The post Planning for the “Plan” Questions on the GMAT Critical Reasoning Section appeared first on Veritas Prep Blog.

]]>Note that a plan question is very similar to a strengthen/weaken/assumption question. The main difference between them is that instead of being given a conclusion, you are asked to strengthen/weaken the possibility of a plan working out or an assumption made in the plan (looking at a few example questions will make this clearer). Let’s look at some examples of each of the three types of “plan” questions you are likely to come across on the GMAT exam:

**Example 1 (t****he most common one): Which of the following will help us in evaluating the success of the plan?**

*In the country of Bedenia, officials have recently implemented a new healthcare initiative to reduce dangerous wait times at emergency rooms in the country’s hospitals. This initiative increases the number of available emergency nurses and doctors in urban settings: scholarships and no-interest loans are being offered to prospective students in these fields if they work in major city hospitals, relocation packages to urban centers are being offered for current emergency practitioners, and immigration rules are being changed to enable foreign emergency doctors and nurses to more easily move to Bedenia’s major cities.*

*Which of the following would be most important to determine in assessing whether the initiative will be successful?*

*(A) What percentage of current nurses and doctors work in emergency medicine.*

*(B) Which hospitals in Bedenia have dangerous wait times in their emergency rooms.*

*(C) Whether a career in emergency medicine pays substantially less than other types of medicine.*

*(D) Whether wait times could be reduced by means other than increasing the number of available nurses and doctors.*

*(E) Whether many foreign doctors and nurses are currently not allowed to enter Bedenia.*

Plan: Reduce the dangerous wait time by increasing the availability of emergency nurses and doctors in urban settings by providing scholarships, offering relocation packages and changing immigration rules.

We need to find out whether this given plan will actually reduce wait time. Note that we are not worried about what else could reduce the dangerous wait time or what else this plan could do. The only point of concern for us is whether this plan will reduce the wait time.

This plan intends to increase the availability of emergency nurses and doctors in urban settings, so ask yourself this question: is this actually what is required? Do the urban hospitals have dangerous wait times? What if only rural hospitals have wait times and that is where the impetus is required? Answer choice B addresses exactly this question and, hence, will allow us to determine whether or not the initiative will be successful. Therefore, the answer is B.

Now look at our second example:

**Example 2: Which of the following provides an argument against the plan?**

*In the last two years alone, nearly a dozen of Central University’s most prominent professors have been lured away by the higher salaries offered by competing academic institutions. In order to protect the school’s ranking, Central University’s president has proposed increasing tuition by 10% and using the extra money to offer more attractive compensation packages to the most talented and well-known members of its faculty.*

*Which of the following provides the most persuasive argument against the university president’s proposed course of action?*

*(A) It is inevitable that at least some members of the faculty will ultimately take jobs at other universities, regardless of how much Central University offers to pay them.*

*(B) Other universities are also looking for ways to provide higher salaries to prominent members of the faculty.*

*(C) Central University slipped in the last year’s ranking of regional schools.*

*(D) The single most important factor in ranking a university is its racial and socioeconomic diversity.*

*(E) The president of Central University has only been in office for 18 months and has never managed such a large enterprise.*

Plan: Protect the school’s ranking by retaining its most prominent members by increasing their compensation.

We need to find a persuasive argument against the given plan – something that leads us to believe the plan should not be implemented. Here, test takers often become confused between options B and D. Let’s break down each answer choice in detail to determine which one is correct:

*(B) Other universities are also looking for ways to provide higher salaries to prominent members of the faculty.*

This option supports the given plan. It is a reason to actually implement the plan since if more disparity gets created, more prominent professors will leave. Remember, we are looking for an option that is against the plan, so B cannot be our answer.

*(D) The single most important factor in ranking a university is its racial and socioeconomic diversity.*

This is an argument against the plan. It states that the single most important factor in ranking is “racial and socioeconomic diversity,” so trying to retain prominent professors is not likely to retain ranking. Hence, the correct answer would be D.

Now let’s look at our final example:

**Example 3: Which of the following is an assumption of the plan?**

*The general availability of high-quality electronic scanners and color printers for computers has made the counterfeiting of checks much easier. In order to deter such counterfeiting, several banks plan to issue to their corporate customers checks that contain dots too small to be accurately duplicated by any electronic scanner currently available; when such checks are scanned and printed, the dots seem to blend together in such a way that the word “VOID” appears on the check.*

*A questionable assumption of the plan is that*

*(A) in the territory served by the banks the proportion of counterfeit checks that are made using electronic scanners has remained approximately constant over the past few years.*

*(B) most counterfeiters who use electronic scanners counterfeit checks only for relatively large amounts of money. *

*(C) the smallest dots on the proposed checks cannot be distinguished visually except under strong magnification.*

*(D) most corporations served by these banks will not have to pay more for the new checks than for traditional checks.*

*(E) the size of the smallest dots that generally available electronic scanners are able to reproduce accurately will not decrease significantly in the near future.*

Plan: To deter counterfeiting, issue checks that contain dots too small to be accurately duplicated (which will form the word VOID) by any electronic scanner currently available.

