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Posted on
07
Jul 2020

GMAT Problem – Speed Distance Problem

Speed and distance problems are among the most complained about problems on the GMAT. Numerous clients come to us and say they have difficulty with speed and distance problems, word problems, or work rate problems. So we’re going to look at a particularly difficult one and see just how easy it can be with the right approach.

The Two Cars Problem

In this problem we have two cars – car ‘A’ and ‘B’. Car ‘A’ begins 20 miles behind car ‘B’ and needs to catch up. Our immediate DSM (Default Solving Mechanism) is to dive in and create an equation for this and that’s exactly what we don’t want to do.

These types of problems are notorious for being algebraically complex, while conceptually simple. If you hold on to the algebra, rather than getting rid of it, you’re going to have a hard time.

Solution Paths

In this problem we’re going to build up solution paths. We’re gonna skip the algebra entirely. We’re going to take a look at an iterative way to get to the answer and then do a conceptual scenario, where we literally put ourselves in the driver’s seat to understand how this problem works. So if we want to take the iterative process we can simply drive the process hour-by-hour until we get to the answer.

Iterative solution path

We can imagine this on a number line or just do it in a chart with numbers. ‘A’ starts 20 miles behind ‘B’ so let’s say ‘A’ starts at mile marker zero. ‘B’ starts at 20. After one hour ‘A’ is at 58, ‘B’ is at 70 and the differential is now -12 and not -20. After the second hour ‘A’ is at 116, ‘B’ is at 120. ‘A’ is just four behind ‘B’. After the third hour ‘A’ has caught up! Now it’s 4 miles ahead. At the fourth hour it’s not only caught up but it’s actually +12, so we’ve gone too far. We can see that the correct answer is between three and four and our answer is three and a half.

Now let’s take a look at this at a higher level. If we take a look at what we’ve just done we can notice a pattern with the catching up: -20 to -12 to -4 to +4. We’re catching up by 8 miles per hour. And if you’re self-prepping and don’t know what to do with this information, this is exactly the pattern that you want to hinge on in order to find a better solution path.

You can also observe (and this is how you want to do it on the exam) that if ‘A’ is going 8 miles an hour faster than ‘B’, then it’s catching up by 8 miles per hour. What we care about here is the rate of catching up, not the actual speed. The 50 and 58 are no different than 20 and 28 or a million and a million and eight. That is, the speed doesn’t matter. Only the relative distance between the cars and that it changes at 8 miles per hour.

Now the question becomes starkly simple. We want to catch up 20 miles and then exceed 8 miles, so we want to have a 28 mile shift and we’re doing so at 8 miles an hour. 28 divided by 8 is 3.5.

Conceptual scenario solution path

You might ask yourself what to do if you are unable to see those details. The hallmark of good scenarios is making them personal. Imagine you’re driving and your friend is in the car in front of you. He’s 20 miles away. You guys are both driving and you’re trying to catch up. If you drive at the same speed as him you’re never going to get there. If you drive one mile per hour faster than him you’ll catch up by a mile each hour. It would take you 20 hours to catch up. This framework of imagining yourself driving and your friend in the other car, or even two people walking down the street, is all it takes to demystify this problem. Make it personal and the scenarios will take you there.

Thanks for the time! For other solutions to GMAT problems and general advice for the exam check out the links below. Hope this helped and good luck!

Found it helpful? Try your hand at some other GMAT problems: Profit & Loss Problem.


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Posted on
11
Jun 2020

Snack Shop GMAT Problem

The Snack shop GMAT problem is an average or a mean problem. A characteristic of many average problems is that one big takeaway right at the outset is that the answer choices are clustered tightly together. We want to refrain from making any calculations.

The problem is below:

snack shop problem

Selecting A Solution Path

If they’re looking for a level of precision, the estimation solution path isn’t available to us. If we dive into the problem, right from the first sentence we have sort of a conclusion that we can create via either a graphic or accounting solution path.

If you were the business owner immediately you’d say to yourself: Well for 10 days and an average of $400 a day I made $4000. 

This is how we want to think about averages. Many times they’ll tell us a parameter about a length of time or over a certain universe of instances and here we want to treat them all as equal.

Solving the Problem

It doesn’t matter if one day we made 420 and another day we made 380. We can treat them in aggregate as all equal and start out with that assumption. That’s a very useful assumption to make on average problems. So, we start out knowing that we made 4,000. 

What I want us to do is do a little pivot and notice from a running count standpoint how much above or below we are on a given day. So we’re told that for the first six days we averaged $360 which means each of those six days we’re short $40 from our average. That means in aggregate we’re short $240. 6 days times $40 –  and this has to be made up in the last 4 days.

