Now, some numbers are spelled out down to the decimals, but other numbers, such as 11!, seem unnecessarily abstract. 11 factorial is a big number, but wouldn’t it be simpler if I had a concrete number in front of me instead of a shorthand notation for 10 multiplications. The answer is: not really. If you wanted to expand 11! To get a longhand answer, you’ll end up with a large concrete number that is no easier to manipulate than the shorthand you had before. For example, 11! is actually 39,916,800. Does that make it any easier to use? Again, the answer is: not really.
In essence, every time you see a big number like this, the GMAT is baiting you into performing tedious calculations that don’t help you in any way. Having a cumbersome number is the GMAT’s way of saying “Don’t try and solve this with brute force, there’s a concept here you should recognize”. While it’s uncommon for the GMAT to actually speak, given that it’s an admissions exam, it actually is telling you loud and clear that concentrating on the number is a trap. There will always be some element that will help highlight the underlying issue without performing tedious math.
There are many concepts that may come into play, and it’s hard to approach these questions with a single standard approach, but some elements repeat more frequently than others. One of the first things to look for is the units digit. The units digit gives away many properties of a number. As an example, 39,916,800 ends with a 0, indicating that it is even, and that it is divisible by 10. Different units digits can yield different number properties, so you can learn a lot from one simple digit. The factors of the number in question can often unlock clues as to which numbers to look for among the answer choices. Finally the order of magnitude can also play a pivotal role in determining how to approach a question.
Since we don’t have one definitive strategy, let’s test our mental agility on an actual GMAT question:
For integers x, y and z, if ((2^x)^(y))^(z) = 131,072, which of the following must be true:
(A) The product xyz is even
(B) The product xyz is odd
(C) The product xy is even
(D) The product yz is prime
(E) The product yz is positive
This question is significantly easier if you recognize which power of two 131,072 is off the bat (I knew that Computer Science degree would be good for something). However, let’s approach this knowing that 131,072 is a multiple of two, but that calculating which one would require more time than the two minutes we have earmarked for this question. Furthermore, simply knowing that 131,072 is a power of 2 gives us all the information we really need to solve this question.
We know x, y and z will combine to form some integer, but we’re not sure which. Let’s call it integer R (as in Ron) for simplicity’s sake. Moreover, the way the equation is set up, the powers will all be multiplied by one another, meaning that their exact order won’t matter. As such, the commutative law of mathematics confirms that if ((2^5)^(3))^(2) is the exact same thing as ((2^3)^(2))^(5). If the order doesn’t matter, then there are a lot of potential situations that could occur. So R will equal x + y + z, but the order won’t change anything. Let’s look at the answer choices, and start from the end because they’re easier to eliminate.
Answer choice E asks us whether y*z must be positive. If y*z gives us some positive number, then x would just be whatever is left over to form R. It doesn’t matter is y*z is positive or negative, as x can just come and make up the difference. Let’s say y*z = 4, then x would just be R – 4. If, instead, y*z = 4, then x would just be R – 12 and there would be no difference. In other words, as long as one variable is unrestricted, it will always be able to make up for the restriction on the other two. If you recognize this, you can eliminate C, D and E for the same reason. Two out of three ain’t bad, but in this case, it ain’t enough.
This brings us down to answer choices A and B, which are complimentary. Either the product of the three numbers is even, or it is odd. One of these, logically, must be true. Unfortunately, the best way to verify this appears to be doing the calculation longhand (like the petals of a flower: she loves me, she loves me not). Herein lays a potential shortcut: the units digit. Since the number is a power of two, we can simply follow the pattern of multiples of two and see what we get. Considering primarily the units digit (underlined for emphasis):
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
You probably don’t have to go this far to notice the pattern, but it doesn’t hurt to confirm if you’re not sure after 2^5. Essentially, the unit digit oscillates in a fixed pattern: 2, 4, 8, 6, and then repeats. This is helpful, because the number in question ends with a 2, and every power of two that ends with a 2 is either 2^1, 2^5, 2^9, etc. All of these numbers are odd powers of 2, repeating every fourth element. With this pattern clearly laid out, it becomes apparent that the answer must be that the product of these three variables must be odd. As such, answer choice B is correct here. We can also probably deduce from order of magnitude that 131,072 is 2^17.
