How-to GMAT: No Calculator? Use These Mental Math Tips Instead
Posted on
07
Sep 2021

How-to GMAT: No Calculator? Use These Mental Math Tips Instead

The GMAT is an exam largely focused on numbers and numerical data. And while doing math on the GMAT should be avoided sometimes it is inevitable. True, the test-taker is given a calculator for the duration of the Integrated Reasoning section but the same cannot be said for the Quantitative Reasoning Section. 

This article is going to provide some smart calculation shortcuts and mental math tips to help you go through some arithmetical questions without losing too much time and help you get a higher score on the GMAT Quant.

The Basics

Before explaining any methods for dividing and multiplying with ease, let’s make sure we have revised a few simple rules:

  • Numbers with an even last digit are divisible by 2 – 576 is and 943 is not;
  • Numbers with a sum of digits divisible by 3 are also divisible by 3 – 3,465 for example (3+4+6+5=18);
  • If the last 2 digits of a number a divisible by 4, the number itself is divisible by 4 – 5,624 for example (because 24/4=6);
  • Numbers with last digit 0 or 5 are divisible by 5;
  • Numbers that can be divided by both 2 and 3 can be divided by 6;
  • Similar to numbers divisible by 3, numbers divisible by 9 must have a sum of digits divisible by 9 – 6,453 for example;
  • If the last digit of a number is 0 it is divisible by 10;

With that out of the way, we can move onto some more advanced mental math techniques.

Avoid division at all costs

Don’t divide unless there is no other option. And that is especially true with long division. The reason why long division is so perilous is that it is very easy to make a careless mistake as there are usually several steps included in the calculation, it takes too much time, and to be honest, few people are comfortable doing it.

Fortunately, the GMAT doesn’t test the candidates’ human-calculator skills but rather their capacity to think outside the box and show creativity in their solution paths, especially when under pressure – exactly what business schools look for.

However, sometimes you cannot avoid division, and when that is the case remember: Factoring is your best friend. Always simplify fractions especially if you’ll need to turn them into decimals. For example, if you have 234/26 don’t start immediately trying to calculate the result. Instead, factor them little by little until you receive something like 18/2 which is a lot easier to calculate.

A tip for factoring is to always start with smaller numbers as they are easier to use (2 is easier to use compared to 4, 6, or 8) and also look for nearby round numbers. 

If you have to calculate 256/4 it would be far less tedious and time-consuming to represent 256 as 240+16 and calculate 240/4+16/4=60+4=64. Another example is 441/3. If we express it like 450-9 it is far easier to calculate 450/3-9/3=150-3=147.

Dividing and Multiplying by 5

Sometimes when you have to divide and multiply by 5 (you’ll have to do it a lot) it would be easier to substitute the number with 10/2. It might not always be suitable for your situation but more often than not it can be utilized in order to save some time.

Using 9s

With most problems, you could safely substitute 9 with 10-1. For example, if you have to calculate 46(9) you can express it as 46(10 – 1) which is a lot more straightforward to compute as 46(10) – 46(1) = 460 – 46 = 414

You can also use the same method for other numbers such as 11, 8, 15, 100, etc:

18(11) = 18(10 + 1) = 180 + 18 = 198

28(8) = 28(10 – 2) = 280 – 56 = 224

22(15) = 22(10 + 5) = 220 + 110 = 330

26(99) = 26(100 – 1) = 2600 – 26 = 2574

Dividing by 7

The easiest way to check if a number is divisible by 7 is to find the nearest number you know is divisible by 7. For instance, if you want to check if you can divide 98 by 7 you should look for the nearest multiple of 7. In this instance either 70, 77, or 84. Start adding 7 until you reach the number: 70 + 7 = 77 + 7 = 84 + 7 = 91 + 7 = 98. The answer is yes, 98 is divisible by 7 and it equals 14

Squaring

When you have to find the square of a double-digit number it might be easier to break the number into components. For example, 22^2 would be calculated like this:

22^2
= (20 + 2)(20 + 2)
= 400 + 40 + 40 + 4
= 484

Similarly, if you have to find the square of 39 instead of calculating (30 + 9)(30 +9) you could express it like this:

