The post Why We Need to Redraw GMAT Geometry Figures appeared first on Veritas Prep Blog.

]]>In Problem Solving questions, our target is to find just one solution. For example, when we have questions involving percentages, we assume some values and get the answer. No matter what values we assume, we will always get the same answer as long as the integrity of the data is maintained.

In Data Sufficiency questions, our target is to find multiple possible solutions after using all the given data and arrive at answer (E). If we are unable to find more than 1 solution using either statement (1) and/or statement (2), we arrive at answers (A), (B), (C) or (D).

The aim is diametrically opposite in the two cases. Therefore, our strategies in the two cases would also be different and they are. Consider Geometry questions with figures in them. In Problem Solving questions, we try to make the figures as symmetrical as possible under the given constraints. With symmetrical figures, it is easier to get an answer. One answer is all we need.

In Data Sufficiency questions, we try to make the figures as extreme as possible. Only the given data should hold in such a figure and no symmetry should exist in the other dimensions. Only then will we be able to really figure out whether the given information is enough to arrive at a unique answer.

Let’s explain this using two examples:

Problem Solving Question

PSvsDSQuesPS1.jpg ********************************

*In the figure above, the area of square PQRS is 64. What is the area of triangle QRT?*

*(A) 48*

* (B) 32*

* (C) 24*

* (D) 16*

* (E) 8*

This is a Problem Solving question.

All we are given is that PQRS is a square. Note that the location of point T is not defined. It is just any point on side PS. We can place it anywhere we like as long as it is on PS. At what point will it be easy for us to calculate the area of triangle QRT? Of course, T could be the middle point of PS (bringing in symmetry) and we could calculate the area of the triangle or we could make it coincide with S so that QRT is a right triangle half of square PQRS. Then, the area of triangle QRT will simply be half of 64, i.e. 32.

Note that we don’t necessarily need to do this. We can assume T to be a random point, drop an altitude from T to QR, find that the length of the altitude will be same as the side of the square, find that side of the square will be √(64) = 8 and area of triangle QRT will be (1/2)*8*8 = 32

We will arrive at the same answer of course! But, assuming a better position for point T (but only because it is not defined) will cut the calculations and help us arrive directly at 32 from 64.

Data Sufficiency Question

PSvsDSQuesDS1.jpg ********************************

*If AD is 6 and ADC is a right angle, what is the area of triangular region ABC?*

*Statement 1: Angle ABD = 60°*

* Statement 2: AC = 12*

Looking at the figure, many test takers are tempted to think that the altitude AD will bisect BC. Note that that may not be the case.

According to the data given in the question stem alone, the figure could very well look something like this:

PSvsDSQuesDS2.jpg ********************************

All we know is that ADC is a right angle and the length of the altitude is 6. We don’t know whether any of the sides are equal, etc. Hence, it is a good idea to redraw the figure with extreme proportions – one side much greater than the other.

Now we can use the given statements to re-adjust the proportions.

Area of triangle ABC = (1/2)*AD*BC

We know that AD is 6. But we don’t know BC. Let’s examine each of the statements separately.

*Statement 1: Angle ABD = 60°*

This statement tells us that triangle ABD is a 30-60-90 triangle. Knowing the length of AD will give us the length of the other two sides too. But here is the problem – to know BC, we need to know length of CD too. That we cannot find from this statement alone. This statement alone is not sufficient to answer the question.

*Statement 2: AC = 12*

We know that ADC is a right angled triangle. Knowing AC and AD, we can find the length of CD using Pythagorean Theorem. But we cannot find BD using this statement and that is needed to get the length of BC. This statement alone is also not sufficient to answer the question.

Using both statements, we can find the lengths of both BD and CD, and hence, can find the length of BC. This will give us the area of the triangle. Therefore, our answer is C.

Note here that if we mistakenly assume that D is the mid point of BC, we might come to the conclusion that each statement alone is sufficient and might mark the answer as D, instead of C. Hence, it is a good idea to redraw the given figure in a Data Sufficiency question to ensure that it has as little symmetry as possible.

*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 Quarter Wit, Quarter Wisdom: To Learn To-Infinitives appeared first on Veritas Prep Blog.

