Real GMAT® Problems - Ep. 47 - Average Speed

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Transcript

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Welcome to the G Math Strategy Podcast. You're here because you believe there's a better way to study for the G Mat, and so do we. We created the GMAT Strategy to maximize your results and minimize your efforts so you can get to the fun parts about business school and life as quickly as possible. My name is Isaac Poolia, and I've been teaching G MAT classes and tutoring privately for the GMAT for over a decade, and I've helped thousands of students get into the business schools of their choice. I'm excited to be a part of your MBA journey since we all at TGS believe that our world can benefit from the best possible business leaders that we can find. If this show is bringing you value, please share it with your friends and family who are studying so that together we can make this process as easy and as painless as it can possibly be. Let's go. Welcome to Real GMAP Problems episode 47. Doing Real GMAP problems is one of the most data-backed ways to make test day 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're welcome to web search them if you would like to follow along. 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 extremely similar to what you'll see on your official GMAT. I'll give you a chance to work through the solution on your own if you'd like, and then I'll discuss my thoughts on what to take away from each problem. I recommend pausing after I read through each question and trying to solve it, or at the very least, visualize how you would solve it. You may not be able to do everything in your head if you're on the go, but just do your best. If you are in a situation where you can write things down the way you would on test day, that's ideal. We're going to start with a lower difficulty level warm-up problem, and then we'll bump up the difficulty a couple times from there. If you find yourself struggling with any of the questions, then I recommend just heading back to an earlier episode in the series where we'll go through more lower level questions, and then you can work your way back up to this level when you are comfortable. Today we're going to focus on average speed questions. This is problem number one. The problem says car X and Car Y traveled the same 80-mile route, 8-0 mile route. If car X took two hours and car Y traveled at an average speed that was 50% faster than the average speed of car X, how many hours did it take car Y to travel the route? I'll read that again. Car X and Car Y traveled the same 80-mile route. If car X took two hours and car Y traveled at an average speed that was 50% faster than the average speed of car X, how many hours did it take car Y to travel the route? Option A is two over three. Option B is one. Option C is four over three. Option D is eight over five. And option E is three. It's worth memorizing that average speed is equal to total distance over total time. Again, average speed equals total distance over total time. It's a simple enough formula. In fact, it's so simple that I think a lot of people don't appreciate how useful it is. So I'm going to try to demonstrate that usefulness for you as we progress through the lesson. For now, though, it's worth memorizing however you like to memorize things. The second piece of the puzzle that I think you're going to find really valuable, and if you've listened to or watched any of our rates content, then you've seen it in action already. And maybe they even teach this method in a course that you might be taking. It's called the rate chart. It's a specific instance of simple rows and columns. And it's a handy way of organizing data in distance and speed questions as well as rate and work questions, which have a lot in common with distance and speed questions. So I'm going to start by organizing the information into the chart so you can visualize it if you're on the pure audio feed here. Or obviously you'll be able to see me draw it here in a moment if you're on the video feed. And if if that video is available on the platform that you're on. So it starts with a row across the top of the chart. And in that row, I'm going to make a column header for the rate, and then a column header for the time, and then a column header for distance. And I'm pretty much just writing rate times time equals distance across the top. It's kind of like it's the top row of a set of rows and columns. And each each rate, time, and distance is its own column. So then within each column, I'm going to give fill in the information that's given in the problem. So first thing I'm told is that both cars travel the same 80 mile route. So under distance, I'm just going to write 80 miles. And to the far left of the chart, in what I would call like the first row or the row right below rate times time equals distance, I'll just write car X as the label of that row. And then under the 80 miles that I just wrote, I will write 80 miles again. And then to the far left of that row, I'll write car Y. And so now I've got a set of rows and columns to help me organize the rate times time equals distance equation for each car. The next thing I'm told is that car X took two hours. So I can write that under the uh the time column in the car X row. And finally I'm told that car Y traveled at an average speed that's 50% faster than car X, but I I don't know car X's speed yet. So that's a good indicator of where I can start doing some math in the question. So the the good news is I have car X's total distance and I have car X's total time. And so I will be able to calculate its average speed. That's just total distance again, 80 miles over total time, two hours. And of course that's 40 miles per hour. So I'm gonna go ahead and write that in the rate column next to car X. Okay, so now I can solve for car Y's average speed, which is 50% greater than car X's. So to