When it comes to large numbers on the GMAT, you should never try to use brute force to solve the problem. The numbers are arbitrarily large to dissuade you from trying to actually calculate the numbers, and they can be made arbitrarily larger on the next question to waste even more of your time. The GMAT is a test of how you think, so thinking in terms of constantly calculating the same numbers over and over limits you to being an ineffective calculator. Your smart phone currently has at least 100 times your computational power (but not the ability to use it independently… yet…). Brute force may break some doors down, but mental agility is a skeleton key.
Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.
]]>Myth 3: Use of ‘being’ is always wrong on GMAT!
Often, the way we use ‘being’ in our daytoday communication, is incorrect. For example,
Being a doctor, he is very well respected.
But there are correct ways of using ‘being’. Since most students believe that ‘being’ is wrong, don’t trust the GMAC to not use this nugget of information to misdirect the test takers. The correct answers of questions at higher ability are worded in such a way that they make the test takers uncomfortable!
So how is ‘being’ used correctly?
‘Being’ is used to express a temporary state.
The little boy started screaming when he saw his dog being impounded.
‘Being impounded’ is a temporary state and would be over – unlike being a doctor. So the use of being is correct here.
Let’s look at one of our own sentence correction questions now:
Question: The data being collected in the current geological survey are providing a strong warning for engineers as they consider the new dam project, but their greatest importance might lie in how they influence the upcoming decision by those same engineers on whether to retrofit 75 bridges in the survey zone.
A. The data being collected in the current geological survey are providing a strong warning for engineers as they consider the new dam project, but their greatest importance
B. The data being collected in the current geological survey provide a strong warning for engineers as they consider the new dam project, but its greatest importance
C. The data collected in the current geological survey is providing a strong warning for engineers as they consider the new dam project, but their greatest importance
D. The data collected in the current geological survey provides a strong warning for engineers in consideration of the new dam project, but its greatest importance
E. The data collected in the current geological survey provide a strong warning for engineers in consideration for the new dam project, but the greatest importance
Solution: Let’s find the decision points:
First decision point: being collected vs collected
‘The data being collected’ is a temporary state here. Data won’t always be collected, but are being collected for a short time right now, so ‘being’ is used properly here. And the sentence makes it very clear that this is temporary; look at the word ‘current’ before ‘geological survey.’ If you were skeptical of the word ‘being’ before, that is understandable, but the word ‘current’ should serve as a clear warning that this is a temporary, ongoing event. (And furthermore, the nonunderlined portion talks about an upcoming decision, even further cementing the idea that this is a temporary survey with data ‘being’ collected for a short period of time). Alas, even with that temporary state, there isn’t really anything wrong with ‘The data collected in …’ either so we retain all answer options.
It’s worth noting that they put the ‘being collected’ vs. ‘collected’ decision early in the sentence/answerchoices as a great place to put a ‘false decision point.’ The authors of these questions know that they want to reward emphases on logical meaning and core grammar rules, and they also know that students like to study quick tips and tricks, so they leave that bait there for the tips/tricks folks while they hold off the bigger reward for those willing to prioritize decision points from most important to least and not just from left to right in order of appearance.
Second decision point: are/is
Technically, data is plural of datum. In academic writing it is almost always treated as plural. It is treated as singular in informal writing but GMAT favors treating it as plural.
Even if you do not know this, the use of “they influence the upcoming” – in the portion of the sentence that is not underlined – should tell you that ‘data’ is used in plural form here.
Hence the use of ‘are’ is appropriate. Hence, options (C) and (D) are eliminated.
Third decision point: Pronouns
There are many pronouns used here. Antecedent of each pronoun is present in the sentence. The usage clarifies which pronoun refers to data and which refers to engineers.
Original Sentence: The data being collected in the current geological survey are providing a strong warning for engineers as they consider the new dam project, but their greatest importance might lie in how they influence the upcoming decision by those same engineers on whether to retrofit 75 bridges in the survey zone.
they – only engineers can consider the new dam project so ‘they’ refers to engineers
their/its  greatest importance will be of data, which is plural, so ‘their’ would be the correct usage. Eliminate option (D)
There is no ambiguity in the use of pronouns. The nouns are present and the usage clarifies the antecedent.
Now we are left with options (A) and (E). In option (E), “in consideration for the new dam project” is bad diction. Also, it doesn’t tell us ‘whose greatest importance?’.
Answer is (A)
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!
]]>What motivates you to be a GMAT instructor?
