GMAT-esque Challenge Question: Today is the Day… To Get Better at GMAT Math

It’s March 15, and now that the hysteria over Pi Day (good ol’ 3-14) is over, the true GMAT math geeks can celebrate an even more exciting use of math and dates (and, no, we don’t mean impressing your dates with math tricks. Not that that’s not a good idea, of course).

Today is 3/15, a not-all-that-unique case of the month being a factor of the day (3 is a factor of 15, so you could factor the date 3/15 into 1/5. It’s almost like it’s January 5th all over again!).
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GMAT Tip of the Week AND GMAT Challenge Question: Two-for-Twentieth!

Hello again readers, and happy May Twentieth!  In honor of that 2 in 5/20 (the day before The End of the World, of course), today’s post is dual-purposed in two ways:  We have a GMAT challenge question and a GMAT tip of the week, and the GMAT tip of the week is actually two in one.  First, take a look at this challenge Critical Reasoning question:

A recent Bloomberg Businessweek article cited that the average GMAT score for 20 and 21-year olds is 575, while for 22 and 23 year olds it is only 539.  Therefore, all potential business school applicants should be advised to take the GMAT while they’re younger so that they can expect a higher score.
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GMAT Challenge Question: Two For Thursday

GMAT PrepBack by popular demand…let’s try another GMAT Challenge Question today, and since we’re doing two consecutive days of challenge questions, let’s just go all out.  As Ernie Banks would say, let’s play two.  The questions, courtesy of San Diego GMAT instructor Matt Douglas, our 2010 worldwide Instructor of the Year, are below; post your answers in the comments field and check back later today for detailed solutions.

1.  Is a = 0?

(1)  a = a^2

(2)  a = a^3 – a^2

2.  Is ab < a?

(1)  a = 0

(2)  b = 0

UPDATE: Solutions

Question 1 is a great example of a GMAT unscored, experimental question.  If you selected B…you should be commended for your work.  Clearly statement 1 is not sufficient, as both 0 and 1 satisfy the given equation, so we cannot determine whether a is exclusively 0.  Statement 2 most likely leads you to “sufficient”, as 0 is the only rational solution.  But further examination leads us to:

a = a^3 – a^2

0 = a^3 – a^2 – a

0 = a (a^2 – a – 1)

Here, 0 is one solution for a, and so are the factors of the quadratic a^2 – a – 1 = 0.  Were you to apply the Quadratic  Formula – which the GMAT will NOT require you to do – you’d note that there are, indeed, two imaginary solutions for the quadratic.  Accordingly, although it’s not likely that you would go as far as to calculate those solutions past the point of determining that they do exist, statement 2 is also not sufficient.

Taking the statements together, we know that:
Statement 1: a = 0 or a = 1

Statement 2: a = 0 or a = one of two imaginary (not 1) numbers

Therefore, as 0 is the only number that satisfies both statements, the correct answer is C.  However, note that because the GMAT does not explicitly require the quadratic formula, this may be the type of question that becomes weeded out due to the GMAT’s sophisticated item analysis via unscored experimental questions.  If statistical analysis on a question demonstrates that there is noise in the data – that a 700 scorer may not be significantly more likely to answer this question correctly than a 600 scorer, for example, an item like this may be removed.  We bring that up because, while B is a tempting but probably a valid incorrect answer given that you can solve this without using the quadratic formula, and just knowing that it does exist, E is also a possible answer that a reasonable high-scorer could confidently suggest.   If you use statement 1 and substitute that into statement 2, you could replace a with a^2 to get:  a^2 = a^3 – a^2.    That also means that 2a^2 = a^3, leaving open the possibility that 2 is a solution in addition to 0, when clearly that solution does not work in the individual solutions.  Because this question can get so dense with the quadratic formula and some unique algebraic rules associated with non-function quadratics…well, like plenty of officially-tested GMAT questions over time, this one is probably a lot more complicated than we intended when we first drew it up.  It’s a great learning tool, but ultimately we predict that the GMAT’s rigorous statistical analysis would weed this one out in the experimental phase.

#2:  Is ab < a?

