How to Use Difference of Squares to Beat the GMAT

GMATIn Michael Lewis’ Flashboys, a book about the hazards of high-speed trading algorithms, Lewis relates an amusing anecdote about a candidate interviewing for a position at a hedge fund. During this interview, the candidate receives the following question: Is 3599 a prime number? Hopefully, your testing Spidey Senses are tingling and telling you that the answer to the question is going to incorporate some techniques that will come in handy on the GMAT. So let’s break this question down.

First, this is an interview question in which the interviewee is put on the spot, so whatever the solution entails, it can’t involve too much hairy arithmetic. Moreover, it is far easier to prove that a large number is NOT prime than to prove that it is prime, so we should be thinking about how we can demonstrate that this number possesses factors other than 1 and itself.

Whenever we’re given unpleasant numbers on the GMAT, it’s worthwhile to think about the characteristics of round numbers in the vicinity. In this case, 3599 is the same as 3600 – 1. 3600, the beautiful round number that it is, is a perfect square: 602. And 1 is also a perfect square: 12. Therefore 3600 – 1 can be written as the following difference of squares:

3600 – 1 = 602 – 12

We know that x– y= (x + y)(x – y), so if we were to designate “x” as “60” and “y” as “1”, we’ll arrive at the following:

60– 1= (60 + 1)(60 – 1) = 61 * 59

Now we know that 61 and 59 are both factors of 3599. Because 3599 has factors other than 1 and itself, we’ve proven that it is not prime, and earned ourselves a plumb job at a hedge fund. Not a bad day’s work.

But let’s not get ahead of ourselves. Let’s analyze some actual GMAT questions that incorporate this concept.

First:

999,9992 – 1 = 

A) 1010 – 2

B) (106 – 2) 2   

C) 105 (106 -2)

D) 106 (105 -2)

E) 106 (106 -2)

Notice the pattern. Anytime we have something raised to a power of 2 (or an even power) and we subtract 1, we have the difference of squares, because 1 is itself a perfect square. So we can rewrite the initial expression as 999,9992 – 12.

Using our equation for difference of squares, we get:

999,9992 – 12  = (999,999 +1)(999,999 – 1)

(999,999 + 1)(999,999 – 1) = 1,000,000* 999,998.

Take a quick glance back at the answer choices: they’re all in terms of base 10, so there’s a little work left for us to do. We know that 1,000,000 = 106  (Remember that the exponent for base 10 is determined by the number of 0’s in the figure.) And we know that 999,998 = 1,000,000 – 2 = 106 – 2, so 1,000,000* 999,998 = 106 (106 -2), and our answer is E.

Let’s try one more:

Which of the following is NOT a factor of 38 – 28?

A) 97

B) 65

C) 35

D) 13

E) 5

Okay, you’ll see quickly that 38 – 28 will involve same painful arithmetic. But thankfully, we’ve got the difference of two numbers, each of which has been raised to an even exponent, meaning that we have our trusty difference of squares! So we can rewrite 38 – 28 as (34)2 – (24)2. We know that 34 = 81 and 24 = 16, so (34)2 – (24)2 = 812 – 162. Now we’re in business.

812 – 162 = (81 + 16)(81 – 16) = 97 * 65.

Right off the bat, we can see that 97 and 65 are factors of our starting numbers, and because we’re looking for what is not a factor, A and B are immediately out. Now let’s take the prime factorization of 65. 65 = 13 * 5. So our full prime factorization is 97 * 13 * 5. Now we see that 13 and 5 are factors as well, thus eliminating D and E from contention. That leaves us with our answer C. Not so bad.

Takeaways:

  • The GMAT is not interested in your ability to do tedious arithmetic, so anytime you’re asked to find the difference of two large numbers, there is a decent chance that the number can be depicted as a difference of squares.
  • If you have the setup (Huge Number)2 – 1, you’re definitely looking at a difference of squares, because 1 is a perfect square.
  • If you’re given the difference of two numbers, both of which are raised to even exponents, this can also be depicted as a difference of squares, as all integers raised to even exponents are, by definition, perfect squares.

*Official Guide question courtesy of the Graduate Management Admissions Council.

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

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

Quarter Wit, Quarter Wisdom: Cyclicity in GMAT Remainder Questions (Part 2)

Quarter Wit, Quarter WisdomLast week, we reviewed the concepts of cyclicity and remainders and looked at some basic questions. Today, let’s jump right into some GMAT-relevant questions on these topics:

 

 

If n is a positive integer, what is the remainder when 3^(8n+3) + 2 is divided by 5?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

In this problem, we are looking for the remainder when the divisor is 5. We know from last week that if we get the last digit of the dividend, we will be able to find the remainder, so let’s focus on finding the units digit of 3^(8n + 3) + 2.

The units digit of 3 in a positive integer power has a cyclicity of: 3, 9, 7, 1

So the units digit of 3^(8n + 3) = 3^(4*2n + 3) will have 2n full cycles of 3, 9, 7, 1 and then a new cycle will start:

3, 9, 7, 1

3, 9, 7, 1

3, 9, 7, 1

3, 9, 7, 1

3, 9, 7

Since the exponent a remainder of 3, the new cycle ends at 3, 9, 7. Therefore, the units digit of 3^(8n + 3) is 7. When you add another 2 to this expression, the units digit becomes 7+2 = 9.

