7 Daily Practices For GMAT Success - GMAT Guide
Posted on
08
Jul 2021

7 Daily Practices For GMAT Success

By: Apex GMAT
Contributor: Ruzanna Mirzoyan
Date: 8th July 2021

7 Things You Need To Do Daily When Preparing For The GMAT (GMAT Guide)

  1. Visualize success and the value you will get in the end
  2. Review a the GMAT sections
  3. Set a time limit for each day
  4. Do not forget to reward yourself
  5. Forget about the target score only focus on improvement
  6. Give yourself a pep talk 
  7. Evaluate Yourself Honestly

     Achieving a great score on the GMAT exam is not an easy task. The overall preparation process is daunting for a majority of test takers, especially for non-native English speakers. It requires diligent work and a daily checklist that you need to follow. So how do you come up with a plan that works? This article covers seven tips for successful GMAT prep which will guide you throughout the entire process. Even though every individual taking the exam has different expectations, experiences and may be approaching the test in a different way, sticking to a daily routine is an integral part of test success; the most difficult thing is adhering to it, avoiding procrastination and maintaining motivation. Therefore, after learning all the exam basics, such as the timing, the sections, and the preparation materials, it is worth creating a checklist to help keep you on track.

Visualize success and the value you will get in the end

The thought of success can create happiness! Once we attain something that seemed difficult initially, the suspense wears off, and the excitement rapidly grows. By taking time every day to imagine achieving your goal you can stay motivated and on the right path. When we experience happiness our brain releases serotonin, the hormone responsible for happiness. By keeping the picture of accomplishment in our mind, this happiness never fades. Hence, if every day contains even a tiny bit of happiness, even the most complex struggles seem simpler to overcome. Whether the GMAT exam is a struggle or not, happiness and motivation are something that one undoubtedly always lacks. Do your best to look at the bigger picture and think of the steps that will expedite reaching the top.

Review the GMAT exam sections

Whether you have a private GMAT tutor or are studying on your own, be sure to review difficult parts of the overall format of the exam every day before going through your study materials, for example the data sufficiency answer choices. You may do a short quiz on quantitative, verbal, or integrated reasoning to keep pace with timing and question types. You can consider this form of revision as stretching your brain muscles before the main exercise. Doing a simple GMAT quiz each time will make you more cautious about time management and remind you about the type of questions that you may have already mastered in previous study sessions.

Set a study time limit for each day

As it is said, time is the only non-redeemable commodity, so proper allocation is a fundamental key to success. We recommend you have a specific time allocation for GMAT prep each day. That can be some time for weekday preparation and extension on the weekends. Ensure the limit you set for yourself is reasonable because procrastinating one day and doubling the hours the next day does not work out. It does not matter how many months you have on your hands; the significant thing is precise allocation. If you want to get a decent score, you must spend approximately 100-120 hours reviewing the materials and practicing. However, top scorers usually  spend 120+ hours studying. Whether you belong to the former or the latter category, remember that time is the most expensive investment you are making. At the same time keep in mind that your study-life balance should be of utmost importance. 

Do not forget to reward yourself

It is not a secret that the GMAT is burdensome and overwhelming, and preparing for it can be stressful and oftentimes disheartening. Not having small rewards to look forward to can lead to demotivation. Rewards are things that rejuvenate your broken concentration. Try something like the Pomodoro Technique. This technique helps break down time into intervals with short breaks. Instead of breaks, you can think of something ‘non-GMAT related’ that will make you regain focus. For example, by grabbing a quick snack, meditating, or walking around the house or even watching a short YouTube video. Whichever works best for you, make use of it; even brief respites retain your stamina. Finally, never forget about the bigger reward; your final score. 

Forget about the target score, only focus on improvement

GMAT preparation practices do generate plight both in physical and mental states. It is crucial to remind oneself of the improvement phases. We agree that everything you are going through is for the final score. But focusing on the final score too much can frustrate you if you are not making big leaps towards it, which in turn can be counter productive. All successful practices dictate that you should focus on one thing at a time, which improves every day until the exam day. When the exam day comes, you will utilize all the knowledge and effort to get the highest GMAT score possible. Keeping daily track of your improvements relieves some of the burden on your shoulders. Even the tiniest advantage acquired can be a game changer. For instance, finishing each section a minute earlier than before will eventually contribute to achieving more significant results on the exam day, or perfecting a solution path which has you approaching a host of GMAT problems in a more efficient manner. These small wins can be the fuel to keep you going. 

Give yourself a pep talk 

I am sure you receive a lot of support from the people surrounding you. However, self-encouragement is of the utmost importance. Look around, see what others are doing at your age and inspire yourself. Choose wisely between the tradeoffs. Such as choosing to study instead of partying. Giving yourself a daily pep talk will make you more enthusiastic about reaching your objectives. A recent scientific study has shown that talking to yourself dwindles anxiety and stress while boosting performance. This is no less true for GMAT test preparation. Give yourself motivational and instructional pep talks. This method promotes positivity as motivational talks cheer you up and keep up the eagerness to study and strive for more, while a self-instructional talk directs detail-orientation and accentuates what exactly you need to do for that particular day. For example, start every day by loudly stating what should be done for the day. It helps with thinking about the mechanisms of every individual task and visualizing methods to complete them correspondingly. 