increase by something by 50%, I take 50% of that thing and then I add it to the original amount of the thing. And if you want to reskill or upskill with percents, whether you're just starting out and this is like your first contact with the exam, or you just might have forgotten, even if you've been studying for a while, then we've got a math basic series and we've got specific episodes on percents. Those are both gonna be linked below, both the full series and the specific percents episode. So you should just be able to click on that and uh reskill with that or upskill with that if you want. 50% of 40 is 20. So I'm gonna add 20 to the original 40 miles per hour in order to get car wise speed, which would be 60 miles per hour. That's great news because now I can see in my chart that I have car wise distance and car wise speed. So I can just write 60 times T, I don't know car wise time yet, equals 80 miles. And I can either pop that out of the chart or I can do the math right in the chart. You'll come up with your own style and what you feel works best for you over time. I don't know that there's a right or wrong there. Uh just try to be consistent once you figure out what does work for you. I'm just gonna use algebra to solve. We've got a couple refresher lessons on algebra and that math basic series. Again, if you're just starting out and it's been a while since you've done percent change or algebra, uh, that's that's where I was when I was first starting out. And yeah, it was a little humbling, but uh honestly, it was probably good for me. And um, we made that math basic series for you so that you can avoid all the friction that I felt relearning math uh in my 20s, which was great. It would just, you know, let's just let's just uh put on some rose-colored glasses there and pretend it was great for me. My hope is it's gonna be great for you since we made that material, but I won't lie. This whole thing was just an enormous struggle for me. And I'll give you more information later if you want to hear about my story of pain and strife and why I'm doing this for you now to hopefully help you avoid all that. So uh T would equal 80 divided by 60. And if I reduce that fraction, which again we've got a math basics episode link for you below, if you want to reskill with reducing fractions, I would get four over three, which leads me to my correct answer of option C, as in Charlie. Okay, so do you absolutely need a rate chart to get that one right? Probably not. It's it's certainly not the hardest GMAC question that you're going to see. So if you can do things like that in your head 100% of the time, or you have a better way that works for you, that's awesome. I'll never argue with your results. Uh, but if you feel like you've been struggling with distance and speed questions, or you often get lost halfway through them, or you often have difficulty setting them up, which is really, really common, I think you're gonna find that rate chart really, really helpful. And I'll try to demonstrate that more as we progress up the difficulty spectrum from here. I'm just using this slightly lower level difficulty question to introduce the topics. Um probably obvious, but just in case it's not, that average speed formula is critical on this question. So if you weren't aware of it, it's great that you are now. And it's going to play a major role in uh the more difficult questions we'll look at momentarily. So again, it's just a warm-up question. You may have found it challenging, you may have found it easy. That doesn't really matter because you're here to grow. So let's just pause and reflect for a sec. Is there anything that you want to recall from that brief discussion on the first question for the future? If there's nothing, that's fine. We'll get into some more complex and difficult material in a moment. But if there is something to take away, I'd recommend just pausing and texting or emailing yourself if you're on the go or if you're at the desk, adding that note to self to your memorization system of choice. Okay. Let's bump up the difficulty just a bit. This is problem number two. The problem says Jill went up a hill at an unknown constant speed. She then immediately tumbled down the hill along the same route, but twice as fast. If Jill's average speed for the round trip was six miles per hour, what was her average speed tumbling down the hill? One more time? Jill went up a hill at an unknown constant speed. She then immediately tumbled down the hill along the same route, but twice as fast. If Jill's average speed for the round trip was six miles per hour, what was her average speed tumbling down the hill? Option A is seven miles per hour, option B is eight miles per hour, option C is nine miles per hour, option D is ten miles per hour, and option E is eleven miles per hour. Again, recommend pausing and getting as far as you can. This is a really good follow-up to the previous problem because there are definitely more layers to this question, but the core approaches remain almost exactly the same. So let's start with the rate chart to make sure that we're organized and give ourselves the maximum odds of success at connecting the many dots that need to be connected to make a good approach on this one. Uh so again, I'm just gonna write rate times time equals distance across the top, and then I'll make a row for up the hill, and then I'll make a row for down. Now, what's interesting here is that I'm given the average speed across the whole journey. And that's a little bit unusual, actually. Uh so what I'd probably do there is make a total row in the rates chart, and that way I can divide the total distance by the total time in order to arrive at the six miles per hour that's fixed in the question. Now, you don't need to see that far and specifically into questions at this stage in in your prep. In fact, you never need to like have that kind of x-ray vision. It's helpful, but you don't need to