Real GMAT® Problems Ep. 45: Algebra (What to Do When You're Stuck)

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Problem 1

Welcome to Real GMAT® Problems Episode 45. Doing Real GMAT® Problems is one of the surest ways to make tests that go well, so let's talk through a few examples. The problems that we're going to go through are from the 11th edition of the Official Guide for GMAT® Review and you can web search them if you'd like to follow along visually.

Just so you know, the 11th edition of the Official Guide is out of print and that's partly why I'm going through questions from that book because that way if you purchase this year's Official Guide there shouldn't be any overlap between the questions that we're going to work through here.

The Official Guide contains retired problems that have appeared on past GMAT® exams so these are going to be super similar to what you're going to see on your official GMAT®. I'll give you a chance to work through the solution on your own if you want and then I'll discuss my thoughts on what to take away from each one. I recommend pausing after I read through each problem and trying to solve it or at the very least visualize how you would solve it.

You might not be able to execute all the steps in your head if you're on the go, but just do your best. If you are in a situation where you can physically write down what you would on test day, that's definitely ideal. We're going to start with a lower difficulty level warm-up problem and then we'll bump up the difficulty level from there.

We're going to go to a bit higher difficulty spectrum than we have in the past if you find yourself struggling at any point you might want to go to an earlier lower-numbered episode and work your way up to this level over time.

Today we're going to focus on algebra specifically what to do when you're stuck on algebra questions. This is problem number one. The problem says if the absolute value of x is equal to the absolute value of y.

And x times y is equal to zero which of the following must be true. I'll read that again. If absolute value of x equals absolute value of y and x times y equals zero which of the following must be true.

Option A is x times y squared is greater than zero. So just to be clear that's not x times y in parentheses and then square the whole thing. It's x by itself times y to the second power.

So it's just y being squared there so x times y squared greater than zero. B is x squared times y by itself so x is being squared y is by itself so x squared times y is greater than zero. Option C is x plus y is equal to zero.

Option D is x divided by y plus one so it's x on the top of a fraction and then y plus one is the denominator of that fraction. x over y plus one equals two. And Option E is one divided by x plus one divided by y equals one half.

So it's the fraction one over x plus the fraction one over y equals one over two. Again, I recommend pausing here and getting as far as you can on your own. It's a really good warm-up problem for a couple reasons actually.

It's a good example of some of the alternative algebra style questions that the GMAT® likes to throw at us. And what I mean by that is we're used to solving equations when we see algebra like isolating a variable using the rules of algebra. And there are plenty of times when doing that makes sense.

We've actually talked at length about standard approaches to algebra on the pod. And we've even got several Math Basics episodes on that topic as well as many Real GMAT® Problems episodes. When we talked about some fairly intricate equations, I would recommend just searching the feed for algebra and equations.

If you want to find some good approaches to more standard equation type situations. This one I want to specifically focus on stuff that's a little different than the standard down the center. Hey, here's an equation and solve that kind of deal.

What's interesting is that I found that most people don't struggle heavily with typical algebra type questions. Certainly there's going to be a handful of you out there who just were never great at algebra. Maybe didn't have the best algebra instruction when you were younger, but I found for folks like yourself.

If you put in the time and work hard at it like those algebra skills will be there for you. This type of question, just like the other ones that we're going to work through today, I think it's a little bit different. And I think solving with algebra the old fashioned way, it's either going to be really tough or just like a bad idea because gauged against alternative approaches is going to be really slow and maybe confusing.

Now, as I always say, if the way you're currently approaching these questions is working well for you, then just keep doing what you're doing. But if you want to explore some alternatives or maybe just consider some alternatives when things get weird, I'm hopeful you're going to find today's discussion really valuable. I want to start by considering what to do if you get stuck on questions like this because the odds are if you knew how to start this question with algebra or you knew how to operate on it well using standard algebra like isolating variables, you probably just didn't struggle with the question at all.

But I have also seen a good number of people make a great algebraic start on questions like this and then they just get stuck part way through the solution. And we've all been there. Remember you're here to learn like this is one of the best places that you could be making mistakes and getting questions wrong because every question you get in wrong in practice is probably something you're going to get right on test day, assuming you have a good review process and we'll talk about that later.

But let's give you some tools to use if the going gets tough. One of the best alternative approaches you can employ when there are variables in the answer choices like we have here is inserting a number or what I've called plugging in numbers and past episodes. So if you want like a ground up build discussing the step by step process of plugging in numbers, if the way I'm talking through it today is not quite landing.

