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: 25 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

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An Introduction to combination math
Posted on
23
Feb 2021

An Intro to Combination Math

By: Rich Zwelling (Apex GMAT Instructor)
Date: 23 Feb 2021

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

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

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

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

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

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

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

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

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

combinations problem

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

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

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

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

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

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

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

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

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

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

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

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

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

6C2 = 15 combinations of two cheeses

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

solving a combinatorics problem

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

combination problem on the gmat answer

That leaves us with:

6C3 = 20 combinations of three cheeses

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

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

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

Permutations and Combinations Intro
A Continuation of Permutation Math
An Intro To Combination Math
Permutations With Repeat Elements

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A Continuation of GMAT Permutation Math
Posted on
18
Feb 2021

A Continuation of Permutation Math

By: Rich Zwelling (Apex GMAT Instructor)
Date: 16 Feb 2021

Review of example from last post

Last time, when we started our discussion of GMAT Combinatorics, we gave a brief example of GMAT permutations in which we had five paintings and asked how many arrangements could be made on a wall with those paintings. As it turns out, no complicated combinatorics formula is necessary. You can create an easy graph with dashes and list five options for the first slot, leaving four for the second slot, and so on:

_5_  _4_ _3_ _2_ _1_

Then multiply 5*4*3*2*1 to get 120 arrangements of the five paintings. Remember you could see this notationally as 5!, or 5 factorial. (It’s helpful to memorize factorials up to 6!)

More permutation math

But there could be fewer slots then items. Take the following combinatorics practice problem:

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

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

First, as a review, how do we know this is a PERMUTATION and not a COMBINATION? Because order matters. In the previous problem, the word “arrangements” gave away that we care about the order in which items appear. In this problem, we’re told that we’re interested in the “orderings” of cheeses. Cheddar followed by gouda would be considered distinct from gouda followed by cheddar. (Look for signal words like “arrangements” or “orderings” to indicate a PERMUTATION problem.)

In this case, we must consider the options of three or four cheeses separately, as they are independent (i.e. they cannot both happen). But for each case, the process is actually no different from what we discussed last time. We can simply consider each case separately and create dashes (slots) for each option. In the first case (three cheeses), there are six options for the first slot, five for the second, and four for the third:

_6_  _5_  _4_

We multiply those together to give us 6*5*4 = 120 possible ways to present three cheeses. We do likewise for the four-cheese case:

_6_  _5_  _4_  _3_

We multiply those together to give us 6*5*4*3 = 360 possible ways to present four cheeses.

Since these two situations (three cheeses and four cheeses) are independent, we simply add them up to get a final answer of 120+360 = 480 possible orderings of cheeses, and the correct answer is D. 

You might have also noticed that there’s a sneaky arithmetic shortcut. You’ll notice that you have to add 6*5*4 + 6*5*4*3. Instead of multiplying each case separately, you can factor out 6*5*4 from the sum, as follows:

6*5*4 + 6*5*4*3

= 6*5*4 ( 1 + 3)

= 6*5*4*4

= 30*16 OR 20*24

= 480

Develop the habit of looking for quick, efficient ways of doing basic arithmetic to bank time. It will pay off when you have to do more difficult questions in the latter part of the test. 

Now that we have been through GMAT permutations, next time, I’ll give this problem a little twist and show you how to make it a COMBINATION problem. Until then…

Permutations and Combinations Intro
A Continuation of Permutation Math
An Intro To Combination Math
Permutations With Repeat Elements

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Featured Video Play Icon
Posted on
17
Feb 2021

Data Sufficiency: Area of a Triangle Problem

Hey guys! Today we’re checking out a geometry Data Sufficiency problem asking for the area of a triangle, and while the ask might seem straightforward, it’s very easy to get caught up in the introduced information on this problem. And this is a great example of a way that the GMAT can really dictate your thought processes via suggestion if you’re not really really clear on what it is you’re looking for on DS. So here we’re looking for area but area specifically is a discrete measurement; that is we’re going to need some sort of number to anchor this down: whether it’s the length of sides, or the area of a smaller piece, we need some value!

