well, dummies in the sense of like placeholders. So when you’re adding .7 and .35, you may want to make it .70 plus .35 to make sure that you keep the seven lined up with the three and not inadvertently make it into something like .45. It actually needs to be 1.05, rather than if you did .35 plus .7, you could get your .42 because you would be adding 7 to 35 incorrectly. So line them up, you may put in dummies to make sure. Subtraction, same thing. .70 minus .35 and…so it ends up being .35. So it’s the same thing with 70 minus 35 but then you have to make sure that the decimal points line up. Multiplication, you have several different options, but basically it doesn’t matter which one you put on top. You just have to make sure that you keep track of all the decimal places at the end. 7 times 5, 35. 21, 22, 23, 24. So you get 245 and you have to account for your 3 decimal places. One, two, three, it’s .245.
Division, you have a couple of ways you can do it. .7 divided by .35, you could actually represent it as a fraction. And then that might make life a little bit easier. You can then also multiply the numerator and the denominator by the same thing, multiply them both by 100 to move the decimal point to the right. And you end up with 700 over 35, and then the answer is 20. You can also just do it as long division. So .35 goes in…because everybody loves long division. You can also just punch it into your dang calculator. So sometimes it’s gonna be faster to do it in your head or on paper, but .7, you have to move the decimal over to for both of them, and it amounts to the same thing. So anyway, a lot of these are easier with the calculator, but sometimes you will also find yourself doing it out on paper because it’s faster than just typing them in. 2.3 falls exactly half way between which two numbers? So we’re gonna have 2.3 in the middle. And we have to decide, and it’s gonna have the same amount, plus and minus on both sides. So is it the same distance from negative two point three and two point three? Well this gonna be clearly impossible because it actually at two point three and that’s not the same distance from the negative two point three. So it’s not A.
And then we have negative two point three. Oh, and we have to determine here, this is negative four point six difference. So this is still negative four point six difference. On the other side here, we have four point six. Which might seem right but four point six is actually only two point three more than two point three. So we have to subtract four point six on the left, add two point three on the right, needs to be the same number. Negative one and three. So if we have negative one here to get from two point three to negative one, we need to subtract two point three and then subtract another one on top of that. So this is minus three point three. And to get to three, we just added point seven. So still not there. Negative point five and five point one. So to get here we subtract two point three, and then another point five on top that. So this is minus two point eight. To get here, it’s very interesting. Here you simply add two point eight. Two point eight plus two point three equals five point one. So let’s keep these separate. Clearly it is this one. Two point three is exactly halfway between negative point five and five point one, even though the two the two numbers don’t look anything alike. We had to correctly add and subtract decimals on both sides to make sure we got it.
Ratios and proportions. You basically can’t avoid these. Ratios are then the comparison of two numbers which will look like a fraction or a division problem. So four to five is equal to four to five. And that’s equal to written out four to five, because it could be written out like that in a word problem. Any of these are interchangeable. And sometimes you can set up, when you’re given a word problem involving ratios, you can take the words and convert them into a chart to help keep things a little bit straighter. So for example, if you have three sisters who split the cost of a plane ticket, if the flight costs $360 and Mary pays twice as much as both Sarah and Pauline, how much did Pauline pay? So we know that Mary paid twice as much as Sarah and Pauline. So Mary equals two S, Mary equals two P. And how much did Pauline pay? It might make it much easier…this also means that Sarah and Pauline paid the same amount. So we know then that what Sarah paid and what Pauline paid and what Mary paid, that equaled 360. But then we also know, we’re kind of setting this up as a chart here. So we really know that Sarah…well, we want do it in terms of Pauline because that’s what we’re doing. Sarah equals Pauline, Pauline equals Pauline. And we know that Mary equals two times Pauline, so that ends up equaling four P equals 360 and P, for Pauline’s value is 90.
So keeping the values straight, you may want to set up a chart initially which keeps things straight. And then if you try to condense things, have it condensed…I’m really unhappy with that P. There, that’s better. This may help you keep it straight in your head which things represent what. So a proportion as differentiated from a ratio is two ratios set equal to each other. So it would be something like one-half equals two-fourths. Now the reason you’ll get a proportion is usually because one of this numbers…one or more of this numbers is actually missing; it’s a variable instead of an actual constant. So you might get something like if Hal can answer four questions in 64 minutes, which we could represent as four per 64, four questions per 64 minutes, that would equal some other…I probably should make it a capital Q, it will be some other number of questions, X number of questions, in 12 minutes. So now we have two ratios that are equal to each other, but we need to figure out what this missing X is. And we can talk about how to do that later on. You actually cross multiply or just multiply both sides by 12 and then both sides by 64, and then divide by whatever that number is.