We've got an episode titled literally plugging in numbers from a few years back and I think you'll find that helpful if you want to listen to that and then come back here. So this may or may not be the best default approach and by the way if you're on a streaming app on your phone and you're not sure how to search the feed just go to the page where you're seeing all the episodes and just pull down on the page a little bit and it should it should give you a field at the top.

To search search the feed or search this podcast or yeah something like that usually works well. Don't hold me to that if it's been like five years since we published this because who knows what's happening with software on that timeline okay but we're probably safe for the next let's call it like 12 months.

So this type of plugging in approach. It may or may not be the best default approach meaning like the thing that you always do on problems like this but again let's just assume you're stuck and therefore unsure how to proceed like your your initial approach did not work.

Just start plugging in some numbers in an organized way and see what happens. For example I'll make a couple columns here one with X at the top and the other with Y at the top to make sure I don't get mixed up about what's what as I get further down the road so really really simple organization technique just columns.

The problem starts by telling me that absolute value of X is equal to the absolute value of Y so I'm thinking of putting in some equivalent values because they have to equal each other. Maybe experimenting with different combinations of positives and negatives that tends to be a good way to go if you're plugging in numbers on absolute value questions because if you if you think about like what are the concepts that they might be testing me on when there's an absolute value in the question you're probably thinking like positives and negatives that's that's kind of like the key thing that happens with.

With absolute values in general and on on a test where it's partly about my math skills but also like partly about my reasoning skills which sometimes could be like a euphemism for like just trying to trick me and get me to miss stuff that I might normally get right in other situations.

It depends on your viewpoint of course I won't take sides on that one today but it's up to you to decide what what's happening here. So on a reasoning test where they're trying to test my reasoning abilities with absolute values they're probably testing my ability to think about positives and negatives so it's just a good thing to put in your back pocket when you see absolute values and you're going to plug in some numbers.

You might think different combinations of positives negatives but let's get in a little ahead of ourselves so let's just start simple here. I'd probably start with numbers that feel easy to work with and compute so to is like generally a good place to begin unless unless it says that numbers are odd or something like that where there's an obvious constraint that prevents to but here we're totally allowed to use to.

In fact it doesn't constrain the numbers we can plug in at all so we can plug in anything we want. So I'll just make X2 and I'll make Y2 and I'll write those under my X column all right to and under my Y column all right to just to keep track of what's going on. So probably to the right of that X column and Y column where I'm tracking what numbers I've picked to the right of that I put absolute value of X equals absolute value of Y to make sure I'm sticking to this constraints of the problem.

And then to the right of that I would put the other constraint which is X times Y has to equals zero those are the two conditions we've been given in the problem. Now if I plug in two for X and two for Y perhaps obviously I can satisfy the absolute value of X equals absolute value of Y requirement. And also maybe obviously I will not satisfy the X times Y equals zero requirement.

So I'm going to want to think about that a little bit deeper but it's a good time to make a point which is when you're plugging in numbers sometimes the reason you're plugging in numbers is you don't understand the question hence the topic of today's lesson like what do I do if I'm stuck what do I do if I'm confused.

Sometimes plugging in some numbers that don't work with what's happening in the question can still be super valuable because it can get you thinking a little bit more deeply about like okay what's happening with the math behind the scenes here.

That might not make a ton of sense right now but I think as you try out the technique a little bit especially when you're stuck you'll start to feel what I'm talking about there of like okay yeah I'm starting to get like a better grip a better intuition about what I'm really being tested on here.

A lot of times when I'm working with folks one on one and they're getting stuck in questions I'm like hey why don't you just plug in a couple numbers see if that can help you reason through the logic a little bit or like bring bring the logic to the forefront and it's not a hundred percent thing perhaps obviously but it's nice it's a lot better than having nothing and being dead in the water.

So here I'll think a little deeper I might try negative two for X and positive two for why that of course satisfies absolute value of X equals absolute value of Y if you're just starting out and you're jumping right in with this episode everybody we've got a whole math basic series on absolute values.

If it's been a little while since you've thought about absolute values and there's no calculator on the GMAT® Quant section so it's a good idea to go through that whole math basic series if you're just beginning because you might just be a little bit rusty on long subtraction and long multiplication for example if you're like most adults walking around in the professional business world you probably not using long division very often.