Begin with Statement #2

Jumping into the introduced information, if we look at number 2, because it seems simpler, we have x = 45 degrees. Now you might be jumping in and saying, well, if x = 45 and we got the 90 degree then we have 45, STOP. Because if you’re doing that you missed what I just said. We need a discrete anchor point. The number of degrees is both relative in the sense that the triangle could be really huge or really small, and doesn’t give us what we need. So immediately we want to say: number 2 is insufficient. Rather than dive in deeply and try and figure out how we can use it, let’s begin just by recognizing its insufficiency. Know that we can go deeper if we need to but not get ourselves worked up and not invest the time until it’s appropriate, until number 1 isn’t sufficient and we need to look at them together.

Consider Statement #1

Number 1 gives us this product BD x AC = 20. Well here, we’re given a discrete value, which is a step in the right direction. Now, what do we need for area? You might say we need a base and a height but that’s not entirely accurate. Our equation, area is 1/2 x base x height, means that we don’t need to know the base and the height individually but rather their product. The key to this problem is noticing in number 1 that they give us this B x H product of 20, which means if we want to plug it into our equation, 1/2 x 20 is 10. Area is 10. Number 1 alone is sufficient. Answer choice A.

Don’t Get Caught Up With “X”

If we don’t recognize this then we get caught up with taking a look at x and what that means and trying to slice and dice this, which is complicated to say the least. And I want you to observe that if we get ourselves worked up about x, then immediately when we look at this BD x AC product, our minds are already in the framework of how to incorporate these two together. Whereas, if we dismiss the x is insufficient and look at this solo, the BD times AC, then we’re much more likely to strike upon that identity. Ideally though, of course, before we jump into the introduced information, we want to say, well, the area of a triangle is 1/2 x base x height. So, if I have not B and H individually, although that’ll be useful, B x H is enough. And then it’s a question of just saying, well, one’s got what we need – check. One is sufficient. Two doesn’t have what we need – isn’t sufficient, and we’re there. So,

I hope this helped. Look for links to other geometry and fun DS problems below and I’ll see you guys soon. Read this article about Data sufficiency problems and triangles next to get more familiar with this type of GMAT question.

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Posted on
16
Feb 2021

Triangles With Other shapes

By: Rich Zwelling (Apex GMAT Instructor)
Date: 16 Feb 2021

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

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

triangles in other shapes GMAT article

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

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

triangles in other shapes gmat article gmat probelm

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

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

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

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

  1. Triangles created using the central angle of a circle

triangle in a circle, gmat geometry article

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

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

2. Inscribed triangles

triangle inscribed in circle, gmat problem

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

3. Squares and rectangles inscribed in circles

rectangle in circle, gmat geometry

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

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

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

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

Explanation

The diagonal splits the rectangular park into two similar triangles:

triangle in other shapes gmat problem

Use SIGNALS to avoid algebra

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

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

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

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

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

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

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

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

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Combinatorics: Permutations and Combinations Intro
Posted on
11
Feb 2021

Combinatorics: Permutations and Combinations Intro

By: Rich Zwelling (Apex GMAT Instructor)
Date: 11 Feb 2021

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

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

Does Order Matter?

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

Permutations example

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

ABCDE
ACDEB
BDCEA

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

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

_5_  ___ ___ ___ ___

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

_5_  _4_ ___ ___ ___

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

_5_  _4_ _3_ _2_ _1_

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

Combinations example

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

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

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

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

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

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

Quantitative connection

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

ABC
ACB
BAC
BCA
CAB
CBA

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

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

Permutations and Combinations Intro
A Continuation of Permutation Math
An Intro To Combination Math
Permutations With Repeat Elements

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similar triangles on the gmat
Posted on
02
Feb 2021

Similar Triangles – GMAT Geometry

By: Rich Zwelling (Apex GMAT Instructor)
Date: 2 Feb 2021

One of the most important things to highlight here is that “similar” does not necessarily mean “identical.” Two triangles can be similar without being the same size. For example, take the following:

similar triangles on the GMAT 1

Even though the triangles are of different size, notice that the angles remain the same. This is what really defines the triangles as similar.

Now, what makes this interesting is that the measurements associated with the triangle increase proportionally. For example, if we were to present a triangle with lengths 3, 5, and 7, and we were to then tell you that a similar triangle existed that was twice as large, the corresponding side lengths of that similar triangle would have to be 6, 10, and 14. (This should be no surprise considering our lesson on multiples of Pythagorean triples, such as 3-4-5 leading to 6-8-10, 9-12-15, etc.)