Statement 1, that a = 0, is sufficient.  If a is 0, then we’d have:  0(b) < 0 ?   Clearly that is not the case; 0 = 0, so ab can never be less than a if a is 0.  The answer is “no” and statement 1 is sufficient.

Statement 2, that b = 0, is not sufficient.  Here we know that the left hand side of the inequality is 0, but we don’t know anything about the right hand side.  a(0) < a ?     That asks whether 0 < a, and it all depends on the value of a.  Because statement 2 is not sufficient, the correct answer is A.

GMAT Challenge Question: Sentence Correction Is the (Tar) Pits

GMAT PrepOnce again, it’s time for a GMAT challenge question!  This installment  features the dreaded maximum-word-count, everything-underlined Sentence Correction device.  Can you avoid getting sucked into a La Brea Tar Pit of wasted time?  Here’s the question, courtesy of Boston GMAT tutor Ashley Newman-Owens; check back later for the solution and some tips for working through a problem like this:

Based on accelerator mass spectrometry, an advanced technique that makes it possible to obtain radiocarbon dates from samples much smaller than that necessary for traditional radiocarbon dating, scholars have been able to determine impressively precise radiocarbon ages for samples of Glyptodon and Holmesina recovered from the La Brea Tar Pits, extinct large mammals similar to armadillos.

(A)       Based on accelerator mass spectrometry, an advanced technique that makes it possible to obtain radiocarbon dates from samples much smaller than that necessary for traditional radiocarbon dating, scholars have been able to determine impressively precise radiocarbon ages for samples of Glyptodon and Holmesina recovered from the La Brea Tar Pits, extinct large mammals similar to armadillos.

(B)       Based on accelerator mass spectrometry, an advanced technique that makes it possible to obtain radiocarbon dates from samples much smaller than necessary for traditional radiocarbon dating, scholars have been able to determine impressively precise radiocarbon ages for samples of Glyptodon and Holmesina, extinct large mammals similar to armadillos, recovered from the La Brea Tar Pits.

(C)       Using accelerator mass spectrometry, an advanced technique that makes it possible to obtain radiocarbon dates from samples much smaller than those necessary for traditional radiocarbon dating, scholars have been able to determine impressively precise radiocarbon ages for samples of Glyptodon and Holmesina, extinct large mammals similar to armadillos, recovered from the La Brea Tar Pits.

(D)       Basing them on accelerator mass spectrometry, which allowed them to obtain radiocarbon dates from samples much smaller than those necessary for traditional radiocarbon dating, scholars have been able to determine impressively precise radiocarbon ages for samples of Glyptodon and Holmesina, extinct large mammals similar to armadillos, recovered from the La Brea Tar Pits.

(E)       The use of accelerator mass spectrometry, an advanced technique that makes it possible to obtain radiocarbon dates from samples much smaller than those necessary for traditional radiocarbon dating, have enabled scholars to determine impressively precise radiocarbon ages for samples of Glyptodon and Holmesina recovered from the La Brea Tar Pits, extinct large mammals similar to armadillos.

UPDATE: Solution.

Since this sentence is underlined in its (incredibly long) entirety, you can save yourself an especially hefty chunk of time by looking for differences that jump out between the answer choices.  Scanning only the ends of the answers choices reveals a split between two possible endings; the choice in the original sentence improperly situates extinct large mammals similar to armadillos, an appositive which should immediately follow Glyptodon and Holmesina. The introductory phrase is a modifier that attempts to link to scholars.  The participle using makes this link legitimate, because it describes the scholars’ activity.

(A)             Based on … incorrectly modifies scholars; the comparison between samples (plural) and that (singular) is awkward; extinct large mammals is not in proper apposition to Glyptodon and Holmesina.

(B)              Based on … incorrectly modifies scholars; the sentence lacks a noun after than to stand in comparison to samples.

(C)       Correct. Using accelerator mass spectrometry correctly modifies scholars;            samples and those stand properly in comparison; extinct large mammals … is in          appropriate apposition to Glyptodon and Holmesina.

(D)       In both of its appearances, the pronoun them lacks a clear reference.

(E)       Verb agreement problem. The singular subject, use, does not agree with the plural verb have enabled.

The correct answer is C.