Evaluate Yourself Honestly

Of course, you need all the encouragement and self-support to reach your goals, but especially during GMAT exam preparation, you need to be hard on yourself if required. If you need a 650+ GMAT score, you should be aware that it will not be a piece of cake. Give yourself credit for what you are doing right, but also consider aspects of the GMAT problems that you need to elaborate on and master additional skills. The dominant thing is separating the action from the person because you are evaluating your actions and not you as a person; you should not upset yourself but rather detect the triggers of low performance and challenges and make yourself accountable for such actions with a plan to move forward from them successfully. Ultimately, the ability to discern your flaws and work on personal evolution is an inherent quality for capacitating your abilities and aptitudes and pulling it off in life. 

We hope that adding these practical and mindful aspects to your daily preparation will be helpful as when you are preparing for an exam like the GMAT, being in the right mind frame can be as important as doing the quant or verbal practice. Whether you have a GMAT private tutor or not, it is on you to maintain motivation during the entire process. We suggest you develop a GMAT test strategy along with these seven tips to attain greater productivity and manifest superb performance. Make studying for the GMAT a daily habit and success will follow. 

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When to Hire a Private GMAT Tutor?
Posted on
08
Jun 2021

When to Hire a Private GMAT Tutor?

By: Dana Coggio
Published: 8th June 2021

Once you have made the decision to get your MBA, the next challenge awaits you: Studying for and taking the GMAT. You’ve purchased the books, set up a study schedule, prepared yourself mentally for the task that lays before you. (This infographic provides you with handy tips for your GMAT study prep). And yet, as you begin studying, you find yourself stuck and unsure of what to do next. First things first, it is important for you to understand that this is perfectly normal. The GMAT is an exam that tests more than just your quantitative and analytical skills, it is meant to score your ability to think critically and outside of the box. If you find yourself rereading sections of your GMAT prep books or googling various study tricks and tips, it might be time to consider getting yourself a private GMAT tutor.

That in and of itself seems a daunting task. Where do you even begin with finding someone capable of helping you achieve your goals? Well, before you look for a private GMAT tutor, it is necessary that you first understand when to get yourself a private tutor. This important step means you, and your tutor, are ready to work together to achieve a well thought out goal.

Establish your Goals

Before you begin looking for a private GMAT tutor, you need to know what your GMAT and MBA goals are. Perhaps you are striving for a 700+ and looking to apply to a top-tier MBA program. Or maybe you are looking to up your score by 50 points to be considered more competitive for your dream MBA. Whatever the reason, you need to have a good grasp on why you are taking the GMAT and what your goals are.

Having a clear mindset means you can search for a private GMAT tutor whose skills match your goals. This also means you are not wasting precious time with your GMAT tutor recapping where you see yourself in 20 years. As GMAT tutors can be pricey, it is important to optimize your time with your tutor. Additionally, it is important to research how private tutors can help you achieve your goal. If you are interested in increasing your score above a 700, for example, our article on how private GMAT tutors help you in this task can be read here: How Can Private Tutoring Help You Score 700+ on the GMAT?

Establish your baseline

One of the most important things you can do when starting to study for the GMAT is to take a practice exam. You can find a free practice GMAT exam HERE. Once you have taken the practice GMAT exam, you will have a better understanding of where your strengths and weaknesses lie. However, knowing how to strengthen your weakness or grow your strengths may seem intimidating. This is why, when looking for a private GMAT tutor, you should go into your first meeting with a clear understanding of where you score on the GMAT and what some of your initial GMAT prep challenges are.

This gives your private tutor a better understanding of what aspects of the GMAT you struggle the most with and which parts require only brief reviews. Even better would be to take the mock GMAT exam with your tutor present so that they provide you with more knowledgeable feedback and study plans. In Apex’s case for instance, in a 2.5 hour assessment session an instructor puts you through your paces to see where you need the most help and where they should focus your efforts to get the most leverage in your allotted prep time.  

Begin Studying

So, you have created achievable goals and established your baseline GMAT score. You are confident about beginning your studying, and yet, as the weeks pass and the GMAT exam comes closer you realize you aren’t anywhere near where you want to be. This is an important realization for you as a student. If you recognize that no matter how many hours you commit to studying or how many practice exams you take your understanding of the material is not increasing then, perhaps, it is time to turn to a professional to support you in your journey.

A private GMAT tutor will not only help you in comprehension of the materials, but they also will give you confidence and the support to achieve your goals while holding you accountable to your studies. This scoring plateau phenomenon is what most GMAT prep students face at one point or another during the GMAT journey. 

Investing in a GMAT tutor

When it is time to look for a private GMAT tutor it is important to know that this decision is an investment. Although they may not always say so outright, numerous MBA students at many top-tier universities invested in a private GMAT tutor to help them study for the exam. Investing in a private GMAT tutor is an investment in your future and can pay off in the long run even after you have been admitted to your dream MBA program. Working with a high achieving tutor can be pricey.

The costs associated with a reputable GMAT tutor should reflect the investment they put into helping you achieve your goals. Your time is valuable, and so looking for a private GMAT tutor shouldn’t be the main objective of your studies but investing a couple hours to find the right fit can pay off in the end as a stellar GMAT score will not only help in your MBA application process, but it could land you quality scholarships and a place at some of the top consulting firms after b-school.  