develop it. If you're looking for a simpler approach, then just put you're just like, okay, well, they told me six miles an hour for the for the average speed. So I I should make another row for that. And if you go back to any of our rate time distance lessons in the past, I've talked about when to make new rows in the chart. It's either when they introduce a new situation or a new a new data point that's not covered by the existing rows, or they change units. So sometimes you'll see these rates questions where they give you miles per hour, but then they ask for the answer in kilometers, or they'll give you dollars per liter, but then they ask you for the total, the total uh time that it would take to travel on a full tank of gas. If if Jimmy has $20, you know, they're kind of swapping between units between the question. So the best way to handle those situations is just make a new row in the chart. It happens somewhat frequently. So that's what I would do with that six miles per hour, because you'll notice, like, okay, I've got up the hill and down the hill. So then where do where does the average speed come in? That's again the total distance over the total time, and that's why I labeled that third row total. So the more of these questions you do, the more of an intuition you'll develop for how to make the chart work well for you. I just trying to give give you the kind of the heads up that it's okay to make new rows. And in general, it's probably better to have too many rows than too few because you can always collapse data from too many rows. But if you try to squeeze too much information into too few few rows, that can get more confusing. Um, again, you'll develop a good intuition for this as you practice over the next uh days with race questions. And there's plenty of sources online. You could you could just get a hundred race questions right now. There's there's so, so many, uh, not least of which is the official guide itself, which I definitely recommend picking up if you can afford it. And the reason I wanted to go long on that at that point is sometimes the way I talk through these problems makes it seem like you should have that X-ray vision of like, oh yeah, I I should be able to see those like next few steps of oh yeah, I'm gonna divide the total distance by the total time. That's totally fine if you don't see that. And and we're gonna talk about some of the leaps of faith that need to be made in these race questions on the next question. So let's just let's just hold that thought. Don't worry if you're like, wait a minute, I wouldn't have seen that to make the total row. You you would have figured it out eventually, of like, okay, six miles an hour, that's not fitting into either of these rows. I need a new row. Uh quick side note. I've found it's it's really valuable to write out total distance over total time equals average speed at some point in these problems, because the temptation to average the two individual rates is really, really strong. The key is to realize that average speed is almost never the simple average of two individual rates from each part of the trip, even though that's that intuitively seems to make sense. Uh, because if we're traveling along the same route at different speeds, then we will definitely spend different amounts of time traveling those distances. So if you're going 60 miles an hour up the hill and you're going 30 miles an hour down the hill, it's gonna take way longer to go down the hill, twice as long. And so if you try to average those things, you're not making a fair comparison of how long the person spends going the slower speed. So the actual average is gonna be closer to the slower speed because the person's spending more of the total time of the trip going that speed. So that gets into a deeper topic called weighted averages, which we're not gonna touch right now. I just want you to get ahead of the whole, like, oh, I'll just average these two individual rates and everything will work out well. Like that, that again almost never works on average speeds questions. And and the best defense against that is just write out at the beginning of the problem, as soon as you see the phrase average speed, total distance over total time in a fraction. And I've found that as simple as as that sounds, it's it's a great defense against that uh pitfall, that very, very common pitfall. Okay, so back to my total row in the chart. Let me write six miles an hour under the rate column. And I've got a pretty empty chart at that point. So let me go back to the problem and see what I might have missed. The key phrase, and you'll probably see this a lot in average speed questions, is quote unquote along the same route. I sort of hinted at this when I was talking about like not averaging the individual speeds. It's not always obvious, but what that means is that the distance up the hill and down the hill are exactly equal. Now, we don't know what that number is, but that's okay. We can make a variable for it. And the key is just gonna be to write that same variable for both the up distance and the down distance. If you've been studying for a while, that might just sound like painfully obvious. Like, why is this dude even saying this? Uh, but if you're just starting out, that might not be very obvious at all. So I just want to try to emphasize the stuff that's gonna help all of you no matter where you are in in the journey. So uh let's just go ahead and write that in now. And I'll just make the variable D for distance miles. So under the distance column in the up row, I would write D miles, and then right underneath that, I'll write D miles again for the down row. So it's looking quite similar to our warm-up problem, except we don't have a number, we have a variable in that placeholder. So if you're on the pure audio feed, just do your best to visualize this. I'm gonna try to spell it out really, really clearly. And uh that should help next time you're