I don’t really need to talk you through it, because you will be doing more practice this later. But one of the things you can do is just multiply this times this…I’ll rewrite it here. So it’s really just 4 times 64 equals X over 12. So then you would say 48 equals 64X, divide both sides by 64 and there’s your answer. And you can do that with your calculator. Exponents and square roots. Another thing that you’ve probably haven’t dealt with for a very long time, but another one of the things that you really are going to have to commit to memory. I’m sorry, today is really a commit to memory day. Remember, you can do practice questions on Grockit afterwards. So your rules of exponents are as follows.
So we have general rules, first off. So first off, anything to the zero power is always going to be equal to one, even zero to the zero power. So anything to the zero power is equal to one, whereas anything to the first power is just equal to itself. Two of the first power is two. 100 to the first power is 100. X to the first power is X. Those are relatively easy to remember. We also have rules for adding exponents. If you have [inaudible 50:22]. So if we have X squared plus X cubed, that’s the other word for X to the second power and X to the third power, this just equals X to the second power plus X to the third power. You cannot add these two together to somehow get two X to the fifth. You don’t multiply the things together to get two X to the sixth or just X to the fifth or anything like that. All of those are naughty. You just have to leave them alone. If they have the same base, different exponents, and you’re adding them, you leave it. Multiplying them on the other hand, you don’t leave them alone. When you’re multiplying two exponents of the same base together, you would add the exponents. So X squared times X cubed equals X to the two plus three, which equals X to the fifth. If ever you’re unsure about this and you’re saying, “Oh, John,”–the chin stroking is optional–“John, I don’t know about that, I don’t think I can remember that,” You can also always write it out with small enough numbers to kind of reverify for yourself that this is right.
So X squared is X times X, and we’re multiplying that times X to the third, which is X times X times X. That’s our X to the third right here, this is the X squared right here. Multiply the two together we have five Xs, that’s X to the fifth. Pretty cool. Now when you raise exponents to exponents, that’s when you finally…people have been aching to multiply those exponents for a while, maybe. When you raise an exponent to an exponent, that’s when you multiply the two together. So this becomes X to the two times three which equals X to the sixth. And again that’s another one which you can write out when the numbers are small. So it’s X squared, cubed. So here is one X squared, here’s another X squared, and here’s our third X squared. When you multiply all these together, that’s six Xs. It’s X to the sixth power. Finally, dividing them. So X over Y to the third power. You aren’t really doing the division at this point but this can really be rewritten X to the third times Y to the third. So that may not be helpful in solving a particular problem, and it may be also helpful to go in the opposite direction. So just know that you can do that little conversion. Finally, when you are dividing an exponent by an exponent, you’re factoring things out.
So you are actually subtracting these exponents from each other. This is X to the two minus three, because you may recall that X to the negative two is one over X to the X squared, X to the second power. So X squared over X cubed is X to the two minus three, which is X to the negative one. And remember, negative powers are not necessarily negative numbers, X to the negative one is equal to one over X, or one over X to the first power. Oops. All right, using those rules, putting them into effect here. Remember how I mentioned that dividing by zero is naughty/ When you see problems that say if X does not equal to zero, that’s probably…if they give you something that X does not equal, they’re doing it so that they’re not dividing by zero. Note that if X did equal zero, X to the fourth power would still be zero, and we’d have zero as the denominator, which is naughty. So the GRE and Grockit seek to avoid breaking the laws of time and space by dividing by zero. So we will go out of our way to avoid it.
We can however follow our other exponent rules here. So first off, we raise these guys to exponents, X squared to the first power. We multiply the exponents, so this becomes X to the sixth times X to the fourth times X squared. And you could do one of two things. You could add all of these up and subtract this, or you could recognize that these two guys vertically, and X to the fourth up here, and X to the fourth down there will cancel each other out. And we’re left with X squared times X to the sixth. When you multiply two exponents of the same base together, you add their exponents which equals X to the eighth. So this is X to the sixth plus two. And there we have choice D.
Finally, your rules of roots, very important. As important, possibly even more important…no, I’d say they’re about equally important as the exponents ones. So something else you should devote to memory. Remember that the square root of something…let’s start with this one down here. This will probably be the easiest. The cube root of X is equal to X to the one-third power. So note that square roots can be converted to regular little number exponents as much as you want, and sometimes that makes life easier. So the square root of something is equal to X to the one-half power. Alright, so when you square a square root, you basically undo the square root. So the square root of two times the square root of two just equals two. The square root X times the square root of X equals X. On the other hand, if you add them together, you’re not undoing the square root, you’re just creating more of the square roots. The square root of X plus the square root of X equals two times the square root of X because, hey look, there’s two of them. Again, when you multiply roots together, it helps to turn them into those exponents. The cube root of X equals X to the one-third power. And that times X to the fourth root of X, which is equal to X to the one-fourth power. And remember, when you have two exponents of the same base, and you are multiplying those exponents together, you add the exponents together.