Where to look for a GMAT tutor

Finding a proper GMAT tutor means finding a tutor who works with you to achieve your goals. There are a lot of GMAT tutors on the market who claim to ‘know the secret’ or can ‘guarantee a score’. Be wary of these tutors, as there is no true way to guarantee success. Success on the GMAT comes down to you as an individual and the time you invest in studying. A private GMAT tutor is there to help guide you and support you on your GMAT study journey. It is important to find a reputable GMAT tutor whose skills and ways of teaching match how you learn and your goals.

On an average search engine, a deluge of offers and potential tutors appear, and this plethora can seem quite overwhelming.  Luckily, Apex GMAT offers a complimentary 30-minute session with one of our instructors. This session gives you the opportunity to gauge whether or not a private GMAT tutor is right for you. To learn more about where to find a fitting GMAT tutor, check out our article on How to Select a GMAT Tutor.

Finally…

…whether you are 6 months into studying or you are just starting the process, it is never too late to invest in a GMAT tutor. The sooner you do, however, the more your tutor can support you and the more you can get out of the experience. We have worked with clients who are 8 months into their GMAT journey and beaten down to newcomers who are looking at the test through fresh eyes, so we have heard it all before.

Knowing when to get a GMAT tutor is vital to your success. We highly suggest signing up for a complimentary consultation with one of our tutors, as they can help you more narrowly define when to find a GMAT tutor and if a GMAT tutor is right for you. You can sign up for a complimentary 30-minute slot HERE. Still unsure, feel free to listen HERE for some testimonials from people, just like you, who invested in a private GMAT tutor and are very glad they did! 

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Posted on
19
May 2021

GMAT Algebra Problem – Parts – Hotdogs & Donuts

GMAT Algebra Problem Introduction

Hi guys. Today I’m here with a classic GMAT Algebra problem, what we call a parts problem. And if you take a look at this problem you’re going to realize that it just looks like a bunch of algebra. But the key here is in how you frame it. We’ve got this diner or whatnot selling hot dogs and then after that point, so imagine like a timeline, they start selling donuts. Then they give us a piece of information about hot dogs to donuts over that course of secondary time but then give us this overarching total number of food products sold.

Distill The Ration

So what we need to do are two steps: the first one is fairly straightforward. We see that we have to get rid of the hot dogs that were sold in advance in order to distill the ratio but then the ratio can seem very, very complex, especially because it just tells us seven times and a lot of times the GMAT will do this as a way to throw us off the scent. So when we have seven times, what that means is we have eight parts. That is it’s saying for every one of these we have one, two, three, four, five, six, seven of these. Meaning in total there are eight. So while seven is kind of a scary number, eight is a number we can divide by easily. You always want to look for that when you’re given a ratio of one thing to another especially when they say something times as many.

Solving the GMAT Problem

We take that thirty thousand two hundred knock off the fifty four hundred and get to twenty four thousand eight hundred and lo and behold that’s divisible by eight meaning each part is going to be 3100. Notice there’s no complex division there, 24 divided by 8, 800 divided by 8 and that’s the sort of mental math we can expect from the GMAT always. Which as you’ve seen before: if you’re doing that you’re doing something wrong.

Each part is 3100 and we’re concerned with the seven parts so we can either scale that 3100 up by seven into 21700, again the math works out super smoothly or we can take the 24800 knock off 3100 and get to that 21700. Notice in the answer choices there’s a few things to address sort of common errors that might be made.

Reviewing the Answer Signals

On one of the answer choices what you’re looking at is dividing the total, the 30 200 by eight and multiplying by seven that is seven eighths of it without getting rid of those first 5400. Another answer is close to our 21700 correct answer and this is also a fairly reliable signal from the GMAT.

When they give you a range of answers but two of them are kind of tightly clustered together a lot of times it’s going to be one of the two and that second one there is to prevent you from too roughly estimating. But at the same time if you’re short on time or just in general you want to hone down and understand what you’re supposed to do that serves as a really strong signal. And then one of the answer choices is the 1/8 of it rather than the 7/8.

Clustered Answer Choices

I want to speak a little more deeply about that signal about those two tightly clustered answer choices because as I said it can help you narrow to a very quick 50/50 when you’re constrained for time or this problem is just one that’s really not up your alley but it also can be leveraged in a really, really neat way.

If we assume that one or the other is the answer choice we can differentiate these two different answer choices by what they’re divisible by and so notice the 21700 is very clearly, with strong mental math is divisible by seven. Where the other one is not. Also neither of them are divisible by eight. We can look at these two say okay one of them is probably right, one of them is divisible by seven, the other one is not, so there’s our right answer and we can move on to the next problem. So I hope this helps. Write your comments and questions below. Subscribe to our channel at Apex GMAT here and give us a call if we can ever help you.

To work on similar GMAT algebra problem/s see this link: Work Rate Problem.

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Posted on
08
Apr 2021

GMAT Factorial Problem: Estimation & Scenario Solution

GMAT Factorial Introduction

Factorials and divisibility, together. Two mathematical kids from opposite sides of the tracks, they come together and fall in love and they create this problem. Here we’re asked what numbers might divide some new number 20 factorial plus 17. As a refresher, a factorial is simply the number times each integer below it. So in this case, 20! is equal to 20 x 19 x 18 …. x 3 x 2 x 1. It’s a huge number. And it’s not at all possible to process in GMAT time. What we want to notice about any factorial is that it has as factors every number that it contains. So 20! is divisible by 17, it’s divisible by 15, it’s divisible by 13, 9, 2, what have you and any combination of them as well.

