GMAT Probability Problems
Posted on
12
Aug 2021

GMAT Probability Problems – How to Tackle Them & What Mistakes to Avoid

By: Apex GMAT
Contributor: Ilia Dobrev
Date: August 12, 2021

The concept of probability questions is often pretty straightforward to understand, but when it comes to its application in the GMAT test it may trip even the strongest mathematicians.

Naturally, the place to find such types of problems is the Quantitative section of the exam, which is regarded as the best predictor of academic and career success by many of the most prestigious business schools out there – Stanford, Wharton, Harvard, Yale, INSEAD, Kellogg, and more. The simple concept of probability problems can be a rather challenging one because such questions appear more frequently as high-difficulty questions instead of low- or even medium-difficulty questions. This is why this article is designed to help test-takers who are pursuing a competitive GMAT score tackle the hazardous pitfalls that GMAT probability problems often create.

GMAT Probability – Fundamental Rules & Formulas

It is not a secret that the Quantitative section of the GMAT test requires you to know just the basic, high-school-level probability rules to carry out each operation of the practical solution path. The main prerequisite for success is mastering the Probability formula:
Probability = number of desired outcomes / total number of possible outcomes

Probability = number of desired outcomes
total number of possible outcomes

We can take one fair coin to demonstrate a simple example. Imagine you would like to find the probability of getting a tail. Flipping the coin can get you two possible comes – a tail or a head. However, you desire a specific result – getting only a tail – which can happen only one time. Therefore, the probability of getting a tail is the number of desired outcomes divided by the number of total possible outcomes, which is ½. Developing a good sense of the fundamental logic of how probability works is central to managing more events occurring in a more complex context.

Alternatively, as all probabilities add up to 1, the probability of an event not happening is 1 minus the probability of this event occurring. For example, 1 – ½ equals the chance of not flipping a tail.

Dependent  Events vs. Independent Events

On the GMAT exam, you will often be asked to find the probability of several events that happen either simultaneously or at different points in time. A distinction you must take under consideration is exactly what type of event you are exploring.

Dependent events or, in other words, disjoint events, are two or more events with a probability of simultaneous occurrence equalling zero. That is, it is absolutely impossible to have them both happen at the same time. The events of flipping either a tail or a head out of one single fair coin are disjoint.

If you are asked to find a common probability of two or more disjoint events, then you should consider the following formula:

Probability P of events A and B   =    (Probability of A) + (Probability of B)

Therefore, the probability of flipping one coin twice and getting two tails is ½ + ½.

If events A and B are not disjointed, meaning that the desired result can be in a combination between A and B, then we have to subtract the intersect part between the events in order to not count it twice:

Probability P of events A and B   =    P(A) + P(B) – Probability (A and B)

Independent events or discrete events are two or more events that do not have any effect on each other. In other words, knowing about the outcome of one event gives absolutely no information about how the other event will turn out. For example, if you roll not one but two coins, then the outcome of each event is independent of the other one. The formula, in this case, is the following:

Probability P of events A and B   =    (Probability of A) x (Probability of B)
How to approach GMAT probability problems

In the GMAT quantitative section, you will see probability incorporated into data sufficiency questions and even problems that do not have any numbers in their context. This can make it challenging for the test taker to determine what type of events he or she is presented with.
One trick you can use to approach such GMAT problems is to search for “buzzwords” that will signal out this valuable information.

  • OR | If the question uses the word “or” to distinguish between the probabilities of two events, then they are dependent – meaning that they cannot happen independently of one another. In this scenario, you will need to find the sum of the two (or more) probabilities.
  • AND | If the question uses the word “and” to distinguish between the probabilities of two events, then they are independent – meaning their occurrences have no influence on one another. In this case, you need to multiply the probabilities of the individual events to find the answer.

Additionally, you can draw visual representations of the events to help you determine if you should include or exclude the intersect. This is especially useful in GMAT questions asking about greatest probability and minimum probability.

If you experience difficulties while prepping, keep in mind that Apex’s GMAT instructors have not only mastered all probability and quantitative concepts, but also have vast experience tutoring clients from all over the world to 700+ scores on the exam. Private GMAT tutoring and tailored customized GMAT curriculum are ideal for gaining more test confidence and understanding the underlying purpose of each question, which might be the bridge between your future GMAT score and your desired business school admissions.

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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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4 Techniques to Ace GMAT Sentence Correction Questions
Posted on
17
Jun 2021

4 Techniques to Ace GMAT Sentence Correction Questions

By: Apex GMAT
Contributor: Andrej Ivanovski
Date: 17th June 2021

 

The GMAT Sentence Correction questions are one of the three question formats that comprise the Verbal section, with the other two being Reading Comprehension and Critical Reasoning. Test takers should expect to come across anything between 11 and 16 sentence correction questions on the exam. Each GMAT Sentence Correction question contains a part that is underlined, and you will be prompted to identify the mistake in the sentence and replace it with one of the five options provided.