trying to organize a rates question. So going back to the problem, we've got a relationship between the rate going up the hill and the rate going down the hill. But again, we do not know what those values are. Uh so we'll need to make another variable. I'm gonna choose R for her rate going up the hill, and then I can use two times R for her rate tumbling down the hill. So let me just write R under the rate column to the right of up, and I'll write two R under the rate column in my down row. Okay, so from here, if you step back and look at the chart, you'll realize we're actually in a pretty good position because we have enough data filled out to solve for the time that she spends going up and down the hill. And that's really good news. Because it seems like if I can find her total distance and her total time, then I'm gonna be able to make an equation with that six they gave me at the beginning. So this is where the leap of faith comes in, I think, for most of us. Some of you might have this X-ray vision into the problem where you can see like four, five, six steps ahead. I'll be honest, like I don't have that. And so that's why I've developed a lot of these frameworks and and scaffolds to just help my uh different intelligence, if you will, uh, to still be able to score high on the test. And I think you'll find, even if you're super, super intelligent, that that scaffolding still works extremely well. It builds, it it helps with speed, even though it might feel slower in the short run. I think you'll find in the long run you'll you'll end up starting questions slower but finishing them faster. A little bit of a paradox there. But um, again, just look at the data. If I'm wrong about that in your specific case, obviously you can make adjustments. But um, just just give it a test, a fair shake, collect some data on it. And um, yeah, I I hope it works as well for you as it's it's worked for many, many folks that I've worked with over the years at this point. So back to the leap of faith thing. Personally, I wouldn't be a hundred percent confident that these moves that I'm about to walk you through are gonna work out, but I don't have any other moves to make. And so I'm thinking, like, well, I've got gaps in the chart. Let me solve for those, and let's just see what happens. I'm I'm not, I haven't convinced myself that I totally know what I'm doing, but I haven't convinced myself that I don't know what I'm doing. So let me just take that leap of faith and I can always stop working on the question and guess and let it go if I run out of gas. So I'm gonna first solve for the time going up the hill, and that would be D divided by R, because again, rate times time equals distance. I can just make an equation, pop that out of the chart if I want to, and uh use D for distance, T for time, and then R for rate. And then to solve for the time, I would just divide both sides of the equation by R to get T equals D over R. Again, we've we've got some math basics on algebra and equations linked for you below as well, if needed. I can perform a very similar move in the down row, dividing d by two r to get the time for that is d divided by two times r. Good news there, my leap of faith paid off because now I can see that if I sum those two times, I can fill in the time for the total row at the bottom. So that's what I'm going to do. Uh I can I can uh first add up the distances, but that's probably even easier because I can see, well. With D miles up and d miles down. So the total has to be two times D. So I'll put that into the total row at the bottom. And then I'll I'll just probably write the sum of D over R plus D over two R, even though I don't have a common denominator yet. So I'm just going to write it like that for the moment. And I found that just taking one step at a time when using the chart is a good bet. So again, you can check out that fractions math basics episode if you if you forgot about common denominators. But I'm actually in a really good spot here if you can visualize just that bottom row of the chart filled out because I've got uh I've got an equation that I can simplify. I've got six for the rate, I've got that sum of fractions for the time, and then I've got two D for the distance. So I'm not in the best position because I've got two variables and and one equation, and that doesn't always work out. But again, I'm just gonna take another leap of faith here and see what I can do. Now, the math here is gonna get maybe a little bit wild with all the variables floating around. So if you're on the pure audio feed, just do your best to track it. But at the end of the day, the real key to these questions is a good framework for the setup. So even if you're not following every single step of this algebra I'm about to do, that's totally fine. And also like the exact way that I do algebra is not particularly important. You can use your own style there. Uh but having a firm and consistent setup, that's that's really, really key. So hopefully I've I've made that crystal clear already. So here I go with my equation. Again, I've got six times d over r plus d over two r equals two times d. So I'm going to start by getting a common denominator for the times because that's kind of ugly and bothersome. Uh I'll multiply d over r times two over two. So that'll get me two d over two r, and then I'll have two d over two r plus d over two r. And that gets me three times d over two times r. And what I will probably do from there is isolate the six on one side of the equation. I'm gonna see if I can get some of the variables to cancel out by dividing. So I'm gonna divide the 2d by the 3d over 2r. If you can imagine 6 times 3d over 2r equals 2d, I'm gonna divide by the time to isolate the rate. So when I'm dividing by a fraction, that's the same as multiplying by the reciprocal. So I'm going to write uh 2d over 1. And then I'll write times 2r divided by 3d. That's the reciprocal of the fraction I'm