What The GMAT is Counting On You Not Knowing

When we’re adding the 17 though, the GMAT is counting on the idea that we don’t know what to do with it and in fact that’s the entire difficulty of this problem. So I want you to imagine 20! as a level and we’re going to take a look at this graphically. So 20! can be comprised by stacking a whole bunch of 15’s up. Blocks of 15. How many will there be? Well 20 x 19 x 18 x 17 x 16 x 14 times all the way down the line. There will be that many 15’s. But 20! will be divisible by 15. Similarly, by 17, by 19, by any number. They will all stack and they all stack up precisely to 20! because 20! is divisible by any of them.

Answer

So when we’re adding 17 to our number all we need to see is that, hey, 15 doesn’t go into 17, it’s not going to get all the way up there. 17 fits perfectly. 19? guess what? It’s too big and we’re going to have a remainder. So our answer here is B, only 17.

For other problems like this, other factorials, and what have you, please check out the links below and we will see you next time. If you enjoyed this GMAT problem, try your hand at this Science Fair Problem.

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Posted on
10
Mar 2021

GMAT Ratio Problem – Mr. Smiths Class

GMAT Ratio DS Problem

Expressing Different Notations

Hey guys!

Expressing different notations is often challenging when you’re first starting out on the GMAT and by different notations mean percentages fractions decimals ratios. We learn all these separately and we tend to of them as separate systems of math when in fact they’re all different expressions of the same math. One half is no different from 0.5 is no different from 50 percent there are different ways of the same thing.

Breaking Down The Problem

In this problem all their testing is our ability to shift notations. We’re being asked what the ratio, keyword ratio, is between boys and girls in the or what do we need is just that a ratio it’s fairly straightforward. So they’re probably going to come to us with weird information that doesn’t quite look like a ratio. The big thing to note before we dive in is that when we’re being asked for a ratio. In fact, when we’re being asked for any sort of relative notation, fractions, percentages, anything that needs a base that is compared to a whole. We don’t need precise numbers.

Possible ways to solve this problem

So this leaves us open either to run scenarios if we want to or to deal entirely in the relative. So we’re looking for an expression of that ratio in a non-ratio sort of language. Number one tells us there are three times as many boys and girls. We can run a scenario with 3 boys, 1 girl, 75 boys, 25 girls, but we’re being given that ratio. It’s being expressed in language rather than with the term ratio or with the two dots : in between but it’s still a ratio. So it’s sufficient!

What Did You Miss?

Correction!! Number one states there are three times as many girls as there are boys. Why do we leave that error in? To point out that here it doesn’t matter. We’re not looking to determine whether the ratio is 1 boy to 3 girls or 3 girls to 1 boy or 3 boys to 1 girl. The only thing that matters, the threshold issue on this problem, is getting to a single specific ratio. What that is or in this case even reversing the boys and girls doesn’t matter because it’s a referendum on the type of information that we have. The moment we have a quantitative comparison of boys and girls coming from number one we know that number one is sufficient. Being able to have flexibility and even focus on the more abstract thing you’re looking for sometimes leads to careless errors on the details though and this is important. Many times those careless errors don’t matter, freeing yourself up to make those and understanding that you don’t have to manage the nitty-gritty once you have the big abstract understanding is very important.

Looking at Statement No. 2

Number two goes fractional, telling us that 1/4 of the total class is boys. We can break that into a ratio by understanding that a ratio compares parts to parts whereas a fraction is part of a whole so one out of four has a ratio of one to three. If this isn’t immediately obvious, imagine a pizza and cut it into four slices. One slice is one quarter of the total pizza the comparison of the one slice to the other three slices is the ratio one to three so if you get one slice and your friends get the other ones. The ratio of your slice to the others is 1:3. You have 1/4 of the total so two is also sufficient. Therefore, the answer choice here is D.

Hope this helped guys! Practice this skill of going in between these different notations because it’s one that pays off in dividends. Check out the links below for other problems and we’ll see you again real soon.

If you enjoyed this GMAT Ratio DS Problem, try your hand at this Triangle DS Problem.

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Permutations with Restrictions GMAT Article
Posted on
02
Mar 2021

Permutations with Restrictions

By: Rich Zwelling, Apex GMAT Instructor
Date: 2nd March, 2021

So far, we’ve covered the basics of GMAT combinatorics, the difference between permutations and combinations, some basic permutation and combination math, and permutations with repeat elements. Now, we’ll see what happens when permutation problems involve conceptual restrictions that can obscure how to approach the math.

To illustrate this directly, let’s take a look at the following Official Guide problem:

The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter?

A) 12
B) 18
C) 24
D) 36
E) 48

Did you catch the restriction? Up until the end, this is a standard permutation with repeats combinatorics problem, since there are five letters and two repeats of the letter ‘I’. However, we’re suddenly told that the two I’s must be separated by at least one other letter. Put differently, they are not allowed to be adjacent.

So how do we handle this? Well, in many cases, it’s helpful to set aside what we want and instead consider what we don’t want. It seems counterintuitive at first, but if we consider the number of ways in which the two I’s can appear together (i.e. what is not allowed) and then subtract that number from the total number of permutations without any restrictions, wouldn’t we then be left with the number of ways in which the two I’s would not appear together (i.e. what is allowed)? 

Let’s demonstrate: 

In this case, we’ll pretend this problem has no restrictions. In the word “DIGIT,” there are five letters and two I’s. Using the principle discussed in our Permutations with Restrictions post, this would produce 5! / 2! = 60 permutations. 