]]>Note that the infinitive is the base form of a verb. The infinitive has two forms:

** • the to-infinitive** = to + base

** • the zero infinitive** = base

We will discuss the to-infinitive form, a verbal. It can work as a noun, an adjective, or an adverb

The to-infinitive form is used in many sentence constructions, often expressing the purpose of something or someone’s opinion about something. The to-infinitive is used following a large collection of different verbs as well such as afford, offer, refuse, prepare, undertake, proceed, propose, promise etc

The function of a to-infinitive in a sentence could be any of the following:

I. To show the purpose of an action: In this case “*to”* has the same meaning as “*in order to”* or “*so as to”*. It follows a verb in this case.

For Example: She has gone to complete her homework.

II. To indicate what something can or will be used for: It follows a noun or a pronoun in this case.

For Example: I don’t have anything to wear. This is the right thing to do.

III. After adjectives

For Example: I am happy to be here.

IV. The subject of the sentence

For Example: To visit Paris is my lifelong dream.

V. With adverbs: It is used with the adverbs *too* and *enough* to express the reasoning behind our satisfaction or dissatisfaction. The pattern is that *too* and *enough* are placed before or after the adjective, adverb, or noun that they modify in the same way they would be without the to-infinitive. We then follow them by the to-infinitive to explain the reason why the quantity is excessive, sufficient, or insufficient.

For Example: He has too many books to carry on his own.

VI. With question words: The verbs ask, decide, explain, forget, know, show, tell, & understand can be followed by a question word such as where, how, what, who, & when + the to-infinitive.

For Example: I am not sure how to use the new washing machine.

We are likely to see infinitive phrases in GMAT sentence correction questions. An infinitive phrase is made up of the infinitive verb with its object and modifiers.

Let’s take a look at how we could see an infinitive in a GMAT question.

*Question: Twenty-two feet long and 10 feet in diameter, the AM-1 is one of the many new satellites that is a part of 15 years effort of subjecting the interactions of Earth’s atmosphere, oceans, and land surfaces to detailed scrutiny from space.*

*(A) satellites that is a part of 15 years effort of subjecting the interactions of Earth’s atmosphere, oceans, and land surfaces*

*(B) satellites, which is a part of a 15-year effort to subject how Earth’s atmosphere, oceans, and land surfaces interact*

*(C) satellites, part of 15 years effort of subjecting how Earth’s atmosphere, oceans, and land surfaces are interacting*

*(D) satellites that are part of an effort for 15 years that has subjected the interactions of Earth’s atmosphere, oceans, and land surfaces*

*(E) satellites that are part of a 15-year effort to subject the interactions of Earth’s atmosphere, ocean, and land surfaces*

Solution:

First let’s try to understand the basic structure of the sentence.

… AM-1 is one of the many new satellites “that/which clause”

“that/which clause” modifies the noun “satellites” in four of the given five options. Note that “satellites” is plural so we need to use the verb “are”. So options (A) and (B) are out.

(C) is also incorrect. It looks like “part of 15 years … from space” is a bad attempt at writing an absolute phrase. Absolute phrases modify the entire clause but here we need to modify “satellites” only. Satellites are a part of a 15 year effort to subject A to detailed scrutiny and hence we should use a that/which clause.

(D) is incorrect too. It uses another “that clause” – that has subjected the interactions …

This “that clause” modifies the noun “effort”, not “15 years”. The effort has subjected A to detailed scrutiny.

There is a better way of writing this sentence such that the “that clause” comes immediately after “effort”

(E) is correct. Note how it uses the infinitive form immediately after the noun “effort” to indicate how the effort is being used. It is being used to subject A to detailed scrutiny.

Hope now you will be able to recognise the different verbals and use them correctly.

*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 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!

*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 Quarter Wit, Quarter Wisdom: Using Ingenuity on GMAT Remainder Questions appeared first on Veritas Prep Blog.

]]>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 How to Answer GMAT Critical Reasoning Questions Involving Experiments appeared first on Veritas Prep Blog.