“I have been teaching the GMAT for 10 years because I absolutely love what the test is designed to assess and how it makes you learn and think. This is not a content regurgitation test, but rather it is one that assesses who is good at taking basic content and using that to solve very difficult problems and reasoning puzzles. I believe that the skills and thinking processes the GMAT assesses are invaluable not only in business but in all walks of life. I really enjoy unlocking this way of thinking for students and teaching them to love a test that they may have at first despised!”
If you could give three pieces of advice to future GMAT test takers, what would they be?
“1) Do not waste 3 months preparing on your own, receive a low score, and THEN sign up for a high quality GMAT prep course. Take our full GMAT course before you even open a book or read about the GMAT. It will save you so much time, energy, and frustration.
Click here for Chris’ other two points of advice.
Is there a common misconception of the GMAT or of what is a realistic GMAT score?
“I think there are many important misconceptions about the test as a whole and the scoring system in particular. As I have intimated earlier, the biggest misconception about the GMAT is that it is a content test in which memorizing all the rules and the underlying content will allow you to do well. This is certainly not the case and it is why so many students get frustrated when they prepare on their own. The GMAT is so different from the tests that you were able to ace in college with memorization ‘allnighters.’ Also, I think people underestimate how competitive and difficult the GMAT really is. Remember that you are competing against a highly selective group of college graduates from around the world who are very hungry to attend a top US business school. This test is no joke and requires an intensive preparation geared toward success in higher order thinking and problem solving.”
Read the rest of the interview here!
Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!
]]>Now, at a restaurant, you may be particularly hungry and decide to order both the soup and the salad (and the frog legs while we’re at it). Similarly, on forms, someone who selects both options is being confusing. Perhaps you’ve smoked once and didn’t like it. Perhaps you smoke only on long weekends when the Philadelphia Eagles have a winning record. Sometimes people decide they don’t want to pick between the two choices given. However, if the question were changed to “have you ever smoked a cigarette?” and then given yes or no options, the decision becomes much easier. You have to be in one camp or the other, there is no sitting on the fence (like Humpty Dumpty).
For questions that set up this kind of duality, the entire spectrum of possibilities is essentially covered in these two options. There is no third option; there is no “It’s Complicated” selection. There isn’t even a section for you to explain yourself in the comments below. On these questions, you have to either be on one side or the other, you cannot be in both. Equally, you cannot be in the “neither” camp either. Necessarily, to this point in your life, you have either smoked a cigarette or you have not. Since one of them must be true, this certainty offers some insight on inference questions in critical reasoning.
As you probably recall, inference questions require that an answer choice must be true at all times. This isn’t always easy to see as many answer choices seem likely, but simply are not guaranteed. Sometimes, on inference questions, you get two answer choices that are compliments of one another. You get two choices that say something to the effect of “Ron is always awesome” and “Ron is not always awesome”. Even I would go for the latter here, but clearly one of these must be correct. They cannot both be correct, but they also cannot both be false. Having two answer choices like this guarantees that one of them must be the correct answer, and makes your task considerably easier.
Let’s look at an example:
A few people who are bad writers simply cannot improve their writing, whether or not they receive instruction. Still, most bad writers can at least be taught to improve their writing enough so that they are no longer bad writers. However, no one can become a great writer simply by being taught how to be a better writer, since great writers must have not only skill but also talent.
Which one of the following can be properly inferred from the passage above?
(A) All bad writers can become better writers
(B) All great writers had to be taught to become better writers.
(C) Some bad writers can never become great writers.
(D) Some bad writers can become great writers.
(E) Some great writers can be taught to be even better writers.
Since this is an inference question, we must read through the answer choices because there are many possible answers that could be inferred from this passage. When reading through the passage, you probably note that answers C and D are somewhat complimentary. Either the bad writers can become great writers, or they can’t. However, some people might be miffed by the fact that “some writers” is vague and could mean different people in different contexts. However, while the term “some writers” is undoubtedly abstract, it can refer to any subset of writers one or greater (and up to the entire group). Any group of bad writers is thus conceivable in this passage, but the answer choice must be true at all times, so the groups comprised of “some writers” can mean anyone, and these two groups can be considered equivalent.