However, we now want to subtract out the permutations that involve the two I’s side by side, since this condition is prohibited by the problem. This is where things become less about math and more about logic and conceptual understanding. Situationally, how would I outline every possible way the two I’s could be adjacent? Well, if I imagine the two I’s grouped together as one unit, there are four possible ways for this to happen:

II DGT

D II GT

DG II T

DGT II

For each one of these four situations, however, the three remaining letters can be arranged in 3*2*1 = 6 ways. 

That produces a total of 6*4 = 24 permutations in which the two I’s appear side by side.

Subtract that from the original 60, and we have: 60 – 24 = 36. The correct answer is D

As you can see, this is not about a formula or rote memorization but instead about logic and analytical skills. This is why tougher combinatorics questions are more likely to involve restrictions.

Here’s another Official Guide example. As always, give it a shot before reading on:

Of the 3-digit integers greater than 700, how many have 2 digits that are equal to each other and the remaining digit different from the other 2 ?

(A) 90
(B) 82
(C) 80
(D) 45
(E) 36

Explanation

This is a classic example of a problem that will tie you up in knots if you try to brute force it. You could try writing up examples that fit the description, such as 717, 882, 939, or 772, trying to find some kind of pattern based on what does work. But as with the previous problem, what if we examine conceptually what doesn’t work?

This will be very akin to how we handle some GMAT probability questions. The situation desired is 2 digits equal and 1 different. What other situations are there (i.e. the ones not desired)?  Well, if you take a little time to think about it, there are only two other possibilities: 

  1. The digits are all the same
  2. The digits are all different

If we can figure out the total number of permutations without restrictions and subtract out the number of permutations in the two situations just listed, we will have our answer. 

First, let’s get the total number of permutations without restrictions. In this case, that’s just all the numbers from 701 up to 999. (Be careful of the language. Since it says “greater than 700”, we will not include 700.)

To get the total number of terms, we must subtract the two numbers then add one to account for the end point. So there are (999-701)+1 = 299 numbers in total without restrictions.

(Another way to see this is that the range between 701 and 999 is the same as the range between 001 and 299, since we simply subtracted 700 from each number, keeping the range identical. It’s much easier to see that there are 299 numbers in the latter case.)

Now for the restrictions. How many of these permutations involve all the digits being the same? Well, this is straightforward enough to brute force: there are only 3 cases, namely 777, 888, and 999. 

How about all the digits being different? Here’s where we have to use our blank (or slot) method for each digit:

___ ___ ___

How many choices do we have for the first digit? The only choices we have are 7, 8, and 9. That’s three choices:

_3_  ___ ___

Once that first digit is in place, how many choices do we have left for the second slot? Well, there are 10 digits, but we have to remove the one already used in the first slot from consideration, as every digit must be different. That means we have nine left:

_3_  _9_  ___

Using the same logic, that leaves us eight for the final slot:

_3_  _9_  _8_

Multiplying them together, we have 3*9*8 = 216 permutations in which the digits are different.

So there are 216+3 = 219 restrictions, or permutations that we do not want. We can now subtract that from the total of 299 total permutations without restrictions to get our final answer of 299-219 = 80. The correct answer is C.

Next time, we’ll take a look at a few examples of combinatorics problems involving COMBINATIONS with restrictions.

Permutations and Combinations Intro
A Continuation of Permutation Math
An Intro To Combination Math
Permutations With Repeat Elements
Permutations With Restrictions
Combinations with Restrictions
Independent vs Dependent Probability
GMAT Probability Math – The Undesired Approach
GMAT Probability Meets Combinatorics: One Problem, Two Approaches

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An Intro to Combination Math GMAT Article
Posted on
23
Feb 2021

An Intro to Combination Math

By: Rich Zwelling, Apex GMAT Instructor
Date: 23rd February, 2021

Last time, we looked at the following GMAT combinatorics practice problem, which gives itself away as a PERMUTATION problem because it’s concerned with “orderings,” and thus we care about the order in which items appear:

At a cheese tasting, a chef is to present some of his best creations to the event’s head judge. Due to the event’s very bizarre restrictions, he must present exactly three or four cheeses. He has brought his best cheddar, brie, gouda, roquefort, gruyere, and camembert. How many potential orderings of cheeses can the chef create to present to the judge?

A) 120
B) 240
C) 360
D) 480
E) 600

(Review the previous post if you’d like an explanation of the answer.)

Now, let’s see how a slight frame change switches this to a COMBINATION problem:

At a farmers market, a chef is to sell some of his best cheeses. Due to the market’s very bizarre restrictions, he can sell exactly two or three cheeses. He has brought his best cheddar, brie, gouda, roquefort, gruyere, and camembert. How many potential groupings of cheeses can he create for display to customers? 

A) 6
B) 15
C)
20
D) 35
E) 120

Did you catch why this is a COMBINATION problem instead of a PERMUTATION problem? The problem asked about “groupings.” This implies that we care only about the items involved, not the sequence in which they appear. Cheddar followed by brie followed by gouda is not considered distinct from brie followed by gouda followed by cheddar, because the same three cheeses are involved, thus producing the same grouping

So how does the math work? Well, it turns out there’s a quick combinatorics formula you can use, and it looks like this: 

combinations problem

Let’s demystify it. The left side is simply notational, with the ‘C’ standing for “combination.” The ‘n’ and the ‘k’ indicate larger and smaller groups, respectively. So if I have a group of 10 paintings, and I want to know how many groups of 4 I can create, that would mean n=10 and k=4. Notationally, that would look like this:

combinatorics and permutations on the GMAT, combination math on the gmat

Now remember, the exclamation point indicates a factorial. As a simple example, 4! = 4*3*2*1. You simply multiply every positive integer from the one given with the factorial down to one. 