This means the units digit of 3^(8n+3) + 2 is 9. When we divide this by 5, the remainder will be 4, therefore, our answer is E.

Not so bad; let’s try a data sufficiency problem:

If k is a positive integer, what is the remainder when 2^k is divided by 10?

Statement 1: k is divisible by 10

Statement 2: k is divisible by 4

With this problem, we know that the remainder of a division by 10 can be easily obtained by getting the units digit of the number. Let’s try to find the units digit of 2^k.

The cyclicity of 2 is: 2, 4, 8, 6. Depending on the value of k is, the units digit of 2^k will change:

If k is a multiple of 4, it will end after one cycle and hence the units digit will be 6.

If k is 1 more than a multiple of 4, it will start a new cycle and the units digit of 2^k will be 2.

If k is 2 more than a multiple of 4, it will be second digit of a new cycle, and the units digit of 2^k will be 4.

If k is 3 more than a multiple of 4, it will be the third digit of a new cycle and the units digit of 2^k will be 8.

If k is 4 more than a multiple of 4, it will again be a multiple of 4 and will end a cycle. The units digit of 2^k will be 6 in this case.

and so on…

So what we really need to find out is whether k is a multiple of 4, one more than a multiple of 4, two more than a multiple of 4, or three more than a multiple of 4.

Statement 1: k is divisible by 10

With this statement, k could be 10 or 20 or 30 etc. In some cases, such as when k is 10 or 30, k will be two more than a multiple of 4. In other cases, such as when k is 20 or 40, k will be a multiple of 4. So for different values of k, the units digit will be different and hence the remainder on division by 10 will take multiple values. This statement alone is not sufficient.

Statement 2: k is divisible by 4

This statement tells you directly that k is divisible by 4. This means that the last digit of 2^k is 6, so when divided by 10, it will give a remainder of 6. This statement alone is sufficient. therefore our answer is B.

Now, to cap it all off, we will look at one final question. It is debatable whether it is within the scope of the GMAT but it is based on the same concepts and is a great exercise for intellectual purposes. You are free to ignore it if you are short on time or would not like to go an iota beyond the scope of the GMAT:

What is the remainder of (3^7^11) divided by 5?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

For this problem, we need the remainder of a division by 5, so our first step is to get the units digit of 3^7^{11}. Now this is the tricky part – it is 3 to the power of 7, which itself is to the power of 11. Let’s simplify this a bit; we need to find the units digit of 3^a such that a = 7^{11}.

We know that 3 has a cyclicity of 3, 9, 7, 1. Therefore (similar to our last problem) to get the units digit of 3^a, we need to find whether a is a multiple of 4, one more than a multiple of 4, two more than a multiple of 4 or three more than a multiple of 4.

We need a to equal 7^{11}, so first we need to find the remainder when a is divided by 4; i.e. when 7^{11} is divided by 4.

For this, we need to use the binomial theorem we learned earlier in this post (or we can use the method of “pattern recognition”):

The remainder of 7^{11} divided by 4

= The remainder of (4 + 3)^{11} divided by 4

= The remainder of 3^{11} divided by 4

= The remainder of 3*3^{10} divided by 4

= The remainder of 3*9^5 divided by 4

= The remainder of 3*(8+1)^5 divided by 4

= The remainder of 3*1^5 divided by 4

= The remainder of 3 divided by 4, which itself = 3

So when 7^{11} is divided by 4, the remainder is 3. This means 7^{11} is 3 more than a multiple of 4; i.e. a is 3 more than a multiple of 4.

Now we go back to 3^a. We found that a is 3 more than a multiple of 4. So there will be full cycles (we don’t need to know the exact number of cycles) and then a new cycle with start with three digits remaining:

3, 9, 7, 1

3, 9, 7, 1

3, 9, 7, 1

3, 9, 7, 1

3, 9, 7

With this pattern, we see the last digit of 3^7^11 is 7. When this 7 is divided by 5, remainder will be 2 – therefore, our answer is C.

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

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!

Quarter Wit, Quarter Wisdom: Cyclicity in GMAT Remainder Questions

Quarter Wit, Quarter WisdomUsually, cyclicity cannot help us when dealing with remainders, but in some cases it can. Today we will look at the cases in which it can, and we will see why it helps us in these cases.

First let’s look at a pattern:

 

20/10 gives us a remainder of 0 (as 20 is exactly divisible by 10)

21/10 gives a remainder of 1

22/10 gives a remainder of 2

23/10 gives a remainder of 3

24/10 gives a remainder of 4

25/10 gives a remainder of 5

and so on…

In the case of this pattern, 20 is the closest multiple of 10 that goes completely into all these numbers and you are left with the units digit as the remainder. Whenever you divide a number by 10, the units digit will be the remainder. Of course, if the units digit of a number is 0, the remainder will be 0 and that number will be divisible by 10 — but we already know that. So remainder when 467,639 is divided by 10 is 9. The remainder when 100,238 is divided by 10 is 8 and so on…

Along the same lines, we also know that every number that ends in 0 or 5 is a multiple of 5 and every multiple of 5 must end in either 0 or 5. So if the units digit of a number is 1, it gives a remainder of 1 when divided by 5. If the units digit of a number is 2, it gives a remainder of 2 when divided by 5. If the units digit of a number is 6, it gives a remainder of 1 when divided by 5 (as it is 1 more than the previous multiple of 5).