If you recognize that either answer choice C or answer choice D must be the answer, then you can easily skip over the other three choices. For completeness’ sake, let’s run through them quickly here. Answer choice A directly contradicts the first sentence of this passage: Some bad writers simply cannot improve their writing. Answer choice B contradicts the major point of this passage, which is that great writers have a combination of skill and talent, and you cannot teach talent. Answer choice E makes sense as an option, but it doesn’t necessarily have to be true. This is a classic example of something that’s likely true in the real world, but not necessarily guaranteed by this particular passage.
This leaves us with two options to consider. Can bad writers become great writers, or can they never become great writers? As mentioned above, great writers are born with some level of talent that cannot be mimicked by practice alone. The passage explicitly states “no one can one can become a great writer simply by being taught how to be a better writer”. Even though some bad writers can improve their writing with some help (perhaps even writing a Twilight Saga), some cannot improve their writing at all. If these bad writers cannot improve their writing, they necessarily will never become great writers. Answer choice C must be true based on the passage.
Looking at answer choice D in contrast, it states: “some bad writers can become great writers”. Perhaps some can, but this cannot be guaranteed in any way from the passage. It’s possible that all the writers are terrible even after year of practice. In fact, since we know that some will never improve (the opposite), this conclusion is certainly is not guaranteed. Answer choice C is supported by the passage, answer choice D seems conceivable in the real world, but it is certainly not assured.
On the GMAT, as in life, when confronted with two complimentary choices, you have to end up making a choice. In this instance, because you typically have five choices to consider, whittling the competition down to two choices already saves you time and gives you confidence. Recognizing which option must always be true is all that’s left to do, and that often comes down to playing Devil’s Advocate. When you’re tackling a decision such as this, consider what has to be true, and you’ll make the right choice.
Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.
]]>Today, we bring another tip for you to help get that dream score of 51 – if you must write down the data given, write down all of it! Let us explain.
If you think that you will need to jot down the data given in the question and then solve it on your scratch pad (instead of in your mind), you must jot down every single detail. It is easy to overlook small things which are difficult to express algebraically such as ‘x is an integer’. These details are often critical and could make all the difference between an ‘unsolvable’ question and a ‘solvable within 2 minutes’ one. Once you start solving the question on your scratch pad, you will not refer back to the original question again and again and hence might forget these details. Have them along with the rest of the data. Read every word of the question carefully, and ensure that it is consolidated on your scratch pad. For example, look at this question:
A set of five positive integers has an arithmetic mean of 150. A particular number among the five exceeds another by 100. The rest of the three numbers lie between these two numbers and are equal. How many different values can the largest number among the five take?
It is a difficult question because it incorporates statistics as well as maxmin – both tricky topics. On top of it, people often overlook the ‘are equal’ part of the question here. The reason for that is that they are actively looking for implications of the sentences and the moment they read “The rest three numbers lie between these two numbers”, they go back to the previous sentence which tells us “A particular number among the five exceeds another by 100”. They then make a note of the fact that 100 is the range of the five positive integers. In all this excitement, they miss the three critical words “and are equal”. Ensure that when you go to the sentence above, you pick the next sentence from the point where you left it. Another thing to note here is that all numbers are positive integers. This information will be critical to us.
Let’s demonstrate how you will solve this question after incorporating all the information given.
Question: A set of five positive integers has an arithmetic mean of 150. A particular number among the five exceeds another by 100. The rest of the three numbers lie between these two numbers and are equal. How many different values can the largest number among the five take?
(A) 18
(B) 19
(C) 21
(D) 42
(E) 59
Solution:
Let’s assume that the 5 natural numbers in increasing order are: a, b, b, b, a+100
We are given that a < b < a+100.
Also, we are given that a and b are positive integers. This information is critical – we will see later why.
The average of the 5 numbers is (a+b+b+b+a+100)/5 = 150
(a+b+b+b+a+100) = 5*150
2a+3b = 650
We need to find the number of distinct values that a can take because a+100 will also take the same number of distinct values.
Now there are two methods to proceed. Let’s discuss both of them.
Method 1: Pure Algebra – Write b in terms of a and plug it in the inequality
b = (650 – 2a)/3