Even though this might sound like a piece of cake at first glance, there is a catch. The reason that most GMAT test takers find the Sentence Correction questions challenging is the fact that the sentences provided are usually several lines long and the grammatical mistakes are not very apparent. If you follow these 4 GMAT Sentence Correction techniques you will find it a lot easier to spot the mistakes and ace the GMAT Sentence Correction questions.

Get rid of the extra information

The GMAC intentionally makes the GMAT Sentence Correction problems long by including a lot of fluff and descriptive information which very often covers up the error and makes it very difficult to spot. Therefore, getting rid of that extra information would not only make the sentence shorter and simpler, but it would also make it easier for you to uncover the mistake. But, how do you know which part of the sentence to get rid of?

 

  • Look for parts of the sentence set off by commas. Oftentimes, the part that is set off the comma only serves to better explain or give more details about the subject, and when removed it would not affect the meaning of the sentence. Here’s what extra information looks like in a sentence (note that there are no mistakes in the given example):

Maria, Stephen’s youngest and most talented daughter, moved to Sweden. 

Maria, Stephen’s youngest and most talented daughter, moved to Sweden. 

In the sentence above, the part set off by commas is not necessary to convey the meaning of the sentence. So, even if you get rid of that part, you would still be left with a complete sentence. However, one caveat to keep in mind is that the extra information does not necessarily have to be separated by two commas, as it can come at the beginning or the end of the sentence (a modifier), in which case it would only be set off with a single comma.

  • Look for adjectives and adverbial phrases. These could be a little more challenging to find, as they are not set off by commas and one needs to understand the meaning of the sentence in order to identify them.

A group of young men coming from Dubai held a conference in New York.

The sentence above can exist without the two underlined parts: of young men and coming from Dubai. Even though they make the sentence more descriptive, they do not convey the main meaning of the sentence, and can therefore be taken out of the sentence for the sake of simplicity and spotting the mistake more easily.

Pay attention to the meaning

We have already established that grammar is vital if you want to do well on the GMAT Sentence Correction problems. Is grammar necessary? Absolutely! Is grammar everything that you need? Definitely not! No matter how good you are at grammar, solely relying on it is guaranteed to get you stuck at one point or another.

It is often the case that GMAT Sentence Correction problems are free of grammatical errors, but contain logical ones. GMAT test-makers are actually hoping that test-takers will only rely upon grammar and would not pay attention to less formal errors, so if you want to do well on this type of question you absolutely need to pay attention to the meaning of the sentence.

In order to do so, you first need to read the sentence carefully and try to understand the meaning behind it. Oftentimes, it might seem that the sentence is perfectly correct and free of grammar mistakes, and you would not be able to find a logical gap or an inconsistency. In that case, you will want to look through the answers provided and try to assess the message that they are trying to convey. When doing that, you might get an idea of what could be wrong with the original sentence and that way find the correct one.

Use “splits”

Another strategy which includes using the answer choices in order to successfully answer the GMAT Sentence Correction problems is the so-called “splits” strategy. This strategy involves trying to find similarities and dissimilarities, or any kind of patterns in the answer choices. In order to explain this strategy, we will use a GMAT Sentence Correction problem from the GMAT Official Guide.

 

The overall slackening of growth in productivity is influenced less by government regulation, although that is significant for specific industries like mining, than the coming to an end of a period of rapid growth in agricultural productivity.

  • the coming to an end of
  • the ending of
  • by the coming to an end of
  • by ending
  • by the end of

 

In a question like this, the mistake might not be apparent at first. Therefore, in order to get an idea of what the mistake could be, we will have a look at the answer choices. In there, we can see two patterns: C, D and E all contain “by”, whereas A and B do not. If we look at the sentence, we can see that the first part of it says “is influenced less by”, which implies that the second part of the questions has to begin with “…than by”. Therefore, the split AB, and we continue looking for the answer in the CDE split. If we try to plug each of these three answers into the sentence, we can see that E is the only one that is grammatically correct and therefore we get E as an answer.

The “splits” technique is especially useful in helping you narrow down the choices and find the right answer more easily.

Learn the most common GMAT idioms

In order to do well on the Sentence Correction GMAT questions, you need to have a good command of idioms. If you have already started preparing you might have come across a GMAT idiom list in the prep materials. So, you might be wondering why it is important to learn them and how they will be tested on the GMAT.

First, let us begin by explaining what an idiom is. Chances are, if you are not a “grammar freak” you might not be sure what the exact meaning of an idiom is. An idiom is a common expression or a grammatical structure in a given language, in this case – English. Oftentimes, the term idiom is used to describe a saying such as “let the cat out of the bag” or “a piece of cake”. Even though these are important to know if you want to sound more fluent and natural in English, they are not tested on the GMAT. In the context of the GMAT, an idiom is a formation of two or more words that are often used together, such as “invest in” or “indicate that”.

So, now that we have gotten the definition out of the way, you might be wondering why it is important to learn some of the most common GMAT idioms, and how they will be tested. In the GMAT Sentence Correction problems, oftentimes you will come across an incorrectly used idiom. The mistake can take several different forms. 

