Posted on
19
Apr 2022

## Cylinders & Spheres In The GMAT

Welcome back to our fifth and final article on GMAT circles. Last time we explored the possibilities of treating a circle’s of a right triangle. This time we will introduce you to the concept of cylinders and spheres — two 3-dimensional shapes built from circles.

## 1. Cylinder

More than likely, you already know what these things are and could describe them. But let’s try to define them in some interesting ways. A cylinder is a “tall circle” or – to use more proper geometric terminology – a circular prism. A prism is the solid shape that results when you take any polygon and “pull it” upward into something like a pillar. The polygon you started with still exists as the “top and bottom” faces of the prism, and the faces around the sides of the prism are rectangles. (Technically they can be parallelograms, which would produce a “leaning” pillar, but this won’t happen on the GMAT.)

Since a circle doesn’t have sides, a cylinder doesn’t have faces – except for the two circles on its top and bottom. In between, there is one smoothly-curving surface. If you need to find the area of this third surface, you can treat it like a rectangle. The length of this rectangle is the height of the cylinder, and the width of this rectangle is the circumference of the circle. The volume of any prism is the area of its base polygon multiplied by the prism’s height. So for a cylinder, the equation is

V = πr²h

## 2. Sphere

Now for spheres. We all know that a sphere is a perfectly round ball. But think about this: a sphere is like a circle “any way you slice it” – quite literally. If you have some citrus fruits in your kitchen, you can try slicing them in different places at different angles, and the faces of the two resulting pieces will always be circles. Another way to say this is that any cross section taken from a sphere will be a circle. No matter how hard you try, you will never be able to produce an elliptical orange slice. Sorry to disappoint you.

Let’s see how the GMAT employs these shapes in some official problems. Some basic cylinder problems focus on one whole cylinder. More challenging cylinder problems compare one cylinder to another or treat a cylinder as a partially-filled tank.

## 3. A Data Sufficiency Problem Featuring Two Cylinders

It costs \$2,250 to fill right circular cylindrical Tank R with a certain industrial chemical. If the cost to fill any tank with this chemical is directly proportional to the volume of the chemical needed to fill the tank, how much does it cost to fill right circular cylindrical Tank S with the chemical?

1. The diameter of the interior of Tanks R is twice the diameter of the interior of Tank S.
2. The interiors of Tanks R and S have the same height.

(A) Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
(B) Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

### Solution

Since the cost to fill any tank (including tanks R and S) with this chemical is directly proportional to the volume of the cylindrical tank, the only thing we care about here is the ratio of the two tanks’ volumes. Remember that for the volume of a cylinder, we need to know (or be able to derive) both the area of the circle and the height of the cylinder.

Statement 1 gives us the ratio of the tanks’ diameters: 2:1. This means that the ratio of the areas of the tanks’ bases is 4:1 (if lost here, review on area, circumference, and pi). This is great, but it is still not enough to know the overall ratio of the tanks’ volumes. Statement 1 is insufficient.

Statement 2 tells us that tanks R and S are the same height, specifying “interior” because we are filling up space with a chemical and can’t count whatever volume is taken up by the tank walls. On its own, this information is insufficient.

Combining statements 1 and 2, we have the ratio of the tanks’ diameters (2:1) and the ratio of their heights (1:1). This means that the overall ratio of the tanks’ volumes is fixed. Statements 1 and 2 together are sufficient, and the correct answer is C.

## 4. Partially Filled Cylinder-as-a-tank Problem

The figures show a sealed container that is a right circular cylinder filled with liquid to 12 its capacity. If the container is placed on its base, the depth of the liquid in the container is 10 centimeters and if the container is placed on its side, the depth of the liquid is 20 centimeters. How many cubic centimeters of liquid are in the container.

(A) 4,000 π
(B) 2,000 π
(C) 1,000 π
(D) 400 π
(E) 200 π

### Solution

This problem is less complex than it might first appear. It all comes together when you realize that the 20cm depth in the second orientation of the tank represents the radius of the circle!  Now you can get the area of the circle in cm² using A = r² and then multiply the result by 10 (the depth in centimeters of the liquid in the upright tank) to get the volume of the liquid in cm³. If you can mentally square 20 and then multiply by 10, you should be just seconds away from selecting correct answer choice A.

