• ## Strategy for Bold-faced GMAT Questions

Students facing the Critical Reasoning section of the GMAT exam often ask for more practice with “Bold-faced Questions.” These questions comprise only a fraction of the questions you’ll see on the exam (less than 8%) – but somehow students who are stumped on these types of questions tend to get all of them wrong.

So why are they such a tough hill to climb for some students?

Firstly, if you’re getting these questions, it’s usually a good sign as they tend to be more difficult.

Secondly, chances are, the weakness underlies are more fundamental problem in the students’ ability to breakdown the parts to an argument. Bold-faced questions are all about structure and as you read the statement, you should be able to quickly identify the components of that statement. What is the argument? What is supporting that argument?

### Identifying the Argument

Look for keywords like:

[ASSUMPTION/SUPPORT] Thus, … [ARGUMENT]
[ASSUMPTION/SUPPORT] Therefore, …[ARGUMENT]
[ASSUMPTION/SUPPORT] As a result, …[ARGUMENT]
[ASSUMPTION/SUPPORT] infer that…[ARGUMENT]
[ASSUMPTION/SUPPORT] Consequently, …[ARGUMENT]
[ASSUMPTION/SUPPORT] Clearly, …[ARGUMENT]
From the observations, it appears that…[ARGUMENT]
Researchers believe that...[ARGUMENT]
Topic X…should be like this…[ARGUMENT]

The conclusion of the statement (ie the argument) will come directly after these key words.

### Identify the Assumption

You may often come across fancy words like “premise” and “assumption”. Here at GMATPill, we simplify it and treat them the same. They are both “assumptions” in our view.

Refer to the picture above.

Within the table-top framework, we have:

1) Table-top, which is the argument.
2) Supporting leg, which is the stated assumption (sometimes called “premise”)

3) Supporting leg, which is the unstated assumption (“assumption”). Sometimes the “screw” connecting the leg and table-top can also be the unstated assumption – it’s what is necessary to link one topic to another in order to make the argument valid.

Others may reference a “premise”/”assumption” as different things, but we at GMATPill simplify it and refer to both the premise and assumption as “assumption”. They both help explain why the argument is valid.

Key words to look for when identifying the assumption:

[ARGUMENT] because…[ASSUMPTION/SUPPORT]
[ARGUMENT] since…[ASSUMPTION/SUPPORT]
[ARGUMENT] on the basis of…[ASSUMPTION/SUPPORT]
[ARGUMENT] the reason is that…[ASSUMPTION/SUPPORT]

CR questions can get more complicated when the statement starts off in one direction, then changes direction. Whatever new direction it takes, the statement still will come down to a conclusion and supporting statement. Key words that signal a change are:

actually, …
despite, …
except, …
although, …
however, …

### Strategy for solving Boldface CR Questions

Step 1) Scan for key word. Determine whether supporting statement comes after the keyword or before it.

Example: ________ since _______
Argument comes before the keyword “since”.
Assumption comes after the keyword “since”.

Example: _________ therefore ______
Argument comes after the keyword “therefore”.
Assumption comes before the keyword “therefore”.

Step 2) Identify where the boldface statements are, and what role they play relative to this initial structure that is identified

### Examples

Example 1

Gotham City currently supplements its funding for the new tunnel project into the neighboring city of Farmville through a 3% unincorporated business tax, for all businesses operating in Gotham City. In place of this system, the city plans to increase the sales tax rate from 5% to 7% on all retail sales made in the city. Critics protest that the two percent increase in sales tax falls short of the amount raised for public works by the unincorporated business tax. The critics are correct on this point. Nevertheless, implementing the plan will probably not reduce the money going to the tunnel project. Several large retailers are opening stores in Gotham City and these stores will certainly attract a large number of shoppers from neighboring towns, where sales tax rates are ten percent and more. Therefore, retail sales in Gotham City are bound to increase significantly.

In the argument given, the two portions in boldface play which of the following roles?

(A) The first is an objection that has been raised against a certain plan; the second is a prediction that, if accurate, undermines the force of that objection.[correct]
(B) The first is a criticism, endorsed by the argument, of a funding plan; the second is a point the argument makes in favor of adopting an alternative plan.
(C) The first is a criticism, endorsed by the argument, of a funding plan; the second is the main reason cited by the argument for its endorsement of the criticism.
(D) The first is a claim that the argument seeks to refute; the second is the mainpoint used by the argument to show that the claim is false.
(E) The first is a claim that the argument accepts with certain reservations; the second presents that claim in a way that is not subject to those reservations.

For Example 1, let’s follow the twists and turns of the passage by identifying the keywords.

Step 1: Scan for keywords. We see “therefore” at the end of the passage, and this is near the 2nd bolded phrase. We know that the 2nd bolded phrase is some kind of conclusion because anything after “therefore” signals a conclusion or argument that is reached.