GMAT Challenge Question: Taxing Sentence Correction

GMAT PrepIt’s time for another GMAT Challenge  Question, and it’s also about time to start thinking about your taxes – April 15 is just over two months away, and remember that February is a short month!  With those two things in mind, we  thought we’d let you audit an IRS-related Sentence Correction question.  Submit your answers in the comments field and we’ll be back later today with the solution and explanation.

Mutual funds, though helpful for personal investors who wish to diversify their portfolios, expose shareholders to additional taxation: not only are taxes on shareholders’ eventual sales of the securities collected by the IRS, but also on reinvested dividend stakes earned by the securities held by the fund itself.
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GMAT Challenge Question: Bet You “Kant” Pick The Correct Idiom…

GMAT PrepHello again, everyone!  It’s time for another GMAT Challenge Question, this time from the verbal side of the exam.  Try your hand at this Sentence Correction question, log your answers and explanations in the comments field, and we’ll be back with the official solution and an important lesson regarding GMAT Sentence Correction.
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GMAT Challenge Question: Blackout Dates

GMAT PrepHappy Thanksgiving, everyone!  With today as one of the busiest travel dates of the year, and undoubtedly a blackout date for your frequent-flier or rewards-miles program, we’ll offer this Data Sufficiency question for which blackouts can play a prominent role.

We’ll explain what we mean later today; for now, please enter your solutions in the comments field and we’ll be back this evening with an explanation and takeaway for which you’ll be thankful!

If y is an odd integer and the product of x and y equals 222, what is the value of x?

(1)  x is a prime number

(2)  y is a three-digit number

Update: Solution.  Much of this problem’s difficulty derives from the fact that it is hard for us to forget what we’ve already been told.   On Data Sufficiency questions, it’s therefore important to practice “Memory Blackout” when reading statement 2 – we must assess statement 2’s sufficiency on its own by blacking out everything we’ve learned from statement 1.
Here, statement 1 tells us that x is prime, and we already know that y is odd.  For an odd y to multiply by x and produce 222, an even number, x must be even.  Therefore, if x is both even and prime, x must be 2, and the statement is sufficient.

Attacking statement 2, we no longer know that x is prime – nor do we even know that x is an integer. We ONLY know that y is a 3-digit odd number.  You may be baited into thinking that x is an integer, bringing all or some of what you’ve learned from statement 1 into the process, but we  don’t know that at all.  y could be 333 and x could be 2/3 and statement 2 is still satisfied. Or, as you were probably trapped into predicting, y could be 111 and x could be 2.  But because we can’t know for certain in this case that x is an integer, the statement is not sufficient, and the correct answer is A.

GMAT Challenge Question: The Squared Circle

GMAT PrepIt’s time again for another GMAT Challenge Question, and this one features a favorite GMAT theme: you’re provided no actual numbers and need to use your conceptual knowledge to determine a proportional relationship.  Please post your answers in the comments field, and we’ll be back later today with a detailed solution:

Circle A is perfectly inscribed in a square, and the square is perfectly inscribed within circle B.  The area of circle B is what percent greater than the area of circle A?

(A)   50%

(B)   100%

(C)   150%

(D)  200%

(E)   250%

UPDATE: Solution!  One important note about inscribed shapes is that both squares and circles are perfectly symmetrical, meaning that if you know one length (radius, diameter, circumference, area; or side, diagonal, area, perimeter) you can solve for everything else.

In this case, is we call the radius of the smaller circle r, then the diameter of that circle is 2r and the area is pi*r^2.

If that circle is perfectly inscribed inside a square, then that means that the length of the diameter will perfectly fit within the boundaries of the square, making the side of the square also equal to 2r.

Now, if a circle is perfectly inscribed around that square, then the circle will hit each corner exactly once, making the diameter of the larger circle equal to the diagonal of the square.  Using what we know about squares (or using Pythagorean Theorem), we know that the diagonal is equal to the side * ?2, making the diameter of the larger circle equal to 2r*?2, and the radius of the larger circle then equal to half that: r*?2.