dividing by. So let's multiply across the top and bottom. That's gonna get me 2d times 2r on top and 3 times d on the bottom. Uh I can cancel the d in the numerator and the denominator. So that bet actually worked out really well. Again, I don't know that I would have had that vision into the problem for the beginning. So if you're like, whoa, like how did that happen? It's it's just um it's just it's just like muscle memory of using the chart and muscle memory of doing algebra. And so if you get a decent number of reps on these, I think that'll gel for you. But uh, if not, we've got tons and tons of episodes about how to select a better provider and how to get more support if if you're struggling with this stuff. Uh, but if you're on the like totally free self-study path, at least give yourself a few reps. If you're on the more structured pay as you go path, uh or you or you have a program that you really like, they've probably got a system for you to reach out to them and get support if you're stuck with rates or how to perform the algebra at any point. But I just want you to know those, that that amount of uncertainty where I have to take a little bit of leap a leap of faith and say, like, hmm, let's just see what happens if I fill out the rest of the chart. Or this equation looks kind of weird, but let me just try to solve it anyway. That that kind of stuff is um, I think part of being a great problem solver in general, but it certainly pays on the GMAT uh because even if it's the wrong move, sometimes it will reveal to you what the right next move is. And I think one of the worst myths out there, I think it's kind of like an unspoken myth, is that there's always going to be like a clean textbook predictable solution to every single GMAT problem that you see. And I actually think that's not a hundred percent true. While there are like clean textbook solutions available for a lot of problems, those aren't always practical in the moment. And it's not always practical, especially if you're on a tight timeline with this, to memorize a specific solution path for every variation of every problem they could create. Uh, so that's a huge reason why we've made the pod and all our free products, uh, which you can find at the link below that says want more free help. It should say want more free help in the description. And you should just be able to click a link and see a bunch of free products that we have. We have we have videos, we have uh ebooks, we have uh the podcast, obviously, we have um we have all kinds of stuff to help you uh if you if you can't afford more support. And uh the the main point that I'm making there is the reason that we put all that stuff out is so you have a simp simplified frameworks that you can use in a wide variety of situations and a realistic expectation that you will you may have to improvise with some frameworks, especially on test day. You'll see versions of stuff you've seen in your practice, but you may not see like problems that are identical just with different numbers. That's very, very rare with the GMAT. That's why they call it a reasoning test. It's not like a, hey, memorize all this stuff and get a great score test. Like some of the tests you might have taken in the past, uh, like Finra stuff, you know, is a little bit more memorization based, less reasoning based. And that makes sense if you think about the purpose of those exams. Whereas the purpose of this exam is to try to become a better strategic thinker, try to become a better resource allocator. That's why there's limited, like very, very limited time, sometimes not enough time to answer every question. You have to like strategically let go of some questions that are bad investments, double down on questions that seem like good investments. You know, there's this whole new layer on top of just, hey, get questions right. Um, and it's it's it's usually good for us to go through that process. Like it makes us better professionals, uh, but it but it it can be trying if you're ex if you're not expecting it, if you're expecting it to just be like a raw memorization test. Like you take one of these programs where they're like, oh yeah, just memorize all our stuff and you'll do great. And then your score isn't that great on test A. That's a bummer. So just trying to help you have a realistic plan of attack so you can get ahead of that stuff and and just just be like, okay, it's okay if my pre-baked framework gets me like 50% through the problem or 80% through the problem, and then I need to just like do a leap of faith and try some stuff and see what happens. Uh so that's totally, totally normal. And uh we'll we'll talk more about the scoring algorithm later, but for now, there's a a free video on our website. You can check it out, and I'll go through in detail about how the scoring algorithm works. That's that's way beyond the scope of this lesson, but that'll help you realize like why you may end up missing more more questions than expected, but still be able to get a great score. And that that was something that really held me back when I was in your shoes, is I just didn't understand that. And I was just trying to get 100% of the questions right 100% of the time. And uh a small, small, small, small percentage of people can do that. Um, but even super, super bright people who have never struggled academically, that's that's really not realistic on the GMAT because of the way the scoring algorithm works. So check out that video if you haven't already. And that'll help you realize why it's okay to strategically miss questions here and there and how you can actually get a higher score by answering fewer questions, uh, which is a real mind blow if you haven't thought about that before. So anyway, that's there for you. Let's let's get back to the straits questions and finish it out. And sorry for the the long aside there, but um I I think it's well worth hammering those points over and over