39^2
= (40 – 1)(40 – 1)
= 1600 – 40 – 40 + 1
= 1521

You can use the same approach when multiplying almost any double-digit numbers, not only squaring. For example 37 times 73:

(40 – 3)(70 + 3)
= 2800 + 120 – 210 – 9
= 2701

Conclusion

This ends the list of mental math tips and tricks you can utilize to make the Quant section a bit less laborious. Keep in mind that no strategy or shortcut would be able to compensate for the lack of proper prep so it all comes down not only to practicing but doing so the right way.

For more information regarding the GMAT Calculator, GMAT Calculator & Mental Math – All You Need To Know, is a very insightful article to read.

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Posted on
01
Sep 2021

Additional Voters – GMAT Quant Problem

Additional Voters – GMAT Quant Problem

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

Additional Voters Problem Introduction

GMAT Quant Problem

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

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

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

Reverse Engineering the Solution

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

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

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

Something to Keep in Mind

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

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

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Posted on
18
Aug 2021

GMAT Prime Factors Problem – GMAT Quant

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

GMAT Prime Factors Problem - GMAT Quant

GMAT Prime Factors Problem – Applied Math Solution Path

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

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

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

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

GMAT Prime Factors Problem – Another Solution Path

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

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

GMAT Prime Factors Problem – Takeaways

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

Hope this helped and good luck!

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

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GMAT Arithmetic
Posted on
17
Aug 2021

GMAT Arithmetic 101- All You Need To Know

By: Apex GMAT
Date: 17 August 2021

While studying and preparing for the GMAT quant section, you might have come across some different types of GMAT arithmetic questions. These are actually quite common in the quantitative reasoning section and can be often intertwined with other GMAT algebra and GMAT geometry questions.

These usually come in 2 different formats: data sufficiency problems and problem-solving. The former have a very particular structure where you will have to determine whether the 2 statements are enough to come up with a solution. The latter type of problem requires you to actually solve the problem and derive a proper solution.

In this article, we’ll tell you more about how to go about solving these GMAT arithmetic questions, so keep on reading to find out more.

Arithmetic Concepts you need to revise

These are the arithmetic concepts you’ll need to know before you start practicing. Make sure to revise these fundamentals:

How to solve a GMAT arithmetic problem?

The number 1 thing you need to keep in mind when dealing with GMAT arithmetic problems is that the concepts that you’ll come across are fairly simple. You can easily revise these concepts because they are all things we study in high-school-level math. But here’s the kicker: the way these concepts are incorporated into the GMAT problems makes them more challenging, especially when the GMAT arithmetic problems are intertwined with GMAT algebra problems or even GMAT geometry problems. That is where things get tricky, as you need to apply your knowledge in a much more complicated setting that incorporates more than one concept. However, it all comes down to knowing the basics of arithmetics, which we can also refer to as the mechanics of the problem. 

In order to help you better understand how to go about a GMAT arithmetic question, we will discuss an arithmetic problem and its solution and solution paths. In this GMAT problem, we are going to see how even the simplest mathematical concepts can become more challenging given the way the problem is formulated and structured.

Problem (GMAT Official Guide 2018) 

When positive integer x is divided by positive integer y, the remainder is 9.
If x/y = 96.12, what is the value of y?

(A) 96
(B) 75
(C) 48
(D) 25
(E) 12

Solution:
In this case, you will have to revise the properties of numbers in order to properly find a solution to the problem.
When x is divided by y, the remainder is 9. So x=yq + 9 for a random positive integer q.
After dividing both sides by y, we get: x/y = q + 9/y.
But, x/y= 96.12 = 96 + 0.12.
Equating the two expressions for x/y gives q + 9/y= 96+0.12.
Thus: 

q=96 and 9/y= 0.12
9=0.12y
y=9/0.12
y=75 

The correct answer is B.

Now that we went over the solution path for an arithmetic problem on the GMAT exam, you are ready to start your prep. Keep in mind that you should not overthink the questions. Some of them might really look challenging and complex. However, the solution paths can be fairly easy and it ultimately comes down to knowing the “mechanics” of the question.