You can also extend this to Perimeter, as perimeter is another one-dimensional measurement. So, if for example we ask:

similar triangles on the GMAT 2

A triangle has line segments XY = 6, YZ = 7, and XZ = 9. If Triangle PQR is similar to Triangle XYZ, and PQ = 18, as shown, then what is the perimeter of Triangle PQR?

Answer: Perimeter is a one-dimensional measurement, just as line segments are. As such, since PQ is three times the length of XY, that means the perimeter of Triangle PQR will be three times the perimeter of Triangle XYZ as well. The perimeter of Triangle XYZ is 6+7+9 = 22. We simply multiply that by 3 to get the perimeter of Triangle PQR, which is 66.

Things can get a little more difficult with area, however, as area is a two-dimensional measurement. If I double the length of each side of a triangle, for example, how does this affect the area? Think about it before reading on…

SCENARIO

Suppose we had a triangle that had a base of 20 and a height of 10:

similar triangles on the GMAT 3

The area would be 20*10 / 2 = 100.

Now, if we double each side of the triangle, what effect does that have on the height? Well, the height is still a one-dimensional measurement (i.e. a line segment), so it also doubles. So the new triangle would have a base of 40 and a height of 20. That would make the area 40*20 / 2 = 400.

Notice that since the original area was 100 and the new area is 400, the area actually quadrupled, even though each side doubled. If the base and height are each multiplied by 2, the area is multiplied by 22. (There’s a connection here to units, since units of area are in square measurements, such as square inches, square meters, or square feet.)

Now, let’s take a look at the following original problem:

Triangle ABC and Triangle DEF are two triangular pens enclosing two separate terrariums. Triangle ABC has side lengths 7 inches, 8 inches, and 10 inches. A beetle is placed along the outer edge of the other terrarium at point D and traverses the entire perimeter once without retracing its path. When finished, it was discovered that the beetle took three times as long as it did traversing the first terrarium traveling at the same average speed in the same manner. What is the total distance, in inches, that the beetle covered between the two terrariums?

A) 25
B) 50
C) 75
D) 100
E) 125

Explanation

This one has a few traps in store. Hopefully you figured out the significance of the beetle taking three times as long to traverse the second terrarium at the same average speed: it’s confirmation that the second terrarium has three times the perimeter of the first. At that point, you can deduce that, since the first terrarium has perimeter 7+8+10 = 25, the second one must have perimeter 25*3 = 75. However, it can be tempting to then choose C, if you don’t read the question closely. Notice the question effectively asks for the perimeters of BOTH terrariums. The correct answer is D.

GMAT Triangle Series Articles:

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

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Intro to GMAT Data Sufficiency- All you’ll need to know
Posted on
29
Jan 2021

GMAT Data Sufficiency Introduction

by Apex GMAT
Contributor: Altea Sulollari
Date: 28 January 2021

 

As a GMAT test-taker, you are probably familiar with data sufficiency problems. These are one of the two question types that you will come across in the GMAT quant section, and you will find up to 10 of them on the exam. The rest of the 31 questions will be problem-solving questions.

The one thing that all GMAT data-sufficiency questions have in common is their structure. That is what essentially sets them apart from the problem-solving questions. 

Keep on reading to find out more about these questions’ particular structures and the topics that they cover:

The question structure:

The GMAT data sufficiency problems have a very particular structure that they follow and that never changes. You are presented with a question and 2 different statements. You will also be given 5 answer choices that remain the same across all data sufficiency problems on the GMAT exam. These answer questions are the following:

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.

Your job would be to determine whether the 2 statements that you are provided with are sufficient to answer the question.

What topics are covered?

Some of the math topics that you will see in this type of question are concepts from high school arithmetic, geometry, and algebra.

Below, you’ll find a list of all concepts you need to know for each math topic:

Geometry

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

Algebra

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

Arithmetic

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

Word problems

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

Common mistakes people make when dealing with this question type

Actually solving the question

This is the #1 mistake most test-takers make with these problems. These problems are not meant to be solved. Instead, you will only need to set up the problem and not execute it. That is also more time-efficient for you and will give you some extra minutes that you can use to solve other questions. 