]]>The logic of these arguments is always rooted in the notion that we can only trust the results of the experiment if we have a legitimate control group, and there aren’t any other confounding variables that we’ve failed to account for. Spoiler alert: typically in GMAT questions, we will find such confounding variables tainting the experiment’s predictive value.

Imagine, for example, that you’re testing a drug designed to alleviate headaches. You have two groups of subjects: a control group that takes a placebo and an experimental group that receives the drug. The results of the experiment show that the control group has a higher rate of headaches than the group receiving the medication. Time to rejoice, notify the delighted shareholders, and move this drug to market as quickly as possible? Well, maybe.

But now imagine that the control group consisted largely of stressed-out, sleep-deprived college students living near construction sites, and the experiment group consisted of retired yoga instructors. Suddenly we’ve got other variables to contend with. Yes, it’s possible that the effectiveness of the drug is what accounts for the differential in headache incidence between the two groups. But it’s just as likely that other environmental factors are responsible. A good experiment would have controlled for these factors.

The upshot: whenever you see a question that involves an experiment with a control group, always ask yourself if there are variables that the experimenters have failed to account for.

Here’s a good example of such an argument:

*In Colorado subalpine meadows, nonnative dandelions co-occur with a native ﬂower, the larkspur. Bumblebees visit both species, creating the potential for interactions between the two species with respect to pollination. In a recent study, researchers selected 16 plots containing both species; all dandelions were removed from eight plots; the remaining eight control plots were left undisturbed. The control plots yielded significantly more larkspur seeds than the dandelion-free plots, leading the researchers to conclude that the presence of dandelions facilitates pollination (and hence seed production) in the native species by attracting more pollinators to the mixed plots. *

*Which of the following, if true, most seriously undermines the researchers’ reasoning? *

*A) Bumblebees preferentially visit dandelions over larkspurs in mixed plots.*

*B) In mixed plots, pollinators can transfer pollen from one species to another to augment seed production. *

*C) If left unchecked, nonnative species like dandelions q**u**i**c**kly crowd out native species. *

*D) Seed germination is a more reliable measure of a species’ ﬁtness than seed production.*

*E) Soil disturbances can result in fewer blooms, and hence lower seed production. *

This is a classic experiment argument. There are two populations: plots that contain both dandelions and larkspurs, and plots that have had all the dandelions removed, and thus contain only larkspurs. We’re told that the plots containing both types of flowers produced more larkspur seeds than the plots containing only larkspurs, thus validating the contention that the presence of dandelions has a positive benefit on larkspur seed yields.

Fortunately, the GMAT is pretty predictable. If we’re trying to weaken the conclusion derived from an experiment comparing two populations – a control group and an experimental group – we’re looking for a confounding variable. The initial hypothesis is that the presence of dandelions promotes seed production in larkspurs. An alternative hypothesis is that an environmental factor we haven’t yet considered accounts for the differential in larkspur seed production in the two groups, so that’s what we’re on the lookout for when we examine each of the answer choices.

A) Which flower bees prefer sheds no light on the validity of the experiment. A is out.

B) This answer option would be entirely consistent with the hypothesis that dandelions promote larkspur seed production. We’re trying to weaken the argument. B is also out.

C) This answer choice makes no sense. We’ve already been told that the plots containing both types of flower produce more larkspur seeds – we never want to contradict a premise. C is no good.

D) This tells us nothing about whether it is the presence of dandelions that’s helping promote larkspur seed production. D gets kicked to the curb.

E) If removing the dandelions *disrupts the soil*, perhaps it’s the disrupted soil, rather than the absence of dandelions, that accounts for the lower larkspur production in the plots where the dandelions have been removed. We’ve got our confounding variable – E is the answer.

Takeaway: On Critical Reasoning questions on the lookout for the tainted experiment. If you’re trying to weaken an argument regarding an experiment containing a control group and an experimental group, the key will be determining which answer choice provides a confounding variable, and thus, an alternative explanation for the conclusion given.

*Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And be sure to follow us on Facebook, YouTube, Google+ and Twitter!*

*By David Goldstein, a Veritas Prep GMAT instructor based in Boston. You can find more articles written by him here.*

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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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