a < (650 – 2a)/3 < a+100
3a < 650 – 2a < 3a + 300
Now split it into two inequalities: 3a < 650 – 2a and 650 – 2a < 3a + 300
Inequality 1: 3a < 650 – 2a
5a < 650
a < 130
Inequality 2: 650 – 2a < 3a + 300
5a > 350
a > 70
So we get that 70 < a < 130. Since a is an integer, can we say that a can take all values from 71 to 129? No. What we are forgetting is that b is also an integer. We know that
b = (650 – 2a)/3
For which values will be get b as an integer? Note that 650 is not divisible by 3. You need to add 1 to it or subtract 2 out of it to make it divisible by 3. So a should be of the form 3x+1.
b = (650 – 2*(3x+1))/3 = (648 – 6x)/3 = 216 – 2x
Here, for any positive integer x, b will be an integer.
From 71 to 129, we have the following numbers which are of the form 3x+1:
73, 76, 79, 82, 85, … 127
This is an Arithmetic Progression. How many terms are there here?
Last term = First term + (n – 1)*Common Difference
127 = 73 + (n – 1)*3
n = 19
a will take 19 distinct values so the last term i.e. (a+100) will also take 19 distinct values.
Method 2: Using Transition Points
Note that a < b < a+100
Since a < b, let’s find the point where a = b, i.e. the transition point
2a + 3a = 650
a = 130 = b
But b must be greater than a. If we increase b by 1, we need to decrease a by 3 to keep the average same. But decreasing a by 3 decreases the largest number i.e. a+100 by 3 too; so we need to increase b by another 1.
We get a = 127 and b = 132. This give us the numbers as 127, 132, 132, 132, 227. Here the average is 150
Since b < a+100, let’s find the point where b = a+100
2a + 3(a+100) = 650
a = 70, b = 170
But b must be less than a+100. If we decrease b by 1, we need to increase a by 3 to keep the average same. But increasing a by 3 increases the largest number, i.e. a+100 by 3 too, so we need to decrease b by another 1.
We get a = 73 and b = 168. This gives us the numbers as 73, 168, 168, 168, 173. Here the average is 150
Values of a will be: 73, 76, 79, ….127 (Difference of 3 to make b an integer)
This is an Arithmetic Progression.
Last term = First term + (n – 1)*Common difference
127 = 73 + (n – 1)*3
n = 19
a will take 19 distinct values so the last term i.e. (a+100) will also take 19 distinct values.
Answer (B)
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!
]]>Watch this video to learn how you can find hidden hints within statements and how that can help you avoid any GMAT traps. You don’t want to leave any points on the table.
Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!
By Brian Galvin
]]>A common strategy in puzzles is to build the outsides or the corners first, as these pieces are more easily identifiable than a typical piece, and then try and connect them wherever possible. Indeed, you are unlikely to have ever solved a puzzle without needing to jump around (except for puzzles with 4 pieces or so).
Similarly, you are often faced with GMAT questions that seem like intricate puzzles, and this same strategy of jumping around can be applied. If you start at the beginning of a question and make some strides, you may find your progress has been jammed somewhere along the way and you must devise a new strategy to overcome this roadblock. Jumping around to another part of the problem is a good strategy to get your creative juices flowing.
Let’s say a math question is asking you about the sum of a certain series. A simplistic approach (possibly one used by a Turing machine) would sequentially count each item and keep a running tally. However, a more strategic approach might involve jumping to the end of the series, investigating how the series is constructed, and finding the average. This average can then be multiplied by the number of terms to correctly find the sum of a series in a couple of steps, whereas the brute force approach would take much longer. Since the GMAT is an exam of how you think, the questions asked will often reward your use of logical thinking and your understanding of the underlying math concepts.
Let’s look at a sequence and see how thinking out of order can actually get our thinking straight:
In the sequence a1, a2, a3, an, an is determined for all values n > 2 by taking the average of all terms a1 through an1. If a1 = 1 and a3 = 5, then what is the value of a20?
(A) 1
(B) 4.5
(C) 5
(D) 6
(E) 9
This question is designed to make you waste time trying to decipher it. A certain pattern is established for this sequence, and then the twentieth term is being asked of us. If the sequence has a pattern for all numbers greater than two, and it gave you the first two numbers, then you could deduce the subsequent terms to infinity (and beyond!). However, only the first and third terms are given, so there is at least an extra element of determining the value of the second term. After that, we may need to calculate 16 intermittent items before getting to the 20th value, so it seems like it might be a time consuming affair. As is often the case on the GMAT, once we get going this may be easier than it initially appears.
If a1 is 1 and a3 is 5, we actually have enough information to solve a2. The third term of the sequence is defined as the average of the first two terms, thus a3 = (a1 + a2) / 2. This one equation has three variables, but two of them are given in the premise of the question, leading to 5 = (1 + a2) /2. Multiplying both sides by 2, we get 10 = 1 + a2, and thus a2 has to be 9. The first three terms of this sequence are therefore {1, 9, 5}. Now that we have the first three terms and the general case, we should be able to solve a4, a5 and beyond until the requisite a20.