Over-calculating

This relates to the first point we made. This question type requires you to determine whether the data you have is sufficient to solve the problem. In that case, calculating won’t help you determine that. On the contrary, over-calculating will eat up your precious minutes.

Rushing

This is yet another common mistake that almost everyone is guilty of. You will have to spend just enough time reading through the question in order to come up with a solution. Rushing through it won’t help you do that, and you will probably miss out on essential details that would otherwise make your life easier. 

Not understanding the facts

What most test-takers fail to consider is that the fact lies in the 2 statements that are included in the questions. Those are the only facts that you have to consider as true and use in your question-solving process. 

3+ tips to master this question type:

Review the fundamentals

That is the first step you need to go through before going in for actual practice tests. Knowing that you will encounter these high school math fundamentals in every single quant problem, is enough to convince anyone to review and revise everything beforehand.

Memorize the answer choices

This might sound a bit intimidating at first as most answer choices are very long sentences that tend to be similar to each other in content. However, there is a way to make this easier for you. What you need to do is synthesize the answer choices into simpler and more manageable options. That way, they will be easier to remember. This is what we suggest:

  1. Only statement 1
  2. Only statement 2
  3. Both statements together
  4. Either statement
  5. Neither statement

Examine each statement separately

That is definitely the way to go with this GMAT question. You will need to determine whether one of the statements, both, either, or neither is sufficient, and you cannot do that unless you look at each of them separately first.

Now that you have read the article and are well-aware of the best ways to solve data sufficiency problems on the GMAT, try your hand at this practice question.

 

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45-45-90 triangles on the gmat
Posted on
06
Jan 2021

45-45-90 Right Triangle – GMAT Geometry Guide

By: Rich Zwelling (Apex GMAT Instructor)
Date: Jan 6 2021

45-45-90 Right Triangle

Another of the commonly tested triangles on the GMAT is the 45-45-90, also known as the isosceles right triangle. Know that term, as it could appear by name in a question.

As shown in the above diagram, the side lengths of this triangle always fit the same ratio (1 : 1 : √2) , where the legs are the same length and the hypotenuse length is √2 times the leg length. For example, if the leg lengths were 3 instead of 1, then the hypotenuse would be 3√2 instead of simply √2.

But likewise, don’t forget that you can go backwards and divide the hypotenuse length by √2 to get to the leg length. It may seem obvious, but it presents an important point: what’s more important than simply memorizing the ratio is understanding the mathematical relationship between the side lengths. This will help you avoid trouble if the GMAT happens to give you a problem that doesn’t conform to expectations.

For example, the following problem fits expectations quite nicely:

A yard in the shape of an isosceles right triangle has a hypotenuse of length 10√2. What is the area of this yard?

From this information, it’s easy enough to deduce that the leg length is 10, and we can draw a diagram that looks roughly like this:


From there, we can easily calculate the area, which is base*height / 2, or in this case 10*10/2 = 50.

But what happens if we give the problem a little twist:

A yard in the shape of an isosceles right triangle has a hypotenuse of length 10. What is the area of this yard?

Did you catch the twist? We’re used to the hypotenuse including a √2. This is what the GMAT will do. They’ll throw you off-center, and you’ll have to adjust. But this is also why we said earlier that what matters more than memorizing the ratio of sides is understanding the relationships between the sides of an isosceles right triangle…

Remember we said that, just as we multiply the leg length by √2 to get to the hypotenuse length, so we must divide the hypotenuse length by √2 to get to the leg length. That must mean each leg has length 10/√2. 

You can then take 10/√2 and multiply it by √2/√2 to de-radicalize the denominator and get (10√2) / 2, or a leg length of 5√2:

Notice again that we have a more unfamiliar form, with the √2 terms in the legs and an integer in the hypotenuse. We can’t count on the GMAT to give us what we’re used to. 

Now we can calculate the area:

Area = (base*height)/2 = (5√2)(5√2)/2 = (5*5)(√2*√2)/2 = (25)*(2) / 2 = 25

 

Problem #1

Now, to try this on your own, take a look at this Official Guide problem:

If a square mirror has a 20-inch diagonal, what is the approximate perimeter of the mirror, in inches?