## 5. Final Cylinder-as-a-tank Problem

A tank is filled with gasoline to a depth of exactly 2 feet. The tank is a cylinder resting horizontally on its side, with its circular end oriented vertically. The inside of the tank is exactly 6 feet long. What is the volume of the gasoline in the tank?

1. The inside of the tank is exactly 4 feet in diameter.
2. The top surface of the gasoline forms a rectangle that has an area of 24 square feet.

(A) Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
(B) Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

### Solution

#### Statement 1.

Evaluating statement 1 is fairly straightforward. Combined with the information from the question stem that the depth of the gasoline in the tank is 2 feet, the additional information that the inside of the tank has a 4-foot diameter means that the tank is filled halfway with gasoline. If 4 is the diameter, then 2 is the radius, and the gas fills the tank up to its center line. This looks just like the half-filled tank in the previous problem. The question stem also gave us the length of the tank (called a length rather than a height since this tank is a cylinder “lying down”), so the cylinder’s total and fractional volumes are calculable. Statement 1 is sufficient.

#### Statement 2.

Statement 2 performs something like a “double flip.” We are told that the top surface of the gasoline is a 24ft² rectangle. Remembering from the question stem that the tank is 6 feet long, you may realize that 24/6 = 4 and think that this tells you the same thing as statement 1: that the tank has a 4-foot diameter. This would be a mistake. The 24ft² rectangle formed by the surface of the gasoline indeed has a length of 6 and a width of 4, but this width of 4 is not necessarily the diameter of the tank. It could just as easily happen in a larger tank that is less than half (or more than half) full.

Does this make statement 2 insufficient? Well so far, yes. But there’s something we’ve left out that makes it sufficient after all! From the question stem, the depth of the gasoline in the tank is 2 feet. Imagine that the circular end of this tank is transparent. Looking at it this way, the top surface of the gasoline makes a horizontal chord across the circle, and this chord has a length of 4. Simultaneously, this chord is a vertical distance of 2 feet from the bottom of the circle (since the depth of the gasoline in the tank is 2 feet). The only way this can happen is if the 4-foot chord is the diameter of the circle!

Therefore the tank is still half full, and the volume of the gasoline is half of the (calculable) volume of the cylinder. Statement 2 is also sufficient, and the correct answer choice is D.

## 6. Sphere Problem

For the final problem in our circles series, we’ll work with spheres. Spheres are less common on the GMAT than cylinders, and you will never have to memorize any of their formulas. If you need a sphere formula for a problem, it will be supplied with the problem.

For a party, three solid cheese balls with diameters of 2 inches, 4 inches, and 6 inches, respectively, were combined to form a single cheese ball. What was the approximate diameter, in inches, of the new cheese ball? (The volume of a sphere is 433, where r is the radius.)

(A) 12
(B) 16
(C) ∛16
(D) 8
(E) 236

### Solution – Long Way

This sounds like a party you don’t want to miss. I don’t know exactly how to combine three solid cheese balls into one, but I do know how to calculate the diameter.

There are two ways to solve this problem: the long way and the best way. The long way is to calculate the volumes of the three original cheese balls, sum your answers into one volume, and then solve for the radius of the combined cheese ball. First you must divide the given diameters of the original cheese balls by 2, since the volume equation uses radius instead.

V = (4π/3)r³
V = (4π/3)1³ + (4π/3)2³ + (4π/3)3³
V = (4π/3)(1³ + 2³ + 3³)
V = (4π/3)(1 + 8 + 27)
V = (4π/3)(36)
V = 48π
V = (4π/3)r³
48π = (4π/3)r³
48 = (4/3)r³
36 = r³
∛36 = r
2(∛36) = D

And the correct answer choice is E.

### Solution – Short Way

That was the long way. The best way is to think logically and exploit the answer choices. Since we are effectively adding some cheese onto a ball that already has a diameter of 6 inches, the diameter of the combined cheese ball will be greater than 6 inches. This means that answer choices C and D are nonstarters. (C is somewhere between 2 and 3, and D is exactly 6.) Let’s think next about choices A and B, since they are integers and easier to evaluate than choice E.

Can the diameter of the combined cheese ball be as great as 12 (choice A) or even 16 (choice B)? No, it can’t. Picture a “cheese ball snowman” made of the three original cheese balls – a cooler idea for a party than smashing them into one ball, I argue. His height is 12 inches, but this is not the same as having a single ball with a 12-inch diameter. Three spheres whose diameters sum to 12 cannot combine their volumes to produce a single sphere with a diameter of 12. Therefore choices A and B are also out, leaving us with correct choice E. If we approximate the value of E, it is greater than 6 but less than 8, since the cube root of 36 is greater than 3 but less than 4. A combined cheese ball this size makes logical sense.