The fourth term, a4 is defined as the average of the first three terms. Since the first three terms are {1, 9, 5}, the fourth term will be a4 = (1 + 9 + 5) / 3. This gives us 15/3, which simplifies to 5. A4 is thus equal to 5. Let’s now solve for a5. The same equation must hold for all an, so a5 = (1 + 9 + 5 + 5) /4, which is 20/4, or again, 5. The third, fourth and fifth terms of this sequence are all 5. Perhaps we can decode a pattern without having to calculate the next fourteen numbers (hint: yes you can!).
A3 is 5 because that is the average of 1 and 9. Once we found a3, we set off to find subsequent elements, but all of these elements will follow the same pattern. We take the elements 1 and 9, and then find the average of these two numbers, and then average out all three terms. Since a3 was already the average of a1 and a2, adding it to the equation and finding the average will change nothing. A4 will similarly be 5, and adding it into the equation and taking the average will again change nothing. Indeed all of the terms from A3 to A∞ will be equal to exactly 5, and they will have no effect on the average of the sequence.
You may have noticed this pattern earlier than element a5, but it can nonetheless be beneficial to find a few concrete terms in order to cement your hypothesis. You can stop whenever you feel comfortable that you’ve cracked the code (there are no style points for calculating all twenty elements). Indeed, it doesn’t matter how many terms you actually calculate before you discover the pattern. The important part is that you look through the answer choices and understand that term a20, like any other term bigger than a3, must necessarily be 5, answer choice C.
While understanding the exact relationship between each term on test day is not necessary, it’s important to try and see a few pattern questions during your test prep and understand the concepts being applied. You may not be able to recognize all the common GMAT traps, but if you recognize a few you can save yourself valuable time on questions. If you find yourself faced with a confusing or convoluted question, remember that you don’t have to tackle the problem in a linear fashion. If you’re stuck, try to establish what the key items are, or determine the end and go backwards. When in doubt, don’t be afraid to skip around (figuratively, literal skipping is frowned upon at the test center).
Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.
]]>In this post, we discussed how to use graphing techniques to easily solve very high level questions on nested absolute values. We don’t think you will see such high level questions on actual GMAT. The aim of putting up the post was to illustrate the use of graphing technique and how it can be used to solve simple as well as complicated questions with equal ease. It was aimed at encouraging you to equip yourself with more visual approaches.
We gave you two questions at the end of that post to try on your own. We have seen quite a bit of interest in them and hence will discuss their solutions today.
The solutions involve a number of graphs and hence we have made pdf files for them.
Question 1: Given that y = x – 5 – 10 5, for how many values of x is y = 2?
Question 2: Given that y = x – 3 – x, for what range of x is y = 3?
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!
]]>The advantage of the matrix box is that it highlights the innate relationships that must be true, but that are not always easy to keep track of. For instance, if a box contains 100 paperclips, some of which are metallic and some of which are plastic, then if we find 40 paperclips made of metal, there must necessarily be 60 that are made of plastic. The binary nature of the information guarantees that all the elements will fall into one of the predetermined categories, so knowing about one gives you information about the other.
The matrix box allows you to catalogue information before it becomes overwhelming. Anyone who’s studied the GMAT for any length of time (five minutes is usually enough) knows that the exam is designed to be tricky. As such, questions always give you enough information to solve the problem, but rarely give you the information in a convenient manner. Setting up a proper matrix box essentially sets you up to solve the problem automatically, as long as you know what to do with the data provided.
Let’s look at an example and what clues us into the fact that we should use a matrix box.
Of 200 students taking the GMAT, all of them have college degrees, 120 have been out of college for at least 3 years, 70 have business degrees, and 60 have been out of college for less than 3 years and do not have business degrees. How many of them have been out of college for at least 3 years and have business degrees.
A) 40
B) 50
C) 60
D) 70
E) 80
The principle determinant on whether we should use Venn diagrams or matrix boxes is whether the data has any overlap. In this example, it’s very hard to believe that a student could both have a business degree and not have a business degree, so it looks like the information can’t overlap and a matrix box approach should be used. Before we set up the matrix box, it’s important to know that the axes are arbitrary and you could put the data on either axis and end up with essentially the same box. We can thus proceed with whichever method we prefer. The box may look like what we have below:
Business Degree 
No Business Degree 
Total 