Step 2: See which answer choices indicate the 2nd bolded phrase to be a “conclusion”.
(A) uses the word “prediction” – that’s possible so hold onto it. Possible.
(B) references the second point as a supporting point. But “therefore” is a pretty strong word indicating conclusion. It’s unlikely that it is simply making a supporting point – unless “retail sales going up” is supporting some other larger argument. But that’s not clear here. Eliminate.
(C) Similar to B, it is referencing the 2nd bolded phrase as “support” – when we are interpreting it to be a “conclusion/argument” instead. Eliminate.
(D) Similar to B and C, it is referencing the 2nd bolded phrase as “support”. Eliminate.
(E) “A claim that the argument accepts” – what does this mean? This is like an “inference”. The claim is the “inferred statement” – the argument is the initial statement. So in this case, the argument has to support the inferred statement. So it’s kind of like saying the inferred statement is the table-top or the “conclusion” that is reached. And the argument is “below” that – supporting it. That is okay, it’s saying the 2nd bolded phrase is an inferred statement and that it is the “table-top”. That’s okay. We can hold onto it. Possible.

So we are left with (A) and (E).

Step 3: Identify “key word” related to the 1st bolded phrase. “Critics protest that…” – this is basically telling us that the 1st bolded phrase is an objection or statement that goes against the very first thing that was presented in the first sentence.

Step 4: See which answer choice makes sense:
(A) this looks okay – it’s saying that the 1st bolded phrase is an objection against a certain plan. The first sentence was that certain plan and the second sentence is essentially against it. Sounds exactly right.
(E) “a claim that the argument accepts with certain reservations”. There is a slight mismatch here. Yes, there is something here that is accepted with some reservations. The “plan” to increase sales tax is great, there are some objections, but “nevertheless”—- this process indicates some hesitation about the plan working. But acknowledges that it will/can still work.
So what exactly is the claim that is being accepted with some hesitation? That claim is really that the plan to increase sales taxes from 5-7% is going to work. But that is not the bolded portion. The bolded portion is the “objection” part – not the “claim” part. So (E) is wrong here.

As you can see, if you follow the structured approach of finding the keyword, then identifying what is “support” and what is “conclusion” before and after this keyword, you can more easily identify the structure of the whole passage.

Example 2

Studies have shown that, initially, amateur body builders who time their daily meals and calorie-count are far more successful at gaining muscle mass than those who don’t track what they consume. Researchers believe that many amateur body builders fail because they are not eating meals frequently enough and drinking enough water to supplement their strength-training workouts. One study followed a group of patients who reported they could not build muscle mass when eating more than 2,500 calories a day. The study found that the group consumed, on average, 41% more than it claimed and actively strength trained for 26% less, in terms of minutes at the gym. In contrast, when successful body builders record what they eat, their actual consumption more closely matches their reported consumption.

The two boldface portions in the argument above are best described by which of the following statements?

(A) The first is a conclusion reached by researchers; the second is evidence that that conclusion is correct.
(B) The first is an explanation of why a certain theory is thought to be true; the second is an example of research results that support this theory.
(C) The first is an example illustrating the truth of a certain theory; the second is a competing theory.
(D) The first is a premise upon which the researchers base their opinion; the second illustrates that their opinion is correct.
(E) The first introduces a theory that the researchers have disproved; the second is the basis for the researchers’ argument.

Step 1: Scan for keywords. In this case, there aren’t any real keywords you can refer to. “In contrast” is used in the last sentence, but it doesn’t seem helpful in identifying whether the phrase before/after is “support” or “conclusion”

Step 2: “Studies have shown that…” – this seems to be some kind of factual data. As we know with facts, they are often used to support some kind of argument. Keep reading to find out whether some actual argument is made.

Step 3: “Researchers believe” – aha! Keyword has been found to indicate an argument that is expressed. What do researchers believe? They believe that many amateur body builders fail because they are not eating enough protein calories and drinking enough water to supplement their strength training workouts.
How is this related to the 1st bolded phrase? Well they are connected. The “argument” says amateurs fail because of not eating frequently enough, the facts say that amateurs who succeed tend to be disciplined about their eating. So the “facts” tend to somewhat support the argument.

One way to check is to ask the challenge question. How do we know that amateurs are failing because they do not eat frequently enough? Let’s try to answer that – because a study showed that amateurs who do succeed tend to be disciplined about their eating. Using the “A vs Not A Framework” – this is implying that those who fail might NOT be disciplined about their eating. THis is exactly the argument that is made by researchers.

So this tells us the 1st bolded phrase is some kind of support.
(A) NO, the 1st bolded phrase is not a conclusion, it is support.
(B) While this is “support”, it’s not an explanation for why a theory is true. It is the results of a factual study.
(C) While this is “support”, it’s not an explanation illustrating the truth of a theory. The theory is about those who fail – while the study is about those who succeed.
(D) YES, this is “support”. It uses the word “premise” which is a support, explicitly stated. The 1st bolded phrase supports the researcher’s belief, and it does so through some fact-based study – and that is explicitly mentioned. The “support” is based on those who succeed, researchers then hypothesize on those who fail. This one is okay.
(E) NO, the first is not a “theory” which is a similar word to “argument or conclusion”. We are looking for “support” – not “argument/conclusion”. So “introducing a theory” is not what the 1st bolded phrase does.