So satisfies my absolute value X equals absolute value Y equation but still does not satisfy X times Y equals zero now you might be thinking how does this help me but when you're stuck again knowing what doesn't work knowing what doesn't work can often lead you to what does work.

Maybe a little counter intuitive there but again if you experiment with it enough I think you I think you'll find that to be very true so this questions a good example of that because if I go down this path one more step and I think to myself OK have to satisfy this X times Y equals zero thing let me try X equals zero.

Then when I'm brainstorming possible values for Y and I'm thinking about absolute value of X has to equal absolute value of Y I'm probably going to realize that if X is zero Y also has to be zero. So let me write that in my columns here just to keep on top of my thinking and make sure I don't miss the question because of something silly like forgetting what I plugged in for Y which. That means it's just such a painful reason to miss these questions and if you've been studying for a while you probably see that happen more often than you wanted to if you're like me when I was studying back and I made way too many of those mistakes before I got my stuff together.

And from here let me just start to experiment with some of the answer options since I finally have a set of values that fits the constraints. If I plug in X equals zero and Y equals zero for option A I'm going to get zero times zero squared which is of course zero and that is not larger than zero you might recall that option A is X times Y squared is greater than zero so if X and Y are both zero that's not going to work.

So it looks like a doesn't have to be true which which is basically what quote unquote must be true questions are really asking us which answer has to be true as long as the conditions of the problem are satisfied.

So if I have absolute value of X equals absolute value of Y and X times Y equals zero then one of these answers has to be true 100% of the time that's what I'm trying to figure out here which one is it.

Let's try B and you'll probably notice that something very similar is going to happen with B just to refresh your memory if you're on the pure audio feed B is X squared times Y is greater than zero and if you want to see how I would work through these problems you can you can grab this episode on YouTube Scott some video of

a wonderful TGST mate writing on the whiteboard after the fact here. So trying to bring some method to the madness which I appreciate by the way. So if I plug X equals zero and Y equals zero in for option A it's not going to work if I plug X equals zero Y equals zero and for option B not going to work.

Let's try C that's X plus Y equals zero and zero plus zero does of course equals zero so C works just going to leave that alone for the moment. Let's look at D because if D and you don't work then I can just pick C and I don't have to think too deeply about it. Refreshing your memory on D that's X divided by the quantity Y plus one so that's going to be zero over one.

And D says X over Y plus one has to equal to zero over one is zero AKA not to we can eliminate D and then E asks us to divide by zero which is not allowed on the GMAT®. So it's it's worth saying if you're ever asked to divide by zero and an answer option on a GMAT® question you can assume it's definitely incorrect. Having said that even if theoretically we could do one plus zero excuse me one over zero plus one over zero it would definitely not equal one half which is what he is saying.

So just in case that previous point was a little weird you can be doubly sure that E does not have to be true. And that should lead us to our correct answer of option C as in Charlie. Okay so I took us down a very very specific path there and again if you had a good solid algebraic solution for this one that's awesome.

That's amazing probably not struggling on this kind of question and I'm not saying this is the greatest example of when 99% of us should should plug in numbers I'm not saying that. But I did want to use a simpler problem to introduce the idea before we get into some more complex questions and just in case you get stuck on more complex questions on test. So are there some clever algebraic solutions here yes for sure but when you see the opportunity to leverage those by definition you won't be struggling with the question.

So if I'm really going to make a contribution to you and your success here I don't think validating what's clearly already working for you is going to qualify like you're probably pretty sure what's working for you that would be my guess.

And that's why I gave you a tool for when things are not going well. I think that's going to help you a lot more than a two step algebra approaches is going to help especially when that two step algebra approach might not apply very broadly that can be part of the difficulty that we're going to confront in some of the next questions here when we get into the algebra.

So with all that being said if there's something that you want to remember from that I recommend you make a quick note to self here by texting or emailing yourself just pause the episode. And that way you can make a note when you're back at the desk if you're already at the desk now would be a good time to pause and make a note of anything you feel could be valuable for you in the future and put that into your flashcard set or your general notes to self said or something that you're looking at on a regular basis.

So that you're actually learning from the questions that you're doing in practice not just completing a lot of questions without learning very much that's that's a unfortunately excellent way to waste a lot of time and that's what I did back in the day and I'm really trying to save you that pain.