  • Preposition

Take, for instance, the expression invest on. Here, the preposition used is on when in fact it should be in. Even though it could be apparent in this case, on the GMAT the mistake can often be subtle and a little more difficult to spot. 

  • Word choice 

This is also a common mistake, especially when it comes to words that are close in meaning. Examples of such words are among/between, fewer/less, whether/if, like/as, and so on.

  • Correlatives

Correlatives are words that are used together to serve a single function in a sentence. Some examples include both/and, either/or and neither/nor. A mistake in correlative pairs is also common, especially when it comes to longer and more complex sentences, as these mistakes could be more difficult to spot in those cases.

Conclusion

Here’s a summary of all of the techniques that we discussed here:

  1. Get rid of extra information
  2. Pay attention to the meaning
  3. Use “splits”
  4. Get familiar with the most common GMAT idioms

These techniques are not mutually exclusive and they can be used in combination with one another. Applying them and putting them into practice can save you a whole lot of work and help you do better on the GMAT Sentence Correction problems. And if you feel like you could use some more guidance, please make sure to check out our highly personalized one-on-one GMAT tutoring. Our tutoring sessions are delivered by 770+ scoring tutors and are available both online and in-person, no matter where in the world you are.

 

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4 techniques to ace gmat sentence correction questions
Posted on
29
Apr 2021

4 Techniques to Ace GMAT Sentence Correction Questions

By: Apex GMAT
Contributor: Andrej Ivanovski
Date: 29th April 2021

The GMAT Sentence Correction questions are one of the three question formats that comprise the Verbal section, with the other two being Reading Comprehension and Critical Reasoning. Test takers should expect to come across anything between 11 and 16 sentence correction questions on the exam. Each GMAT Sentence Correction question contains a part that is underlined, and you will be prompted to identify the mistake in the sentence and replace it with one of the five options provided. Even though this might sound like a piece of cake at first glanceglimpse, there is a catch. The reason that most GMAT test takers find the Sentence Correction questions challenging is the fact that the sentences provided are usually several lines long and the grammatical mistakes are not very apparent. If you follow these 4 GMAT Sentence Correction techniques you will find it a lot easier to spot the mistakes and ace the GMAT Sentence Correction questions.

Get rid of the extra information

The GMAC intentionally makes the GMAT Sentence Correction problems long by including a lot of fluff and descriptive information which very often covers up the error and makes it very difficult to spot. Therefore, getting rid of that extra information would not only make the sentence shorter and simpler, but it would also make it easier for you to uncover the mistake. But, how do you know which part of the sentence to get rid of?

  • Look for parts of the sentence set off by commas. Oftentimes, the part that is set off the comma only serves to better explain or give more details about the subject, and when removed it would not affect the meaning of the sentence. Here’s what extra information looks like in a sentence (note that there are no mistakes in the given example):

Maria, Stephen’s youngest and most talented daughter, moved to Sweden. 

Maria, Stephen’s youngest and most talented daughter, moved to Sweden. 

In the sentence above, the part set off by commas is not necessary to convey the meaning of the sentence. So, even if you get rid of that part, you would still be left with a complete sentence. However, one caveat to keep in mind is that the extra information does not necessarily have to be separated by two commas, as it can come at the beginning or the end of the sentence (a modifier), in which case it would only be set off with a single comma.

  • Look for adjectives and adverbial phrases. These could be a little more challenging to find, as they are not set off by commas and one needs to understand the meaning of the sentence in order to identify them.

A group of young men coming from Dubai held a conference in New York.

The sentence above can exist without the two underlined parts: of young men and coming from Dubai. Even though they make the sentence more descriptive, they do not convey the main meaning of the sentence, and can therefore be taken out of the sentence for the sake of simplicity and spotting the mistake more easily.

Pay attention to the meaning

We have already established that grammar is vital if you want to do well on the GMAT Sentence Correction problems. Is grammar necessary? Absolutely! Is grammar everything that you need? Definitely not! No matter how good you are at grammar, solely relying on it is guaranteed to get you stuck at one point or another.

It is often the case that GMAT Sentence Correction problems are free of grammatical errors, but contain logical ones. GMAT test-makers are actually hoping that test-takers will only rely upon grammar and would not pay attention to less formal errors, so if you want to do well on this type of question you absolutely need to pay attention to the meaning of the sentence.

In order to do so, you first need to read the sentence carefully and try to understand the meaning behind it. Oftentimes, it might seem that the sentence is perfectly correct and free of grammar mistakes, and you would not be able to find a logical gap or an inconsistency. In that case, you will want to look through the answers provided and try to assess the message that they are trying to convey. When doing that, you might get an idea of what could be wrong with the original sentence and in that way find the correct one.

Use “splits”

Another strategy which includes using the answer choices in order to successfully answer the GMAT Sentence Correction problems is the so-called “splits” strategy. This strategy involves trying to find similarities and dissimilarities, or any kind of patterns in the answer choices. In order to explain this strategy, we will use a GMAT Sentence Correction problem from the GMAT Official Guide.