and over again because it's just so different than what we would naturally expect. And I don't want you to lose belief in yourself just because someone didn't didn't give you the full picture of what this test is really about. I really don't want that. Okay, so back to our equation. If we simplify, we would now be at a place where we have four times r over three equals six. And that's really good news because now I can solve for r. Uh to do that, I'll just multiply both sides by three. That gets me four r equals eighteen, and then r equals eighteen divided by four. So again, a couple of leaps of faith involved there. It's worth stepping back and noting, again, I wasn't 100% sure that was going to work out, but I had enough confidence to make the bet because I had a fairly organized rate table, and I also knew the cornerstone of average speed questions is total distance over total time. So I'm like, okay, I got the table, I got the formula. It's highly likely I'm doing the right thing here. Um so uh while the rate chart might have been a little bit of overkill on the previous question, um I still recommend using it 100% of the time because it has a lot of value in these situations where it can give me the confidence to proceed, even if I'm like, wait, am I on the right track? So that's a kind of a side benefit or unexpected benefit of being organized. Uh so we have R equals 18 over four, but that's that's her rate going up the hill, and that's not what the question's asking us for. The question's asking us for her rate going down the hill, which is double the rate going up the hill. So we need to multiply R times two. Two R would equal 18 over four times two over one. That's 36 over four, and that would get us nine miles per hour, which leads us to our correct answer of option C, again, as in Charlie. Okay, at the risk of over-explaining or just like reiterating these points too much. My best advice there is use the rates table 100% of the time and then write out the average speed formula, total distance over total time 100% of the time when you see average speed. If you're super crazy robotically consistent with that, it's going to solve tons and tons and tons of problems before they happen. And it's extremely unlikely to hurt your performance on any rates questions, just taking, I don't know, five seconds to write those things out. Um, obviously, in terms of filling out the rates chart, it might take longer, but again, that comes back to the like slow start. Usually you'll fit uh have a fast finish. Now, you might have struggled with the algebra a little bit, but that's very learnable with raw repetition in my experience. So don't be afraid to put in those hours if if you need them. And I've got some advice for you in the Math Basic series around how to get those reps in intelligently if you need it. And if you need support or you're not sure how to engage with that material or you're stuck with your algebra and you're like, wait, how do I do this? Um, I'm really struggling to get this to connect. Uh, you can always DM us at the G Mat Strategy on current social channels. You can email us, we're contact C O N T A C T at the Gmat Strategy.com. Or you can click on that want more free help link I was telling you about in the description. Click on that, and there'll be a contact us section near the bottom. And uh you should be able to just click those links to get in touch with us if you're on mobile and you don't want to type all that stuff into uh anything. So, what I tried to give you there is again connecting threads and simple approaches that'll work for a wide variety of situations and a wide variety of people. That's important. A lot of us have different performance strategies. Some of us are gonna perform with uh approach A, some of us are gonna perform better with approach B. That's totally natural and fine. But just those two things I talked about, the rates chart and writing out the formula 100% of the time, that solves like 99% of the bottlenecks that I see most people experience on these questions. So again, if there's a note to self there, I recommend pausing now and making that note, and then we'll jump into one final question that's a little bit harder. Last problem. The problem says during a trip on an expressway, Don drove a total of X miles. His average speed on a certain five mile section of the expressway was thirty miles per hour. And his average speed for the remainder of the trip was sixty miles per hour. His travel time for the X mile trip was what percent greater than it would have been if he had traveled at a constant rate of sixty miles per hour for the entire trip. I'll read that again. During a trip on an expressway, Don drove a total of X miles. His average speed on a certain five-mile section of the expressway was thirty miles per hour, and his average speed for the remainder of the trip was sixty miles per hour. His travel time for the X mile trip was what percent greater than it would have been if he had traveled at a constant rate of sixty miles per hour for the entire trip. Option A is eight point five percent, option B is fifty percent, five zero, option C is X divided by twelve percent, option D is sixty divided by X percent, and option E is five hundred divided by X percent. As always, I recommend pausing and I'll jump right in. This one can be a little intense. But the great news is that the the rate chart and the average speed formula are gonna do most of the heavy lifting on the setup. There's still an extra layer with the whole percent change thing, and we'll we'll get to that in a moment. But the basic foundation is gonna be remarkably similar to the first two questions we did. So let's start with a good setup. Let's write rate times time equals distance across the top. We're gonna make a row for the first five miles, and I'm just gonna fill in five miles for the distance there, and then fill in 30 miles per hour for the speed. Then we'll