With this in mind:

20/5 gives a remainder of 0 (as 20 is exactly divisible by 5)

21/5 gives a remainder of 1

22/5 gives a remainder of 2

23/5 gives a remainder of 3

24/5 gives a remainder of 4

25/5 gives a remainder of 0 (as 25 is exactly divisible by 5)

26/5 gives a remainder of 1

27/5 gives a remainder of 2

28/5 gives a remainder of 3

29/5 gives a remainder of 4

30/5 gives a remainder of 0 (as 30 is exactly divisible by 5)

and so on…

So the units digit is all that matters when trying to get the remainder of a division by 5 or by 10.

Let’s take a few questions now:

What is the remainder when 86^(183) is divided by 10?

Here, we need to find the last digit of 86^(183) to get the remainder. Whenever the units digit is 6, it remains 6 no matter what the positive integer exponent is (previously discussed in this post).

So the units digit of 86^(183) will be 6. So when we divide this by 10, the remainder will also be 6.

Next question:

What is the remainder when 487^(191) is divided by 5?

Again, when considering division by 5, the units digit can help us.

The units digit of 487 is 7.

7 has a cyclicity of 7, 9, 3, 1.

Divide 191 by 4 to get a quotient of 47 and a remainder of 3. This means that we will have 47 full cycles of “7, 9, 3, 1” and then a new cycle will start and continue until the third term.

7, 9, 3, 1

7, 9, 3, 1

7, 9, 3, 1

7, 9, 3, 1

7, 9, 3

So the units digit of 487^(191) is 3, and the number would look something like ……………..3

As discussed, the number ……………..0 would be divisible by 5 and ……………..3 would be 3 more, so it will also give a remainder of 3 when divided by 5.

Therefore, the remainder of 487^(191) divided by 5 is 3.

Last question:

If x is a positive integer, what is the remainder when 488^(6x) is divided by 2?

Take a minute to review the question first. If you start by analyzing the expression 488^(6x), you will waste a lot of time. This is a trick question! The divisor is 2, and we know that every even number is divisible by 2, and every odd number gives a remainder 1 when divided by 2. Therefore, we just need to determine whether 488^(6x) is odd or even.

488^(6x) will be even no matter what x is (as long as it is a positive integer), because 488 is even and we know even*even*even……(any number of terms) = even.

So 488^(6x) is even and will give remainder 0 when it is divided by 2.

That is all for today. We will look at some GMAT remainders-cyclicity questions next week!

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

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!

GMAT Tip of the Week: The Detroit Lions Teach How NOT to Take the GMAT

GMAT Tip of the WeekIf you’re applying to business schools in Round 2, you’re looking for good news (acceptance!) or a chance to advance to the next round (you’ve been invited to interview!) or even just a lack of bad news (you’re on the waitlist…there’s still a chance!) in January or February. Well if those who don’t learn from history are condemned to repeat it, you’d be well served to avoid the pitfalls of the Detroit Lions, an NFL franchise that hasn’t had January/February good news or a chance to advance since 1991.

Any Detroit native could write a Grishamesque (same thing year after year, but we keep coming back for more) series of books about the many losing-based lessons the Lions have taught over the years, but this particular season beautifully showcases one of the most important GMAT lessons of all:

Finish the job.

Six weeks ago, this lesson was learned as Calvin Johnson took the game-winning touchdown within inches of the goal line before having the ball popped out by Seattle Seahawks star Kam Chancellor. And last night this lesson was learned as Green Bay Packers star Aaron Rodgers completed a 60+ yard Hail Mary pass on an untimed final down.

On the GMAT, you have the same opportunities and challenges as the Detroit Lions do: stiff competition (there are Rodgerses and Chancellors hoping to get that spot at Harvard Business School, too) and a massive penalty for doing everything right until the last second. Lions fans and GMAT instructors share the same pain — our teams and our students are often guilty of doing absolutely everything right and then making one fatal mistake at the finish and not getting any credit for it. Consider the example:

A bowl of fruit contains 14 apples and 23 oranges. How many oranges must be removed so that 70% of the pieces of fruit in the bowl will be apples?

(A) 3

(B) 6

(C) 14

(D) 17

(E) 20

Here, most GMAT students get off to a great start, just like the Lions did going up 17-0. They know that the 14 apples (that number remains unchanged) need to represent 70% of the new total. If 14 = 0.7(x), then the algebra becomes quick. Multiply both sides by 10 to get rid of the decimal: 140 = 7x. Then divide both sides by 7 and you have x = 20. And you also have your first opportunity to “Lion up”: 20 is an answer choice! But 20 doesn’t represent the number of oranges; that’s the total for pieces of fruit after the orange removal. So 20 is a trap.

You then need to recognize that 14 of that 20 is apples, so you have 14 apples and 6 oranges in the updated bowl. But Lions beware! 6 is an answer choice, but it’s not the right one: you have 6 oranges LEFT but the question asks for the number REMOVED. That means that you have to subtract the 6 you kept from the 23 you started with, and the correct answer is D, 17.