Step 4:
We’ve eliminated all answer choices except for (D) by just looking at the first bolded phrase. Let’s confirm that it’s correct by looking at the 2nd bolded phrase.

“the second illustrates that their opinion is correct”

Let’s go back and find our argument:
“Researchers believe that….”

From here, the sentence that follows is most likely supporting that argument, unless a contradictory key word is used.

“One study followed…” — identify this as “support”

“The study found that…” — these are the results, also “support”. The study is about those who failed. The researcher’s argument or belief is also about those who failed. We can see that these supporting sentences are illustrating that the researchers’ opinion is correct.

Thus, the 2nd bolded phrase seems to be accurate in (D) as well.

Now go ahead and apply the same strategy to these two questions and let me know what you get!

Example 3

Learning to play a musical instrument can be an integral part to enjoying life, developing self-expression, and releasing stress. The mind needs beats and music to be stimulated and motivated to live life’s ups and downs. However, much practice is required to master the art of playing a musical instrument and over-practicing can lead to fatigue and despair. A practice schedule should always maintain a progression that slowly builds up and fits in time for rest and recovery.

In the argument given, the two portions in boldface play which of the following roles?

(A) The first is an opinion; the second is a conclusion based on that opinion.

(B) The first is a factual possibility; the second is an opinion that opposes that possibility.

(C) The first is a general opinion; the second is a conclusion that supports that opinion.

(D) The first is a factual possibility; the second is a conclusion that presents a method of preventing the occurrence of that possibility.

(E) The first is a possible event of cause and effect; the second denies the possibility of such an event to occur.

Example 4

Hiring managers: Large corporations tend to set up high cubicles for employee work spaces – allowing each person to have his/her privacy. Unfortunately, such a set up discourages employees from brainstorming and interacting with each other, which is essential especially for companies that hope to innovate. Nevertheless, it is possible to create the appropriate corporate culture under which cubicle walls are kept high during quiet project periods and are kept low during periods of open brainstorming, thus enabling employees and company to benefit on both fronts.

In the argument given, the two portions in boldface play which of the following roles?

A. The first describes a behavior which the organizational consultant deems problematic in certain circumstances; the second explains why this behavior can be problematic.

B. The first is a strategy advocated in order to alleviate an undesirable condition; the second describes that condition.

C. The first describes a certain condition which the organizational consultant deems undesirable; the second is a strategy advocated in order to alleviate that condition.

D. The first describes a certain condition which the organizational consultant deems undesirable; the second is a suggestion on how to preserve that situation.

E. The first is an opinion expressed by the organizational consultant; the second is evidence which supports this opinion.

Are you getting (D) and then (A)?

Remember, on bold-faced GMAT questions, your job is not to make sense of the argument and figure out how to support or weaken it. Nor do you have to choose among the answer choices to find something that helps you better evaluate the argument or to better explain it.

No, all those types of CR questions require a lot of brain thinking.

You don’t have to do that for bold-faced questions. You ONLY have to pay attention to keywords in the passage. Identify where the “support” and “argument/conclusion” are by focusing in on KEY words. Then take into consideration other key words that change the direction of the argument.

Do NOT spend your time trying to make sense of the argument and determine whether it’s valid or not. It does not matter whether the argument made is sound or not sound. Your job is not to determine it’s strength, but merely to determine the structure of the argument.

Don’t waste your time. Do NOT try to make sense of bold-faced CR questions.

### More GMAT Practice

For more practice, try the CR questions on the GMATPill Practice Pill Platform.

If you have the OG13, try #18, #28, #34, #63, #76, #78, #84, #85, #89, #98, #104, #116, #123.

Verbal Questions: Sentence Correction | Critical Reasoning | Reading Comprehension
Quant Videos: Problem Solving | Data Sufficiency

• ## Prime Numbers: Know the pitfalls for the GMAT

Prime numbers are the funny numbers – and the GMAT always throws some question related to prime numbers on their exam.

When you hear “prime number” – you should be thinking firstly about these numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23

Then, you should be thinking about the properties of these numbers to help you figure out what makes a prime number a prime number.
Simple prime number rule:

Prime Number = Divisible ONLY by 1 and itself

### Are all prime numbers odd?

No, not all prime numbers are odd. They almost all are since even numbers generally cannot be prime (“2″ is the exception). Starting with 4, the even numbers are 4, 6, 8, 10, 12, etc… All of these numbers are divisible by 1 and itself…BUT they are also divisible by 2 since by definition an even number is divisible by 2.

So if the factors for even numbers are:
1, 2, itself….we already know that number violates the prime number rule of ONLY “1 + itself” as divisors.

Example:
4: Factors are 1, 2, itself
8: Factors are 1, 2, itself, and others
24: Factors are 1, 2, itself, and others

### Is “2″ a prime number?

2 is the only exception to even numbers NOT being a prime number.
Because “2″ in fact IS a prime number. It is divisible by 1 and itself.

The problem with other even numbers greater than 2 is that the factors are 1, 2, and itself. They cannot be prime.
But 2 is prime because its factors are 1 and itself. Here, the value of “itself” happens to be the same as 2. But for the rule of being divisible by 1 and itself, this rule is satisfied and thus 2 IS a prime number.