This concludes our fifth and final article on GMAT circles. Cheers.

Contributor: Elijah Mize (Apex GMAT Instructor)

Posted on
05
Apr 2022

## Radius As Hypotenuse – Problems & Solutions

Welcome back to our fourth article on GMAT circles. Last time we considered inscribed angles and learned that where there is a 90-degree inscribed angle, there is a hypotenuse that is also a diameter of the circle. This time we will explore a class of problems where the radius, rather than the diameter, pulls double duty as a hypotenuse. Let’s dive right in with the following official problem.

## 1. Radius as Hypotenuse  – GMAT Official Problem

The figure above represents a semicircular archway over a flat street. The semicircle has a center at O and a radius of 6 feet. What is the height h, in feet, of the archway 2 feet from its center?

A. √2
B. 2
C. 3
D. 4√2
E. 6

This problem is a straightforward application of the Pythagorean theorem. Since we are told that the radius of the semicircle is 6 feet, we can draw a 6-foot radius from center O to the point where height h meets the semicircle. Voila – a right triangle.

h = √(62 – 22)
h = √(36 – 4)
h = √32

This is where you should stop and mark answer choice D since we are taking the square root of a number that is not a perfect square. When we simplify this radical, something will get left inside. Therefore answers B, C, and E are out (Answer A is out because √2 =/= √32), and the correct choice is D.

## 2. Radius as Hypotenuse Problem 1

Let’s try something a little different:

In the xy-plane, point (r,s) lies on a circle with center at the origin. What is the value of + s²?

1. The circle has radius 2.
2. The point (2,-2) lies on the circle.

A. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are not sufficient.

This is the first problem we’ve seen where a circle is placed on the xy-plane. In such problems, it is usually helpful to remember the basic circle principle that every point on the circle (meaning on its edge or perimeter) is equidistant from its center.

### Solution

If you’re unfamiliar with these problems, statement 1 may trip you up. Is the radius of the circle sufficient to determine + ? Yes, it is. If you are concerned about the unknown positivity/negativity of the coordinates r and s, recall that the square of any number (except 0) is positive. This means that for any positive/negative combination of r and s, the sum + will have the same value.

But what you really need here is to see that the expression + matches the famous + from the Pythagorean theorem, and in fact, it functions the exact same way.

In this setup, the radius is the hypotenuse of the right triangle with legs r and s. Therefore, applying the Pythagorean theorem, the value + represents the square of the radius. So if we know the value of the radius (2), we know the value r² + s², and statement 1 is sufficient.

Statement 2 offers that the point (√2, -√2) lies on the circle. This statement should be “easier” to evaluate than statement 1. Seeing the radicals in the coordinates ought to help you make the connection to the Pythagorean theorem if you didn’t already while evaluating statement 1. But using the principle that every point on a circle is equidistant from its center, we know that this given point (√2, -√2) is the same distance from the center as the point (r, s) in the question. Therefore if we sum the squares of √2 and -√2, the result (4) will also represent the value r² + s² we were asked about.

## 3. Problem 2

Let’s try one more:

In cross section, a tunnel that carries one lane of one-way traffic is a semicircle with radius 4.2 m. Is the tunnel large enough to accommodate the truck that is approaching the entrance to the tunnel?

1. The maximum width of the truck is 2.4 m.
2. The maximum height of the truck is 4 m.

A. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are not sufficient.

This one is a little more complex. Sometimes on GMAT quant problems, it is helpful to ask why certain details were specified. In this case, we are told that the tunnel “carries one lane of one-way traffic.” This is important because if it were not the case, the truck would have to drive on one side or the other, and there’s no way it would be able to get through the tunnel. Since there is only one lane going through the tunnel, the truck can “center up” to give itself the best chance of fitting through.

### Solution

This is one of those less-common DS problems where each statement on its own is clearly insufficient. If all we know is that the truck is 2.4m wide at its widest point (statement 1), it may still be too tall to fit through the tunnel. If all we know is that the truck is 4m tall at its tallest point, we don’t know whether the truck is narrow enough to make it through the tunnel while being this tall.

But if we combine statements 1 and 2, we can use the Pythagorean theorem to calculate the max distance of a point on the “centered up” truck from the point at the “center” of the semicircle.