What befalls many GMAT students is that ticking clock and the pressure to move on to the next problem. By succumbing to that time/pace pressure — or by being so relieved, and maybe even surprised, that their algebra is producing numbers that match the answer choices — they fail to play all the way to the final gun, and like the Lions, they tragically lose a “game” (or problem) that they should have won. Which, as any Lions fan will tell you, is tragic. When you get blown out in football or you simply can’t hack the math on the GMAT, it’s sad but not devastating: you’re just not good enough (sorry, Browns fans). But when you’ve proven that you’re good enough and lose out because you didn’t finish the job, that’s crushing.

Now, like Lions fans talking about the phantom facemask call last night, you may be thinking, “That’s unfair! What a dirty question to ask about how many ‘left over’ instead of how many remaining. I hate the GMAT and I hate the refs!” And regardless of whether you have a fair point, you have to recognize that it’s part of the game.

The GMAT won’t give you credit for being on the right track — you have to get the problem right and be ready for that misdirection in the question itself. So learn from the Lions and make sure you finish every problem by double-checking that you’ve answered exactly the question that they asked. Finish the job, and you won’t have to wait 24 years and counting to finally have good news in January.

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!

By Brian Galvin.

Is Your GMAT Score More Important Than Ever?

GMAT ReasoningThe dreaded GMAT has long been one of the most feared components of the MBA application process. For many years the importance of the GMAT has been a bit overvalued by applicants, with too much focus being placed on the score and not enough on other areas of the application process. Just as admissions committees’ consistent message of their reliance on holistic reviews of candidate profiles has begun to sink in, a shift has seemingly started back the other way.

Although there has been a consistent upward trends over the last few decades in GMAT scores across the board, over the last year or two in particular the average GMAT scores at top MBA programs like Northwestern’s Kellogg School, Chicago’s Booth School and Pennsylvania’s Wharton School have risen by record percentage points. These record averages should signal to prospective applicant’s the increased importance of the GMAT.

Now, GMAT scores have always been important aspects of the MBA admissions process, but should applicants be more concerned with the rising scores at these top MBA programs?  The quick answer is no!  But you do want to accept this answer with a bit of a caveat: with dramatically rising GMAT scores across the board, it is even more important for applicants to target programs that are a clear fit for their background and showcased aptitude (GPA/GMAT). More specifically, applying to programs where your GMAT score falls below the average score has become a riskier option.

The typical candidate should make sure they hit or are very close to the listed averages. Now for candidates coming from a more competitive applicant pool like the Indian male, White male, and Asian male, it is important to target a score above schools’ listed averages to ensure you stand out from the pack. For non-traditional applicants, a strong GMAT score can be a way to stand out in the face of rising scores and increased competition.

The main takeaway from this trend for all applicants should be to really focus up front on creating the right list of target schools. Mind you, this list should not simply be one of the top 10 programs. Instead, create a list where your academic aptitude, professional goals, and other data points all align with the programs you plan to apply to so that you are able to maximize your chances of gaining admission to your target schools.

Applying to business school? Call us at 1-800-925-7737 and speak with an MBA admissions expert today, or click here to take our free MBA Admissions Profile Evaluation for personalized advice for your unique application situation! As always, be sure to find us on Facebook, YouTube, Google+ and Twitter.

Dozie A. is a Veritas Prep Head Consultant for the Kellogg School of Management at Northwestern University. His specialties include consulting, marketing, and low GPA/GMAT applicants. You can read more of his articles here.

Quarter Wit, Quarter Wisdom: Cyclicity of Units Digits on the GMAT (Part 2)

Quarter Wit, Quarter WisdomAs discussed last week, all units digits have a cyclicity of 1 or 2 or 4. Digits 2, 3, 7 and 8 have a cyclicity of 4, i.e. the units digit repeats itself every 4 digit:

Cyclicity of 2: 2, 4, 8, 6

Cyclicity of 3: 3, 9, 7, 1

Cyclicity of 7: 7, 9, 3, 1

Cyclicity of 8: 8, 4, 2, 6

Digits 4 and 9 have a cyclicity of 2, i.e. the units digit repeats itself every 2 digits:

Cyclicity of 4: 4, 6

Cyclicity of 9: 9, 1

Digits 0, 1, 5 and 6 have a cyclicity of 1, i.e. the units digit is 0, 1, 5, or 6 respectively.

Now let’s take a look at how to apply these fundamentals:

What is the units digit of 813^(27)?

To get the desired units digit here, all we need to worry about is the units digit of the base, which is 3.

Remember, our cyclicity of 3 is 3, 9, 7, 1 (four numbers total).

We need the units digit of 3^(27). How many full cycles of 4 will be there in 27? There will be 6 full cycles because 27 divided by 4 gives 6 as quotient and 3 will be the remainder. So after 6 full cycles of 4 are complete, a new cycle will start:

3, 9, 7, 1

3, 9, 7, 1

… (6 full cycles)

3, 9, 7 (new cycle for remainder of 3)

7 will be the units digit of 3^(27), so 7 will be the units digit of 813^(27).

Let’s try another question:

What is the units digit of 24^(1098)?

To get the desired units digit here, all we need to worry about is the units digit of the base, which is 4.

Remember, our cyclicity of 4 is 4 and 6 (this time, only 2 numbers).