### Is “1″ a prime number?

1 is NOT a prime a number. If you ask yourself whether it passes the rule, is 1 divisible by 1 and itself?
If you answer the question individually, the answer to each of those questions is YES.

However, the question should not be answered individually, but rather collectively. This means that the answer to “itself” only counts if it is a value different from “1″.

So is 1 divisible by “1″ and “itself” — the latter being a value different from the former value?

1 only satisfies the first statement, not the second one there.

1 is NOT a prime number.

### Tricky Prime Numbers – from 50 -100

Knowing the definition to prime numbers is pretty basic, but the GMAT still manages to trip students up in different ways. For large numbers, it’s difficult to tell whether a number is prime as some numbers might look prime, but actually aren’t.

Example: 51

51 actually is NOT prime. It may look like nothing divides into it, but actually it’s factors are 1, 3, 17, 51.

Example: 91

91 actually is NOT prime. You may think that the multiples of 3 go to 90 and then 93. It skips 91 so 91 must be prime. But hey, if you build out the factor tree, notice that it’s factors are 1, 7, 13, 91.

So when it comes to large prime numbers, just be very careful!

### Prime Number Examples

How many prime numbers are there between 50-70?

Well, even numbers are not prime numbers. So let’s list out the odd:
51, 53, 55, 57, 59, 61, 63, 65, 67, 69

Now let’s remove the ones that are multiples of 5: (55, 65)
51, 53, 57, 59, 61, 63, 67, 69

And then remove the ones that are multiples of 3: (51, 57, 63, 69)

53, 59, 61, 67

Test out multiples of 7? 70, 63, 56. Looks like these numbers are not in our list.

So after a few rounds, it looks like there are 4 prime numbers remaining. There are 4 prime numbers in that original list of numbers between 50 and 70.

Prime Numbers Example 1
B is a prime number. If 6b is between 15 and 95, which of the following can be a value of 7b + 2?

A) 13
B) 77
C) 125
D) 93
E) 19

So the range of possible b that can satisfy 15 < 6b < 95 is:

2.5 < b < ~16

But we have another constraint. "b" is prime. So within this range, what are possible prime b values?

b = 3, 5, 7, 11, 13

Now, try a value for 7b + 2 using these prime numbers:

3 => 7*3 + 2 = 23
5 => 7*5 + 2 = 37
7 => 7*7 + 2 = 51
11 => 7*11 + 2 = 79
13 => 7*13 + 2 = 93

93 is the only one that satisfies the constraint for the range of 6b (6*13=78; this value is between 15 and 95). It also takes on a value for b that is prime (13). And it is the only possible value for 7b+2 in the list.

Prime Numbers Example 2
If n is equal to 11!, how many different prime factors greater
than 1 does n have?

(A) Four
(B) Five
(C) Six
(D) Seven
(E) Eight

This question is asking you to find the prime factors in a large number. That large number is broken down into factors that are multiplied with each other:

11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

“How many different prime factors greater than 1 does n have?”

Ok, let’s break this question down.

First, “how many different prime factors greater than 1…”

Then continue reading, “…does n have?”

11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

What are the factors? Which of them are both greater than 1 and are prime?

Well, these would be 2, 3, 5, 7, 11

Total, there are 5 prime factors. The answer use be (B).

Data Sufficiency Example

Prime Numbers Example 3
If the sum of the digits of b are not divisible by 7, What is the value of the integer b?

(1) b is a prime number.

(2) 51 ≤ b ≤ 65

If we only look at (1), b can have many different values. Valid values for “b” are 2, 3, 5, 7, 11. Using this, if we answer the question “what is the value of b?” – well, we have multiple answers. When our answer is NOT DEFINITIVE, we have insufficient information to give a definitive answer.

So (1) is not sufficient.

(2) Same idea here. b has multiple values here and we get multiple answers to our question. Not sufficient.

(1+2) Overlap of prime numbers between 51 and 65. OK, so here the prime numbers are 53, 59, 61.

Now we have multiple values here, but we still didn’t account for the extra constraint inserted at the beginning of the question stem. The sum of the digits of b must be divisible by 7.

So let’s check each of the values for b and see which ones pass the test.

53 => 5+3 = 8; not divisible by 7
59 => 5+9 = 14; YES divisible by 7
61 => 6+1 = 7; YES divisible by 7

So the only valid value for b is then b=53. So we see that together 1+2 provide sufficient information to answer the question. (C) is the answer.

Prime Numbers Example 4
Does the integer n have at least three different positive prime factors?

(1) n/6 is an integer.
(2) n/14 is an integer.

Step 1: Think of positive prime factors in your head. {2, 3, 5, …} We are looking for these primes to be factors in some number n.

Step 2: Rewrite constraints (1) and (2)

(1) n = 6 * k
= 2 * 3 * k
Factors are 2, 3, and something else.

(2) n = 14 * k
= 2 * 7 * k
Factors are 2, 7, and something else.