If you want to know about how horrible my situation and story was you can check out the video on our website it's called how to reach a dream GMAT® score and half the normal time which if some saintly person had been for me back in the day would have really helped me out so I hope I hope it helps you too.

Problem 2

Okay let's bump up the difficulty level a little bit here this is question two question says. X plus one so it's the quantity X plus one in parentheses divided by X minus one also in parentheses so it's on top of a fraction X plus one on the bottom of the fraction X minus one that whole fraction is in parentheses.

And we're taking that whole fraction to the second power okay so it's X plus one over X minus one squared that whole fraction squared okay that's the beginning of the problem is just that. There's no equal sign or anything else then the rest of the problem says if X does not equal zero and X does not equal one and if X is replaced by one divided by X everywhere in the expression above then the resulting expression is equivalent to.

Blank and then they give us five different answer options that could be equivalent okay let me read that again because perhaps obviously a little bit more complicated than the previous one. So we've got the fraction X plus one over X minus one the numerator is X plus one the denominator is X minus one that entire fraction is squared. Text on the question says if X does not equal zero and X does not equal one and if X is replaced by one divided by X so the fraction one over X goes in for.

X everywhere in the expression above then the resulting expression is equivalent to blank option A is a fraction numerator the fraction is X plus one denominator the fraction is X minus one that whole fraction squared so it's literally identical to the given expression above that's option A option B is the fraction X minus one in the numerator divided by X plus one in the denominator that whole fraction squared.

Option C is just the variable X squared and then plus one that's the numerator of a fraction the denominator of that fraction is one minus X to the second power okay so just X squared in the denominator as well so C is a fraction the numerator is the variable X squared first and then add one to that the denominator of the fraction is one minus just X squared by itself no parenthesis and C.

Option D is another fraction we've got just the variable X squared on top minus one and then in the denominator we've got X squared plus one. Then option E is the fraction X minus one in the numerator divided by X plus one in the denominator that whole thing squared and then multiplied by negative one. Okay so don't worry if you're losing track of all that on the audio feed that's okay you can still get value out of grappling with this one a little bit and thinking about how you would solve it again you might not be able to execute all your steps in your head the way you would be able to if you were writing things out one step at a time on test day I probably wouldn't be able to.

But you can still think about okay how would I approach a question like this what are some of the patterns I'm seeing can I associate this with any questions I've seen in the past those are really great questions to ask yourself if you're at the gym or you're on the bike or you're on in the car or something like that and you're just trying to level up with the GMAT®.

Okay recommend pausing getting as far as you can excellent follow up to the previous question because I think a lot of the same points a lot probably even more so since this question is a lot more complicated.

Some of you may have taken the algebra route here and I've definitely seen that work well in about 20% of cases people get to the right answer in a reasonable amount of time on this one using algebra but for a lot of folks even if they can get the algebra to work it takes a long long time like over five minutes that's what I've seen about 30% of cases end up that way.

And for the majority of folks who start with the algebra they get the first few steps to happen fine and then they really struggle to connect that work that they've done to the right answer choice like they've done a lot of correct algebra but it's but their their result is not lining up with what the answer choice says about 50% of cases end up in that situation.

And that that's roughly equivalent to the previous question it's just that more more people get the previous question right about 20% more of us get the previous question right somewhere or another versus this one I have doubts about that stat team just going to double check that.

Yeah yeah three times as many of us miss this question versus the previous question. So I guess we need a Math Basics lesson here behind the scenes at TGS got a good podcast for you. Now's not the time to be talking trash to me guys okay so quite a few more of us miss this question let's put it that way.

So I'm not going to go through all the steps on the algebra here because I don't I don't think the values going to be there again if you want to hear me do a ton of algebra just search equations and algebra in the feed and you can listen to me do lots and lots of steps of algebra on earlier episodes.

But just imagine for a second let's just let's just like imagine a couple steps imagine using foil we've done a lot of foil on the pod again if you're just starting out you might want to check out the Math Basics episode on quadratic equations to reskill with foil you probably remember or knew what foil was at some point in your academic career might just have to shake off the rest a little bit even if you do all the math correctly with foil.

And you can correctly substitute one over X you'll wind up with one over X squared plus X plus one in the numerator and then in the denominator one over X squared minus X plus one. And where do I go from there that's the tough thing you didn't do any math wrong but it still didn't yield the answer that you're looking for it's not in the same form as the answer choices. And that can be confusing that can be really really confusing so if you want to use algebra to get this question right you would have to resist what most of our natural instincts would be meaning you would have to not foil and you would have to do the math inside the parentheses first and even then even then there's no guarantee that you're going to simplify things properly in order to arrive at the answer the way they've expressed it not saying you're going to do some math wrong just saying one of the risks in these algebra questions.