We need the units digit of 24^(1098) – every odd power of 24 will end in 4 and every even power of 24 will end in 6.

Since 1098 is even, the units digit of 24^(1098) is 6.

Not too bad; let’s try something a little harder:

What is the units digit of 75^(25)^5

Note here that you have 75 raised to power 25 which is further raised to the power of 5.

25^5 is not the same as 25*5 – it is 25*25*25*25*25 which is far more complicated. However, the simplifying element of this question is that the last digit of the base 75 is 5, so it doesn’t matter what the positive integer exponent is, the last digit of the expression will always be 5.

Now let’s take a look at a Data Sufficiency question:

Given that x and y are positive integers, what is the units digit of (5*x*y)^(289)?

Statement 1: x is odd.

Statement 2: y is even.

Here there is a new complication – we don’t know what the base is exactly because the base depends on the value of x and y. As such, the real question should be can we figure out the units digit of the base? That is all we need to find the units digit of this expression.

When 5 is multiplied by an even integer, the product ends in 0.

When 5 is multiplied by an odd integer, the product ends in 5.

These are the only two possible cases: The units digit must be either 0 or 5.

With Statement 1, we do not know whether y is odd or even, we only know that x is odd. If y is odd, x*y will be odd. If y is even, x*y will be even. Since we don’t know whether x*y is odd or even, we don’t know whether 5*x*y will end in 5 or 0, so this statement alone is not sufficient.

With Statement 2, if y is even, x*y will certainly be even because an even * any integer will equal an even integer. Therefore, it doesn’t matter whether x is odd or even – regardless, 5*x*y will be even, hence, it will certainly end in 0.

As we know from our patterns of cyclicity, 0 has a cyclicity of 1, i.e. no matter what the positive integer exponent, the units digit will be 0. Therefore, this statement alone is sufficient and the answer is B (Statement 2 alone is sufficient but Statement 1 alone is not sufficient).

Finally, let’s take a question from our own book:

If n and a are positive integers, what is the units digit of n^(4a+2) – n^(8a)?

Statement 1: n = 3

Statement 2: a is odd.

We know that the cyclicity of every digit is either 1, 2 or 4. So to know the units digit of n^{4a+2} – n^{8a}, we need to know the units digit of n. This will tell us what the cyclicity of n is and what the units digit of each expression will be individually.

Statement 1: n = 3

As we know from our patterns of cyclicity, the cyclicity of 3 is 3, 9, 7, 1

Plugging 3 into “n”, n^{4a+2} = 3^{4a+2}

In the exponent, 4a accounts for “a” full cycles of 4, and then a new cycle begins to account for 2.

3, 9, 7, 1

3, 9, 7, 1

3, 9

The units digit here will be 9.

Again, plugging 3 into “n”, n^{8a} = 3^{8a}

8a is a multiple of 4, so there will be full cycles of 4 only. This means the units digit of 3^{8a} will be 1.

3, 9, 7, 1

3, 9, 7, 1

3, 9, 7, 1

3, 9, 7, 1

Plugging these answers back into our equation: n^{4a+2} – n^{8a} = 9 – 1

The units digit of the combined expression will be 9 – 1 = 8.

Therefore, this statement alone is sufficient.

In Statement 2, we are given what the exponents are but not what the value of n, the base, is. Therefore, this statement alone is not sufficient, and our answer is A (Statement 1 alone is sufficient but Statement 2 alone is not sufficient).

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

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!

GMAT Tip of the Week: What Test-Takers Should Be Thankful For

GMAT Tip of the WeekIf you’re spending this Thanksgiving weekend studying for the GMAT in hopes of a monster score for your Round 2 applications, there’s a good chance you’re feeling anything but grateful. At the very least, that practice test kept you inside and away from the hectic horror that has become Black Friday, but it’s understandable that when you spend the weekend thinking more about pronouns than Pilgrims and modifiers than Mayflowers, your introduction to the holiday season has you saying “bah, humbug.”

As you study, though, keep the spirit of Thanksgiving close to your heart. Those who made the first pilgrimage to New England didn’t have it easy, either – Thanksgiving is about being grateful for the small blessings that allowed them to survive in the land of HBS, Yale, Sloan, and Tuck. And the GMAT gives you plenty to be thankful for as you attempt to replicate their journey to the heart of elite academia. This Thanksgiving, GMAT test-takers should be thankful for:

1) Answer Choices

While it’s normal to dislike standardized, multiple-choice tests, those multiple choices are often the key to solving problems efficiently and correctly. They let you know whether you can get away with an estimate, allow you to backsolve or pick numbers to test the choices, and offer you insight into how you should attack the problem (that square root of 3 probably came from a 30-60-90 triangle if you can find it). On the Verbal Section, they allow you to use process of elimination, and particularly on Sentence Correction, to see what the true Decision Points are. A test without answer choices would mean that you’d have to do every problem the long way, but those who know to be thankful for answer choices will often find a competitive advantage.