The overall slackening of growth in productivity is influenced less by government regulation, although that is significant for specific industries like mining, than the coming to an end of a period of rapid growth in agricultural productivity.

  • the coming to an end of
  • the ending of
  • by the coming to an end of
  • by ending
  • by the end of

In a question like this, the mistake might not be apparent at first. Therefore, in order to get an idea of what the mistake could be, we will have a look at the answer choices. In there, we can see two patterns: C, D and E all contain “by”, whereas A and B do not. If we look at the sentence, we can see that the first part of it says “is influenced less by”, which implies that the second part of the questions has to begin with “…than by”. Therefore, the split AB, and we continue looking for the answer in the CDE split. If we try to plug each of these three answers into the sentence, we can see that E is the only one that is grammatically correct and therefore we get E as an answer.

The “splits” technique is especially useful in helping you narrow down the choices and find the right answer more easily.

Learn the most common GMAT idioms

In order to do well on the Sentence Correction GMAT questions, you need to have a good command of idioms. If you have already started preparing you might have come across a GMAT idiom list in the prep materials. So, you might be wondering why it is important to learn them and how they will be tested on the GMAT.

First, let us begin by explaining what an idiom is. Chances are, if you are not a “grammar freak” you might not be sure what the exact meaning of an idiom is. An idiom is a common expression or a grammatical structure in a given language, in this case – English. Oftentimes, the term idiom is used to describe a saying such as “let the cat out of the bag” or “a piece of cake”. Even though these are important to know if you want to sound more fluent and natural in English, they are not tested on the GMAT. In the context of the GMAT, an idiom is a formation of two or more words that are often used together, such as “invest in” or “indicate that”.

So, now that we have gotten the definition out of the way, you might be wondering why it is important to learn some of the most common GMAT idioms, and how they will be tested. In the GMAT Sentence Correction problems, oftentimes you will come across an incorrectly used idiom. The mistake can take several different forms. 

  • Preposition

Take, for instance, the expression invest on. Here, the preposition used is on when in fact it should be in. Even though it could be apparent in this case, on the GMAT the mistake can often be subtle and a little more difficult to spot. 

  • Word choice 

This is also a common mistake, especially when it comes to words that are close in meaning. Examples of such words are among/between, fewer/less, whether/if, like/as, and so on.

  • Correlatives

Correlatives are words that are used together to serve a single function in a sentence. Some examples include both/and, either/or and neither/nor. A mistake in correlative pairs is also common, especially when it comes to longer and more complex sentences, as these mistakes could be more difficult to spot in those cases.

Conclusion

Here’s a summary of all of the techniques that we discussed here:

gmat sentence correction

These techniques are not mutually exclusive and they can be used in combination with one another. Applying them and putting them into practice can save you a whole lot of work and help you do better on the GMAT Sentence Correction problems. And if you feel like you could use some more guidance, please make sure to check out our highly personalized one-on-one GMAT tutoring. Our tutoring sessions are delivered by 770+ scoring tutors and are available both online and in-person, no matter where in the world you are.

Read more
GMAT Combinations with Restrictions Article
Posted on
04
Mar 2021

Combinations with Restrictions

By: Rich Zwelling, Apex GMAT Instructor
Date: 4th March, 2021

In our previous post, we discussed how GMAT combinatorics problems can involve subtracting out restrictions. However, we discussed only PERMUTATIONS and not COMBINATIONS.

Today, we’ll take a look at how the same technique can be applied to COMBINATION problems. This may be a bit more complicated, as you’ll have to use the formula for combinations, but the approach will be the same.

Let’s start with a basic example. Suppose I were to give you the following problem:

The board of a large oil company is tasked with selecting a committee of three people to head a certain project for the following year. It has a list of ten applicants to choose from. How many potential committees are possible?

This is a straightforward combination problem. (And we know it’s a COMBINATION situation because we do not care about the order in which the three people appear. Even if we shift the order, the same three people will still comprise the same committee.)

We would simply use the combination math discussed in our Intro to Combination Math post:

                         10!
 10C3 =       ————-
                     3! (10-3!)

 

   10!
———
3! (7!)

 

10*9*8
———
3!

 

10*9*8
———
3*2*1

= 120 Combinations 

However, what if we shifted the problem slightly to look like the following? (As always, give the problem a shot before reading on…):

The board of a large oil company is tasked with selecting a committee of three people to head a certain project for the following year. It has a list of ten applicants to choose from, three of whom are women and the remainder of whom are men. How many potential committees are possible if the committee must contain at least one woman?

A) 60
B) 75
C) 85
D) 90
E) 95

In this case, there’s a very important SIGNAL. The language “at least one” is a huge giveaway. This means there could be 1 woman, 2 women, or 3 women which means we would have to examine three separate cases. That’s a lot of busy work. 

