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Posted on
01
Sep 2021

Additional Voters – GMAT Quant Problem

Additional Voters – GMAT Quant Problem

Hey guys, today we’re going to look at a particularly challenging GMAT Quant problem that just about everyone resorts to an algebraic solution path on, but there’s a very elegant part solution path. When we take a look at this problem we observe immediately that the difficulty is that we have no baseline for the number of voters that we start with. That’s the confusing part here and this is one of the ways that the GMAT modulates difficulty; when they give us a problem without fixed numbers, and where we’re not free to run a scenario because there are add-on numbers that change the relative values.

Additional Voters Problem Introduction

GMAT Quant Problem

Here they’re adding the 500 and the 600 which means there exist fixed values at the beginning, but we don’t know what they are. What we want to do here is remove ourselves a bit from the problem and let the ratios that they give us guide our way.

We start out with three parts Republicans, five parts Democrats. These eight parts constitute everything, but we don’t know how many voters are in each part – it could be one voter in each, or a hundred, or a thousand, and we can’t speculate yet. So, what we need to do is not worry about it, and this is where a lot of people get really uncomfortable. Let it go for a second, and notice that, after we add all the new voters, we end up with an extra part on the Republican side and the same number of parts on the Democrat side.

What does this mean? Well, the parts are obviously getting bigger from the before to the after. But because we have an overall equivalence between the number of parts we can actually reverse engineer the solution out of this.

Reverse Engineering the Solution

We’re adding 500 Democrats and we’re maintaining five parts from the before to the after. This means that each part is getting an extra 100 voters for the total of plus 500. On the Republican side, we’re adding 600 voters. We already know, from the Democratic side, that each part needs to increase by 100 to keep pace with all the other parts. So, 300 voters are used in the three republican parts, leaving 300 extra voters to constitute the entirety of the fourth part.

Now we know that each part after we add the voters equals 300 and therefore each part before we added the voters was 200. From there we get our answer choice. I forget what they were asking us at this point, and this is actually a really great moment because it’s very common on these complex problems to get so caught up, even if you’re doing it mentally, with a more conducive solution path, to forget what’s being asked. When you’re doing math on paper, which is something we really don’t recommend, it’s even easier to do so because you get so involved processing the numbers in front of you that you lose conceptual track of what the problem is about.

So, they’re asking for the difference between the Democratic and Republican voters after the voters are added. Now we know there’s one part difference and we know that after voters are added a part equals 300 voters so the answer choice is B, 300.

Something to Keep in Mind

This one is not easy to get your head around, but it’s easier than dealing with the mess of algebra that you’d otherwise have to do.
Review this one again. This is a GMAT Quant problem you may have to review several days in a row. It’s one where you might attain an understanding, and then when you revisit it four hours later or the next day, you lose it and you have to fight for it again. It’s in this process of dense contact and fighting that same fight over and over again that you will slowly internalize this way of looking at it, which is one that is unpracticed. The challenge in this problem isn’t that it’s so difficult. It’s that it utilizes solution pads and way of thinking that we weren’t taught in school and that is entirely unpracticed. So, much of what you see as less difficult on the GMAT is less difficult only because you’ve been practicing it in one form or another since you were eight years old. So, don’t worry if you have to review this again and I hope this was helpful.

Check out this link for another super challenging GMAT Quant problem.

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Posted on
18
Aug 2021

GMAT Prime Factors Problem – GMAT Quant

Hey guys, check out this problem. This is an example of a problem that requires daisy-chaining together or linking together several key algebraic insights in order to answer it.

GMAT Prime Factors Problem - GMAT Quant

GMAT Prime Factors Problem – Applied Math Solution Path

Notice there’s an applied math solution path. We want prime factors of 3⁸ - 2⁸, and it’s just reasonable enough that we can do the math here. And the GMAT will do this a lot, they’ll give us math that’s time-consuming, but not unreasonably time-consuming in order to just draw us into an applied math solution path. We’ll take a look at this really quickly.