The problem in 1 is that the 3 factors are “2, 3 and something else”. “Something else” could just be 1 – which is not a prime factor. If k=1, then the value of n is just n = 6, in which case there are only two prime factors: 2 and 3. We don’t necessarily know that there is a 3rd factor that is prime.

We may have n = 2 * 3 * 1 = 6 2 different prime factors
n = 2 * 3 * 2 = 12 2 different prime factors
n = 2 * 3 * 3 = 18 2 different prime factors
n = 2 * 3 * 5 = 30 3 different prime factors
n = 2 * 3 * 6 = 36 2 different prime factors

So we can see that we don’t necessarily have 3 different prime factors when looking at constraint (1).

Constraint (2) has the same problem.

We know that 2 and 7 are definitely factors for n, but that the third factor could be some number that is not prime (ie 1 or 4).

We may have n = 2 * 7 * 1 = 14 2 different prime factors
n = 2 * 7 * 2 = 28 2 different prime factors
n = 2 * 7 * 3 = 42 3 different prime factors
n = 2 * 7 * 4 = 56 2 different prime factors
n = 2 * 7 * 5 = 70 3 different prime factors

We don’t necessarily have 3 different prime factors when considering constraint (2).

Now consider combining constraints (1) + (2)

(1+2) n = 6 * k1 AND n = 14 * k1
Factors must include all the factors of the two above. If divisible by both 6 and 14, the number should be divisible by 2, 3, AND 7. Potentially others as well but at least these numbers are factors. The question is asking whether there are at least 3 different positive prime factors. Yes, here there are at least 3.

n = 2 * 7 * 3 * k

Regardless of the value of k, n has at least 3 different prime numbers as factors so (1+2) is sufficient.

Verbal Questions: Sentence Correction | Critical Reasoning | Reading Comprehension
Quant Videos: Problem Solving | Data Sufficiency

• ## Sum of Sequence: Consecutive Integers and Multiples

Sum of sequences are inevitable on the GMAT Exam.

Between Arithmetic and Geometric sequences, you only need to know sum of sequences for Arithmetic sequences. These are the sequences where the numbers in the sequence are separated by a fixed value. That means you either add or subtract a fixed value from one number to get to the next. Geometric sequences on the other hand “multiply/divide” by a fixed value to go from one number to the next.

Here is a quick example:

Arithmetic Sequence
{1, 2, 3, 4, 5, 6, 7} (add 1)
{16, 13, 10, 7, 4, 1} (subtract 3)

Geometric Sequence
{1, 2, 4, 8, 16, 32} (multiply 2)
{100, 50, 25, 12.5, 6.25} (divide 2)

In the Problem Solving section, you’ll need to make some calculations for summing a sequence, but it will only be for the arithmetic sequence. You won’t be asked to sum a geometric sequence (unless it goes between positive and negative and approaches a particular value; more on this later). Summing a sequence might be easier for a few values, but as the number of terms in the sequence goes up, it’s going to be more difficult to sum a sequence. So there are a few key tips to follow.

### General Strategy Sum of Arithmetic Sequence

[ sum of sequence ] = [ # of terms / 2 ] x [ sum of first and last ]

Let’s first try an example, then explain why it works.

{1, 2, 3, 4, 5 }
Sum of Sequence = (5/2) * (1 + 5) = 15

{11, 12, 13, 14, 15}
Sum of Sequence = (5/2) * (11 + 15) = 65

Here’s Why It Works

If you take the sequence {1, 2, 3, 4, 5, 6, 7}, summing the first and last actually is done multiple times and the sum is the same each time.

1+7 = 8
2+6 = 8
3+5 = 8
4 = 4

So what’s going on here? The sum of the first and last happens 3 times. Then the last one is the middle item and that value is half sum. In other words, we adding 8 to itself 3.5 times. How do we get 3.5 times? # of terms in this sequence is 7. We simply divide it by 2 and say “7/2″. Don’t worry that the two is there…it will cancel out with the sum of the first and last.

7/2 * 8 = 28

Got it?

Whenever you sum an arithmetic sequence, it’s all about finding the # of n terms and then finding the sum of the first+last. From here, it’s simply (n/2) * (first+last).

Step 1: Find n
Step 2: Find first + last
Step 3: Solve n/2 * (first + last).

Example 1: Sum of the sequence {51, 52, 53, 54, …, 99, 100}

Step 1: n = 50. How do we know this? Turns out, it’s (last – first)/1 + 1. I’ll explain in a little bit.
Step 2: (100+51) = 151
Step 3: 50/2 * 151 = 3,775

Example 2: Sum of the sequence {51, 57, 63, 69, 75, 81, …, 105}
Step 1: n=? (105 – 51)/6 + 1 = 9 + 1 = 10
Step 2: (105+51) = 156
Step 3: 10/2 * 156 = 780

As you can see in these advanced examples, the numbers are bigger and the arithmetic sequence gets more complex. In the first example, we have consecutive numbers starting from 51 (rather than a simple small number). In the second example, we have an example where the number starts at 51 AND the distance between each term in the sequence is not just 1 but it is 6. So we go from 51 to 57 to 63. Immediately, we can see that if this sequence continues on, it may be difficult to quickly find out how many #’s are in this sequence (the all important value for “n”).