The area of the larger circle is the pi*(r*?2)^2, or 2*pi*r^2.  Therefore, the area of the larger circle (2*pi*r^2) is twice that of the smaller circle (pi*r^2).  But keep in mind that you must answer the right question!  The question asks “how much GREATER is the larger circle than the smaller” not “the larger circle is what percent OF the smaller”.  To get to twice the size, we only ADD 100%, so the correct answer is B.

GMAT Challenge Question: Solve This Stats Problem, Stat!

Set A consists of integers -9, 8, 3, 10, and J; Set B consists of integers -2, 5, 0, 7, -6, and T. If R is the median of Set A and W is the mode of set B, and R^W is a factor of 34, what is the value of T if J is negative?

(A) -2
(B) 0
(C) 1
(D) 2
(E) 5

Please include your answers in the comments field and we’ll be back later today with the solution!

UPDATE: Solution.

This problem demonstrates a helpful note about statistics problems – quite often the key to solving a stats problem is something other than stats: number properties, divisibility, algebra, etc. The statistics nature of these problems is often just a way to make a simpler problem look more difficult.

Here, the phrase “factor of 34” should stand out to you, as there are only four factors of 34, so you can narrow down the possibilities pretty quickly to 1, 2, 17, and 34. And because the number in question must be an exponential term that becomes a factor of 34, it’s even more limited: 2, 17, and 34 can only be created by one integer exponent – “itself” to the first power.

The base of that exponent is going to be the median of Set A, and because we know that the median of Set A will be 3 (a negative term for variable J means that 3 will be the middle term), the question becomes that much clearer. 3^W can only be a factor of 34 if it’s set equal to 1, and the only way to do that is for W to be 0. REMEMBER: anything to the power of 0 is equal to 1, a great equalizer on the GMAT!

Therefore, the correct answer is 0.

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GMAT Challenge Question: Prime Time

It’s time again for another GMAT challenge question, and this one focuses on one of the quantitative section’s favorite themes: prime factors.

Please submit your answers in the comments field, and check back later today for the solution and a more-thorough explanation of prime factors!

What is the greatest prime factor of 12!11! + 11!10!?

(A) 7
(B) 11
(C) 13
(D) 17
(E) 19

UPDATE: Solution!

While it’s quite common for students to simply look at the numbers 12!, 11!, and 10! and recognize that the highest naturally-occurring prime number is 11, it’s important to recognize that this is an addition problem – the numbers 12!11! and 11!10! are combined to create a new number that may well have a higher prime factor than its factorial components.

When adding large numbers like factorials and exponents, as we discussed in this space last week, it’s often quite helpful to factor out common terms. In this case, it’s particularly important, because our entire goal is to break out the large sum into prime factors so that we can determine which is biggest. Each term has a common 11!, so by factoring that out we can get from:

12!11! + 11!10!

to

11! (12! + 10!)

Now, 12! includes a 10! – it’s essentially 12 * 11 * 10!, so we have a common 10! within the parentheses that can also be factored out, going from:

11! (12*11*10! + 10!)

to

11!10! (12*11 + 1)

At this point, the largest prime factor must be either the 11 outside the parentheses or a factor of the number within it, so it’s necessary to check the number within. 12*11 + 1 = 132 + 1 = 133. 133 is the product of 7*19, so 19 is a prime factor of 12!11! + 11!10!, and therefore the largest prime factor. Accordingly, E is the correct answer.

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GMAT Challenge Question: Your Opponent is the Exponent

It’s time again for another Veritas Prep Challenge Question. Once again you’ll find that exponents will play a fairly significant role in this question. Stay tuned to the GMAT Tip of the Week post tomorrow for an explanation of this question and a quick checklist for everything you need to know about GMAT exponents.

For integers x, y, and z, if (3^x) (4^y) (5^z) = 3,276,800,000 and x + y + z = 15, what is the value of xy/z?

(A) undefined
(B) 0
(C) 3
(D) 5
(E) 15

Please submit your answers in the comments field, and check back tomorrow morning for an explanation and some critical exponent strategies.

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GMAT Challenge Question: Too Many Twos

Right at this second it’s September 2nd, with 2 days until the college football season starts, once again with too many teams in the Big Ten. In honor of all of these twos and toos, we present you a GMAT problem that features too many twos:

What is the value of 2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8?