We need to find an assumption that this given plan makes. Note that the plan is based on the capabilities of the currently available scanners and assumes that their capabilities will not improve in the near future. Hence, E is an assumption.

Some test takers get confused with answer choice C:

*(C) the smallest dots on the proposed checks cannot be distinguished visually except under strong magnification*

This option is actually not an assumption. Even if the dots can be distinguished visually, they don’t form the word VOID. Only when current scanners scan the checks and then we print them do the dots merge to form the word. Thus, our answer is E.

We hope you have understood how to handle various “plan” questions on the GMAT. The most important aspect of such questions to remember is to first identify the plan and what one hopes to achieve through it.

*Getting ready to take the GMAT? Check out one of our many free GMAT resources to get a jump start on your GMAT prep. And as always, be sure to follow us on Facebook, YouTube, Google+, and Twitter for more helpful tips like this one!*

*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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]]>The post All About “That” on the GMAT appeared first on Veritas Prep Blog.

]]>- Demonstrative Determiner
- Demonstrative Pronoun
- Relative Pronoun

**Demonstrative Determiner** – In this role, “that” specifies the specific person/thing about which we are talking. It is followed by a noun.

Can I have some of that cake, please?

I have never been to that part of Italy.

When we are talking about a plural noun, “that” becomes “those”.

**Demonstrative Pronoun** – In this role, “that” replaces a noun.

That is beautiful.

Look at that!

When we replace a plural noun, “that” becomes “those”.

**Relative Pronoun** – “that” introduces a defining/restrictive clause. This clause is essential to the sentence.

Loki is on the team that lost.

The produce that is sourced locally is environment-friendly.

There is no “that”/“those” distinction in this case. The clause is always introduced by “that”.

Hope these simple examples clarified the various roles “that” can play in a sentence. Not understanding this distinction could lead to a lot of confusion. The words around “that” will help you understand exactly what role it is playing in each case.

Let’s take a look at one of our own questions in which knowing this distinction comes in handy.

*Question: In nests across North America, the host mother tries to identify their own eggs and weed out the fakes, but the brown-headed cowbird – a brood parasite that sneaks its eggs into other birds’ nests – produces eggs that look very similar to those of the host, making that task surprisingly difficult.*

*(A) the host mother tries to identify their own eggs and weed out the fakes, but the brown-headed cowbird – a brood parasite that sneaks its eggs into other birds’ nests – produces eggs that look very similar to those of the host, making that task surprisingly difficult*

*B) the host mother tries to identify its own eggs and weed out the fakes, but the brown-headed cowbird – a brood parasite that sneaks its eggs into other birds’ nests – produces eggs that look very similar to that of the host, making it surprisingly difficult*

*C) host mothers try to identify their own eggs and weed out the fakes, but the brown-headed cowbird – a brood parasite that sneaks its eggs into other birds’ nests – produces eggs that look very similar to the host’s, making that task surprisingly difficult*

*D) host mothers try to identify their own eggs and weed out the fakes, but the brown-headed cowbird – a brood parasite that sneaks its eggs into other birds’ nests – produces eggs that look very similar to that of the host’s, making it surprisingly difficult*

*E) host mothers try to identify its own eggs and weed out the fakes, but the brown-headed cowbird – a brood parasite that sneaks its eggs into other birds’ nests – produces eggs that look very similar to those of the host’s, making that task surprisingly difficult*

Solution:

This is a complicated sentence and unfortunately, almost the entire sentence is underlined. That just makes it harder and more time consuming.

- … the host mother tries to identify their own eggs…

In the beginning itself, we see that the subject is “host mother” which is singular and the pronoun that refers back to it – “those” – is plural. Hence this sentence is incorrect. We just move on.

(B) … produces eggs that look very similar to that of the host …

We have two instances of the use of “that” here. The first “that” is used as a relative pronoun to introduce the clause “that look very similar to ….”

The second “that” is used as a placeholder for “eggs” hence we need to use “those” – the plural form – here.

(C) All correct

(D) … produces eggs that look very similar to that of the host’s…

The explanation is the same as that of (B). The second “that” is used as a placeholder for “eggs” hence we need to use “those” – the plural form – here.