2) Right Triangles

Right triangles are everywhere on GMAT geometry problems, and learning to use them to your advantage gives you a huge (turkey?) leg up on the competition. Right triangles:

  • Provide you with side ratios, or at least the Pythagorean Theorem
  • Make the base-height combination for the area of a triangle easy (just use the two sides adjacent to the right angle as your base and height)
  • Allow you to use the Pythagorean Theorem to solve for the distance between any two points in the coordinate plane
  • Let you make the greatest difference between any two points in a square, rectangle, cylinder, or box the hypotenuse of a right triangle
  • Help you divide strange shapes into easy-to-solve triangles

Much of GMAT geometry comes down to finding and leveraging right triangles, so thankful that you have that opportunity.

3) Verbs

When there are too many differences between Sentence Correction answer choices, it can be difficult to determine which decision points are most important. One key: look for verbs. When answer choices have different forms of the same verb – whether different tenses or singular-vs.-plural – that’s nearly always a primary decision point and a decision that you can make well using logic. Does the timeline make sense or not? Is the subject singular or plural? Often the savviest test-takers are the ones who save the difficult decisions for last and look for verbs first. Whenever you see different versions of the same verb in the answer choices, be thankful – your job just got easier.

4) “The Other Statement”

Data Sufficiency is a challenging question type, and one that seems to always feature a very compelling trap answer. Very often that trap answer is tempting because:

A statement that didn’t look to be sufficient actually is sufficient.

A statement that looked sufficient actually isn’t.

And that, “Is this tricky statement sufficient or not?” decision is an incredibly difficult one in a vacuum, but the GMAT (thankfully!) gives you a clue: the other statement. When one statement is obvious, its role is often to serve as a clue (“you’d better consider whether you need to know this or not when you look at the other statement”) or a trap (“you actually don’t need this, but when we tempt you with it you’ll think you do”). In either case, the obvious statement is telling you what you need to consider – why would that piece of information matter, or not? So be thankful that Data Sufficiency doesn’t require you to confirm your decision on each statement alone before you get to look at them together; taking the hint from one statement is often the best way to effectively assess the other.

5) Extra Words in Critical Reasoning Conclusions

If you spend any of this holiday weekend watching football, watch what happens when the offense employs the “man in motion” play (having one of the wide receivers run from one side of the offense to the other). Either the defensive player opposite him follows (suggesting man coverage) or he doesn’t (suggesting zone). With the “man in motion”, the offense is probing the defense to see, “What kind of defense are you playing?”. On GMAT Critical Reasoning, extra words in the conclusion serve an almost identical purpose – if you’re looking carefully, you’ll see exactly what’s important to the problem:

Country X therefore has to increase jobs in oil refinement in order to avoid a surge in unemployment. (Why does it have to be refinement? The traps will be about other jobs related to oil but not specifically refinement.)

Therefore, Company Y needs to cut its marketing expenses. (Why marketing and not other kinds of expenses?)

The population of black earthworms is now almost equal to that of the red-brown earthworm, a result, say local ecologists, solely stemming from the blackening of the woods. (Solely? You can weaken this conclusion by finding just one alternate reason)

For much of the Verbal Section, the more words you have to read, the more difficult your job is to process them all. But on Critical Reasoning, be thankful when you see extra words in the conclusion – those words tell you exactly what game the author is playing.

6) The CAT Algorithm

For many test-takers, the computer-adaptive scoring algorithm is something to be angry or frustrated about, and certainly not something to be thankful for. But if you look from the right angle (and you know we’re already thankful for right angles…) there’s plenty to be happy about, including:

  • You’re allowed to miss questions and make mistakes. The CAT system ensures that everyone sees a challenging test, so everyone will make mistakes. You don’t have to be perfect (and probably shouldn’t try).
  • You get your scores immediately. Talk to your friends taking the LSAT and see how they feel about turning in their answer sheet and then…waiting. In an instant gratification society, the GMAT gives you that instant feedback you crave. Do well and celebrate; do worse than you thought and immediately start game-planning the next round while it’s fresh in your mind.
  • It favors the prepared. You’re reading a GMAT blog during your spare time… you’ll be among those who prepare! The pacing is tricky since you can’t return to problems later, but remember that everyone takes the same test. If you’ve prepared and have a good sense of how to pace yourself, you’ll do better than those who are surprised by the setup and don’t plan accordingly. An overall disadvantage can still be a terrific competitive advantage, so as you’re looking for GMAT-themed things to be thankful for, keep your preparation in mind and be thankful that you’re working harder than your competition and poised to see the rewards!

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!

By Brian Galvin.

Quarter Wit Quarter Wisdom: Cyclicity of Units Digits on the GMAT

Quarter Wit, Quarter WisdomIn our algebra book, we have discussed finding and extrapolating patterns. In this post today, we will look at the patterns we get with various units digits.

The first thing you need to understand is that when we multiply two integers together, the last digit of the result depends only on the last digits of the two integers.

For example:

24 * 12 = 288

Note here: …4 * …2 = …8

So when we are looking at the units digit of the result of an integer raised to a certain exponent, all we need to worry about is the units digit of the integer.

Let’s look at the pattern when the units digit of a number is 2.

Units digit 2:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32

2^6 = 64

2^7 = 128

2^8 = 256

2^9 = 512

2^10 = 1024

Note the units digits. Do you see a pattern? 2, 4, 8, 6, 2, 4, 8, 6, 2, 4 … and so on

So what will 2^11 end with? The pattern tells us that two full cycles of 2-4-8-6 will take us to 2^8, and then a new cycle starts at 2^9. 