make a row for the remaining distance, and then write x minus five under the distance column. Uh, that's because he traveled a total of X miles, but he had a different speed for the first five miles. And so I'm gonna subtract away that first five miles to get the total distance that he goes the 60 miles an hour, not the 30. So then I'm gonna write 60 miles an hour under the rate column in a row that I'll call like remaining or something like that. So the first row I'd label as like first five, and then the second row I'd label as remaining, something like that. So now I've got a nice visual of the different speeds and distances of the two legs of the trip. And from there, similar to the first two questions, I'm able to solve for the times that he took in each leg of the journey. Uh, and that's great because I want to know the percent change in the times. Uh percent percent greater is just a specific type of percent change. Again, let's just table that for a moment. We'll come back to that. So let me solve for the times. The the first five miles isn't too bad. That's just five divided by 30. Again, I'm just doing rate times time equals distance, popping that outside of the chart, doing some simple algebra, or just intuiting if I can do that in my head. Um usually recommend doing as much as possible on paper and as minimal as possible in your head. I know that's a very polarizing recommendation. A lot of people who claim to be very successful, um I know have different opinions about that. And I have not verified their claims about their success, uh, but I have verified that for 99% of people I've worked with, they get significantly better results by just writing stuff down rather than doing stuff in their head. So look, if if you're getting amazing results by doing all your math in your head, then don't stop. Keep doing what you're doing. But it but if you're like, man, I've been trying to work on this mental math thing for a whole month and it's still not taking me where I want to go, I would say just stop doing that and just write it out. That's that's my advice there. So the remaining X minus five miles would take me X minus five over 60 hours. And I'm going a little faster through the math operations because we've done a ton of that. Rate times time equals distance, divide both sides by the rate to solve for the time. We've done that like, I don't know, six, seven times now in the lesson. So that's why I'm going a little faster there. But um, feel free to check out the video feed. Uh, if there's no video available on the platform you're on, we should have a YouTube link in there. Uh, but we're making a big push for video, like I was talking to you about a couple weeks back. So hopefully that is in the feed on the platform you're you're on, and you could just switch that on, even if you're in the gym or something, and uh see what I'm talking about there. So I think I would probably once again make a total row in the chart to capture the the total time that he takes, which would be five over 30 plus x minus five over 60. Very similar to that previous question where we added the the two times. So let's um let's find a common denominator and then we'll add. So 10 over 60 is the same as 5 over 30. So I'm gonna use that. And I'm gonna add that to x minus five over 60, and I'll skip over all the computation. That's gonna be x plus five over 60. Again, don't worry about the specific math here. Just focus more on the process workflow and the tools that you can use on future problems. Now, this is good news, uh, but I am still missing the time for the hypothetical situation that the problem's asking about. What if he went 60 miles an hour the whole time? So that first thing I just solved for x plus five over 60, that's the actual time that it took him, given that he went 30 miles an hour for a little bit and then 60 miles an hour for the rest. So now let's calculate the hypothetical thing. And I'd probably just put hypothetical or HYP for another row in the chart below that total row I just made. Again, don't fear making too many rows in the chart. You can always collapse them later if you make too many. So I'll write 60 under the rate column, perhaps obviously, because he's going 60 miles an hour for the whole spiel there, and then X miles under the distance column, because that's the full uh full distance. And perhaps obviously when I solve for the time there, I'm gonna get X over 60 hours. Okay, so it's not looking too simple here, but because I have the whole chart filled out now, I still feel enough confidence to be like, maybe I can make this work. So I'm just gonna proceed to the final step. Again, it might be a little bit of a leap of faith here. But I'm looking at it and thinking, okay, all I have to do is calculate the percent change. So as long as I haven't made a big mistake here, things should work out in my favor. Now, you may or may not recall the percent change formula is new value minus old value divided by old value, and then that whole thing times 100. If you didn't know that, that's worth memorizing as well. We've also got a bit of instruction on that in the percents section of the math basics program linked below. So feel free to pause and check that out in return here when you're ready. But most of you will have been sitting for a little bit and you probably memorize the percent change formula at this point, but just in case you're just getting started, no big deal. We're gonna make the new value x plus five over 60. That's the time it takes him to make the journey as is, like we said. And we are specifically told in the problem that that is greater. So I'm just gonna make that the new value so that my uh percent change is positive. You still would get the right answer if you made that the old value. Um actually don't think about it that way. I'm I'm just gonna redact that. Let's make this really simple for you. Um, when you have greater than or less than, make the