Also, the correct comparison is:

either

“A’s eggs look very similar to those of B” (where “those” stands for eggs)

or

“A’s eggs look very similar to B’s” (where eggs is implied at the end).

But “A’s eggs look very similar to those of B’s” is incorrect since it implies

“A’s eggs look very similar to eggs of B’s eggs”

(E) … host mothers try to identify its own eggs…

The subject is “host mothers”, which is plural, but the pronoun is “its”, which is singular.

Hope this clarifies the various ways in which “that” can be used.

*Getting ready to take the GMAT? Check out one of our many free GMAT resources to get a jump start on your GMAT prep. And as always, be sure to follow us on Facebook, YouTube, Google+, and Twitter for more helpful tips like this one!*

*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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]]>The post The Third Type of GMAT Quant Question appeared first on Veritas Prep Blog.

]]>But, if we look carefully, we will see a third type of question – combination of the two. There are a few statements given in them (like in Data Sufficiency questions) and five options to choose from (like in Problem Solving questions). But since we know how to solve both these question types, we shouldn’t really have a problem in solving this third type, or so one would think!

In any GMAT question, it is very important to know two things:

- What is given
- What is asked

Now, one might think that it is a very obvious distinction and why are we even trying to discuss it in a post. In this third question type, this exact distinction is far harder to explain because here the statements do NOT represent the data given. Here the statements actually ask “Is this true?” and many test-takers find it hard to make that switch. To clarify, let’s discuss the structure of the three question types.

**Problem Solving Question:**

Question: A and B are given, what is X?

(A) X is …

(B) X is …

(C) X is …

(D) X is …

(E) X is …

**Data Sufficiency Question:**

Question: A and B are given, what is X?

I. We are given that X and Y are related.

II. We are given that X and Z are related.

**“Which of the following must be true?” Question:**

Question: A and B are given, which of the following must be true about X?

I. Is this true about X?

II. Is this true about X?

III. Is this true about X?

(A) I is true

(B) I and II are true

and so on…

We hope you see that the statements in a Data Sufficiency question are different from the statements in this third type of question.

We will elaborate with the help of an example now:

*Question: If |x| > 3, which of the following must be true?*

*I. x > 3*

*II. x^2 > 9*

*III. |x – 1| > 2*

*(A) I only*

*(B) II only*

*(C) I and II only*

*(D) II and III only*

*(E) I, II, and III*

Solution:

We are given that |x| > 3

This implies that x is a point at a distance of more than 3 from 0. So x could be greater than 3 or less than -3. Before we go any further, let’s think about the values x can take: 3.00001, 3.5, 4.2, 5.7, 67, 1000, -3.45, -4, -8, -100 etc. The only values it cannot take are -3 <= x <= 3

Which of the following must be true?

I. x > 3

This is a question even though it looks like a statement.

Is it necessary that x > 3?

For every value that x can take, must x be greater than 3? No. As discussed above, x could take values such as 3.00001, 3.5, 4.2, 5.7, 67, 1000 but it could also take values such as -3.45, -4, -8, -100.

So this is not necessarily true.

II. x^2 > 9

Again, this is a question even though it looks like a statement.

Taking square root on both sides since they are positive, we get

Sqrt(x^2) > Sqrt(9)

|x| > 3

This is what we are given, hence it certainly is true.

III. |x-1|>2

Yet again, we are asked: Is |x – 1| > 2?

What does |x – 1|> 2 imply?

The distance of x from 1 must be greater than 2. So x is either greater than 3 or less than -1. Now, recall all the values that x can take.

So this is the question now: Is every value that x can take greater than 3 or less than -1?

Recall the values that x can take (discussed above)

3.00001 : x is greater than 3

3.5 : x is greater than 3

4.2 : x is greater than 3

5.7 : x is greater than 3

67 : x is greater than 3

1000 : x is greater than 3

-3.45 : x is less than -1

-4 : x is less than -1

-8 : x is less than -1

-100 : x is less than -1

For every value that x can take, x will be either greater than 3 or less than -1. Note that we are not saying that every value less than -1 must be valid for x. We are saying that every value that is valid for x (found by using |x| > 3) will be either greater than 3 or less than -1 since any value less than -3 is obviously less than -1 too. Hence |x-1|>2 must be true for every value that x can take.

Answer (D)

We hope you are quite clear about how to handle this third question type now!

*Getting ready to take the GMAT? Check out one of our many free GMAT resources to get a jump start on your GMAT prep. And as always, be sure to follow us on Facebook, YouTube, Google+, and Twitter for more helpful tips like this one!*

*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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]]>The post The Importance of Context in Verb Tenses appeared first on Veritas Prep Blog.