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GMAT Probability Problems
Posted on
12
Aug 2021

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

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

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

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

GMAT Probability – Fundamental Rules & Formulas

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

Probability = number of desired outcomes
total number of possible outcomes

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

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

Dependent  Events vs. Independent Events

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

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

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

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

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

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

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

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

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

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

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

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

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

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Posted on
11
Aug 2021

GMAT Markup Problem – GMAT Data Sufficiency

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

GMAT Markup Problem Introduction

GMAT Markup Problem

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

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

Statement 1

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

Statement 2

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

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

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GMAT Quant Syllabus 2021-2022
Posted on
22
Jul 2021

GMAT Quant Syllabus 2021-2022

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

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

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

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

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

Data Sufficiency Questions

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

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

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

Problem Solving Questions

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

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

The main concepts you should focus on

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

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

Algebra

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

Geometry

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

Word problems

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

Check out this Profit and Loss question.

Arithmetic

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

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

5 tips to improve your GMAT quant skills?

  1. Master the fundamentals! This is your first step towards acing this section of the GMAT. As this section only contains math that you have already studied thoroughly in high-school, you’ll only need to revise what you have already learned and you’ll be ready to start practicing some real GMAT problems. 
  2. Practice time management! This is a crucial step as every single question is timed and you won’t get more than 2 minutes to spend on each question. That is why you should start timing yourself early on in your GMAT prep, so you get used to the time pressure. 
  3. Know the question types! This is something that you will learn once you get enough practice with some actual GMAT questions. That way, you’ll be able to easily recognize different question types and you’ll be able to use your preferred solution path without losing time.
  4. Memorize the answer choices for the data sufficiency questions! These answers are always the same and their order never changes. Memorizing them will help you save precious time that you can spend elsewhere. To help you better memorize them, we are sharing an easier and less wordy way to think of them:
  5. Make use of your scrap paper! There is a reason why you’re provided with scrap paper, so make sure to take advantage of it. You will definitely need it to take notes and make calculations, especially for the problem-solving questions that you will come across in this GMAT question.
  • Only statement 1
  • Only statement 2
  • Both statements together
  • Either statement
  • Neither statement
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Posted on
21
Jul 2021

GMAT 3D Geometry Problem – GMAT Math – Quant Section

GMAT 3D Geometry Problem 

In this problem we’re going to take a look at 3D objects and in particular a special problem type on the GMAT that measures the longest distance within a three-dimensional object. Typically, they give you rectangular solids, but they can also give you cylinders and other such objects. The key thing to remember about problems like this one is that effectively we’re stacking Pythagorean theorems to solve it – we’re finding triangles and then triangles within triangles that define the longest distance.

This type of problem is testing your spatial skills and a graphic or visual aid is often helpful though strictly not necessary. Let’s take a look at how to solve this problem and because it’s testing these skills the approach is generally mathematical that is there is some processing because it’s secondary to what they’re actually testing.

gmat 3d geometry question

GMAT 3D Geometry Problem Introduction

So, we have this rectangular solid and it doesn’t matter which way we turn it – the longest distance is going to be between any two opposite corners and you can take that to the bank as a rule: On a rectangular solid the opposite corners will always be the longest distance. Here we don’t have any way to process this central distance so, what we need to do is make a triangle out of it.

Notice that the distance that we’re looking for along with the height of 5 and the hypotenuse of the 10 by 10 base will give us a right triangle. We can apply Pythagoras here if we have the hypotenuse of the base. We’re working backwards from what we need to what we can make rather than building up. Once you’re comfortable with this you can do it in either direction.

Solving the Problem

In this case we’ve got a 10 by 10 base. It’s a 45-45-90 because any square cut in half is a 45-45-90 which means we can immediately engage the identity of times root two. So, 10, 10, 10 root 2. 10 root 2 and 5 makes the two sides. We apply Pythagoras again. Here it’s a little more complicated mathematically and because you’re going in and out of taking square roots and adding and multiplying, you want to be very careful not to make a processing error here.