But as we discussed in the previous post, why not instead look at what we don’t want and subtract it from the total? In this case, that would be the case of 0 women. Then, we could subtract that from the total number of combinations without restrictions. This would leave behind the cases we do want (i.e. all the cases involving at least one woman). 

We already discussed what happens without restrictions: There are 10 people to choose from, and we’re selecting a subgroup of 3 people, leading to 10C3  or 120 combinations possible. 

But how do we consider the combinations we don’t want? Well, we want to eliminate every combination that involves 0 women. In other words, we want to eliminate every possible committee of three people that involves all men. So how do we find that?

Well, there are seven men to choose from, and since we are choosing a subgroup of 3, we can simply use 7C3 to find the number of committees involving all men:

                       7!
7C3 =       ————-
                 3! (7-3!)

 

  7!
———
3! (4!)

 

7*6*5
———
    3!

= 7*5 = 35 Combinations involving all men

So, out of the 120 committees available, 35 of them involve all men. That means 120-35 = 85 involve at least one woman. The correct answer is C. 

Next time, we’ll return to probability and talk about how the principle of subtracting out elements that we don’t want can aid us on certain questions. Then we’ll dovetail the two and talk about how probability and combinatorics can show up simultaneously on certain questions.

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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What happens when Permutations have repeat elements?
Posted on
25
Feb 2021

What happens when Permutations have repeat elements?

By: Rich Zwelling, Apex GMAT Instructor
Date: 25th February, 2021

Permutations With Repeat Elements

As promised in the last post, today we’ll discuss what happens when we have a PERMUTATIONS situation with repeat elements. What does this mean exactly? Well, let’s return to the basic example in our intro post on GMAT combinatorics:

If we have five distinct paintings, and we want to know how many arrangements can be created from those five, we simply use the factorial to find the answer (i.e. 5! = 5*4*3*2*1 = 120). Let’s say those paintings were labeled A, B, C, D, and E. 

Now, each re-arrangement of those five is a different PERMUTATION, because the order is different:

ABCDE
EBADC
CADBE


etc

Remember, there are 120 permutations because if we use the blank (or slot) method, we would have five choices for the first blank, and once that painting is in place, there would be four left for the second blank, etc…

_5_  _4_  _3_  _2_  _1_ 

…and we would multiply these results to get 5! or 120.

However, what if, say we suddenly changed the situation such that some of the paintings were identical? Let’s replace painting C with another B and E with another D:

ABBDD

Suddenly, the number of permutations decreases, because some paintings are no longer distinct. And believe it or not, there’s a formulaic way to handle the exact number of permutations. All you have to do is take the original factorial, and divide it by the factorials of each repeat. In this case, we have 5! for our original five elements, and we now must divide by 2! for the two B’s and another 2! for the two D’s:

  5!
——
2! 2!     

= 5*4*3*2*1
   ————-
  (2*1)(2*1)

= 5*2*3
= 30 permutations

As another example, try to figure out how many permutations you can make out of the letters in the word BOOKKEEPER? Give it a shot before reading the next paragraph.

In the case of BOOKKEEPER, there are 10 letters total, so we start with a base of 10! 

We then have two O’s, two K’s and three E’s for repeats, so our math will look like this:

   10!
———
2! 2! 3! 

Definitely don’t calculate this, though, as GMAT math stays simple and likes to come clean. Remember, we’ll have to divide out the repeats. You are extremely unlikely to have to do this calculation for a GMAT problem, however, since it relies heavily on busy-work mechanics. The correct answer choice would thus look like the term above. 

Let’s now take a look at an Official Guide question in which this principle has practical use. I’ll leave it to you to discover how. As usual, give the problem a shot before reading on:

A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys?

(A) 3/8
(B) 1/4
(C) 3/16
(D) 1/8
(E) 1/16

Quick Probability Review

Remember from our post of GMAT Probability that, no matter how complicated the problem, probability always boils down to the basic concept of:

    Desired Outcomes
———————————–
Total Possible Outcomes

In this case, each child has two equally likely outcomes: boy and girl. And since there are four children, we can use are blank method to realize that we’ll be multiplying two 4 times:

_2_  _2_  _2_  _2_   =  16 total possible outcomes (denominator)

This may give you the premature notion that C or E must be correct, simply because you see a 16 in the denominator, but remember, fractions can reduce! We could have 4 in the numerator, giving us a fraction of 4/16, which would reduce to 1/4. And every denominator in the answer choices contains a factor of 16, so we can’t eliminate any answers based on this. 

Now, for the Desired Outcomes component, we must figure out how many outcomes consist of exactly two boys and two girls. The trick here is to recognize that it could be in any order. You could have the two girls followed by the two boys, vice versa, or have them interspersed. Now, you could brute-force this and simply try writing out every possibility. However, you must be accurate, and there’s a chance you’ll forget some examples. 