]]>Let’s take a look at an official GMAT question to better understand this concept:

*A recent study has found that within the past few years, many doctors had elected early retirement rather than face the threats of lawsuits and the rising costs of malpractice insurance.*

*(A) had elected early retirement rather than face*

*(B) had elected early retirement instead of facing*

*(C) have elected retiring early instead of facing*

*(D) have elected to retire early rather than facing*

*(E) have elected to retire early rather than face*

So the first decision point is “have” vs. “had”. What is correct here? We know that we use past perfect tense when there are two actions in the past. So do we have two actions in the past here – “finding” and “electing” – of which, it may seem, “electing” would have happened before “finding?” Sure, we have two actions but here is the catch – we use past perfect only when the previous action takes place completely before the recent past action. Here, we know that “within the past few years” implies the recent years. The study shows that most probably, doctors are still electing early retirement. So the use of past perfect is incorrect here. In this context, we will use present perfect only.

The other error that helps us to arrive at the right answer is lack of parallelism. “retire” and “face” need to be parallel while rising should not be parallel to them because it is a sub-list under “face”.

They elected to retire … rather than face A and B.

A – the threats

B – the rising costs

“[R]ising” is a present participle that is modifying the noun “costs” in the non underlined part. So our verbs “retire” and “face” should not be in the -ing form. Answer choice E satisfies all these criteria and hence is the right answer.

Note that the correct answer uses present perfect for both verbs since the context requires us to.

Let’s look at a rewrite of this question:

*A recent article in The Economic Times reported that many recent MBA graduates had decided on taking a job rather than face the uncertainty of entrepreneurship.*

*(A) had decided on taking a job rather than face*

*(B) had decided on taking a job instead of facing*

*(C) have decided to take a job instead of facing*

*(D) had decided to take a job rather than facing*

*(E) have decided to take a job rather than face*

How does the solution change now? Again we have two verbs “report” and “decide”. The reporting has already happened so the simple past “reported” has been used in the non-underlined part. Which tense will we use with “decide”? Again, the concept is still the same. We are talking about recent MBA graduates and it shows a trend. It is something that is not completely over, hence the use of past perfect is not justified. We should use the present perfect tense only though it may seem a bit counterintuitive since “report” is in the past tense.

“take” and “face” should be parallel to each other so out of (C) and (E), (E) fits. This is the reason making sweeping statements in grammar is dangerous – a lot depends on the context.

**free GMAT resources** to get a jump start on your GMAT prep. And as always, be sure to follow us on **Facebook**, **YouTube**, **Google+**, and **Twitter **for more helpful tips like this one!

*GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

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]]>The post Is It Incorrect to Use Multiple Verb Tenses in a Sentence? appeared first on Veritas Prep Blog.

]]>Take a look at this example sentence:

*I have heard that Mona left Manchester this morning, and has already arrived in London, where she will be for the next three weeks.*

Here, we have present perfect tense, simple past tense and simple future tense all in the same sentence, but they all make sense together to create a logical sequence of events.

The confusion over using multiple verb tenses in one sentence probably arises because we have heard that we need to maintain verb tense consistency. These two things are different.

Tense Consistency – We do not switch one tense to another unless the timing of the action demands that we do. We do not switch tenses when there is no time change for the actions.

Let’s take a look at some examples to understand this:

Example 1: During the match, my dad **stood** up and **waved** at me.

These two actions (“stood” and “waved”) happen at the same time and hence, need to have the same tense. This sentence could take place in the present or future tense too, but both verbs will still need to take on the same tense. For example:

Example 2: During my matches, my dad **stands** up and **waves** at me.

Example 3: During the match tomorrow, my dad **will stand** up and **wave** at me.

On the other hand, a sentence such as…

Example 4: During the match, my dad **stood** up and **waves** at me.

This sentence is grammatically incorrect. Since both actions (“stood” and “waves”) happen at the same time, we need them to be in the same tense, as shown in the variations of this sentence above. Consider this case, however:

Example 5: My dad **reached** for the sandwich after he had already **eaten** a whole pizza.

Here, the two actions (“reached” and “eaten”) happen at different times in the past, so we use both the simple past and past perfect tenses. The shift in tense is correct in this context.

**Takeaway: The tenses of verbs in a sentence must be consistent when the actions happen at the same time. When dealing with actions that occur at different points in time, however, we can use multiple tenses in the same sentence.**

Let’s look at an official GMAT question now to see how multiple tenses can be a part of the same sentence:

*For the farmer who takes care to keep them cool, providing them with high-energy feed, and milking them regularly, Holstein cows are producing an average of 2,275 gallons of milk each per year.*

*(A) providing them with high-energy feed, and milking them regularly, Holstein cows are producing*

*(B) providing them with high-energy feed, and milked regularly, the Holstein cow produces*

*(C) provided with high-energy feed, and milking them regularly, Holstein cows are producing*

*(D) provided with high-energy feed, and milked regularly, the Holstein cow produces*

*(E) provided with high-energy feed, and milked regularly, Holstein cows will produce*

This is a very tricky question. Let’s first shortlist our options based on the obvious errors.