Now here’s the key step: don’t calculate! Running the Pythagorean theorem with our values here would be a waste of time. As long as the value p [from the graphic] is less than 4.2 (the radius of the tunnel), the truck will fit. But for DS, we don’t have to know whether the truck will fit. All we have to know is whether the value p can be calculated, and in this case, it can be. Statements 1 and 2 together are sufficient, and the correct answer choice is C.

This concludes our fourth article on the GMAT’s treatment of circles. Next time we will look at circles in two different 3-dimensional shapes: cylinders and spheres.

Contributor: Elijah Mize (Apex GMAT Instructor)

Posted on
09
Nov 2021

## Successful GMAT Prep – 5 Things You Need to Know To Ace The GMAT

By: ApexGMAT
Date: October 28, 2021

The GMAT is one of the greatest challenges that many people face on the road to their MBA acceptance, but it doesn’t have to be. For many, the anxiety surrounding the GMAT is due to it being a largely misunderstood challenge. Contrary to what you might think, the GMAT represents an opportunity to illustrate your creativity and improve your critical and creative thinking skills, not just revise your knowledge of high school math and grammar. When properly preparing for the exam you’ll develop:

• new ways to approach solving problems of all sorts
• novel techniques for organizing and characterizing information
• the ability to curate your own thought process to become a more effective thinker

With this in mind, I’d like to discuss five key points to help you get into the correct mindset for a successful (read: transformative) and low-stress GMAT preparation experience.

## 1. You are not your GMAT.

Many people use their GMAT score to define their abilities across a range of fields, their value as an applicant, or, even more insidiously, in a greater self-esteem context.

Your GMAT score doesn’t represent how smart you are or how capable you are as a person, student, or professional. It certainly doesn’t deliver the distinct mix of characteristics that make you, well, you. What admissions committees are seeking when they look at your GMAT score is a set of skills that are valuable in a number of ways (more on this later), but tying your self-worth up in a number is perilous, to say the least.

Putting the self-esteem aspect aside for a moment, identifying yourself with your GMAT means that you are giving short shrift to who you are as a person outside of a testing environment – you know who I’m talking about, the badass who has already achieved so much and is on track for so much more. There is no need to put additional pressure on yourself to perform well on the GMAT to prove to yourself, or to your family, friends, or an admissions committee how “valuable” you are, how smart you are, or how capable you are.

From our perspective as teachers, we also see this occur frequently in the other direction, with tutors who apply to work with us. They define themselves by their GMAT success rather than their ability as educators. We reject many potential tutors out of hand, despite their having a 770+ score, because a score is simply a number on a piece of paper; we seek people who understand others, are strong communicators, and who are always growing as educators.

Takeaway: By focusing on your score, rather than developing stronger critical and creative thinking skills, you’re missing the point of the GMAT.

## 2. The GMAT is both easier and harder than you think.

I know this sounds counterintuitive, but bear with me.

The stigma of the GMAT – that it’s a terribly difficult exam – affects the performance of most test takers. This hyperbole can cause you to freeze up and underperform. The people who make the GMAT out to be more difficult than it is, in the end, hold themselves back by placing it on a pedestal and treating it with too much reverence.

The GMAT is certainly an exceptionally challenging exam that will push you to your limits. There is no mistaking that. Further, it compares you to your peers – people who have similar levels of skill and experience, hence gaining a competitive edge seems nearly impossible without working harder. However, because most people make it out to be harder than it is, they end up holding themselves back.

Conversely, the GMAT is easier than you think because it rewards informality and creative thinking, especially on the math side. A successful GMATter can use intuition and clear, logical reasoning in order to solve the most intractable problems.

Because of this seeming dichotomy, test takers bring to the exam a paradigm of thought that is very restrictive. By not looking for an accessible or intuitive answer – the most efficient answer of methodology to solve a problem – they restrict their options and make their task all the more challenging.

Once you free yourself of the academic restraints that come from the burden of too formal an education, whether with math or language, and utilize your intuitive reasoning mind, all of a sudden GMAT problems become much more simple and straightforward.

Let’s look at an example:

Since implementing new work protocols at the start of 2020, every employee’s efficiency in the factory has increased by 33%, leading to layoffs of 25% of the workforce. Assuming no other changes, and that each worker has the same level of productivity, if the factory produced \$20 m worth of widgets in 2019, what value of widgets did it produce in 2020?