That you run when you're defaulting to algebra or your only approach is algebra is that you do a lot of correct algebra but then the answer choices not in the right form that that can be really really frustrating.

The point is even more of us eventually get stuck on this question the previous one that's that's really the point so just just forgetting about the whole foil thing for a second so you may find that plugging in numbers here really helps.

Let's quickly talk through the optimal algebraic approach to this question and then and then compare so I'm going to quickly go over this you can sub in one over X in the original fraction that gets you one over X plus one on top and then one over X minus one on the bottom and then remember that whole things being squared.

You can then find a common denominator on top so instead of one over X plus one I'll do one over X plus X over X and just check out the Math Basics episode on fractions if you want to re-skill with the fraction math and that gets you X plus one over X on top.

Wait a minute we got another type of there sorry just forget I said that team or no that's right that's right oh man a story of redemption here today everybody. Okay so if you do that math right that gets you X plus one oh I see I see why that was weird that gets you X plus one on the top of a fraction and then X on the bottom so we have one over X plus X over X.

If we add that if we're just focusing on the numerator here that gets us X plus one on the top of a fraction the numerator of the fractions X plus one and then X is the denominator of that fraction. And then that's inside of a larger fraction where we haven't dealt with the bottom yet so you can already see like for certain types of people this is just not not going to go very well. They're either not going to find the common denominator or they're not going to think about it or they're going to make a math mistake.

I'm clearly I'm in that boat to be fair. So if you perform a similar operation on the denominator you would get X minus one in the numerator of that fraction and then X in the denominator of the fraction and again if you've already lost me on the pure audio feed that's okay that's okay.

Probably wouldn't recommend doing the question this way unless you're like oh this is amazing I love this approach. So then we can divide the fractions dividing by a fraction is the same as multiplying by the reciprocal so we would get X plus one over X times X over X minus one and that's all inside the parentheses that are being squared so we haven't squared anything yet.

Then the X cancels on the top and the bottom and we wind up with X plus one over X minus one inside the parentheses and then that is being squared and so we actually get exactly the same thing that we started with which is option A as an alpha and that is the correct answer.

Now again if you can see from the beginning that the foil approaches is going to be bad and then make the required adjustments and then execute that math without making a mistake that's amazing that is truly truly awesome and all those skills are definitely worth practicing and developing because a lot of times that is the best way to do one of these questions and it's good to at least be able to do one of those but again going back to the point of the previous question and I'm going back to the point of the previous question.

And this episode what if you get stuck what if you start trying to execute that way and you just don't see the next connection you don't the next step just is not jumping off the screen actually happens all the time that's a great time to plug in a number you've got variables in the answers here.

Let's see what happens if we go that route so let's plug into the original expression and let's see what would I choose I'm just going to go X equals to again. It's a really really good default again if the problem doesn't say like X has to be negative or X has to be zero or something like that obviously obey those constraints but if it just says X has to be positive or something like that I think in this question yeah in this question we just say X is not zero and it's not one okay great I'll make it too.

So unfortunately we had a significant tech issue during this portion of the video the other day so I'm back in the studio and we're just going to film this for you real quick. So let's say that you're stuck you decide to plug in a number let's choose something that will make the computation simple like X equals to. So I'm going to plug that into the original expression first and recall that in the original expression I need to replace X with one divided by X.

So instead of just plugging into the original expression I'm actually going to plug in one over two as if I'm plugging in one over X. And on top that's going to get me one over two plus one and on the bottom it's going to get me one over two minus one and of course the whole quantity is squared. I want some common denominators there so on top I'll do one over two plus two over two and that'll get me three over two and on the bottom I'll do one over two minus two over two to get negative one half.

So my fraction would be three over two and then divided by negative one over two and that whole thing squared. Once again when I divide by a fraction that's the same as multiplying by the reciprocal so I'll multiply three over two by negative two over one. And that gets me negative six over two again feel free to check on that Math Basics fraction episode if it's been a while since you've multiplied some fractions and negative six over two is negative three.