thing that comes after than the old value in the percent change formula. Just do that 100% of the time. That's gonna solve any confusion or any problems that you might have with consistency there. Because like technically, mathematically, there's a way to make the other way work, but it's just too convoluted. Don't think about it that way unless you have like a lot of advanced math under your belt. Okay. So what I said there, just to repeat that, if you want to pause, make a note to self here. If you haven't heard this before, if you have a greater than or a less than, use the percent change formula and make whatever follows than the word than, there will be a value after the word than. Make that the old value. So here it says blah, blah, blah, was what percent greater than the time it would have taken if he went 60 miles an hour? So now I know the hypothetical 60 mile an hour the whole time thing. That's the old value. It comes after the word than in greater than. Okay. Again, sorry for the long uh asides here, but it's it's really important. I'm never gonna go long on something that's not super, super, super important for you. I value my time too much, I value your time too much. But there is a time and a place to be like, hey, let's let's really pin this down. This is important. Okay, so our entire formula of new minus old over old times 100 would be x plus 5 over 60 minus x over 60, that whole thing divided by x over 60, and then the whole entire fraction times 100. So let's do one step at a time. Let's start by performing the subtraction in the numerator. That's gonna get us five over 60, because uh x plus five over 60 minus x over 60, that's just x plus five minus x divided by 60. Again, go back to the fractions thing if you're unsure how that works. Uh so we've got five over 60 on top, let's divide that by x over 60 on the bottom. That's the same as multiplying by 60 over x. That's gonna get us five over 60 times 60 over x. The 60s cancel in the top and bottom, and that gets us five over x. Then we just need to multiply by 100 to complete the formula, and that gets us 500 divided by x percent. And that's the same as option E as an echo, which is our correct answer. So again, just speeding through a little bit of the math there, because the actual computation, usually not the bottleneck for people on these questions. If you need help with computation, we did a full episode on how to improve your computation skills as the previous episode in the feed to this one. Now, stepping back, not too hard to see why that question is a little bit tougher than the first two. About 10% of us missed the first one, 20% of us missed the second one approximately. I think it's like 22%, and then about 30% of us missed this one. So we've got multi layers of algebra, we've got a uh larger rate chart with more rows, we've got a percent change layer as the Final step. So there's many more connections to make, and by virtue of that, more places where we could make mistakes. Uh, but it but I just want to re-emphasize this here. If you're comfortable with the rate chart and the concept of average speed, you should have a very, very good chance at getting this question right. Maybe not perfect. Yeah, maybe you forgot the percent change thing or set it up wrong. Hopefully I gave you some tools to fix those. But at the end of the day, like you're hitting that last layer of the question in a very, very strong position. And um sometimes that's as good as it gets. So one final time, if there's something you feel that's valuable for you to recall from that question or any of the questions we discussed today, I recommend pausing here and again just make that note to self however you like. And then we'll wrap up. Again, if you have questions about anything we presented here or really anything about your study process, you can reach us anytime at the GMAT Strategy on Current social channels. You can email us c O N T A C T at the G Mat Strategy.com, or tap the link in the description that says want more free help, and there'll be clickable links to our contact channels there if you're on mobile. If you are struggling in your preparation and you're not seeing results at all, like you've seen score drops or you're at a score plateau, or you're not seeing results as fast as you would like with your current provider or your current plan, we specialize in accelerating your progress faster than anyone else can. And we've got a ton of data to back up our ability to do that. If you're interested in working with us directly, just head to our website, thegmatstrategy.com, and check out our free presentation on how you can reach your dream score faster and easier. If you feel like the material in that video resonates, you'll be able to book a call to speak with us live. And we will figure out together if working together is a good idea. 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Um, and that's why we do the calls so that we can customize that for you and make sure that we're giving you the right advice. Uh but if that does not sound good to you or doesn't sound like that's a fit for your situation, that's completely fine. Or you're seeing results already uh and and you're getting them as fast as you want to, that's amazing. And there's definitely no need for you to reach out to us. Um if you're curious, we we do look forward to connecting. We always love speaking with all of you who we do get to speak with. It's uh a really awesome thing uh because we struggled with the process quite a bit. Like I said, you can learn more about my story of pain and strife uh by watching that video on the website if you are curious. And if you're not, that's cool too. And if none of this is sounding good to you, that's totally fine. We'll be back soon to provide more free advice for everybody here. In the meantime, please just stay positive and stay consistent with your studies, everybody. You can do it. Talk to you all soon.