Now let’s try some more. Calculate the sum of this sequence.

Sum of the first 50 consecutive integers?

{1, 2, 3, …, 48, 49, 50}

Step 1: n=50
Step 2: (50+1) = 51
Step 3: 50/2 * 51 = 1,275

Sum of first 30 even positive integers

{2, 4, 6, …, 56, 58, 60}

Step 1: n= (60-2)/2 + 1 = 30
Step 2: (60+2) = 62
Step 3: 30/2 * 62 = 15 * 62 = 930

Sum of numbers between 25 and 75 in multiples of 5?

Step 1: n = (75-25)/5 + 1 = 11
Step 2: (75+25) = 100
Step 3: 11/2 * 100 = 550

### Calculating the Number of Terms in a Sequence

The formula is simply: # of terms in sequence = [(Last - First) / Distance between each term ] + 1

Why? Well, let’s take a small example: {1, 2, 3, 4, 5}

How many terms are in this sequence? Well, easy to say that it’s 5. But if you look at difference between last and first, we have 5-1 = 4. But it’s really 5 terms we have. 4 represents the number of spaces between the terms in the sequence, but we are interested in the “number” of terms in the sequence. So, adding 1 more makes sense and we get 5.

{2, 4, 6, 8, 10, 12} The number of terms using the same approach: (12-2) = 10. But each term jumps by 2. So the number of jumps is actually 5. But 5 only represents the number of spaces between each term. We add 1 more to capture the “number” of terms and we get 6. So overall, it’s (12-2)/2 + 1 = 5 + 1 = 6 terms in the sequence

So if you simply follow this example, you can always find “n”, the number of terms in an arithmetic sequence, when you know the last term, the first term, and the distance between each term.

### Sum of a Geometric Sequence?

Realistically, you will not see a “sum of geometric sequence” question on the GMAT Exam. At least you won’t have to calculate one. So something like this:

{1, 2, 4, 8, 16, 32}

You will NOT be asked to sum this geometric sequence. Yes, there is a formula for summing a geometric sequence but it is a bit complicated to remember. The GMAT Exam tests you on reasoning, not memorizing formulas. So you will not need to do extra work here and memorize the formula for a geometric sequence.

That said, there may be some geometric sequences where the “sum of geometric sequence” may make sense on the GMAT Exam. For example:

Sequence: {4, -2, 1, -1/2, 1/4, -1/8, 1/16, -1/32}

If you look at the cumulative sum as you go through this sequence:

{4, 2, 3, 2.5, 2.75, 2.625, 2.6875, …}

you can see that the sum of the geometric sequence, because it jumps back and forth between positive and negative and the numbers get smaller and smaller, approaches a target value. In each case, you know the next number will be between the last 2.

So the 3rd term will be between 4 and 2.
The 4th term will be between 3 and 2.
The 5th term is between 3 and 2.5
The 6th term is between 2.75 and 2.5.

In the end, the number gets close to ~2.65.

On the GMAT, you won’t be asked to calculate this “approaching” value, but you may be given some answer choices pinpointing the sum of the sequence to be between 2.5 and 3. You may be asked to calculate this range, but not actually solve the question.

This sum of geometric sequence may show up on the Problem Solving section of the exam as such: What is a possible range for the sum of the above geometric sequence?

They will only ask this type of geometric sequence question if they provide a sequence that hops back and forth between an increasingly smaller positive and then increasingly smaller negative number.

So, I simply wanted to point this out as a possibility on the exam. For the most part, your focus should be on the sum of an arithmetic sequence.

Try One Advanced examples of an Arithmetic Sequence Sum on your own.
{1, 6, 11, 16, …, 101, 106, 111}
1) How many terms are in the sequence?
2) What is the sum of the first and last?
3) What is the sum of the sequence?

Verbal Questions: Sentence Correction | Critical Reasoning | Reading Comprehension
Quant Videos: Problem Solving | Data Sufficiency

• ## Tackling Absolute Inequalities

Absolute Inequalities will appear on the GMAT Exam – specifically on the Data Sufficiency section of the quant section.

The other quant section (Problem Solving) does not really have many questions targeted specifically at absolute inequalities. It is primarily a Data Sufficiency-only topic and often involves trial and error, plugging in numbers, and then testing out different outcomes. This is a very common question type on Data Sufficiency and the GMAT folks love it. It’s a chance for you to critically think through different scenarios and to assess whether a statement is still true after considering various subsegments of or constraints to the question.

Thinking through various scenarios happens all the time in business. Many business decisions have no clear answer and so each strategic business step requires a double check for all of the assumptions in various dimensions. If a business leader fails to consider some important scenarios, it can lead to faulty business decisions, millions of dollars poorly spent, and a failing business.

So stay awake and pay attention!

### Two Strategies to Understand

Strategy #1
For absolute inequality, when absolute variable is on one side PLUS constant on the other, USE the 2-equation approach:
1) Remove absolutes then use equation
2) Remove absolutes, reverse sign, negate constant. Use this equation

Strategy #2: PLUG-IN TABLE
Example: Constraint of x>0. Question: is x even?