If I square that I'll get positive nine and at that point I can then start testing the answers and seeing if any of them equal positive nine so let's try a. Now keep in mind here in the answers I'm actually plugging in X I'm not plugging in one over X the questions asking if I plug in one over X to the top thing what's the equivalent expression on the bottom.

Not putting in one over X just putting in X so I'm actually going to plug in two for a so that's going to be two plus one on top and it's going to be two minus one on the bottom and that gets me three divided by one squared recall that Option A is literally the same expression as what's given to us before so we got two plus one on top two minus one on the bottom three over one squared is nine and that's great that's a match so I'm going to keep a and then apparently the tech issue just resolved at this point.

So I'm going to kick you back over to the original recording let's try be just to be safe and that's going to get us one I'm not going to re go through the math well let's let's go ahead and do it. Let's go ahead and do it just in case it's awful so B is X minus one over X plus one squared so that would get me one over three squared so obviously that's not going to be nine one third times one third is not going to be nine.

C is X squared plus one so that's going to be four plus one that's five on top over one minus X squared that's negative three on the bottom obviously that's not going to be nine. D is X squared minus one on top that's three and then on the bottom I have X squared plus one that's five three over five is definitely not nine. And then the last one's negative so no matter what I do is going to be negative so there's no way that's going to equal positive time so where does that get me gets me that.

Let's reflect on that for a moment. Which way feels easier for you to execute I know that maybe we are not going to make this question quote unquote easy for some of you maybe a lot of you but we can make it easier by having good default approaches and good backup approaches.

That is the ideal situation when you're entering into solving one of these questions hopefully you've got something you're comfortable with that you lean on that you know works most of the time and then if that does not work for any reason hopefully you've got something you can fall back on it still get the question right that usually goes really really well in the

quant section. In fact which of these feels easier for me to execute is always a good question to ask ourselves after working through multiple approaches to a question like we just did. It's also good to ask yourself in what different situations might I use each approach it's kind of a bonus question and some of you might default plugging in numbers when you see variables in the answer choices maybe that

approach just felt way easier made way more sense maybe you're like man I'm sold. Sign me up I'm doing I'm doing that others you might have felt the algebra was easier or maybe you want to start with algebra and then pivot to plugging in numbers or maybe you just wanted to plug in numbers for a couple options.

There's no right or wrong here it's all about what's going to help you get the best results in your situation the way your brain works the way you like to manage the exam. So try to judge these strategies based on the results they produce for you. I guess we should judge all strategies that way having said all that I have found that at the very least plugging in numbers is a great way to go if you find yourself getting stuck on algebra questions.

Quick aside if there are not variables in the answers plugging in numbers may not work you still might be able to like use it to wrap your mind around the question like I was talking about earlier but it might not get you all the way to the solution.

But the great news is if you do see variables in the answer choices plugging in numbers is for sure an option so you can lean into that. Okay hopefully at the very least we've given you a backup approach here and maybe even given you a good frontline approach for questions that are similar in the future so as usual I'd recommend pausing making any notes to self that you feel could be helpful and then we'll finish one final problem that's a little bit tougher.

Problem 3

This is problem three. The problem says if x and y are positive which of the following must be greater than one divided by the square root of in parentheses x plus y. So I'll read that again if x and y are positive which the following must be greater than and then we get a fraction one is the numerator of the fraction and the denominator of the fraction is x plus y in parentheses and then you're taking the square root of that whole denominator you're not taking the square root of the whole fraction just the square root of the denominator so it's one over square root of the sum of x and y.

Okay so that's that's our question and then we get three Roman numerals that we are supposed to compare with here okay so we're saying which the following must be greater than what I just read. Roman numeral one is numerator square root of sum of x and y divided by denominator two times x okay so square root of x plus y over two x. Roman numeral two is the square root of x individually added to the square root of y individually that's the numerator of a fraction the denominator of that fraction is x plus y.

Roman numeral three is the square root of x on its own minus the square root of y on its own that's the numerator of the fraction is root x minus root y the denominator of the fraction is x plus y. And in these Roman numeral questions I'll read all those again in a moment. These are options are going to be some combination of the Roman numerals.

So option A here is none of them are greater. Option B is only the first Roman numeral is greater. Option C is only the second Roman numeral is greater, and I'll come back to those in

a second. Okay. D is 1 and 3.

E is 2 and 3. Okay. So Roman numeral 1.