Notice how we’re driving this problem with the story rather than with an equation. In the last four days we need to outperform our 400 by 240. 240 divided by 4 is 60. 60 on top of the 400 target 

that we already have is 460. Therefore, our answer is D.

Graphical Solution Path

If we are more comfortable with graphic solution paths, imagine this in terms of 10 bars each representing $400. Lowering six of those bars down by 40 and taking the amount that we push those first six down and distributing it among the last four bars gives us our $460 total per day.

snack shop graphic solution path

If you enjoyed this Snack Shop GMAT Problem, watch “The Gas Mileage GMAT Problem” next.

 

 

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Posted on
14
Mar 2020

Remainder Number Theory Problem

Today we’re going to be looking at this problem and our big question is that originally we’re given this unknown number N and we know we just have a remainder 3. So the problem is presenting us information in a way that we’re not used to seeing it and what we need to do is work backwards from this to drive the core insights.

Sorting Through the Information

So if we have a remainder of 3 on 23 this means that the chunk that isn’t remainder is 20. So what can our n be in those cases that will allow us to divide out by 20 and leave this remainder 3.

Well first we know that n has to be greater than 3 because in order to have a remainder the amount we’re dividing by has to be something greater. The moment the remainder equalizes the thing we’re dividing by of course we get one more tick in the dividing by box and the remainder goes back down to zero.

Solving

So with 23 and a remainder of 3 our key number to look at is 20. Our factors of 20, that is the things that divide evenly into 20, are 1, 2, 4, 5, 10 and 20. Of course 1 and 2 are below 3 and so they’re not contenders. So we end up with n being 4, 5, 10 or 20.

Check Against the Statements

So for number 1: Is N even? If N can be 4 but can also be 5 then we’re not assured that it’s even. Notice the data sufficiency problem type embedded here. So N is not necessarily even.

Is N a multiple of 5? Once again N is not because N could be 4 or 5. Finally, is in a factor of 20? And in this case it is because 4, 5, 10 and 20 as we just said are all factors of that 20 that we’re looking for. So our answer here is 3 alone, answer choice A.

More Practice

Now here’s a more challenging problem at the same form, see if you can do it and we’re going to come back and in the next video talk about the solution and give you another problem.

So if 67 is divided by some integer N the remainder is 7. Our three things to look at are whether:

    • N is even?
    • If N is a multiple of 10?
    • Or N is a factor of 120?

So give this one a try and see if you can use the principles from the easier problem on this more challenging one to make sure that you actually understand what’s going on. If not, re-watch this video and see if a review might allow you to answer this question.

If you enjoyed using this video for practice, try this one next: Wedding Guest Problem.

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Posted on
21
Jun 2019

Number Theory Problem Form – Wedding Guest

Today we’re going to be looking at what at first seems to be an allocation problem. But on further reflection, actually turns out to be a much simpler number theory problem.

If we take a look at the problem stem these numbers 143 and 77. They stick out to us and they stick out not just because they don’t seem to have any relative association but also because they’re sort of odd-looking numbers, they don’t look like most the numbers were used to seeing. Say 48 or 24 or 36 something easily divisible clearly breakable into factors. Here were given these two disparate numbers and we’re being asked to formulate not what the tables are made up of but how many tables there are.

Solving the Problem

So we look at these two numbers and we examine first the 77 because it’s a simpler lower number. 77 breaks into factors of 7 and 11. This clues us in as to what to look for out of the 143. 143 must have a factor of 7 or 11. And in fact 143 is evenly divisible by 11 and it gives us 13.

Which means that the maximum number of tables is 11. Each one has 13 people from the bride’s side and 7 people from the groom’s side. 13 plus 7 there’s 20 people at each table. Times the 11 tables is 220.

And we can back check our math, 143 plus 77 is 220. We don’t need to go that far but that might help deliver some comfort to this method. So in reality this is a very creative clever way the GMAT is asking us for a greatest common factor.

Graphical Solution Path

Another way to think about this is that we need an equal number of groups from the bride and groom side. The number of people on the bride’s side doesn’t have to equal the number of people on the groom’s side. We just need them broke in into the same number of equal groups. Graphically, the illustration shows us how a certain number of different sized groups combined into this common table. So 13 and 7 and we have 11 groups of each.

Problem Forming

This is a great problem to problem form. It will give you some additional mental math or common result experience by forcing you to figure out numbers that you can present that at first don’t look like they match but in fact do have a common factor. You’ll notice that if they had given you 16 people and 36 people finding that common factor might be easier.