 Random X Valid X Expression Value -1 NA NA 0 0 NA 1 1 FALSE 2 2 TRUE

Then only look at the last column (FALSE/TRUE). If all answers are consistent, then we have SUFFICIENT information. If we have a mix of FALSE and TRUE, then we do NOT have sufficient information.

### Absolute Inequality Manipulation

The math for absolute inequalities can be puzzling. Let’s break it down and first look at just inequalities.

1) Variable < Constant
x < 2
x + y < 2

No issue here. We have variables on one side and constant on the other. We can use simple math and move variables to the other side to get:

x < 2 - y

2) Variable < Variable

x + y < 3x

Still no issue here. We can combine the x's and get:

y < 2x

3) | Variable | < Constant

| x | < 2

Now we introduce the inequality. The value inside the inequality ("x" in this case) can be either the positive form or negative form and still satisfy the expression. So basically, there are 2 basic equations that result from this absolute inequality.

x < 2 OR x >-2

Use Strategy #1: SAME + REVERSE NEGATE
For absolute inequality, when absolute variable is on one side PLUS constant on the other, USE the 2-equation approach:
1) Remove absolutes then use equation
2) Remove absolutes, reverse sign, negate constant. Use this equation

4) | Variable + Variable | < Constant

Here we have two variables inside the absolute expression. This is the same as #3 except that we treat (x+y) as 1 single entity.

| x + y | < 2

Use Strategy #1: SAME + REVERSE NEGATE
For absolute inequality, when absolute variable is on one side PLUS constant on the other, USE the 2-equation approach:
1) Remove absolutes then use equation
2) Remove absolutes, reverse sign, negate constant. Use this equation

(x+y) < 2
(x+y) > -2

5) | Variable | < Variable

|x+4| < 3x

This is where we come into a problem. We cannot apply Strategy #1 in the cases where the same variable is on both sides of the absolute inequality.

If we applied Strategy #1 (which is wrong), this is what we get:
(x+4) < 3x
(x+4) > -3x

– 4 < 2x ---> x>2
– 4x > -4 —-> x > -1

Think about what this is saying. It’s saying that x can be greater than negative 1 OR it can be greater than 2. Well, to capture all bases, this simplifies to x > -1. But we see that this is incorrect.
Let’s try x=0.
|(0+4)| < 3(0)
4 < 0
INCORRECT

So we CANNOT use Strategy #1 (SAME + REVERSE NEGATE). When it comes to absolute inequalities that have the same variable on both sides, we CANNOT use SAME + REVERSE NEGATE.

Use Strategy #2: Plug in Values (use table to help)

|x+4| < 3x

 Random X Valid X Expression Value -3 -3 1 < 3(-3) FALSE -1/2 -1/2 3.5 < -1.5 FALSE 0 0 4 < 0 FALSE 1/2 1/2 4.5 < 1.5 FALSE 3 3 7 < 9 TRUE 4 4 8 < 12 TRUE

As we can see from the table, if this question had no constraints, we would try the infinite range of x and see if we get a consistent answer. It’s usually a good idea to check
1) Negative Value
2) Negative Fraction
3) 0
4) Positive Fraction
5) Positive Value

As you can see here, if you try the full range of valid x, you get answers to the question that conflict. Sometimes, the answer is FALSE. Sometimes the answer is TRUE.

Since this is a DATA SUFFICIENCY question, our job is not to answer the question. It’s to determine whether or not we have SUFFICIENT information to provide a definitive answer to the question. And that question is “Is this expression true? |x+4| < 3x ?"

Well, sometimes that expression is TRUE.
Sometimes that expression is FALSE.

Since our answer is not consistent, our answer to this Data Sufficiency question is that we do NOT have sufficient information.

So as you can see here, whenever we have the same variable on BOTH sides of an absolute inequality, we cannot do a simple Strategy #1 of “SAME + REVERSE NEGATE”. We have to use Strategy #2, use a plug-in table of a wide range of valid x values.

6) | Variable | < | Variable |
| x + y | < |2x|

This has the same problem as the example before this one. With absolutes on BOTH sides and the x variable on BOTH sides, we definitely cannot use the "SAME + REVERSE NEGATE" strategy. You have to use Strategy #2, plug in values into a table.

Let's apply a constraint as an example here:
Let's say constraint #1 and #2 are
1) x > 0
2) y = 0

If we check both constraints (which is answer choice C), we plug in values into the table that satisfy these constraints.

Use Strategy #2: Plug in Values (use table to help)

 Random X Valid X Expression Value -3 NA NA -1/2 NA NA 0 NA NA 1/2 1/2 1/2 < 1 TRUE 3 3 3 < 6 TRUE 4 4 4 < 8 TRUE

Based on the two constraints, the only valid values for x were the ones where x>0. And when we plugged in those values, we found that our answer to the expression | x + y | < |2x| was ALWAYS TRUE. Thus, we have sufficient information to get a definitive answer.