(A)   40
(B)   60
(C)   80
(D)   100
(E)   120

Explanation:

This is a nice change-up, because it involves another shape. Did you notice that splitting a square along its diagonal creates two isosceles right triangles

Once you realize this, you can divide 20 by √2 to get 20/√2, then multiply top and bottom by √2 to get x=10√2.

Since the question asks for perimeter, we can multiply this by four to get 40√2. 

The final step is to realize that √2 is approximately 1.4. If we multiply 40 by 1.4, the only answer choice that possibly makes sense is 60, and thus the correct answer is B

 

After reviewing the 45-45-90 triangle identity, these further articles in the triangle geometry series will take you through more identities, each of the specific triangles and how the GMAT uses them to test your critical and creative solving skills:
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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gmat 101 : what to expect on the gmat test
Posted on
28
Jul 2020

GMAT 101: What to Expect on the GMAT Test

by ApexGMAT

Contributor: Svetozara Saykova

July 28th, 2020

The GMAT is a challenging exam, and in this article we’ll provide both a broad overview of how it works as well as a deep dive into its nuances to put you on a surer footing for preparing, and ultimately conquering, the exam. There’s a good chance that you’ve already decided to apply to several MBA programs, and that they all require a GMAT score, so let’s get started!

What is the GMAT?

   The GMAT (short for Graduate Management Admissions Test) is an advanced examination that is a requirement for admission to most MBA (Masters of Business Administration) programs. The GMAT consists of four sections – Quantitative, Verbal, Integrated Reasoning and the Analytical Writing Assessment. Each part examines a particular set of skills vital in the business world. A candidate’s performance on the exam helps admission officers assess their suitability for the rigorous curriculum and challenges of an MBA program. 

The GMAT requires knowledge of high school level math as well as English language and grammar. The catch is this: they’re not testing your knowledge, but rather your creative application of that knowledge. In that sense, success on the GMAT boils down to two things – your critical assessment of information and your ability to reason

How does a single exam measure whether or not a candidate has the skills to excel in a top MBA program and, by extension, thrive in the business world? The thing is that the GMAT is not a standard standardized test, but it is a CAT.

What the heck is a CAT?

CAT stands for computer-adaptive test, which means that the test adapts to your skill level. It does this by modifying the questions according to your performance. The first question will typically have a moderate level of difficulty, then the difficulties of the second and subsequent questions are based upon your performance on previous questions. The algorithm selects which problems to deliver depending upon your collective performance so far. If you continue to answer correctly, the difficulty of the questions will rise and vice versa.

   On the GMAT three of the sections are computer-adaptive – the Quantitative, the Verbal and the Integrated Reasoning. 

   No two people have ever taken the same exact GMAT test. What’s more, the test is challenging for everyone, even top 700+ performers. Why? First, each candidate gets a unique mix of questions as the test adapts to your performance in real time. This pushes each candidate to the edge of their capabilities, making the GMAT feel like it’s more difficult than it is, and making you feel that you’re not doing as well as you are. The test can continue to toss increasingly challenging questions at you until it reaches your limit. 

The CAT model has another interesting feature. The test taker is presented with one question at a time and cannot go back and forth within the exam. Once an answer is provided and the test taker proceeds to the next question, they cannot return. This is understandably quite  nerve-racking and can contribute to stress-based under performance. Overcoming anxiety surrounding the GMAT can be a daunting task, but it is vital for excellent performance. That is why having effective strategies on how to manage the GMAT related stress is a must in order to enhance your performance.

GMAT Results

Immediately After Taking  the GMAT Test

Right after you sit the GMAT you will see four out of your five scores: The Quantitative, Verbal, Integrated Reasoning and your aggregate score out of 800. Those will be your unofficial scores and you will have two minutes to accept or cancel your results. If you do not decide, your score will be automatically cancelled. The AWA/Writing section is graded by an actual human and so that score comes in with your Official Score Report. 