So we need a common denominator. Remember, we covered fractions also. So one-third plus one-fourth, we need a common denominator. That’s gonna be 12. So this is four-twelfths plus three-twelfths equals seven-twelfths. So this is gonna be equal to X to the seven-twelfths. Crazy, huh? Finally, when you raise exponents, two exponents, you probably still want to convert them to fractions. So again, let’s just say fourth root of X which is X to the one-fourth, to the third power. And when you raise exponents to exponents, you multiply them. So this is X to the one-fourth times three, which equals X to the three-fourths. Memorize these, commit them to memory. Put them up on next to the bathroom mirror, or by your pillow at night so you see it first thing when you wake up in the morning, until you have these down cold. They really are important for the test. So here’s an actual problem: Q is an integer and also the square root of P. Then which of the following is also an integer? So we have Q equals an integer. And we know that Q is the square root of P, so Q times Q equals P.
And an integer times an integer is always going to be an integer. Like three times two, four times five, whatever. As long as both are integers, and we already know they are, then we know that P is an integer. So which of the following is also an integer? Indicate all of them. And if you’re not sure about this, go ahead and pick numbers. Just say that Q equals two, and P equals four. Q is an integer, and it’s the square root of P. So just pick numbers. And then so, well, P over Q is four over two, which equals two over one, which is an integer. So there’s one. Q over P, that’s two over four, which equals one-half, which is not an integer. The square root of Q would be the square root of 2. That’s not gonna work. But the square root of P we already established is Q. We already knew that that was an integer. So D is also definitely an integer. So finally, your strategies for these exponents and square roots, notorious for traps, even just in the difference like raising an exponent to an exponent, and multiplying two exponents of the same base. That’s the most common one. So watch put for those. When in doubt try to rederive the rules with simple numbers. You know, default to what you do know. You can even write it all out this way, if you need to. Or practice it with number, real numbers on you calculator. So you can do that.
Know you calculator ahead of time. So practice things that you use the calculator like it is on the actual test. Positive fractions. This is one of the traps or the exceptions to rules. When you raise a fraction like one-half to a higher power, it becomes smaller. Half of a million dollars is a bigger amount of money than half of half of half of half of half of a million dollars, which is a higher power of the fraction one-half. So note that they get smaller. Note that negative numbers also alternate between being positive and negative. If X equals negative one, let’s just say, X squared is negative one times negative one. Negative times negative is positive. X squared equals one, X cubed equals negative one. X to the fourth goes back to being one again. So just know that negative numbers behave in funny ways when they’re the base of the exponent. The square of four is two, not plus or minus two. So the radical sign here, always means to the positive square root. Always, always, always on the GRE. On the other hand, like we did here, X squared equals four, this is X equals two or negative two, because either one of these squared becomes four. So just keep track of the difference. And then finally, fractions negative zero and one to test particular statements relating to exponents. That’s a lot to practice, but of course there’s homework.
Okay, the old clock on the wall says that’s all for GRE Math one. As I think I alluded to earlier, there is some homework. I mean, I guess I’m not going to come to your house and scold you if you don’t do it. But if the whole point of this is to do better on the test, you really should do this. And if possible, do this before the next video and the next class. Because, aside from practice making perfect, we do build on these concepts as we go. If any of these cause you problems, you can do targeted practice on number properties, arithmetic, decimals, ratios and proportions, and exponents and square roots. We have questions tagged with these same tags, these same phrases. We have these cover questions in the Grockit database tagged with these. So whichever of these you need practice on, and you should practice all of them to make sure that you are in good shape on all fronts, these tags allow you to do targeted practice. And since these are the fundamentals on which the later stuff is built, I guess I don’t want to beat you over the head with this. But really, please do you homework and pay attention to it. I personally recommend doing some practice every day, even if you don’t have time for an hour and a half. Doing two problems before bed keeps you from doing the thing where you say, oh, I’ll just do it all on the weekend, and then you try to commit to eight hours of GRE studying at once. And then when you can’t do it because the prospect of doing eight hours of GRE study drives you mad, then you don’t feel guilty. If you study a little bit every day, you’ll do much better than if you put it all off and promise yourself that you’ll do twice as much next time. Anyway, targeted practice on these for next time.