3⁸ is the same as 9⁴.
3⁸ = 3²*⁴= (3²)= 9
9 = (9 * 9)² = 81²
81 * 81 = 6,561

9 * 9 is 81² – about 6,400 or if we want to get exact, which we do need to do here because we’re dealing with factors, 81 * 81 is 6,561. Don’t expect you to know that, it can be done in 20 seconds on a piece of paper or mentally. And then 2⁸, that one you should know, is 256. And then, 6,561 – 256 = 6,305.

So now we need to break down 6,305 into prime factors. You know how to do that using a factor tree, so I’m going to zoom us right into a better solution path because I don’t want to give away the answer.

GMAT Prime Factors Problem – Another Solution Path

Notice that 3⁸  and 2⁸ are both perfect squares so we have the opportunity to factor this into (3– 2) * (3 + 2). Once again, the first term is a difference of two squares, the second term we can’t do anything with. So we break down that term, and lo and behold, (3² – 2²) * (3² + 2²) * (3 + 2), and once again we can factor that first term out into (3 + 2), (3 – 2), and so on. We work these out mathematically, and they’re much easier and more accessible mathematically, and we get 3 – 2 = 1 which obviously is a factor of everything. 3 + 2 = 5, 3² + 2² = 9 + 4 = 13, and then 3 + 2⁴ = 81 + 16 = 97.

So now we’ve eliminated everything, except B and C, 65 and 35. This is where the other piece of knowledge comes in. Since we have factors of 5 and 13. 65 must also be a factor because it’s comprised of a 5 and a 13. 35 requires a 7. We don’t have a 7 anywhere, so the correct answer choice is C, 35. 

GMAT Prime Factors Problem – Takeaways

So the big takeaways here are, that, when provided with some sort of algebraic expression like this, look for a factoring pattern. And, when it comes to prime factorization, remember, that if you break it down into the basic prime factor building blocks, anything that is a product of those building blocks also exists as a factor.

Hope this helped and good luck!

Found it helpful? Try your hand at this GMAT problem, GMAT Prime Factorization (Anatomy of a Problem).

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Posted on
11
Aug 2021

GMAT Markup Problem – GMAT Data Sufficiency

Hey guys, today we’re going to take a look at a typically characteristic data sufficiency problem that gives us a relationship, and then asks us if we have enough to compute the final value of that relationship. There’s an algebraic solution path here, where they give us the equation and we need to see if we have all but one of the variables, that final variable being the one that they’re asking for. We can also do this via parts, scenario, and graphically, and we’ll take a look at all those as well.

GMAT Markup Problem Introduction

GMAT Markup Problem

This problem describes to us the relationship between the selling price, the cost, and the markup. And notice that, while we’re going to sketch it out here, the actual relationship doesn’t matter to us – all that matters is that if they’re asking for one term in terms of the rest if we have the other terms, that’ll be enough.

Algebraically we have selling price S equals the cost C plus the markup M. So this is giving us the markup in, let’s say dollar terms, whereas we might also set this up as selling price equals cost times one plus the markup percentage. And here we just have that notational shift. So, what we’re looking for, if we want to know the markup relative to the selling price, is an understanding of it either relative to the selling price or relative to the cost. That is, these two things are associated and the markup, when associated with the cost, gives us a ratio. Where the markup, when associated with the selling price, is a fraction. And if you’ll remember notationally these things are expressed differently, but conceptually there’s the same math behind it.

Statement 1

Number one gives us in percentage terms the mark up compared to the cost. So, here we can see it as 25% more and this is where it ties into that second version of the algebraic one we just looked at. The cost we can break up into four parts of 25% so that when we add the markup that’s a fifth part. Therefore, the markup comprises one-fifth or 20% of the selling price.