So, how does this work for our problem? Let’s take a look:

At a farmers market, a chef is to sell some of his best cheeses. Due to the market’s very bizarre restrictions, he can sell exactly two or three cheeses. He has brought his best cheddar, brie, gouda, roquefort, gruyere, and camembert. How many potential groupings of cheeses can he create for display to customers? 

A) 6
B) 15
C)
20
D) 35
E) 120

The process of considering the two cases independently will remain the same. It cannot be both two and three cheeses. So let’s examine the two-cheese case first. There are six cheese to choose from, and we are choosing a subgroup of two. That means n=6 and k=2:

combinations and permutation on the gmat, combination math on the gmat

Now, let’s actually dig in and do the math:

combinatorics and permutations on the GMAT, combination math on the gmat

combinatorics and permutations on the GMAT, combination math on the gmat

From here, you’ll notice that 4*3*2*1 cancels from top and bottom, leaving you with 6*5 = 30 in the numerator and 2*1 in the denominator:

combinatorics and permutations on the GMAT, combination math on the gmat That leaves us with:

6C2 = 15 combinations of two cheeses

Now, how about the three-cheese case? Similarly, there are six cheeses to choose from, but now we are choosing a subgroup of three. That means n=6 and k=3:

solving a combinatorics problem

From here, you’ll notice that the 3*2*1 in the bottom cancels with the 6 in the top, leaving you with 5*4 = 20 in the numerator:

combination problem on the gmat answer

That leaves us with:

6C3 = 20 combinations of three cheeses

With 15 cases in the first situation and 20 in the second, the total is 35 cases, and our final answer is D. 

Next time, we’ll talk about what happens when we have permutations with repeat elements.

In the meantime, as an exercise, scroll back up and return to the 10-painting problem I presented earlier and see if you can find the answer. Bonus question: redo the problem with a subgroup of 6 paintings instead of 4 paintings. Try to anticipate: do you imagine we’ll have more combinations in this new case or fewer?

Permutations and Combinations Intro
A Continuation of Permutation Math
An Intro To Combination Math
Permutations With Repeat Elements
Permutations With Restrictions
Combinations with Restrictions
Independent vs Dependent Probability
GMAT Probability Math – The Undesired Approach
GMAT Probability Meets Combinatorics: One Problem, Two Approaches

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Triangle With Other Shapes on the GMAT
Posted on
16
Feb 2021

Triangles With Other Shapes

By: Rich Zwelling, Apex GMAT Instructor
Date: 16th February, 2021

As discussed before, now that we’ve talked about the basic triangles, we can start looking at how the GMAT can make problems difficult by embedding triangles in other figures, or vice versa. 

Here are just a few examples, which include triangles within and outside of squares, rectangles, and circles:

triangles in other shapes GMAT article

Today, we’ll talk about some crucial connections that are often made between triangles and other figures, starting with the 45-45-90 triangle, also known as the isosceles right triangle.

You’ve probably seen a rectangle split in two along one of its diagonals to produce two right triangles:

triangles in other shapes gmat article gmat problem

But one of the oft-overlooked basic geometric truths is that when that rectangle is a square (and yes, remember a square is a type of rectangle), the diagonal splits the square into two isosceles right triangles. This makes sense when you think about it, because the diagonal bisects two 90-degree angles to give you two 45-degree angles:

triangles in other shapes gmat article, 45 45 90 degree angle

(For clarification, the diagonal of a rectangle is a bisector when the rectangle is a square, but it is not a bisector in any other case.)

Another very common combination of shapes in more difficult GMAT Geometry problems is triangles with circles. This can manifest itself in three common ways:

  1. Triangles created using the central angle of a circle

triangle in a circle, gmat geometry article

In this case, notice that two of the sides of the triangle are radii (remember, a radius is any line segment from the center of the circle to its circumference). What does that guarantee about the triangle?

Since two side are of equal length, the triangle is automatically isosceles. Remember that the two angles opposite those two sides are also of equal measure. So any triangle with the center of the circle as one vertex and points along the circumference as the other two vertices will automatically be an isosceles triangle.

2. Inscribed triangles

triangle inscribed in circle, gmat problem

An inscribed triangle is any triangle with a circle’s diameter as one of its sides and a vertex along the circumference. And a key thing to note: an inscribed triangle will ALWAYS be a right triangle. So even if you don’t see the right angle marked, you can rest assured the inscribed angle at that third vertex is 90 degrees.

3. Squares and rectangles inscribed in circles

rectangle in circle, gmat geometry

What’s important to note here is that the diagonal of the rectangle (or square) is equivalent to the diameter of the circle.

Now that we’ve seen a few common relationships between triangles and other figures, let’s take a look at an example Official Guide problem:

A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

A) 19,200
B) 19,600
C) 20,000
D) 20,400
E) 20,800

Explanation

The diagonal splits the rectangular park into two similar triangles:

triangle in other shapes gmat problem

Use SIGNALS to avoid algebra

It can be tempting to then jump straight into algebra. The formulas for perimeter and diagonal are P = 2L + 2W an D2 = L2 + W2, respectively, where L and W are the length and width of the rectangle. The second formula, you’ll notice, arises out of the Pythagorean Theorem, since we now have two right triangles. We are trying to find area, which is LW, so we could set out on a cumbersome algebraic journey.