2-4-8-6

2-4-8-6

2-4

The next digit in the pattern will be 8, which will belong to 2^11. 

In fact, any integer that ends with 2 and is raised to the power 11 will end in 8 because the last digit will depend only on the last digit of the base. 

So 652^(11) will end in 8,1896782^(11) will end in 8, and so on…

A similar pattern exists for all units digits. Let’s find out what the pattern is for the rest of the 9 digits. 

Units digit 3:

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6 = 729

The pattern here is 3, 9, 7, 1, 3, 9, 7, 1, and so on…

Units digit 4:

4^1 = 4

4^2 = 16

4^3 = 64

4^4 = 256

The pattern here is 4, 6, 4, 6, 4, 6, and so on… 

Integers ending in digits 0, 1, 5 or 6 have the same units digit (0, 1, 5 or 6 respectively), whatever the positive integer exponent. That is:

1545^23 = ……..5

1650^19 = ……..0

161^28 = ………1

Hope you get the point.

Units digit 7:

7^1 = 7

7^2 = 49

7^3 = 343

7^4 = ….1 (Just multiply the last digit of 343 i.e. 3 by another 7 and you get 21 and hence 1 as the units digit)

7^5 = ….7 (Now multiply 1 from above by 7 to get 7 as the units digit)

7^6 = ….9

The pattern here is 7, 9, 3, 1, 7, 9, 3, 1, and so on…

Units digit 8:

8^1 = 8

8^2 = 64

8^3 = …2

8^4 = …6

8^5 = …8

8^6 = …4

The pattern here is 8, 4, 2, 6, 8, 4, 2, 6, and so on…

Units digit 9: 

9^1 = 9

9^2 = 81

9^3 = 729

9^4 = …1

The pattern here is 9, 1, 9, 1, 9, 1, and so on…

Summing it all up:

1) Digits 2, 3, 7 and 8 have a cyclicity of 4; i.e. the units digit repeats itself every 4 digits.

Cyclicity of 2: 2, 4, 8, 6

Cyclicity of 3: 3, 9, 7, 1

Cyclicity of 7: 7, 9, 3, 1

Cyclicity of 8: 8, 4, 2, 6

2) Digits 4 and 9 have a cyclicity of 2; i.e. the units digit repeats itself every 2 digits.

Cyclicity of 4: 4, 6

Cyclicity of 9: 9, 1 

3) Digits 0, 1, 5 and 6 have a cyclicity of 1.

Cyclicity of 0: 0

Cyclicity of 1: 1

Cyclicity of 5: 5

Cyclicity of 6: 6

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

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!

Is Technology Costing You Your GMAT Score?

Veritas Prep GMAT Prep Books on iPadI recently read Sherry Turkle’s Reclaiming Conversation: The Power of Talk in a Digital Age. While the book isn’t about testing advice, per se, its analysis of the costs of technology is so comprehensive that the insights are applicable to virtually every aspect of our lives.

The book’s core thesis – that our smartphones and tablets are fragmenting our concentration and robbing us of a fundamental part of what it means to be human – isn’t a terribly original one. The difference between Turkle’s work and less effective screeds about the evils of technology is the scope of the research she provides in demonstrating how the overuse of our devices is eroding the quality of our education, our personal relationships, and our mental health.

What’s amazing is that these costs are, to some extent, quantifiable. Ever wonder what the impact is of having most of our conversations mediated through screens rather than through hoary old things like facial expressions? College students in the age of smartphones score 40% lower on tests measuring indicators of empathy than college students from a generation ago. In polls, respondents who had access to smartphones by the time they were adolescents reported heightened anxiety about the prospect of face-to-face conversations in general.

Okay, you say. Disturbing as that is, those findings have to do with interpersonal relationships, not education. Can’t technology be used to enhance the learning environment as well? Though it would be silly to condemn any technology as wholly corrosive, particularly in light of the fact that most schools are making a concerted effort to incorporate laptops and tablets in the classroom, Turkle makes a persuasive case that the overall costs outweigh the benefits.

In one study conducted by Pam Mueller and Daniel Oppenheimer, the researchers compared the retention rates of students who took notes on their laptops versus those who took notes by hand. The researchers’ assumption had always been that taking notes on a laptop would be more beneficial, as most of us can type faster than we can write longhand. Much to their surprise, the students who took notes by hand did significantly better than those who took notes on their laptops when tested on the contents of a lecture a week later.

The reason, Mueller and Oppenheimer speculate, is that because the students writing longhand couldn’t transcribe fast enough to record everything, they had to work harder to filter the information they were provided, and this additional cognitive effort allowed them to retain more. The ease of transcription – what we perceive as a benefit of technology – actually proved to be a cost. Even more disturbing, another study indicated that the mere presence of a smartphone – even if the phone is off – will cause everyone in its presence to retain less of a lecture, not just the phone’s owner.

I’ve been teaching long enough that when I first started, it was basically unheard of for a student’s attention to wander because he’d been distracted by a device. Smartphones didn’t exist yet. No one brought laptops to class. Now, if I were to take a poll, I’d be surprised if there were a single student in class who didn’t at least glance at a smartphone during the course of a lesson. One imagines that the same is true when students are studying on their own – a phone is nearby, just in case something important comes up. I’d always assumed the presence of these devices was relatively harmless, but if a phone that’s off can degrade the quality of our study sessions, just imagine the impact of a phone that continually pings and buzzes as fresh texts, emails and notifications come in.