Numerator of the fraction, sum of x and of y, x plus y in parentheses, the square root of that whole numerator, square root of x plus y on top, on the bottom, just 2 times x. Roman numeral 2, square root of x plus square root of y is the numerator of the fraction.

Okay. I've got root x plus root y, both being square root independently. And then the denominator of that fraction is x plus y.

Roman numeral 3 is square root of x minus square root of y. That's the numerator of a fraction. Denominator is x plus y.

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Option A is none of the Roman numerals are greater. Option B is 1 only is greater. Option C is 2 only is greater.

And D is 1 and 3 are greater. And option E is 2 and 3 are greater. Okay.

So get as far as you can there. Again, if you're on the go, couldn't hold all that in your head, that's okay. Just think about how would I strategize here?

What might I do? What might be appealing to me? Have I seen questions like this in the past?

One more time. I'd recommend pausing here and getting as far as you can. Okay.

Quite a bit more complex than either of the first two questions here. And that's definitely reflected in the accuracy numbers as well. With about 35% more of us missing this one versus the previous one.

Let me check that as well since we had an error in the earlier stat. Okay. So let's see.

Yeah, that's right. That's right. Thanks, team.

Got me covered. I love it. Well, clearly it's a more difficult question for us as a group, not a huge surprise since

Roman numeral questions like this one are effectively like three questions in one and we need to get all three of them right or our odds of getting the whole problem right go way way down.

But what I find really interesting here is that most people who I coached plug in numbers on a question like this end up getting it right faster than the two previous questions. So not only is there accuracy about 40% higher, they're also about 60% faster than those

who choose algebra here. I'm not saying that to you. I'm just saying that's what I've seen.

So again, if you're loving algebra on a question like this, keep doing what you're doing. But if you get a little lost or you get stuck or you didn't feel like you had the speed that you wanted with your algebraic approach to this one, it's a great time to plug in

some numbers. And it's a little bit different than the previous questions with the whole Roman numeral thing.

But to address that specific situation, because it's only the Roman numerals that have variables, not technically the answer options. And I found that some folks can get a little confused by that.

So another trigger that you can use for plugging in numbers, if you're a little unsure, is the phrase must be. You may recall we saw that in question one earlier as well.

Anytime you're asked for something that must be true, that's a good time to plug in numbers. I'm actually not even going to deal with the algebra here because it's really a bear

in my opinion. And much like the first question, if you rocked this one with the algebra, you probably don't need very much help on questions like this at all.

But suffice to say, it's a worthwhile review strategy in general to at least to attempt to understand the algebra behind these kinds of questions as you go forward. Even if you don't plan to use algebra as a strategy on these questions on test day, it's

just a good workout for your algebra skills to at least try the problem with algebra, reflect on how would I solve this if I had to do algebra. Really good exercise for future unseen questions.

Having said all that, let's explore what plugging in numbers could look like. I'd like to use numbers that make my computation as easy as possible like I did before. So looking at the original expression, one on the top of a fraction and the denominator

of the fraction being square root of x plus y, I would like to get a perfect square under the square root there so I can deal with an integer and the denominator there. And so I'd probably just select x is 2 and y is 2.

Going back to the constraints, all we're told is x and y are positive. Maybe I could have even gotten away with like one there for x and one for y, but then I'd get root 2 and the denominator, I'm not feeling as good about that.

And stick with x is 2 and y is 2. And just again on the clear scratch work thing, which I've harped on so, so, so, so much in the past, I'd just write x equals 2, y equals 2.

And that creates a perfect square under my square root of the denominator there. That gets me 1 divided by square root of 4 because 2 plus 2 is 4. And that's going to be 1 over 2.

So that gives me a fairly simple fraction to deal with when I'm comparing to the Roman numerals. So now I'm going to plug into the Roman numerals and see which ones are bigger than 1 over

2 because that's what the question's asking me. First one works out pretty well. You might recall that Roman numeral 1 is x plus y and then the square root of that.

So I'd get root 4 on the top. And on the bottom I have 2 times x, which would just be 2 times 2. So that's going to get me square root of 4 on top.

That's 2. On the bottom, 2 times 2 is 4. So I'd get 2 over 4, which is exactly 1 half.

So apparently, Roman numeral 1 does not have to be larger than the original expression. And that's what the question's asking in a must be true. Which one has to be true 100% of the time?