However, let’s try to use some SIGNALS the problem gives us and our knowledge of how the GMAT operates to see if we can short-circuit this problem.

We know the GMAT is fond of both clean numerical solutions and common Pythagorean triples. The large numbers of 200 for the diagonal and 560 for the perimeter don’t change that we now have a very specific rectangle (and pair of triangles). Thus, we should suspect that one of our basic Pythagorean triples (3-4-5, 5-12-13, 7-24-25) is involved.

Could it be that our diagonal of 200 is the hypotenuse of a 3-4-5 triangle multiple? If so, the 200 would correspond to the 5, and the multiplying factor would be 40. That would also mean that the legs would be 3*40 and 4*40, or 120 and 160.

Does this check out? Well, we’re already told the perimeter is 560. Adding 160 and 120 gives us 280, which is one length and one width, or half the perimeter of the rectangle. We can then just double the 280 to get 560 and confirm that we do indeed have the correct numbers. The length and width of the park must be 120 and 160. No algebra necessary.

Now, to get the area, we just multiply 120 by 160 to get 19,200 and the final answer of A.

Check out the following links for our other articles on triangles and their properties:

A Short Meditation on Triangles
The 30-60-90 Right Triangle
The 45-45-90 Right Triangle
The Area of an Equilateral Triangle
Isosceles Triangles and Data Sufficiency
Similar Triangles
3-4-5 Right Triangle
5-12-13 and 7-24-25 Right Triangles

Read more
Combinatorics: Permutations and Combinations Intro On the GMAT
Posted on
11
Feb 2021

Combinatorics: Permutations and Combinations Intro

By: Rich Zwelling, Apex GMAT Instructor
Date: 11th February, 2021

GMAT Combinatorics. It’s a phrase that’s stricken fear in the hearts of many of my students. And it makes sense, because so few of us are taught anything about it growing up. But the good news is that, despite the scary title, what you need to know for GMAT combinatorics problems is actually not terribly complex.

To start, let’s look at one of the most commonly asked questions related to GMAT combinatorics, namely the difference between combinatorics and permutations

Does Order Matter?

It’s important to understand conceptually what makes permutations and combinations differ from one another. Quite simply, it’s whether we care about the order of the elements involved. Let’s look at these concrete examples to make things a little clearer:

Permutations example

Suppose we have five paintings to hang on a wall, and we want to know in how many different ways we can arrange the paintings. It’s the word “arrange” that often gives away that we care about the order in which the paintings appear. Let’s call the paintings A, B, C, D, and E:

ABCDE
ACDEB
BDCEA

Each of the above three is considered distinct in this problem, because the order, and thus the arrangement, changes. This is what defines this situation as a PERMUTATION problem. 

Mathematically, how would we answer this question? Well, quite simply, we would consider the number of options we have for each “slot” on the wall. We have five options at the start for the first slot:

_5_  ___ ___ ___ ___

After that painting is in place, there are four remaining that are available for the next slot:

_5_  _4_ ___ ___ ___

From there, the pattern continues until all slots are filled:

_5_  _4_ _3_ _2_ _1_

The final step is to simply multiply these numbers to get 5*4*3*2*1 = 120 arrangements of the five paintings. The quantity 5*4*3*2*1 is also often represented by the exclamation point notation 5!, or 5 factorial. (It’s helpful to memorize factorials up to 6!)

Combinations example

So, what about COMBINATIONS? Obviously if we care about order for permutations, that implies we do NOT care about order for combinations. But what does such a situation look like?

Suppose there’s a local food competition, and I’m told that a group of judges will taste 50 dishes at the competition. A first, a second, and a third prize will be given to the top three dishes, which will then have the honor of competing at the state competition in a few months. I want to know how many possible groups of three dishes out of the original 50 could potentially be selected by the judges to move on to the state competition.

The math here is a little more complicated without a combinatorics formula, but we’re just going to focus on the conceptual element for the moment. How do we know this is a COMBINATION situation instead of a permutation question? 

It’s a little tricky, because at first glance, you might consider the first, second, and third prizes and believe that order matters. Suppose that Dish A wins first prize, Dish B wins second prize, and Dish C wins third prize. Call that ABC. Isn’t that a distinct situation from BAC? Or CAB? 

Well, that’s where you have to pay very close attention to exactly what the question asks. If we were asking about distinct arrangements of prize winnings, then yes, this would be a permutation question, and we would have to consider ABC apart from BAC apart from CAB, etc. 

However, what does the question ask about specifically? It asks about which dishes advance to the state competition? Also notice that the question specifically uses the word “group,” which is often a huge signal for combinations questions. This implies that the total is more important than the individual parts. If we take ABC and switch it to BAC or BCA or ACB, do we end up with a different group of three dishes that advances to the state competition? No. It’s the same COMBINATION of dishes. 