So as you problem form this try and do it in a way that sort of obscures the common factor. Either try it maybe where they have multiple common factors and you tweak things like what is the greatest number what is the fewest number of tables. Or even do a perspective shift and take a look at say how many guests are represented at each table. Or on the bride’s side or on the groom’s side. Of course there are conceptual shifts to this and you can make this story about anything. Once you control the story or rather the structure of the story this problem becomes very very straightforward.

Hope this helped, I look forward to seeing you guys soon.

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Posted on
12
Feb 2019

If you’re doing math on the GMAT, watch this.

I want to discuss one of the core tenents of Apex’s quantitative philosophy on the GMAT. “If you’re doing math, you’re doing something wrong.” Meaning, if you find yourself doing math, that’s a signal from the exam that you’re using a sub-optimal solution path. By math I don’t mean any calculation whatsoever, but any calculations that aren’t reasonable. That don’t come out easily, neatly and cleanly, once you’re well practiced with mental math. So it’s not that we’ll never do a calculation, but every calculation we do should be deliberate and smooth.

The Most Overused Solution Path

Let’s go a little deeper into this, because it’s a really important concept. Many, many people preparing for the GMAT spend way too much time worrying about the math. Being freaked out about the math and on the exam doing the math. The applied mathematical solution path is the most over used solution path on the quantitative side of the GMAT. Particularity among engineers, and with people who do a lot of self-prepping. They look to the back of the book or look to previous experience as students. And get caught up in the idea that their answer needs to be precise. This gets in the way of using our estimation solution path or other higher solution paths, which can get us to the correct answer much more quickly.

The GMAT isn’t Testing Your Math Skills

How do we know that math is not what the GMAT wants us to do? It’s quite simple. If the GMAT was the referendum on how well you can do mental math, then the scores would reflect your ability to do so. MBA programs at top business schools would be filled with people with extraordinary, almost savant like mental math abilities. We know this isn’t the case.

Actually, as we improve on our mental math, we get diminishing returns with it. So we see a lot of clients getting up to the 70th, 80th, or 90th percent level even, on the quantitative side of things. Then, all of a sudden they plateau; they can’t get any higher. The reason is they are so focused on the math. They are missing the bigger logical reasoning picture or the structure of quantitative problems that doesn’t rely on doing math that allows both quick and accurate solutions.

Key Things to Avoid

While math has its place, we want to be sure that we’re not putting it on a pedestal. And that when we’re performing computations, we’re doing so with great deliberation, intentionality, and that we have a good reason for doing any computation we’re doing. If you find yourself diving into the equation or doing a lot of processing, stop. Say “Wait a minute, there must be a better way to do this.”

Another option is that sometimes you make a basic error early on and that leads to ugly numbers and math. But you should never, never, never be multiplying decimals out to the fourth decimal. That sort of math is the true trigger, the true signal, that there’s a better way to solve the problem. When you’re self prepping, this is what you want to look for.

So by the time you get to the exam, you’re not catching yourself doing math, but you’ve already incorporated it into your process, the fact that math shouldn’t be your default.

So, remember, guys, if you’re doing math, you’re doing something wrong and you can take this one to the bank.

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Posted on
12
Feb 2019

Profit & Loss Problem Form

The profit and loss problem form that this problem fits into is one that has strong DSM’s into mathematics. Here we are tempted to do the math in part because that’s so easy. It’s so available to us.

This is characteristic of a mid-level arithmetic problem where there’s some shifts and shimmies but overall it’s a fairly straightforward problem that utilizes no more than the four basic operations. So on the one hand this profit and loss problem is pre-algebra or even sort of grade school math. On the other this makes the solution path much more elusive.

Solving the Problem Using Math

So of course we can follow the math. We can add up all the costs, five thousand plus two dollars, times twenty thousand. Then contrast that with the revenue that comes in which is 12×20,000. But then we’re left with the ugly division problem that brings us to the profit per t-shirt. This is where the GMAT sticks us.

Instead of handling this in aggregate it’s strongly preferable to handle it with a higher level solution path. Let’s take a look at a few:

Higher Level Solution Path: Distribution

One way to do this is to distribute the fixed cost over the cost per t-shirt. This is actually a lot easier than it seems. Twenty thousand t-shirts, five thousand dollars, five over twenty is one-quarter.

Therefore, it costs one-quarter per t-shirt in addition to the two dollars in variable cost. So twelve minus two is equal to ten dollars, minus one quarter is equal to nine dollars and seventy-five cents.

Higher Level Solution Path: Graphical Equalization

We can also use a graphic equalization method in order to get to the same conclusion. If the numbers were more complicated, understanding that that shift is one-quarter down. That is the fixed cost is one-quarter down.