Careless errors abound particularly when we’re distracted from the math and yet we need to do some processing. So, this is a point where you just want to say “Okay, I’ve got all the pieces, let me make sure I do this right.” 10 root 2 squared is 200 (10 times 10 is 100, root 2 times root 2 is 2, 2 times 100 is 200). 5 squared is 25. Add them together 225. And then take the square root and that’s going to give us our answer. The square root of 225 is one of those numbers we should know. It’s 15, answer choice A.

Okay guys for another 3D and Geometry problem check out GMAT 680 Level Geometry Problem – No Math Needed! We will see you next time.

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Posted on
14
Jul 2021

GMAT 680 Level Geometry Problem – No Math Needed!

GMAT Geometry Problem

Hey guys, top level geometry problems are characterized typically by stringing a whole bunch of different rules together and understanding how one thing relates to the next thing, to the next thing. Until you get from the piece of information you started with to the conclusion. We’re going to start out by taking a look at this problem using the z equals 50° and seeing how that information goes down the line.

top level geometry problem

But afterwards we’re going to see a super simple logical pathway utilizing a graphic scenario that makes the z equals 50° irrelevant. To begin with we’re being asked for the sum of x and y and this will come into play on the logical side. We need the sum not the individual amounts but let’s begin with the y. We have a quadrilateral and it has parallel sides which means the two angles z and y must equal 180°. That’s one of our geometric rules. If z is 50° that means y is 130° and we’re halfway there.

Next we need to figure out how x relates and there are several pathways to this. One way we can do it is drop. By visualizing or dropping a third parallel line down, intersecting x, so on the one hand we’ll have 90 degrees. We’ll have that right angle and on the other we’ll have that piece. Notice that the parallel line we dropped and the parallel line next to z are both being intersected by the diagonal line going through which means that that part of x equals z. So we have 50° plus 90° is 140°. 130° from the y, 140° from the x, gives us 270°.

Another way we can do this is by taking a look at the right triangle that’s already built in z is 50° so y is 1 30°. now the top angle in the triangle must then be 180° minus the 130° that is 50°. it must match the z again we have the parallel lines with the diagonal coming through then the other angle the one opposite x is the 180° degrees that are in the triangle minus the 90° from the right triangle brings us to 90° minus the 50° from the angle we just figured out means that it’s 40° which means angle x is 180° flat line supplementary angles minus the 40° gives us 140° plus the 130° we have from y again we get to 270°.

Graphic Solution Path

Now here’s where it gets really fun and really interesting. We can run a graphic scenario here by noticing that as long as we keep all the lines oriented in the same way we can actually shift the angle x up. We can take the line that extends from this big triangle and just shift it right up the line until it matches with the y. What’s going to happen there, is we’re going to see that we have 270° degrees in that combination of x and y and that it leaves a right triangle of 90°, that we can take away from 360° again to reach the 270°.

Here the 50° is irrelevant and watch these two graphic scenarios to understand why no matter how steep or how flat this picture becomes we can always move that x right up and get to the 270°. That is the x and y change in conjunction with one another as z changes. You can’t change one without the other while maintaining all these parallel lines and right angles. Seeing this is challenging to say the least, it requires a very deep understanding of the rules and this is one of those circumstances that really points to weaknesses in understanding most of what we learn in math class in middle school, in high school. Even when we’re prepping only scratches the surface of some of the more subtle things that we’re either allowed to do or the subtle characteristics of rules and how they work with one another and so a true understanding yields this very rapid graphic solution path.

Logical Solution Path

The logical solution path where immediately we say x and y has to be 270° no matter what z is and as you progress into the 80th, 90th percentile into the 700 level on the quant side this is what you want to look for during your self prep. You want to notice when there’s a clever solution path that you’ll overlook because of the rules. Understand why it works and then backtrack to understand how that new mechanism that you discovered fits into the framework of the rules that we all know and love. Maybe? I don’t know if we love them! But they’re there, we know them, we’re familiar with them, we want to become intimate. So get intimate with your geometry guys put on some al green light some candles and I’ll see you next time.