Quantitative connection

It’s interesting to note that there will always be fewer combinations than permutations, given a common set of elements. Why? Let’s use the above simple scenario of three elements as an illustration and write out all the possible permutations of ABC. It’s straightforward enough to brute-force this by including two each starting with A, two each starting with B, etc:

ABC
ACB
BAC
BCA
CAB
CBA

But you could also see that there are 3*2*1 = 3! = 6 permutations by using the same method we used for the painting example above. Now, how many combinations does this constitute? Notice they all consist of the same group of three letters, and thus this is actually just one combination. We had to divide the original 6 permutations by 3! to get the correct number of permutations.

Next time, we’ll continue our discussion of permutation math and begin a discussion of the mechanics of combination math. 

Permutations and Combinations Intro
A Continuation of Permutation Math
An Intro To Combination Math
Permutations With Repeat Elements
Permutations With Restrictions
Combinations with Restrictions
Independent vs Dependent Probability
GMAT Probability Math – The Undesired Approach
GMAT Probability Meets Combinatorics: One Problem, Two Approaches

Read more
Isosceles Triangles and Data Sufficiency title
Posted on
26
Jan 2021

Isosceles Triangles and Data Sufficiency

By: Rich Zwelling, Apex GMAT Instructor
Date: 21st January, 2021

Although we’ve already discussed isosceles triangles a bit during our discussion of 45-45-90 (i.e. isosceles right) triangles, it’s worth discussing some other contexts in which you may see isosceles triangles on the GMAT, specifically on Data Sufficiency problems. 

As we discussed before, an isosceles triangle is any triangle that features two equal sides and thus two equal opposite angles:

Isosceles Triangles and Data Sufficiency picture 1

That’s an easy enough definition to remember, but how does the GMAT turn this into more challenging problems? For that, let’s take a look at the following Official Guide problem. Try to solve before reading the explanation below the problem:

Isosceles Triangles and Data Sufficiency picture 2

In the figure above, what is the value of x + y ?
(1) x = 70
(2) ABC and ADC are both isosceles triangles

Explanation

In this case, it’s straightforward enough to determine that each statement alone will be insufficient. Statement (1) gives us a definitive value for x, but no information about y, thus we cannot answer the question (the value of x+y). And although Statement (2) labels each triangle in the diagram as isosceles, we have no way of knowing the specific angles involved nor their relationships. 

However, as with many Data Sufficiency problems, especially those involving Geometry, things can get thorny when we have to combine the statements. The two statements look very complimentary, and that could lead us to prematurely conclude the answer is C (i.e. the two statements are sufficient when combined). But we must do a thorough check. 

Reframing the question

Remember that at any point during a Data Sufficiency problem — beginning, middle, or end — you can reframe the question for simplicity. The question asks for the value of x+y. But now that we are combining the statements, we already know that x=70. In terms of sufficiency, then, what information do we need? The only thing missing is a definitive value of y. The question now might as well be “What is the value of y?”

Now, here’s where the GMAT thinking really comes into play. It’s one thing to understand what an isosceles triangle is. It’s quite another to judge what a diagram of an isosceles triangle does or does not tell you and what you can or cannot extrapolate from it. 

One of my personal favorite things about Geometry Data Sufficiency problems is that they tend to be very intuitive visually. You can often answer them by manipulating figures. 

We know that triangle ADC is isosceles, but is that enough to give us definitive measurements? Visually, which of these does it look like?  

Isosceles Triangles and Data Sufficiency picture 3

Without any numerical evaluations, we can see that we can’t get a definitive measure for the angle at D, which in this case is our y. So even when we combine the statements, we cannot get an answer to our question. The correct answer is E

Here’s another case of a tricky Data Sufficiency problem involving isosceles triangles:

In isosceles triangle RST, what is the measure of angle R?

  • The measure of angle T is 100 degrees
  • The measure of angle S is 40 degrees

Again, give the problem a shot before reading the answer and explanation.

Explanation

This is one for which you can draw a diagram, but it’s not necessary. The trick here is to remember another key property of triangles, namely that all angles in the triangle must sum to 180 degrees.

Since the triangle is isosceles, and since each statement gives you only one angle of three, the temptation can be to say that each statement is insufficient on its own. This is certainly the case for Statement (2), because the 40-degree angle could be one of a pair (in which case we would have a 40-40-100 triangle) or the 40-degree angle could be the odd angle out (in which case we would have a 40-70-70 triangle). 

Because the problem asks for the value of R, and since R could be 40, 70, or 100 depending on the situations outlined above, Statement (2) is INSUFFICIENT.

However, there’s a catch when evaluating Statement (1). Notice that angle T is an obtuse angle, meaning it is greater than 90 degrees. Is it possible that there are two 100-degree angles in a triangle? This would produce a total of 200 degrees, which would exceed the 180-degree total for any triangle. As such, the only possibility is that the 100 degree angle is the odd angle out, and the other two angles are equal acute angles (specifically, we have a 40-40-100 triangle). 

Now we know R must be 40 degrees. Statement (1) is sufficient, and the correct answer is A.

But notice how the GMAT sets the statements up to bait you into thinking that you must combine the two statements to figure out the value of angle R. 

Now that we’ve finished talking about the basic triangle types, we can move on to talking about what happens when triangles are used within different shapes. In the meantime, here are links to our other triangle articles:

A Short Meditation on Triangles
The 30-60-90 Right Triangle
The 45-45-90 Right Triangle
The Area of an Equilateral Triangle
Triangles with Other Shapes
Isosceles Triangles and Data Sufficiency
Similar Triangles
3-4-5 Right Triangle
5-12-13 and 7-24-25 Right Triangles

Read more