The Unofficial Score Report 

When given your scores, you will have two minutes to decide whether you want to keep them. If the time expires before you make a decision the score will be automatically cancelled. Rest assured, if you cancel them they can be reinstated within 4 years and 11 month from your exam date. You can also cancel them within 72 hours for a fee  if you change your mind later on. If you decide to accept your results, an Unofficial score report will be issued. You will receive it prior to leaving the test center. The report will contain your Quantitative, Verbal, Integrated Reasoning and Total scores, as well as some personal information. The unofficial score report can help you determine whether you are a competitive applicant for your desired program and whether you need to retake the GMAT, though you should have a sense of what score you are seeking before entering the testing center, so that you can make a good decision about cancelling/keeping scores. 

Although the unofficial report can be very helpful to you, it cannot be used for your MBA applications. Only the Official score report that comes in the mail a few weeks later and is send separately to Business Schools can be used for your application and admissions.

   Within Three Weeks After the Exam

   You will be sent a notice that your Official score report is ready. Besides the scores from your unofficial report, it will contain your Analytical Writing Assessment score, your GMAT percentile rankings – it shows where your score is on the scale compared to your peers, the personal data you provided at registration, and scores from other GMAT tests you have taken within the past five years. 

   Your official score is valid for five years, which gives you the flexibility to send it out to universities when you are ready, or to defer application to another year.

   In addition to the official report, an applicant can request an Enhanced score report for a fee of $30. It contains a comprehensive performance analysis by section and question type, and can provide the candidate with an understanding of their strengths and weaknesses as well as how they rank among their peers. 

GMAT Scoring

   When you receive your Score Report you will see scores for each section ranging as follows:

  • the Quantitative score
  • range: from 0 to 51 points  in 1.0 increments
  • average: 40.2 (for the period 2015 – 2018)
  •  the Verbal score
  • range: from 0 to 51 points in 1.0 increments
  • average: 27.08 (for the period 2015-2018)
  •  the Total GMAT score
  • range: from 200 to 800 points in 1.0 increments 
  • average: 563.43 (for the period 2015-2018)

 

  • the Integrated reasoning score 
  • range: from 1 to 8 points in 1.0 increments 
  • average: 4.41 (for the period 2015 – 2018)
  •  the AWA score 
  • range:  0 to 6 points in 0.5 increments
  • average: 4.49 (for the period 2015-2018)

Source: GMAC.com   

   The major difference between non-adaptive tests and the GMAT is that the GMAT score is derived not by how many problems you answer correctly, but by the relative difficulty of the problems that you answer correctly

   In standard assessments, like the SAT or the TOEFL for instance, each problem has a firm percentage that contributes to the final grade. These tests demand a certain approach that we are all familiar with from high school:  dedicate time to each question and try to get everything right. This approach is ineffective, however, when it comes to computer adaptive tests like the GMAT. In fact, due to the adaptive nature of the exam, regardless of how well they perform, most test takers only answer correctly between 40-60% of the questions. The critical point is that your score depends on the most challenging questions that you can answer correctly on a consistent basis. In essence, the higher the overall difficulty level at which you get 60% of the questions right, the higher you will score.

The best way to perform well on the GMAT is to be properly prepared. This means not only knowing the material on which you are being tested, but being able to effectively allocate scarce resources like time, attention, and focus. Since you are unable to jump backwards or forwards and because each question depends on your answer to the previous one, you need to be able to accurately assess how much of these resources each question deserves in the context of the greater exam. You should be able to balance spending more time on hard questions while not running out of time on any particular section. It is imperative to note that there are harsh penalties for incomplete sections, so be sure to answer each question before time runs out, even if you must guess at random.

What are the GMAT sections?

   The GMAT test is comprised of four distinct sections. Each section assesses a particular area of subject matter expertise and each has its own unique problem types; however, critical thinking and analytical reasoning are the core skills that will get you through each section and through the whole exam. 

The GMAT can be broken down to:

  • Verbal
  • Quantitative
  • Integrated Reasoning
  • Analytical Writing Assessment

  The student sitting the exam has the opportunity to choose with which part to start. There are three variations:

  • AWA & Integrated Reasoning (break) Quantitative (break) Verbal;
  • Quantitative (break) Verbal (break) Integrated Reasoning & AWA;
  • Verbal (break) Quantitative (break) Integrated Reasoning & AWA;

You will be able to choose the order following the computer tutorial you will be given at the test center just before you start your exam. 