So Roman numeral 1 does not have to be true 100% of the time. It's in this case when x is 2 and y is 2, it's actually equal to the given expression. So that's great news because now I can eliminate any answer choices that have Roman numeral

1 in them. So I'm going to go ahead and eliminate b and I'll also eliminate d. Let's try Roman numeral 2.

That's a little bit weirder because I can't easily get rid of the square roots with my number. So I'm going to have root x plus root y on top, so I'll have root 2 plus root 2.

And then on the bottom, I've got x plus y, 2 plus 2 is 4. So I'd want to put root 2 plus root 2 over 4. But let's just say I add root 2 plus root 2 in the numerator to get 2 times root 2 in

the numerator. 2 times root 2 in the numerator divided by 4 in the denominator, then I can cancel the 2 in the numerator with the 4 in the denominator.

And that would get me square root of 2 over 2. Again, scope the Math Basics, fractions less than if you want a little up-skilling on fractions if it's been a minute since you canceled some stuff.

So root 2 over 2. This is a great situation if I've memorized an approximation for square root of 2. And just side note, it does come up a good amount, and it's roughly equal to 1.4.

So it's definitely worth knowing, probably get a lot of mileage out of that, but even if I did not know that, I can probably figure out that it's going to be bigger than 1. Square root of 1 is exactly 1.

So it's very likely that square root of 2 is bigger than 1, even if I didn't know the exact value. And if I'm trying to figure out if root 2 over 2 is bigger than 1 over 2, which you might

recall is the original value for the given expression, I could probably conclude that root 2 over 2 is larger than 1 over 2. I could probably figure that out, even if I didn't memorize that root 2 as approximately

1.4. So I want to eliminate 2, because at least it can be larger. I'm not 100% sure if it has to be larger, but it at least could be larger.

So I'm going to leave that be, and I'll try Roman numeral 3 now. This is interesting, because now I get square root of 2 minus square root of 2 in the numerator. Of course, that makes the entire fraction zero, and that's clearly less than 1 half.

So I can eliminate any options that have Roman numeral 3 in them. So we'll go ahead and eliminate E. So now we've eliminated B, D, and E, based on the fact that Roman numeral 1 and Roman numeral

3 do not have to be larger than the original expression. So I only have to choose between A, which is none of them work 100% of the time, and C, which is 2 works 100% of the time.

Now, Roman numeral questions tend to be time intensive. So at this point, if you're short on time, you can just make a bet. 50% odds of guessing right here are a lot better than where we started, which was 20%

odds of guessing randomly. So that's not a bad return on our time so far. But also consider that if you wanted to use algebra and you had time, you could now implement

that algebra on only one of the Roman numerals, which would be probably quicker than trying to do it on all three. It depends on your algebra skills and where you're at there.

So that could be a great mixed approach if you're into that. Also, if you had the time and you're comfortable testing a few different values, like plug it in 3 for X or plug it in 4 for X or plug it in 16 for X or whatever, and you have that

time, then you could build your certainty that Roman numeral 2 is always going to work by testing a few more cases. But that is one of the pitfalls of plugging in numbers when there's a none option.

Most of the time there won't be an unoption. But if there is one, you may not get 100% certainty. So there's a little bit of risk there.

Now, it's up to you whether you're willing to incur that risk, but I can say confidently that I've very rarely seen someone miss this question when plugging in numbers. Very, very rarely.

They might be stressed out about it. They might not be 100% sure, but they'll generally get it right. And just for the record, too, does always work.

So the answer is C, as in Charlie. OK, so let's step back and evaluate how much harder is it to get to 100% certainty on a question like this versus quickly getting to 80% certainty?

Think about that. What would it take to get 100% certainty versus what would it take to get to 80% certainty? And then maybe you just mark it and return at the end of the section if you have extra

time. Something to consider. Something you should personally do that, all I'm saying is that it would be great to have

that option just in case. It's kind of a bummer when you look at it and you're like, well, I can't do algebra here. Therefore, I cannot do the question.

That's kind of a bummer. And that's why I'm emphasizing the plug-in numbers approach. Sure.

It's not for everyone, but it is a great tool. And I don't see a lot of programs teaching how and when to use it very well, unfortunately. So hopefully I've helped you out there a bit.

At the very least, hopefully it was good to get some practice in and also have the chance to consider what you might choose for each approach and when you might implement each approach.

Hopefully you got some value out of that as well. One final time. I recommend pausing there and making any desired notes to self for what you'd like to remember

from that problem.

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