The non-underlined part of the sentence uses the pronoun “them” to refer to the cows, so using “the Holstein cow” (singular) as the antecedent will be incorrect. The antecedent must be “Holstein cows” (plural) – this means answer choices B and D are out.

Also, we know for sure that “provide” and “milk” are parallel elements in the sentence, so they should take the same verb tense. Hence, answer choice C is also out.

Let’s look at A now. If we assume this option is correct, “providing” and “milking” act as modifiers to “keep them cool”. That certainly does not make sense since “providing with high energy feed” and “milking regularly” are not ways of keeping cows cool.

This means the correct answer is E, but we need to see how.

*For the farmer who takes care to keep them cool, provided with high-energy feed, and milked regularly, Holstein cows will produce an average of 2,275 gallons of milk each per year.*

Let’s break down the sentence:

*For the farmer who takes care to keep them…*

- cool,
- provided with high-energy feed,
- milked regularly,

*…Holstein cows will produce an average of 2,275 gallons of milk each per year.*

Note that we use two different tenses here: “For the farmer who takes care…” and “cows will produce…”. The word “takes” is the present tense while “will produce” is the future, but that does not make this sentence incorrect. The context of the author could very well justify the use of the future tense. Perhaps the farmers have obtained Holstein cows recently, and hence, will see the produce of 2,275 gallons in the future, only.

A shift in the tense certainly doesn’t make the sentence incorrect. When you’re presented with multiple verbs in various tenses in a problem, check to determine whether the verbs convey a logical sequence of events.

**free GMAT resources** to get a jump start on your GMAT prep. And as always, be sure to follow us on **Facebook**, **YouTube**, **Google+**, and **Twitter **for more helpful tips like this one!

*GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

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]]>The post Dreaded Data Sufficiency Questions That Will Test Your Knowledge of Number Properties appeared first on Veritas Prep Blog.

]]>Here is our advice – when solving number properties questions, imagine a number line. It reminds us that numbers behave differently “between 0 and 1”, “between -1 and 0”, “less than -1”, and “more than 1”, and that integers occur only at regular intervals and that there are infinite numbers in between them. The integers are, in turn, even and odd. Also, 0, 1 and -1 are special numbers, hence it is always a good idea to consider cases with them.

Let’s see how thinking along these lines can help us on a practice Data Sufficiency question:

*If a and b are non-zero integers, is a^b an integer?*

*Statement 1: b^a is negative*

*Statement 2: a^b is negative*

The answer to this problem does not lie in actually drawing a number line. The point is that we need to think along these lines: -1, 0, 1, ranges between them, integers, negatives-positives, even-odd, decimals and how each of these comes into play in this case.

What we know from the question stem is that *a* and *b* are non-zero integers, which means they occur at regular intervals on the number line. To answer the question, “Is *a*^*b* an integer?”, let’s first look at Statement 1:

*Statement 1: b^a is negative*

For a number to be negative, its base must be negative. But that is not enough – the exponent should not be an even integer. If the exponent is an even integer, the negative signs will cancel out. Since *a* and *b* are integers, if *a* is not an even integer, it must be an odd integer.

We know that the sign of the exponent is immaterial as far as the sign of the result is concerned (since *a*^(-*n*) is just 1/*a*^*n*). For *b*^*a* to be negative, then we know that *b* must be a negative integer and *a* must be an odd integer. Does this help us in deducing whether *a*^*b* is an integer? Not necessarily!

If *b* is negative, say -2, *a*^(-2) = 1/*a*^2. *a* could be 1, in which case 1/*a*^2 = 1 (an integer), or *a* could be 3, in which case 1/*a*^2 = 1/9 (not an integer). Because there are two possible answers, this statement alone is not sufficient.

Let’s look at Statement 2:

*Statement 2: a^b is negative*

Again, the logic remains the same – for *a* number to be negative, its base must also be negative and the exponent should not be an even integer. If the exponent is an even integer, the negative signs will cancel out. Since *a* and *b* are integers, if *b* is not an even integer, it must be an odd integer. Again, we know that the sign of the exponent is immaterial as far as the sign of the result is concerned (since *a*^(-*n*) is just 1/*a*^*n*).

For *a*^*b* to be negative, then we know that a must be a negative integer and* b* must be an odd integer. *a* could be -1/-2/-3/-4… etc, and* b* could be 1/3/5… or -1/-3/-5.