Then we know we’re looking for something that ends in a seventy-five cents. That allows us to eliminate all the answer choices that don’t end in 0.75. Then we can use scale to determine that 9.75 is the correct answer.

Practice Problems

There are more complicated versions of this problem form. In particular, I’d encourage you to explore being told that the t-shirt company is breaking even. Then determining the amount of variable costs or fixed cost that’s there or even the production run. Similarly, you can be given a target profit or loss, the break-even just being the zero, so it’s a bit easier and have to reverse engineer the relationships.

Once again, this doesn’t have to be done algebraically. As you begin to appreciate the subtlety of the ratio between costs production run and total P&L all of these problems should be simplified and should be very straightforward.

Continue your GMAT practice with the Wedding Guest GMAT problem.

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Posted on
12
Feb 2019

Six Things That GMAT Preppers Get Wrong

I’m Mike Diamond Head Instructor for Apex GMAT, here to talk about the top six things GMAT preppers get wrong.

1. Thinking that a correct answer means you’re done with the problem.

When you arrive at a correct answer, that should mark the beginning of your preparation, not the end of it. There are almost always better solution paths that are more time efficient. They work better with the way your brain engages the problem. Or they will add understanding either to the content or more importantly to the underlying structure of the examination.

So, when you arrive at a correct answer look for alternative solution paths, and for shortcuts. Give yourself the latitude to explore. Moreover, try to identify what permitted you get to get the problem correct in the first place. A lot of times people focus much more on the problems they get wrong; on what they’re doing wrong than on what they’re doing right. And what you’re doing right can often inform those problems where you are struggling. So remember, once you arrive at the correct answer, that’s your starting point.

2. Overusing practice tests.

Practice exams are a crucial part of GMAT preparation, but they’re often misused and overused. Most people use a practice exam to see how they’re doing. But being focused on your score is absolutely the wrong way to approach the GMAT.

Rather, you want to be focused on your process and if your process is tight, if your process is correct. Then the score is going to take care of itself. Practice tests are best used for a number of reasons, none of which have to do with your score.

They can be used to calibrate your timing decisions. They can be used to identify weak points in your conceptual understanding. Finally, they can be used to identify where you DSM, default solving mechanism, back into old time consuming and unconstructive solution pathways. So, the next time you have an urge to do a test remember that this is going to rob you of two to three hours of valuable prep time. When you’re doing a practice test, you’re not learning, you’re doing.

3. Caring about your score.

I know it’s counter-intuitive, you want that 700-plus score. It’s all you think about; it haunts your dreams. And yet caring about your score is the quickest way to a test anxiety problem and it’s actually entirely unconstructive. Rather, you need to focus entirely on your process and let the score handle itself.

Imagine you’re running a race and you’re running as fast as you can. Whether you’re a super fit marathon runner or a couch potato, you can only run as fast as you can. And the time on that race is going to reflect that. So don’t sweat the score, sweat your fitness! Understand what things you can do to improve your GMAT fitness and the score will take care of itself.

4. Studying under a time constraint.

Time trials are really important as you mature in your GMAT progress. But at the start, you want to focus on the mastery of skills in an un-timed environment. Only once you’ve achieved mastery try to do them ever more quickly.

By focusing on the time before you have the underlying process conquered you end up rushing yourself in a way that exacerbates your mistakes rather than allows you to correct them. So as you’re prepping, focus on total mastery and understanding first and then begin putting them under time pressure.

5. Low-yield self-prep.

Most people spend entirely too much time preparing from the GMAT. They do so because they’re not getting enough out of their prep time.

Does this sound familiar? Okay, I’m going to do a group of 10 questions, maybe on a timer for 20 minutes. Afterwards I’m going to look in the back of the book. When I get the problem right I’m going to say, “yeah, I never have to deal with this problem again.” When I get it wrong in going to go a little bit further and normally I’m going to find something that I knew but I sort of forgot. I’ll say, “You know what I won’t forget that, I’m going to get that right next time.”

But it doesn’t happen that way does it? That’s a very low yielding strategy. Instead, you need to become responsible and accountable for your learning and Apex shows you the way to do so by not just being reactive to problems but proactively creating problems of your own.

6. Doing the math.

We have a saying around here and you may have heard it on some of our materials or online videos. If you’re doing math, then you’re doing something wrong. Most of the GMAT quantitative section requires little to no processing and if you’re scribbling tons of stuff on paper it means you’re missing the bigger picture. So remember, if you’re doing math there’s always a better way!

Enjoyed “Six things that GMAT preppers get wrong?”, find more videos here.

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