What if we instead write out an example as GGBB for two girls and two boys? Does this look familiar? Well, this should recall PERMUATIONS, as we are looking for every possible ordering in which the couple could have two girls and two boys. And yes, we have two G’s and two B’s as repeats. Here’s the perfect opportunity to put our principle into play:

We have four children, so we use 4! for our numerator, then we divide by 2! twice for each repeat:

  4!
——
2! 2! 

This math is much simpler, as the numerator is 24, while the denominator is 4. (Remember, memorize those factorials up to 6!)

This yields 6 desired outcomes of two boys and two girls. 

With 6 desired outcomes of 16 total possible outcomes, our final probability fraction is 6/16, which reduces to 3/8. The correct answer is A.

Next time, we’ll look into combinatorics problems that involve restrictions, which can present interesting conceptual challenges. 

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 Inequality Rule on the GMAT
Posted on
09
Feb 2021

Triangle Inequality Rule

By: Rich Zwelling, Apex GMAT Instructor
Date: 9th February, 2021

One of the less-common but still need-to-know rules tested on the GMAT is the “triangle inequality” rule, which allows you to draw conclusions about the length of the third side of a triangle given information about the lengths of the other two sides.

Often times, this rule is presented in two parts, but I find it is easiest to condense it into one, simple part that concerns a sum and a difference. Here’s what I mean, and we’ll use a SCENARIO:

Suppose we have a triangle that has two sides of length 3 and 5:

triangles inequalities 1

What can we say about the length of the third side? Of course, we can’t nail down a single definitive value for that length, but we can actually put a limit on its range. That range is simply the difference and the sum of the lengths of the other two sides, non-inclusive.

So, in this case, since the difference between the lengths of the other two sides is 2, and their sum is 8, we can say for sure that the third side of this triangle must have a length between 2 and 8, non-inclusive. [Algebraically, this reads as (5-3) < x < (5+3) OR 2 < x < 8.]

If you’d like to see that put into words:

**The length of any side of a triangle must be shorter than the sum of the other two side lengths and longer than the difference of the other two side lengths.**

It’s important to note that this works for any triangle. But why did we say non-inclusive? Well, let’s look at what would happen if we included the 8 in the above example. Imagine a “triangle” with lengths 3, 5, and 8. Can you see the problem? (Think about it before reading the next paragraph.)

Imagine a twig of length 3 inches and another of length 5 inches. How would you form a geometric figure of length 8 inches? You’d simply join the two twigs in a straight line to form a longer, single twig of 8 inches. It would be impossible to form a triangle with a side of 8 inches with the original two twigs.

triangle inequalities 2

 

If you wanted to form a triangle with the twigs of 3 and 5, you’d have to “break” the longer twig of 8 inches and bend the two twigs at an angle for an opportunity to have a third side, guaranteed to be shorter than 8 inches:

triangle inequalities 3

The same logic would hold for the other end of the range (we couldn’t have a triangle of 3, 5, and 2, as the only way to form a length of 5 from lengths of 2 and 3 would be to form a longer line segment of 5.)

Now that we’ve covered the basics, let’s dive into a few problems, starting with this Official Guide problem:

If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k?
(A) one
(B) two
(C) three
(D) four
(E) five

Strategy: Eliminate Answers

As usual with the GMAT, it’s one thing to know the rule, but it’s another when you’re presented with a carefully worded question that tests your ability to pay close attention to detail. First, we are told that two of the lengths of the triangle are 2 and 7. What does that mean for the third side, given the triangle inequality rule? We know the third side must have a length between 5 (the difference between the two sides) and 9 (the sum of the two sides).

Here, you can actually use the answer choices to your advantage, at least to eliminate some answers. Notice that k is specified as an integer. How many integers do we know now are possible? Well, if k must be between 5 and 9 (and remember, it’s non-inclusive), the only options possibly available to us are 6, 7, and 8. That means a maximum of three possible values of k, thus eliminating answers D and E.

Since the GMAT is a time-intensive test, you might have to end up guessing now and then, so if you can strategically eliminate answers, it increases your chances of guessing correctly.

Now for this problem, there’s another condition given, namely that 2 < k < 7. We already determined that k must be 6, 7, or 8. However, of those numbers, only 6 fits in the given range 2 < k < 7. This means that 6 is the only legal value that fits for k. The correct answer is A.

Note:

It’s important to emphasize that the eliminate answers strategy is not a mandate. We’re simply presenting it as an option that works here because it is useful on many GMAT problems and should be explored and practiced as often as possible.

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
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

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Posted on
15
Jan 2021

Counting Primes Exponent Inequality – GMAT Problem

Today, we are looking at counting a primes exponent inequality problem. Despite all those scary terms, this one is actually fairly straightforward once you master the ability to count prime factors.

Counting primes is all about understanding how many versions of each prime are necessary to construct the entire prime factorization of an integer. In this problem, we are comparing 25s and 5s and we are being asked how many 25s versus how many 5s there are.

Notice how we are not diving into the math immediately. We are first putting this in terms of counting only. 5 to the 12th means that we are actually multiplying 5 by itself 12 times. Like this: 5x5x5x5x5x5x5x5x5x5x5x5. We can now say we have 12 fives. The question then becomes: how many 25s is this equivalent to?