(A) 2^9
(B) 2^10
(C) 2^11
(D) 3(2^10)
(E) 3(2^11)

Please post your answers in the comments field and we’ll post the solution later today!

Afternoon Update:

Great solutions, everyone. This question brings up an important point about exponents – we only have a few “core competencies” when it comes to performing with algebra, and those are:

-Multiplying/dividing exponents with common bases
-Finding patterns (units digits, relationships between adding/subtracting common terms, etc.)
-Setting common bases equal to equate exponents

Outside of that, there’s very little that we can do without the use of a calculator. So, in order to take advantage of what we do well, we should find ways when we see exponents to:

-Find common bases
-Multiply (using factorization to turn addition/subtraction into multiplication)

Here, we’re asked to add several terms together…that’s not something that we do well with exponents. However, by blending our abilities to factor terms (to get to multiplication) and to see patterns, we can attack this question relatively efficiently:

2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

Combine the 2s to be 4, or 2^2, and you have:

2^2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

Now we can add them together, and we have 2(2^2), or 2^3, simplifying the entire statement to:

2^3 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

Notice that we’ll be able to combine two more terms, the two 2^3 terms, to be 2(2^3) or 2^4, leaving:

2^4 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

By now hopefully you’ve seen a pattern (patterns come up frequently in exponent questions) – the first two terms will add to the third, and then adding those will add to the fourth:

2^4 + 2^4 (the first two) = 2^5 (so now we have two of the third term):

2^5 + 2^5 + 2^6 + 2^7 + 2^8

Do that again and we’ll have:

2^6 + 2^6 + 2^7 + 2^8

If we repeat the pattern, we’ll end up with:

2^8 + 2^8 = 2(2^8) = 2^9. Therefore, the correct answer is A.

When approaching exponent problems, keep your core competencies in mind: factor, multiply, find common bases, and look for patterns. These strategies will help you turn complicated problems into efficient solutions.

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GMAT Challenge Question: The Red Stapler

Studying for the GMAT does not have to be a chore — it can certainly be made enjoyable through fun challenge problems! That’s why we try to keep things lighthearted (yet effective) when it comes to your GMAT preparation.

With that in mind, try today’s Cheerful Challenge Problem, channeling your inner Peter Gibbons in your pursuit of an MBA to become the next Bill Lumbergh. Do it right, and you just may have The Bobs eating out of the palm of your hand. Check back later today for the answer to this problem!

GMAT Critical Reasoning Question

Milton: I believe that you have my red stapler.

Boss: Yeah…we switched from the Swingline to the Boston staplers some time ago. So I am just going to have to go ahead and confiscate this…

Milton: But I was told that I could keep this stapler, it is a better stapler and it does not jam.

Boss: Sorry, Milton. Oh and I’m going to have to ask you to go ahead and move your desk to the basement…

Milton and the Boss are committed to disagreeing about whether_______________

A) Milton should move his desk to the basement.

B) The red Swingline stapler jams less.

C) Milton should be allowed to keep his red stapler.

D) The company previously switched to Boston staplers.

E) Milton needs to go ahead and come in on Saturday.

What do you think? Post your response (along with an explanation!) in the comments field below, and we’ll add the solution later today!

UPDATE: SOLUTION

This is a unique variation of an Inference question, as the question asks for the subject on which the two parties must be in disagreement. They may well disagree on each of the answer choices, but only one of them is definite based on the passage.

The correct answer is C, as both parties mention the stapler, and each has a different opinion. Milton provides reasons that he should be allowed to keep it, and his boss denies him that opportunity and takes it. Because both parties explicitly express an opposite opinion about Milton’s ability to keep his stapler, they must be in disagreement about it.

As in any Inference problem, the other answers could be true, but are not necessarily true. It’s very likely that Milton does not want to come in on Saturday (who would?) and that he wouldn’t want to move his desk to the basement, but because there is no explicit evidence of either, neither is correct. Correct Inference answers on the GMAT must be true, so use that burden of proof to your advantage as you approach these questions on the exam.

Follow these and other steps to GMAT success, and you may one day be in a position to have as many as…four people working directly under you (or get a high-paying consulting job at McKinsey or Bain & Company as one of the Bobs).

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