Pro tip: Choose the order of the exam based upon your comfort levels. Most people like to put their most challenging section first so that they can optimize their performance by tackling the difficult section while one’s brain is still crisp. Others may opt to start off with a stronger section, or the less important AWA/IR to get into a “flow” state before tackling the sections that they find most challenging or important. Ultimately, the best advice is to experiment, and go with what makes you most comfortable, because a strong performance can only come with comfort.

Verbal

Verbal section of the GMAT

   The Verbal section permits test-takers to present their reasoning skills, critical thinking, and command of English grammar. It measures the test taker’s ability to read and comprehend written materials, reason and evaluate subtle arguments, and correct written sentences to match standard written English.

There are three types of questions in the Verbal section:

Reading comprehension

   These questions test your ability to read critically. More specifically, you should be able to:

  • summarize the text and derive the key idea;
  • distinguish between ideas stated directly in the text and ideas implied by the author;
  • come up with conclusions based on the information in a given passage;
  • analyze the logical structure of the argument;
  • deduce the author’s attitude towards the topic. 
Critical reasoning 

   You will be presented with a short argument and asked to select a statement which either represents the conclusion, strengthens or weakens the argument, or analyzes how the argument is constructed. In order to excel in Critical reasoning one should be familiar with logical reasoning, common fallacies and assumption, and structural connections between evidence and conclusion. We all use reasoning daily but more often than not our thought process is not logically precise or rigorous and that is what the GMAT test writers count upon. Examining your own thought process and understanding where you are susceptible to imprecise thinking can be a good start for prepping.

Sentence correction

These questions test your knowledge of English grammar and accurate expression. On sentence correction you’ll be shown a somewhat complex sentence, part of which or the whole of which is underlined. You will be asked to select the best version of the underlined portion, whether the original or one of four alternatives presented.

After getting familiar with the specifics of the Verbal section, you might wonder whether native speakers have an unfair advantage. That is a fair contention, however the answer is nuanced. The GMAT does not test particularly one’s command of English, as opposed to some other language, but their understanding of language construction. If one has a strong eye and ear for grammar in another language, they will likely perform well on Sentence Correction. Bottom line: there can be advantages and disadvantages for both native and non-native English speakers. The key is to learn to use them to your advantage.

Quantitative

Quantitative Section of the GMAT   The Quant section on the GMAT is designed to evaluate the candidate’s analytical knowledge and depth of understanding of basic mathematical concepts like algebra, geometry, number properties and arithmetic. More to the point, the expectation is that you know the math typical for any high school student, but the GMAT is using that as a base of knowledge to test your creativity.

  There are two types of problems in the Quantitative section: 

Data sufficiency 

   These problems consist of a single question and two statements of truth. The task is to determine if each of the statements (or both together) contain enough data to answer the question definitively. DS questions test your ability to promptly identify what information is crucial to answer a particular question and how well you ignore or eliminate unnecessary or insufficient data. It is important to note that you are not being asked to solve the problems, and often it is preferable to not solve the problem. Pro Tip: Insufficient data will often lead you to multiple possible answers – Be Careful!

Problem solving 

PS problems are somewhat generic, and very much what you may be used to from your school days. Each presents a candidate with a problem that they need to solve, and the answer is multiple choice. The knowledge required is high school level maths up to algebra and geometry, with a smattering of statistics and combinatorics, but nothing terribly high level. Once again, in this part as in the GMAT test as a whole, the main skill that is evaluated is your ability to critically assess information. In fact, it is particularly important to avoid doing the actual math but rather pick apart the problem and reduce it to a much simpler question. 

Integrated reasoning

Integrated reasoning   The Integrated reasoning section was added to the GMAT exam in 2012 and is increasingly becoming a more important part of the exam. 

The IR contains both verbal and quantitative topics, weaved together into a challenging problem landscape. This section assesses the ability of a candidate to comb through a significant quantity of data, often delivered in a complicated fashion, and identify a particular piece of information or derive a specific insight. 

   There are four types of questions in the Integrated reasoning section: 

Multi-source reasoning

This problem type offers a combination of text, tables and graphs, and then asks you to identify discrepancies among different sources of data or ask you to draw conclusions or derive inferences by taking tidbits from various sources and combining them together. The key skill  here is adaptability to structurally different content and being able to draw associations between the various content types. Keep in mind that most of the data is not relevant – with multiple sources comes plenty of unnecessary information, so being deliberate with the information you choose to analyze more deeply is essential. 