We are now looking for inequality by forming a baseline of equivalents. We now understand how much too many or too few would be. The key question here is how many 5s make up 25? The answer is not 5: we are not dividing or multiplying. 2 prime factors of 5 make 25. 5×5. That is 25=5 square. We wouldn’t know how many 25 it takes to hate more than 12 5s. Where each 25 is the equivalent of 2 5s, 6 25s is the same as 12 5s. So, we need now a 7th 25 in order to have more 5s than the 12 5s on the other side.

And that is our answer: 7. Answer choice B.

For additional problems like this, especially counting primes and number theory problems, check out these videos. 

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The Area of an Equilateral Triangle
Posted on
14
Jan 2021

The Area of an Equilateral Triangle

By: Rich Zwelling, Apex GMAT Instructor
Date: 14th January, 2021

As promised, we will now connect the 30-60-90 triangle to the equilateral triangle, specifically its area. There is a formula for the area of an equilateral triangle as it relates to the length of its side s, and it is as follows:

Equilateral triangles GMAT picture 1

But more likely than not for the GMAT, you’ll need to understand how this formula is derived. And the √3 term in the area is a big clue.

First, it helps to remember that an equilateral triangle has all equal angles as well as all equal sides. And given that the angles in a triangle must sum to 180 degrees, each angle must be 60 degrees:

Equilateral triangles GMAT picture 2

Now, what happens when we take such a triangle and split it down the middle?

Equilateral triangles GMAT picture 3This should look familiar. Because the line segment down the middle acts as an angle bisector, the 60 degree angle at the top vertex becomes two 30-degree angles. Take a moment to consider what this produces and what the implications are.

As you might have guessed, this line segment produces two 30-60-90 right triangles:

Equilateral triangles GMAT picture 4

Not only that, but we can then use s to denote the side length of the equilateral triangle and map out each segment of the 30-60-90 right triangles. Before viewing the diagram below, take a moment to consider what the height of the triangle would be.

Remember that the ratio of side lengths is 1 : √3 : 2. If we fill in all of the appropriate lengths, we would get the following:

Equilateral triangles GMAT picture 5Now, we’re very close to deriving the area of the triangle, which is simply base*height/2. In this case, the base is s, while the height is s√3/2.

This is how we finally get the universal formula for an equilateral triangle:

Area = base * height / 2
Area = (s) * (s√3/2) / 2
Area = (s) * (s√3/4)
Area = (s2√3) / 4.

Now that we’ve seen the relationship between equilateral and 30-60-90 triangles, let’s see how it plays out in an official GMAT problem:

The figure shown above consists of three identical circles that are tangent to each other. If the area of the shaded region is 64√3 – 32π, what is the radius of each circle?

Equilateral triangles GMAT picture 6

A. 4
B. 8
C. 16
D. 24
E. 32

Using signals

This is a complex problem that seems intimidating at first. However, if we use signals the problem is giving us, we can get to the answer more quickly than we might initially think. What signals does the area of the shaded region give us? Think about it before reading on…

If we look closely at the diagram, we see that an equilateral triangle is involved. We know this because each side of the triangle consists of two radii of each circle (i.e. the distance from the center to the outer edge of the circle), thus each side of the triangle must be equal. That’s a big hint that the √3 term is linked to the area formula we’ve been talking about.

Likewise, although it is not the subject of this post, the term using π is associated with circles in this case, the areas of the identical circles. (For reference, the area of a circle is πr2, and the circumference of a circle is 2πr.)

Conceptually, we should be able to see that 64√3 – 32π represents the area of the equilateral triangle minus the area of the three small sectors from the circles. 

Now, rather than do any unnecessarily complicated math, we should take notice that the question asks for the radius of each circle, and each side of the equilateral triangle is 2r:

Equilateral triangles GMAT picture 7

We already know that the area of the equilateral triangle is 64√3, and we have the formula for that area, so we are just a few steps away from solving for the radius.

Remember the formula, where s is the length of the side of the equilateral triangle:
Area = (s2√3) / 4

Substitute:
64√3 = (s2√3) / 4

Since √3 is common to both sides, you can divide it out:

64 = s2 / 4
256 = s2

Now, normally, you would say that s could be 16 or -16, but since this is a geometric quantity, we only deal in nonnegative quantities. Therefore:

s = 16, giving us the length of each side of the equilateral triangle.

Be careful, however. This could trap you into picking answer choice C. Remember to check exactly what the question asks for. We were asked for the radius of the circle, which as we see in the above diagram is half the length of s. The correct answer is B.

Again, it’s very important to notice that we didn’t do anything with the circles. The 64√3 term and the equilateral triangle were enough to get us the length of each side and thus the radius. Look for signals to help short-circuit problems and avoid lengthy solution paths.

Now that we’ve reviewed all of the basic triangles, we’ll do a little more next time on how triangles can appear in other shapes, such as circles and rectangles. We got a little taste today, so hopefully that will give you a good idea.