At least 3 years 

Less than 3 years 

Total 
Without filling out any information, it’s important to note that the “Total” column and row will be the most important parts. They allow us to determine missing information using simple subtraction. If we have the total figures, as little as one piece of information in the inside squares would be enough to solve every missing square (like the world’s simplest Sudoku). Let’s populate the total numbers provided in the question:
Business Degree 
No Business Degree 
Total 

At least 3 years 
120 

Less than 3 years 

Total 
70 
200 
With these three pieces of information, we can fill out the remaining “Total” squares by simply subtracting the given totals.
Business Degree 
No Business Degree 
Total 

At least 3 years 
120 

Less than 3 years 
80 

Total 
70 
130 
200 
Now all we would need to reach the correct answer is one piece of information: any of the remaining four squares. Luckily the question stem will always provide at least one of these, as the problem is unsolvable otherwise. Problems may be tricky and convoluted on the GMAT, but they will never be impossible. Looking back at the question, there are 60 students who have been out of college for less than 3 years and do not have business degrees. Plugging in this value we get:
Business Degree 
No Business Degree 
Total 

At least 3 years 
120 

Less than 3 years 
60 
80 

Total 
70 
130 
200 
Using a little bit of basic math we can turn this into:
Business Degree 
No Business Degree 
Total 