Graphic interpretation 

Graphic interpretation is exactly what it sounds like. You may be presented with a more traditional graph like a pie or bar chart, but you might also be provided an unusual diagram. The test-taker should be able to accurately interpret the information, recognize relationships among the data and draw conclusions from the graphics provided. It’s crucial to remember that you shouldn’t get carried away trying to understand or interpret all of the information but that the core task is to focus on what you are being asked and finding that needle in the haystack of data provided. 

Two-part analysis 

These types of questions measure one’s ability to solve complex problems – quantitative, verbal or a combination of both. Each question has two sub-questions which can be dependent upon one another. Irrespective of whether they’re related, like other Integrated Reasoning questions, you’ll need to answer both parts correctly to get credit for the question. The format of the problems in this section is intentionally diverse in order to cover a wide range of content and test your ability to synthesize knowledge from different fields.

Table analysis

This question type presents a table of data, but that’s just the beginning. The challenging part of these problems is determining what’s being asked for, and then using the provided tables in an efficient manner.

Table analysis requires not just reading information from the tables provided, but requires one to understand the question, and organize the data in such a way so that it can be efficiently sorted. The candidate is tasked to determine what from the given information is relevant or meets certain criteria. 

Analytical Writing Assessment

Analytical Writing section of the GMAT   The Analytical Writing Assessment (AWA) or the “essay” section provides admission officers 

with an idea of your writing skill. The AWA section is scored separately and does not count towards the Combined (200 to 800 points) score. The essay is checked twice – once by a human reader and once by a computer algorithm. The final grade is an average from both scores. If the scores differ greatly, then the writing sample is reviewed by another human reader and after that the final grade is decided.

For this task you will be presented with a passage similar to those from Critical reasoning in the Verbal section of the GMAT. You will be asked to provide a well-supported critique of the author’s argument, to analyze their strong points and identify the weaknesses in their line of reasoning. What’s more, the AWA section measures the candidate’s ability to express themselves and their ideas clearly and with precision in written form. 

Now that you have a thorough understanding of what to expect on the GMAT you might be concerned with the practical side of things like how, when and where

How?

The GMAT test is administered by the global testing network Pearson VUE. They have 600+ centers all around the world where you can sit the exam. The GMAT is facilitated through a computer system available at the designated Pearson VUE centers, which means that you can take the exam only at those centers.  

As of the COVID-19 pandemic the GMAT centers closed so the GMAC provided an online version GMAT. In case there is still an option to take the GMAT online when you are reading this and you are interested in doing so, check out our videos on how it is administered and what you need to know prior to sitting the online GMAT. 

When and where?

    First of all, you should make sure you know your chosen MBA programs’ application deadlines and from there coordinate accordingly. Consider how much time you will need for preparation. You should also plan to take the exam more than once; even with a strong score, there’s always room for continued improvement, and you might as well take it a second time after putting all that effort into preparing. So plan to factor in a re-take or two, just in case – also good if you do well… you can always do better! This is important because the GMAC has rules regarding re-takes: they must be at least 16 days apart, there cannot be more than 5 within a year and there’s a lifetime limit of 8 total attempts at the exam. You can take the GMAT at any time of the year, and appointments are generally availab;e if you plan a few months ahead, so you can launch your plan without worrying about the precise exam date and then midway through make an appointment based on your progress and practice exam results. 

And last but not least how much does it cost?

   The total price of the GMAT is $250 – as of July 2020 230 Euro/203 GBP. This amount includes sending your official score to up to five universities or MBA programs of your choice. You can of course request your results to be sent to additional programs; each one will cost you an additional $35

This all might seem a little overwhelming, which is reasonable given how important the exam is, and all the idiosyncrasies of the GMAT. Growing familiar with the exam is a challenge in itself. With determination and the proper guidance, you can unleash your full potential and obtain admission to your dream MBA programs. Set yourself up for success by learning how to select the right tutor to begin your GMAT journey. 

 

We are excited to announce that the Apex GMAT Blog is rated as one of the top 10 GMAT blogs in 2020 by Feedspot.

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