If *a* = -1 and *b* = 1, then *a*^*b* = -1 (an integer). If* a* = -2 and *b* = -3, then *a*^*b* = (-2)^(-3) = 1/(-2)^3 = -1/8 (not an integer). This statement alone is also not sufficient.

We hope you see how we are using values of 1 and -1 to enumerate our cases. Now, let’s consider using both statements together:

*a* is a negative, odd integer, so it can take values such as -1, -3, -5, -7, …

*b* is a negative, odd integer too, so it can also take values such as -1, -3, -5, -7, …

If *a* = -1 and* b* = -1, then *a*^*b* = -1 (an integer)

If* a* = -3 and *b* = -3, then *a*^*b* = (-3)^(-3) = -1/27 (not an integer)

Even using both statements together, we do not know whether *a*^*b* is an integer or not. therefore, our answer is E.

Thinking of a number line and knowing what it represents will help you tackle many Data Sufficiency questions that are about number properties.

*Getting ready to take the GMAT? We have free online GMAT seminars running all the time. And, be sure to follow us on Facebook, YouTube, Google+, and Twitter!*

*GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

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]]>The post 3 Ways to Solve a 750+ Level GMAT Question About Irregular Polygons appeared first on Veritas Prep Blog.

]]>*The hexagon above has interior angles whose measures are all equal. As shown, only five of the six side lengths are known: 10, 15, 4, 18, and 7. What is the unknown side length?*

*(A) **7*

*(B)**10*

*(C) 12*

*(D) 15*

*(E) 16*

There are various ways to solve this question, but each takes a bit of effort. Note that the polygon we are given is not a regular polygon, since the side lengths are not all equal. The angles, however, are all equal. Let’s first find the measure of each one of those angles using the formula discussed in this previous post.

(n – 2)*180 = sum of all interior angles

(6 – 2)*180 = 720

Each of the 6 angles = 720/6 = 120 degrees

Though we would like to point out here that if you see a question such as this one on the actual GMAT exam, you should already know that if each angle of a hexagon is equal, each angle must be 120 degrees, so performing the above calculation would not be necessary.

**Method 1: Visualization**

This is a very valid approach to obtaining the correct answer on this GMAT question since we don’t need to explain the reasoning or show our steps, however it may be hard to comprehend for the beginners. We will try to explain it anyway, since it requires virtually no work and will help build your math instinct.

Note that in the given hexagon, each angle is 120 degrees – this means that each pair of opposite sides are parallel. Think of it this way: Side 4 turns on Side 18 by 120 degrees. Then Side 15 turns on Side 4 by another 120 degrees. And finally, Side 10 turns on Side 15 by another 120 degrees. So Side 10 has, in effect, turned by 360 degrees on Side 18.

This means Side 10 is parallel to Side 18.

Now, think of the 120 degree angle between Side 4 and Side 15 – it has to be kept constant. Plus, the angles of the legs must also stay constant at 120 degrees with Sides 10 and 18. Since the slopes of each leg of that angle are negatives of each other (√3 and -√3), when one leg gets shorter, the other gets longer by the same length (use the image below as a visual of what we’re talking about).

Hence, the sum of the sides will always be 15 + 4 = 19. This means 7 + Unknown = 19, so Unknown = 12. Our answer is C.

If you struggled to understand the approach above, you’re not alone. This method involves a lot of intuition, and struggling to figure it out may not be the best use of your time on the GMAT, so let’s examine a couple of more tangible solutions!

**Method 2: Using Right Triangles**

As we saw in Method 1 above, AB and DE are parallel lines. Since each of the angles A, B, C, D, E and F are 120 degrees, the four triangles we have made are all 30-60-90 triangles. The sides of a 30-60-90 triangle can be written using the ratio 1:√(3):2.

AT = 7.5*√3 and ME = 2*√3, so the distance between the sides of length 10 and 18 is 9.5*√3. We know that DN = 3.5*√3, so BP = (9.5*√3) – (3.5*√3) = 6*√3.

Since the ratios of our sides should be 1:√(3):2, side BC = 2*6 = 12. Again, the answer is C. Let’s look at our third and final method for solving this problem:

**Method 3: Using Equilateral Triangles**

First, extend the sides of the hexagon as shown to form a triangle:

Since each internal angle of the hexagon is 120 degrees, each external angle will be 60 degrees. In that case, each angle between the dotted lines will become 60 degrees too, and hence, triangle PAB becomes an equilateral triangle. This means PA = PB = 10. Triangle QFE and triangle RDC also become equilateral triangles, so QF = QE = 4, and RD = RC = 7.