Tricky Trap

When it comes to inequalities and negative signs, you have to be very careful of the normal arithmetic operations you might do.
If you’re doing simple addition/subtraction, there’s usually NO problem.

x + y > -1
x > -1 – y

But once you start doing multiplication/division, then the usual operations will be wrong.

Example #1
x / y > 2
CLAIM: x > 2y

Let’s check a possible value. (-3, -1) gets us (-3) / (-1) > 2. TRUE.
But is the claim true? x>2y using (-3, -1) gets us -3 > -2. FALSE!

So you can see, even without a negative sign, doing a cross multiplication here would be WRONG.

Example #2
x / y > -1
CLAIM: x > -y

Let’s check a possible value. (3, 1) gets us (3) / (1) > -1. TRUE.
But is the claim true? x>-y using (3, 1) gets us 3 > -1. TRUE!
But is the claim true? x>-y using (1, -4) gets us 1 > 4. FALSE!
But is the claim true? x>-y using (-1, 4) gets us -1 > -4. TRUE!

Once you swap the variables for x and y, you can see how the statement switches between TRUE and FALSE.

So when it comes to multiplication OR division of inequalities, whether the opposite side is positive or negative doesn’t matter. Doing cross multiplication is simply going to be WRONG for inequalities.

### Try Yourself

Now, how about we try solving this data sufficiency question

Is x > y?

(1) |x+y| < |x| + |y|

(2) |y| > |x| – 1

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked;
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

HINT:
OK, I’ll give you a hint to get started. Do not use Strategy #1 (SAME + REVERSE NEGATE). We have variables on both sides so we want to use Strategy #2 (Plug in table).
(5,-3) satisfies (1). Is 5>3? YES
(3,-5) satisfies (1). Is 3>-5? YES.
(-3,5) satisfies (1). Is -3>5? NO.
(-5,3) satisfies (1). Is -5>3? NO.

So immediately we can eliminate answer choice (A). Constraint from #1 does not get you a consistent answer in the plug-in table.

Go ahead and check (2) and see what answer you get. Remember, when there are multiple variables, it’s usually easiest to organize the scenarios by following a table approach.

Verbal Questions: Sentence Correction | Critical Reasoning | Reading Comprehension
Quant Videos: Problem Solving | Data Sufficiency

• ## GMAC Reports IR Stats – .55 Correlation

The Graduate Management Admission Council (GMAC) announced stats on student performance on the new Integrated Reasoning section of the GMAT exam. Data was available for the 105,000 exams that have been taken since the IR section launched on June 5, 2012 (6 months of data).

### So far, the data has pointed out some surprising results.

1) Factors such as native vs non-native English speakers, US vs non-US, white vs non-white, etc – generally don’t affect one’s score too much. In terms of covariance analysis, the variation between these subgroups was always less than 1/4 standard deviation.

So effectively, these factors don’t play much of a role within these demographics.

2) Students who do well on the 800 score (quant and verbal) – do not necessarily do well on IR. The correlation between these scores is only .55.

3) 85% of students and alumni find IR skills relevant. And 97% of employers said these skills were important.

See GMAC”s announcement here.

### What does this mean?

It means the IR score tests something completely different from what quant and verbal test.

This low correlation also means that IR scores will likely carry greater significance going forward. Since the type of skills IR tests is much more relevant to the business world—AND the IR score is not really correlated with a student’s Quant and Verbal score—it is now legitimate for GMAC folks to say that IR “adds value” for business school admissions committees in terms of evaluating students’ potential.

### So is this good news for GMAC?

Kind of. As mentioned above, it shows that the IR score does “add value.”

On the flip side, it’s still a bit difficult for business schools to treat the score as seriously as they currently do with the 800 score.

Why?

Because the IR score is not published and business schools generally don’t (yet) compare their student body’s average IR score with another school’s average IR score. Without the competitive pressure to accept students with higher IR scores, it may be difficult for the IR score to carry as much weight as the 800 score.

Nevertheless, the results do show that the IR score is an additional data point. Given that the IR score does test more relevant skills, some may argue that one’s IR score is a better predictive indicator of how well a student will do in business school and post graduation.

Regardless of how business schools view the IR score – and how their thinking will change over time – you as an MBA candidate should take the IR score seriously–not just for admissions purposes, but also for your future career development.

GMAT Pill has released an “Integrated Reasoning Pill” for this new IR section- complete with 100+ questions and video explanations.

### IR = Integrated Reasoning, more details

Recall that the IR section is the second section on the GMAT Exam. It comes right after the AWA Analysis of an Argument Essay (30 min). Your next 30 min will be spent on the Integrated Reasoning section which has 12 questions, scored out of 8. Some of these questions (2-3) may be experimental – there is no way to tell which ones they are. Then the remaining 9-10 questions *actually* count and are used to calculate your IR raw score out of 8. For more details about the IR scoring, read the GMATPill post on IR Scoring.

Since there are 4 new question types on the IR section (some have 2 answers per question, others have 3), it’s usually best to familiarize yourself with a few of the questions. Try some of the questions in the IR e-book or IR practice question on the GMATPill Practice Pill Platform.