If you enjoyed this problem, try other geometry problems here: GMAT Geometry.

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The basics of GMAT Combinatorics
Posted on
24
Jun 2021

The Basics of GMAT Combinatorics

By: Apex GMAT
Contributor: Svetozara Saykova
Date: 24th June 2021

Combinatorics can seem like one of the most difficult types of questions to come across on the GMAT. Luckily there are not many of them within the exam. Still these questions make up the top level of scoring on the test and therefore it is best if you are well equipped to solve them successfully, especially if you are aiming for a 700+ score. The most important rule to follow when considering this question type is the “Fundamental Counting Principle” also known as the “Counting Rule.” This rule is used to calculate the total number of outcomes given by a probability problem. 

The most basic rule in Combinatorics is “The Fundamental Counting Principle”. It states that for any given situation the number of overall outcomes is equal to the product of the number of each discrete outcome.

Let’s say you have 4 dresses and 3 pairs of shoes, this would mean that you have 3 x 4 = 12 outfits. The Fundamental Counting Principle also applies for more than 2 options. For example, you are at the ice cream shop and you have a variety of 5 flavors, 3 types of cones and 4 choices for toppings. That means you have 5 x 3 x 4 = 60 different combinations of single-scoop ice creams. 

The Fundamental Counting Principle applies only for choices that are independent of one another. Meaning that any option can be paired with any other option and there are no exceptions. Going back to the example, there is no policy against putting sprinkles on strawberry vanilla ice cream because it is superb on its own. If there were, that would mean that this basic principle of Combinatorics would not apply because the combinations (outcomes) are dependent. You could still resort to a reasoning solution path or even a graphical solution path since the numbers are not so high. 

Let’s Level Up a Notch

The next topic in Combinatorics is essential to a proper GMAT prep is  permutations. A permutation is a possible order in which you put a set of objects.

Permutations

There are two subtypes of permutations and they are determined by whether repetition is allowed or not.

  • Permutations with repetition allowed

When there are n options and r number of slots to fill, we have n x n x …. (r times) = nr permutations. In other words, there are n possibilities for the first slot, n possibilities for the second and so on and so forth up until n possibilities for position number r.

The essential mathematical knowledge for these types of questions is that of exponents

To exemplify this let’s take your high school locker. You probably had to memorize a 3 digit combination in order to unlock it. So you have 10 options (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) for 3 available slots. The total number of locker passwords you can have is 103 = 1,000. 

  • Permutation without repetition allowed 

When repetition is restricted in the given GMAT problem, we would have to reduce the number of available choices for each position. 

Let’s take the previous example and add a restriction to the password options – you cannot have repeating numbers in your locker password. Following the “we reduce the options available each time we move to the next slot” rule, we get 10x9x8 = 720 options for a locker combination (or mathematically speaking permutation). 

To be more mathematically precise and derive a formula we use the factorial function (n!). In our case we will take all the possible options 10! for if we had 10 positions available  and divide them by 7!, which are the slots we do not have. 

10! =  10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 

7! =  7 x 6 x 5 x 4 x 3 x 2 x 1 

And when we divide them (7 x 6 x 5 x 4 x 3 x 2 x 1) cancels and we are left with 10 x 9 x 8 = 720. 

Pro tip: Taking problems and deeply examining them by running different scenarios, and changing some of the conditions or numbers is a great way to train for the GMAT. This technique will allow you to not only deeply understand the problem but also the idea behind it, and make you alert for what language and piece of information stands for which particular concept. 

So those are the fundamentals, folks. Learning to recognize whether order matters and whether repetition is allowed is essential when it comes to Combinatorics on the GMAT. Another vital point is that if you end up with an endless equation which confuses you more than helps, remember doing math on the GMAT Quant section is not the most efficient tactic. In fact, most of the time visualizing the data by putting it into a graph or running a scenario following your reasoning are far more efficient solution paths. 

Feeling confident and want to test you GMAT Combinatorics skills? Check out this GMAT problem and try solving it. Let us know how it goes!

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