Now note that since angles P, Q, and R are all 60 degrees, triangle PQR is also equilateral, and hence, PQ = PR.

PQ = 10 + 15 + 4 = 29

PR = 10 + BC + 7 = 29

BC = 12 (again, answer choice C)

Note the geometry concepts that we used to solve this problem: regular polygon, parallel lines, angles, 30-60-90 right triangles, and equilateral triangles. We know all of these concepts very well individually, but applying them to a GMAT question can take some ingenuity!

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]]>Say** “x” gives you a remainder of 2 when divided by 6**. What will be the remainder when x + 1 is divided by 6?

Go back to the divisibility concepts discussed above. When x balls are split into groups of 6, we will have 2 balls leftover. If we are given 1 more ball, it will join the 2 balls and now we will have 3 balls leftover. The remainder will be 3.

What happens in the case of** x + 6** – what will be the remainder when this is divided by 6? This additional 6 balls will just make an extra group of 6, so we will still have 2 balls leftover.

What about the case of** x + 9**? Now, of the extra 9 balls, we will make one group of 6 and will have 3 balls leftover. These 3 balls will join the 2 balls leftover from x, giving us a remainder of 5.

Now, what about the case of** 2x**? Recall that 2x = x + x. The number of groups will double and so will the remainder, so 2x will give us a remainder of 2*2 = 4.

On the other hand, if x gives us a remainder of 4 when divided by 6, then 2x divided by 6 will have a remainder of 2*4 = 8, which gives us a remainder of 2 (since another group of 6 will be formed from the 8 balls).

Let’s consider the tricky case of** x^2** now. If x gives us a remainder of 2 when it is divided by 6, it means:

x = 6Q + 2

x^2 = (6Q + 2)*(6Q + 2) = 36Q^2 + 24Q + 4

Note here that the first and the second terms are divisible by 6. The remainder when you divide this by 6 will be 4.

We hope you understand how to deal with these various cases of remainders. Let’s take a look at a GMAT sample question now:

*If z is a positive integer and r is the remainder when z^2 + 2z + 4 is divided by 8, what is the value of r?*

*Statement 1: When (z−3)^2 is divided by 8, the remainder is 4.*

*Statement 2: When 2z is divided by 8, the remainder is 2.*

This is not our typical, “When z is divided by 8, r is the remainder” type of question. Instead, we are given a quadratic equation in the form of z that, when divided by 8, gives us a remainder of r. We need to find r. This question might feel complicated, but look at the statements – at least one of them gives us data on a quadratic! Looks promising!

*Statement 1: When (z−3)^2 is divided by 8, the remainder is 4*

(z – 3)^2 = z^2 – 6z + 9

We know that when z^2 – 6z + 9 is divided by 8, the remainder is 4. So no matter what z is, z^2 – 6z + 9 + 8z, when divided by 8, will *only* give us a remainder of 4 (8z is a multiple of 8, so will give remainder 0).

z^2 – 6z + 9 + 8z = z^2 + 2z + 9

z^2 + 2z + 9 when divided by 8, gives remainder 4. This means z^2 + 2z + 5 is divisible by 8 and would give remainder 0, further implying that z^2 + 2z + 4 would be 1 less than a multiple of 8, and hence, would give us a remainder of 7 when divided by 8. This statement alone is sufficient.

Let’s look at the second statement:

*Statement 2: When 2z is divided by 8, the remainder is 2*

2z = 8a + 2

z = 4a + 1

z^2 = (4a + 1)^2 = 16a^2 + 8a + 1

When z^2 is divided by 8, the remainder is 1. When 2z is divided by 8, the remainder is 2. So when z^2 + 2z is divided by 8 the remainder will be 1+2 = 3.

When z^2 + 2z + 4 is divided by 8, remainder will be 3 + 4 = 7. This statement alone is also sufficient. Because both statements alone are sufficient, our answer is D.

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]]>The post Using Parallel Lines and Transversals to Your Advantage on the GMAT appeared first on Veritas Prep Blog.

]]>**The ratios of the intercepts of two transversals on parallel lines is the same.**

Consider the diagram below:

Here, we can see that:

- “a” is the intercept of the first transversal between L1 and L2.
- “b” is the intercept of the first transversal between L2 and L3.
- “c” is the intercept of the second transversal between L1 and L2.
- “d” is the intercept of the second transversal between L2 and L3.