1. \$10 m
2. \$13.3 m
3. \$16.75 m
4. \$20 m
5. \$33.25 m

It’s very easy to dive into doing a lot of math here, but the real skill is finding what’s important, and realizing that there’s little math to be done.

First, focus on only the important information: Efficiency +33% and Workforce -25%.

Second, realize that you’re not constrained to using percentages: Efficiency +⅓ and Workforce -1/4.

Finally, understand that these changes are built upon the existing base. Efficiency 4/3 as much and Workforce ¾ as much. These changes cancel out! The more problems you do, the more sensitive you become to the ways that simple truths can be communicated in unnecessarily complex ways, but if you just keep hitting the math you’ll never get there.

Takeaway: The most challenging part of the GMAT is dehabituating the solutions paths that you’ve locked in through your training at school and allowing yourself the mental flexibility to really explore, be creative, and go with your gut.

## 3. Don’t force it. It’s not a knowledge test.

There is a great misconception that the GMAT is just about knowing how to solve every problem that they might throw at you, and knowing how to do so before you’re actually sitting in the exam.

In fact, while you need to know all the concepts that are being tested, the exam is not testing your knowledge of these mechanics. Rather, the exam tests your depth of knowledge. The contextual relationship between the rules and the correct answer is often hidden in the space between two concepts, as in the example above. Examining how those rules can be bent, or broken, or how they relate to other rules, can lead to new insights that you wouldn’t think were otherwise there.

Takeaway: It’s a conversation, not a play. There is no script. Being prepared means being able to handle the unknown challenges that will come your way, not knowing exactly what to say in advance. You’ll never be totally prepared, because you’ll never know what the other person will say.

## 4. Most performance issues are not intellectual.

Many high achievers come to the GMAT and find themselves plateauing in the mid-upper 600s or low 700s. They think that a lack of fluency or a deeper understanding of the material is what’s holding them back.

True GMAT success is governed by the recognition that it is a test of acuity, confidence, and temperament. For example, being comfortable in uncertainty, making decisions quickly, and finding out of the box solutions are all highly rewarded skills in this exam.

A general understanding of the dynamics of a problem, rather than a precise answer, are often the characteristics that allow people to truly excel, especially on the most challenging questions. So much of success on the GMAT at the highest levels is about managing the emotional and behavioural stresses, not the intellectual challenge. Being able to regulate your anxiety, self-confidence/questioning, and overall comfort can impact your GMAT score significantly once you’re past 700, where each second and every unique approach can mean extra points.

Takeaway: Once you’re in the upper 600s, improvement comes from focusing on non-intellectual elements. Preparing for these challenges from the start is what makes for the most rapid, fluid, and meaningful preparation.

## 5. Most people don’t do it alone

The dirty little secret that no one talks about is that nearly every high-achiever seeks assistance to obtain a great GMAT score. This is all the more true in those places where the smartest people congregate. People don’t speak about getting help because they are usually in environments, whether academic or professional, where they are valued for their intellectual ability and feel that it is a mark of shame to not be able to “go it alone.”

We have so many clients that come to us from McKinsey and BCG, Goldman Sachs and Morgan Stanely, Google, Apple, et cetera, who are not comfortable sharing with their peers or family the fact that they have sought help. This is because they fear that their admission will in some way diminish their achievements or their cachet in the eyes of those they respect most.

There is no shame in seeking help, even if it is the first time you’ve ever needed to (for many of our top performing clients, we’re the first tutor they’ve ever needed in their lives). You may have found yourself at a great school or already landed your first job and thus consider yourself exceptionally successful. But the GMAT is pitting you against those who are of a similar ilk and so going it alone is fraught with difficulties. One of these difficulties being the ability to gain a competitive edge after being homogenized for so long in academic or corporate environments.

This can often lead to frustration, sadness, and sometimes missing the boat entirely on the next stage of your life. It is important to recognize that everyone, all those people that you respect and admire most, at one point or another, have needed help, and have had to ask for help.

Takeaway: Don’t hesitate to ask for help. That’s what strong people do. It’s what leaders do. It’s what those who are the most successful do. Never go it alone.

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 P  = 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.

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
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.

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:

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.

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.

Posted on
25
Feb 2021

## What Happens When Permutations Have 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

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.

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

Posted on
09
Feb 2021

## Triangle Inequality Rule

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:

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.

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:

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

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:

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