Find more articles in our triangle series here:
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

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5-12-13 and 7-24-25 triangles on the gmat
Posted on
12
Jan 2021

The 5-12-13 and 7-24-25 Right Triangles

By: Rich Zwelling, Apex GMAT Instructor
Date: 12th January, 2021

The 5-12-13 and 7-24-25 Right Triangles

Although the 3-4-5 right triangle is by far the most common of the so-called “Pythagorean triples” tested on the GMAT, there are a few others worth knowing. First, a little review: 

You’ll recall that the Pythagorean Theorem ( + b² = c²) holds for any right triangle where a and b are the two legs and c is the hypotenuse, and that the 3-4-5 triangle represents the smallest such triangle with all integer side lengths:

5-12-13 and -7-24-25 Triangle Identities problem 1

This works not only for 3-4-5 but also for 6-8-10, 9-12-15, or any other multiples of each side length.

5-12-13 and -7-24-25 Triangle Identities problem 2No matter what positive integer n you choose for the figure above, you will produce a valid right triangle.

So now we come to the main topic: what are some other common “Pythagorean triples” the GMAT may test? The next base triples that fit the Pythagorean Theorem are 5-12-13 and 7-24-25. These work because if you check the arithmetic, 5² + 12² = 13² and 7² + 24² = 25²:

5-12-13 and -7-24-25 Triangle Identities problem 3

As we’ve continually discussed, however, your success on more difficult GMAT problems will require you to go beyond mere rote memorization. Let’s take a look at an Official Guide Data Sufficiency problem that illustrates how the test can force you to engage some higher-level reasoning skills:

5-12-13 and -7-24-25 Triangle Identities problem 4

If A is the area of a triangle with sides of lengths x, y, and z as shown above, what is the value of A?

(1) z = 13

(2) A = 5y/2

Give it a try on your own before reading any further.

As with any Data Sufficiency question, let’s identify what we’re asked to find. A represents the area of the triangle, which is found by multiplying base by height and dividing by 2. That means A = xy/2, since x and y represent the height and base, respectively. 

Remember, it helps to frame Data Sufficiency questions in terms of what information you need to get to the answer. We need to know the individual values of x and y. Or, as a matter of fact, we could have sufficiency if we knew xy as a product, even if we didn’t know the values of x and y, individually. For example, on a different problem with the same question, if the test had said that the product of the base and height were 30, that would have been sufficient, as that would be enough for us to deduce that the area is 15.  

You can save yourself much time and mental energy by having a solid idea of what information you need from the statements for sufficiency before you actually view the statements. 

Now that we know what information we need for sufficiency, let’s examine each statement on its own. Statement (1) should get you thinking about the 5-12-13 right triangle, as it tells us that the hypotenuse is 13. But be careful: this is where rote memorization only goes so far (and may actually get in the way). 

Does knowing that the hypotenuse is 13 guarantee that the other sides are 5 and 12? For all we know, they could be non-integers that fit + b² = 13². In fact, a and b could be equal — remember that we can’t assume that the figures are drawn to scale. Without a clear idea of what the base and height are, we cannot get a consistent product for xy. Statement (1) is INSUFFICIENT on its own.

Statement (2) is more complicated, as we have two variables, one of which is the area. But we already discussed that A = xy/2, so we can do a substitution:

A = 5y/2
xy/2 = 5y/2

At this point, we can see that the sides are identical, except that the x on the left has been replaced by a 5 on the right. Therefore, x must be 5. Again, this should get us thinking about the 5-12-13 triangle. But we should again remember that this alone does not guarantee that the other sides are 12 and 13. Even though x is 5, there could be multiple values for y, and that means multiple values for the product xy. Statement (2) is also INSUFFICIENT on its own.

This narrows the answer choices down to C (statements sufficient together) and E (statements insufficient together).

This is where previous knowledge of the 5-12-13 triangle helps. Ideally, once you see that the statements together tell you that x=5 and z=13, you will know without much thought that y must be 12. You won’t bother using the Pythagorean theorem and you certainly won’t wonder if y could have multiple values.

Without knowledge of the 5-12-13, one trap a test-taker could possibly fall into is viewing the two statements and noticing that there are 3 variables and only 2 equations. We need a full 3 equations with 3 variables if we’re going to solve for all 3 variables, and that may lead some to prematurely conclude that the answer is E. 

However, why is that a false conclusion?

Well, we’re not trying to solve for all variables. We’re only solving for one. It’s possible to solve for one variable, even if there are fewer equations than variables. 

In this case, now that we know that x=5 and y=12, we have our base and height, and we can solve for A, the area of the triangle. Note that I’m not going to bother solving, because for sufficiency, I don’t need to. I only care that I CAN solve. The final answer is (C).

We’ve now talked about the various Pythagorean triples and special right triangles. Next time, we’ll talk about how triangles can appear within OTHER shapes. And to tide yourself over, you can also link to our other article about triangles:

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

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