Therefore, the ratios of a/b = c/d. Let’s see how knowing this property could be useful to us on a GMAT question. Take a look at the following example problem:

*In triangle ABC below, D is the mid-point of BC and E is the mid-point of AD. BF passes through E. What is the ratio of AF:FC ?*

*(A) 1:1*

* (B) 1:2*

* (C) 1:3*

* (D) 2:3*

* (E) 3:4*

Here, the given triangle is neither a right triangle, nor is it an equilateral triangle. We don’t really know many properties of such triangles, so that will probably not help us. We do know, however, that AD is the median and E is its mid-point, but again, we don’t know any properties of mid-points of medians.

Instead, we need to think outside the box – parallel lines will come to our rescue. Let’s draw lines parallel to BF passing through the points A, D, and C, as shown in the diagram below:

Now we have four lines parallel to each other and two transversals, AD and AC, passing through them.

Consider the three parallel lines, “line passing through A”, “BF”, and “line passing through D”. The ratio of the intercepts of the two transversals on them will be the same.

AE/ED = AF/FP

We know that AE = ED since E is the mid point of AD. Hence, AE/ED = 1/1. This means we can say:

AE/ED = 1/1 = AF/FP

AF = FP

Now consider these three parallel lines: “BF”, “line passing through D”, and “line passing through C”. The ratio of the intercepts of the two transversals on them will also be the same.

BD/DC = FP/PC

We know that BD = DC since D is the mid point of BC. Hence, BD/DC = 1/1. This means we can also say:

BD/DC = 1/1 = FP/PC

FP = PC

From these two calculations, we will get AF = FP = PC, and hence, AF:FC = 1:(1+1) = 1:2.

Therefore, the answer is B. We hope you see that Geometry questions on the GMAT can be easily resolved once we bring in parallel lines.

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]]>The post Learn How to Begin a GMAT Problem by Focusing on Keywords in the Question Stem appeared first on Veritas Prep Blog.

]]>Take a look at this example Quant question:

*The length and width of a rectangle are integer values. What is the area of the smallest such rectangle that can be inscribed in a circle whose radius is also an integer?*

*(A) 12 *

*(B) 24 *

*(C) 36 *

*(D) 48 *

*(E) 60*

Now here is the problem – the question stem does not give us any numbers! We don’t know any dimensions of the rectangle or the circle, yet the answer choice options are very specific numbers! So how do we begin? The smallest positive integer is 1, so should we start by testing the radius of the circle as 1, and then try to go on from there? And if 1 doesn’t work, then move on to 2, 3, 4… etc?

No – we are not a computer algorithm and on top of that, the GMAT only gives us around 2 minutes to figure out the answer. With this in mind, the question should enough clues to make all all of that trial and error testing unnecessary. So if plugging in numbers isn’t the way to go, how should we start solving this problem?

Now, the moment we read “rectangle inscribed in a circle”, what comes to mind is that a rectangle has 90 degree angles, and hence, the diagonal of the rectangle is the diameter of the circle (an arc that subtends a 90 degree angle at the circumference is a semicircle). The rectangle inside of the circle will look something like this:

Now we can see that we have a circle with a diameter (AB) and 90 degree angles subtended in each semicircle (angle AMB and angle ANB).

Essentially then, we have two right triangles (triangle AMB and triangle ANB) that share the hypotenuse AB. Also, it’s important to note that each side of these triangles is an integer – since we know the radius of the circle is an integer, the diameter has to be an integer too. This should make us think of Pythagorean triples!

Whenever all three sides of a right triangle are integers, they will form a Pythagorean triple. Can you have a right triangle with all integer sides such that the length of one side is 1? No. There are no Pythagorean triples with 1 as a side. The smallest Pythagorean triple we know of is 3, 4, 5 (so there can be no right triangle with all integer sides such that the length of one side is 2, either).

We already know Pythagorean triples are the lengths of the sides of right triangles where all sides are integers. What we need to internalize is that ONLY Pythagorean triples are the lengths of sides of right triangles where all sides are integers. You cannot have a right triangle with all integer sides but whose sides are not a Pythagorean triple.

This means that the smallest right triangle with all integer sides is a 3, 4, 5 triangle.

Now note that in the given question, the hypotenuse is the diameter of the circle. We are given that the radius of the circle is an integer, so the diameter will be twice an integer, i.e. an even integer.

So we know the hypotenuse is an even integer, but as we discussed last week, the hypotenuse of a primitive Pythagorean right triangle must be odd. So this triangle must be a non-primitive Pythagorean triple. The smallest such triple will be twice of 3, 4, 5, i.e. the triangle will have sides with lengths 6, 8, 10.

This means the sides of the rectangle must be 6 and 8, while its diagonal must have a length of length 10. The area of the rectangle, then, must be 6*8 = 48. The answer is D.

Finally, at the end of the post we have figured out that this post is a continuation of last week’s post on properties of Pythagorean triples! We hope you enjoyed it!

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