Statement 2

Number two provides us a concrete selling price but doesn’t tell us anything about the markup or the mix of cost versus markup as a percentage of the total selling price. Two is insufficient on its own, and as we’ve seen in many other data sufficiency problems, what they’re trying to do here is fool us into thinking we need a specific price, a discrete value to get sufficiency. When the question stem is asking us only for a relative value and when we’re being asked for a relative value, a percentage, a fraction, a ratio be on the lookout for fooling yourself into thinking that you need an anchor point a specific discrete value.

I hope this helps. If you enjoyed this GMAT Markup Problem, try your hand at this Triangle DS Problem.

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GMAT Quant Syllabus 2021-2022
Posted on
22
Jul 2021

GMAT Quant Syllabus 2021-2022

Author: Apex GMAT
Contributor: Altea Sollulari
Date: 22 July, 2021

We know what you’re thinking: math is a scary subject and not everyone can excel at it. And now with the GMAT the stakes are much higher, especially because there is a whole section dedicated to math that you need to prepare for in order to guarantee a good score. There is good news though, the GMAT is not actually testing your math skills, but rather your creative problem solving skills through math questions. Furthermore, the GMAT only requires that you have sound knowledge of high school level mathematics. So, you just need to practice your fundamentals and learn how to use them to solve specific GMAT problems and find solution paths that work to your advantage. 

The Quantitative Reasoning section on the GMAT contains a total of 31 questions, and you are given 62 minutes to complete all of them. This gives you just 2 minutes to solve each question, so in most cases, the regular way of solving math equations that you were taught in high school will not cut it. So finding the optimal problem solving process for each question type is going to be pivotal to your success in this section. This can seem a daunting start, so our expert Apex GMAT instructors recommend that you start your quant section prep with a review of the types of GMAT questions asked in the test and math fundamentals if you have not been using high school math in your day to day life. 

What types of questions will you find in the GMAT quant?

There are 2 main types of questions you should look out for when preparing to take the GMAT exam:

Data Sufficiency Questions

For this type of GMAT question, you don’t generally need to do calculations. However, you will have to determine whether the information that is provided to you is sufficient to answer the question. These questions aim to evaluate your critical thinking skills. 

They generally contain a question, 2 statements, and 5 answer choices that are the same in all GMAT data sufficiency questions.

Here’s an example of a number theory data sufficiency problem video, where Mike explains the best way to go about solving such a question.

Problem Solving Questions

This question type is pretty self-explanatory: you’ll have to solve the question and come up with a solution. However, you’ll be given 5 answer choices to choose from. Generally, the majority of questions in the quant section of the GMAT will be problem-solving questions as they clearly show your abilities to use mathematical concepts to solve problems.

Make sure to check out this video where Mike shows you how to solve a Probability question.

The main concepts you should focus on

The one thing that you need to keep in mind when starting your GMAT prep is the level of math you need to know before going in for the Quant section. All you’ll need to master is high-school level math. That being said, once you have revised and mastered these math fundamentals, your final step is learning how to apply this knowledge to actual GMAT problems and you should be good to go. This is the more challenging side of things but doing this helps you tackle all the other problem areas you may be facing such as time management, confidence levels, and test anxiety

Here are the 4 main groups of questions on the quant section of the GMAT and the concepts that you should focus on for each:

Algebra

  • Algebraic expressions
  • Equations
  • Functions
  • Polynomials
  • Permutations and combinations
  • Inequalities
  • Exponents

Geometry

  • Lines
  • Angles
  • Triangles
  • Circles
  • Polygons
  • Surface area
  • Volume
  • Coordinate geometry

Word problems

  • Profit
  • Sets
  • Rate
  • Interest
  • Percentage
  • Ratio
  • Mixtures

Check out this Profit and Loss question.

Arithmetic

  • Number theory
  • Percentages
  • Basic statistics
  • Power and root
  • Integer properties
  • Decimals
  • Fractions
  • Probability
  • Real numbers

Make sure to try your hand at this GMAT probability problem.

5 tips to improve your GMAT quant skills?