At least 3 years 
70 
120 

Less than 3 years 
20 
60 
80 
Total 
70 
130 
200 
And finally the completed:
Business Degree 
No Business Degree 
Total 

At least 3 years 
50 
70 
120 
Less than 3 years 
20 
60 
80 
Total 
70 
130 
200 
The question was asking for how many students have been out of college for at least 3 years and have business degrees, but using this method we could solve any potential question (Other than “What is the meaning of life”?). Since the number of students with business degrees who have been out of college three years or more is 50, the correct answer will be answer choice B.
In matrix box problems, setting up the question is more than half the battle. Correctly setting up the parameters will ensure that the rest of the problem gets solved almost automatically, and all you have to do is avoid silly arithmetic mistakes or getting ahead of yourself too quickly. Remember that if the information doesn’t overlap, it will likely make for a good matrix box problem. On these types of questions, don’t be afraid to think inside the box.
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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.
]]>Before you begin, you might want to review the post that discusses standard deviation: Dealing With Standard Deviation
So here goes the question.
Question: Given that set S has four odd integers and their range is 4, how many distinct values can the standard deviation of S take?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
Solution: Recall what standard deviation is. It measures the dispersion of all the elements from the mean. It doesn’t matter what the actual elements are and what the arithmetic mean is – the standard deviation of set {1, 3, 5} will be the same as the standard deviation of set {6, 8, 10} since in each set there are 3 elements such that one is at mean, one is 2 below the mean and one is 2 above the mean. So when we calculate the standard deviation, it will give us exactly the same value for both sets. Similarly, standard deviation of set {1, 3, 3, 5, 6} will be the same as standard deviation of {10, 12, 12, 14, 15} and so on. But note that the standard deviation of set {25, 27, 29, 29, 30} will be different because it represents a different arrangement on the number line.
Let’s look at the given question now.
Set S has four odd integers such that their range is 4. So it could look something like this {1, x, y, 5} when the elements are arranged in ascending order. Note that we have taken just one example of what set S could look like. There are innumerable other ways of representing it such as {3, x, y, 7} or {11, x, y, 15} etc.
Now in our example, x and y can take 3 different values: 1, 3 or 5
x and y could be same or different but x would always be smaller than or equal to y.
 If x and y were same, we could select the values of x and y in 3 different ways: both could be 1; both could be 3; both could be 5
 If x and y were different, we could select the values of x and y in 3C2 ways: x could be 1 and y could be 3; x could be 1 and y could be 5; x could be 3 and y could be 5.
For clarification, let’s enumerate the different ways in which we can write set S:
{1, 1, 1, 5}, {1, 3, 3, 5}, {1, 5, 5, 5}, {1, 1, 3, 5}, {1, 1, 5, 5}, {1, 3, 5, 5}
These are the 6 ways in which we can choose the numbers in our example.
Will all of them have unique standard deviations? Do all of them represent different distributions on the number line? Actually, no!
Standard deviations of {1, 1, 1, 5} and {1, 5, 5, 5} are the same. Why?
Standard deviation measures distance from mean. It has nothing to do with the actual value of mean and actual value of numbers. Note that the distribution of numbers on the number line is the same in both cases. The two sets are just mirror images of each other.
For the set {1, 1, 1, 5}, mean is 2. Three of the numbers are distance 1 away from mean and one number is distance 3 away from mean.
For the set {1, 5, 5, 5}, mean is 4. Three of the numbers are distance 1 away from mean and one number is distance 3 away from mean.
The deviations in both cases are the same > 1, 1, 1 and 3. So when we square the deviations, add them up, divide by 4 and then find the square root, the figure we will get will be the same.
Similarly, {1, 1, 3, 5} and {1, 3, 5, 5} will have the same SD. Again, they are mirror images of each other on the number line.
The rest of the two sets: {1, 3, 3, 5} and {1, 1, 5, 5} will have distinct standard deviations since their distributions on the number line are unique.
In all, there are 4 different values that standard deviation can take in such a case.
Answer (B)
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!
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