  1. Master the fundamentals! This is your first step towards acing this section of the GMAT. As this section only contains math that you have already studied thoroughly in high-school, you’ll only need to revise what you have already learned and you’ll be ready to start practicing some real GMAT problems. 
  2. Practice time management! This is a crucial step as every single question is timed and you won’t get more than 2 minutes to spend on each question. That is why you should start timing yourself early on in your GMAT prep, so you get used to the time pressure. 
  3. Know the question types! This is something that you will learn once you get enough practice with some actual GMAT questions. That way, you’ll be able to easily recognize different question types and you’ll be able to use your preferred solution path without losing time.
  4. Memorize the answer choices for the data sufficiency questions! These answers are always the same and their order never changes. Memorizing them will help you save precious time that you can spend elsewhere. To help you better memorize them, we are sharing an easier and less wordy way to think of them:
  5. Make use of your scrap paper! There is a reason why you’re provided with scrap paper, so make sure to take advantage of it. You will definitely need it to take notes and make calculations, especially for the problem-solving questions that you will come across in this GMAT question.
  • Only statement 1
  • Only statement 2
  • Both statements together
  • Either statement
  • Neither statement
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Posted on
13
May 2021

GMAT Factors Problem – 700 Level GMAT Question

GMAT Factors Problem

Hey guys! Today we’re going to take a look at one of my favorite problems. It’s abstract, it’s oddly phrased and in fact the hardest part for many folks on this problem is simply understanding what’s being asked for. The difficulty is that it’s written in math speak. It’s written in that very abstract, clinical language that if you haven’t studied advanced math might be new to you.

How this breaks down is they’re giving us this product from 1 to 30, which is the same as 30!. 30*29*28 all the way down the line. Or you can build it up 1*2*3*……*29*30.

The Most Difficult Part of The GMAT Problem

And then they’re asking this crazy thing about how many k such that three to the k. What they’re asking here is how many factors of three are embedded in this massive product. That’s the hard part! Figuring out how many there are once you have an algorithm or system for it is fairly straightforward. If we lay out all our numbers from 1 to 30. And we don’t want to sit there and write them all, but just imagine that number line in your head. 1 is not divisible by 3. 2 is not divisible by 3, 3 is. 4 isn’t. 5 isn’t. 6 is. In fact, the only numbers in this product that concern us are those divisible by 3. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

Important Notes About Factors

Here it’s important to note that each of these components except the three alone has multiple prime factors. The three is just a three. The six is three and a two. The nine notice has a second factor of three. Three times three is nine and because we’re looking at the prime factors it has two. It’s difficult to get your head around but there are not three factors of three in nine when you’re counting prime factors.

Three factors of three would be 3 by 3 by 3 = 27. So notice that 3 and 6 have a single factor. 9 has a double factor. Every number divisible by 3 has one factor. Those divisible by 9 like 9, 18 and 27 are going to have a second factor and those divisible by 27, that is 3 cubed, are going to have a third factor. If we lay it out like this we see ten numbers have a single factor. Another of those three provide a second bringing us to thirteen. Finally, one has a third bringing us to fourteen. Answer choice: C.

GMAT Problem Form

So let’s take a look at this problem by writing a new one just to reinforce the algorithm. For the number 100 factorial. How many factors of seven are there? So first we ask ourselves out of the 100 numbers which ones even play? 7, 14… 21 so on and so forth. 100 divided by 7 equals 13. So there are 13 numbers divisible by 7 from 1 to 100. Of those how many have more than one factor of 7? Well we know that 7 squared is 49. So only those numbers divisible by 49 have a second factor. 49 and 98. There are none that have three factors of 7 because 7 cubed is 343. If you don’t know it that’s an identity you should know. So here our answer is 13 plus 2 = 15.

Try a few more on your own. This one’s great to do as a problem form and take a look at the links below for other abstract number theory, counting prime type problems as well as a selection of other really fun ones. Thanks for watching guys and we’ll see you soon.

If you enjoyed this GMAT factors problem, here is an additional number theory type problem to try next: Wedding Guest Problem.

 

 

 

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