The GMAT is a four-hour test that requires intense focus and concentration, so anything that hampers our ability to focus is a potential drag on our scores. There’s no easy solution here. I’m certainly not advocating that anyone throw away their smartphone – the fact that certain technology has costs associated with it is hardly a reason to discard that technology altogether. There are plenty of well-documented educational benefits: one can use a long train ride as an opportunity to do practice problems or watch a lecture. We can easily store data that can shed light on where we need to focus our attention in future study sessions. So the answer isn’t a draconian one in which we have to dramatically alter our lifestyles. Technology isn’t going anywhere – it’s a question of moderation.

Takeaways: No rant about the costs of technology is going to be terribly helpful without an action plan, so here’s what I suggest:

  • Put the devices away in class and take notes longhand. Whether you’re in a GMAT prep class, or an accounting class in your MBA program, this will benefit both you and your classmates.
  • If you aren’t using your device to study, turn it off, and make sure it’s out of sight when you work. The mere visual presence of a smartphone will cause you to retain less.
  • Give yourself at least 2 hours of device-free time each day. This need not be when you’re studying. It can also be when you’re out to dinner with friends or spending time with family. In addition to improving your interpersonal relationships, conversation actually makes you smarter.

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

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

What to Do if You’re Struggling with GMAT Solutions

stressed-studentOne of the most misleading parts of the whole GMAT experience is the process of reading the solution to a math problem in the Quant section. When you try the problem, you struggle, sweat, and go nowhere; when they explain the problem, they wave a snooty, know-it-all magic wand that clears everything up. But how did they think of that? What can you do to think like them (or barring that, where do they keep that magic wand, and how late do we have to break into their house to be sure they’re asleep when we steal it)?

The short answer is that they struggled just like you did, but like anybody else, they wanted to make it look easy. (Think of all the time some people spend preening their LinkedIn or their Instagram: you only ever see the flashy corporate name and the glamour shot, never the 5 AM wake up call or the 6 AM look in the mirror.) Solution writers, particularly those who work for the GMAC, never seem to tell you that problem solving is mostly about blundering through a lot of guesswork before hitting upon a pattern, but that’s really what it is. Your willingness to blunder around until you hit something promising is a huge part of what’s being tested on the GMAT; after all, as depressing as it sounds, that’s basically how life works.

Here’s a great example:

I haven’t laid eyes on it in thirty years, but I still remember that the rope ladder to my childhood treehouse had exactly ten rungs. I was a lot shorter then, and a born lummox, so I could only climb the ladder one or two rungs at a time. I also had more than a touch of childhood OCD, so I had to climb the ladder a different way every time. After how many trips up did my OCD prevent me from ever climbing it again? (In other words, how many different ways was I able to climb the ladder?)

A) 55       

B) 63       

C) 72       

D) 81        

E) 89

Just the thought of trying 55 to 89 different permutations of climbing the ladder has my OCD going off like a car alarm, so I’m going to look for an easier way of doing this. It’s a GMAT problem, albeit one on the level of a Google interview question, so it must have a simple solution. There has to be a pattern here, or the problem wouldn’t be tested. Maybe I could find that pattern, or at least get an idea of how the process works, if I tried some shorter ladders.

Suppose the ladder had one rung. That’d be easy: there’s only one way to climb it.

Now suppose the ladder had two rungs. OK, two ways: I could go 0-1 then 1-2, or straight from 0-2 in a single two step, so there are two ways to climb the ladder.

Now suppose that ladder had three rungs. 0-1, 1-2, 2-3 is one way; 0-2, 2-3 is another; 0-1, 1-3 is the third. So the pattern is looking like 1, 2, 3 … ? That can’t be right! Doubt is gnawing at me, but I’m going to give it one last shot.

Suppose that the ladder had four rungs. I could do [0-1-2-3-4] or [0-1-3-4] or [0-1-2-4] or [0-2-4] or [0-2-3-4]. So there are five ways to climb it … wait, that’s it!

While I was mucking through the ways to climb my four-rung ladder, I hit upon something. When I take my first step onto the ladder, I either climb one rung or two. If I climb one rung, then there are 3 rungs left: in other words, I have a 3-rung ladder, which I can climb in 3 ways, as I saw earlier. If my step is a two-rung step instead, then there are 2 rungs left: in other words, a 2-rung ladder, which I can climb in 2 ways. Making sense?

By the same logic, if I want to climb a 5-rung ladder, I can start with one rung, then have a 4-rung ladder to go, or start with two rungs, then have a 3-rung ladder to go. So the number of ways to climb a 5-rung ladder = (the number of ways to climb a 3-rung ladder) + (the number of ways to climb a 4-rung ladder). Aha!

My pattern starts 1, 2, 3, so from there I can find the number of ways to climb each ladder by summing the previous two. This gives me a 1-, 2-, 3-, … rung ladder list of 1, 2, 3, 5, 8, 13, 21, 34, 55, and 89, so a 10-rung ladder would have 89 possible climbing permutations, and we’re done.

And the lesson? Much like a kid on a rope ladder, for a GMAT examinee on an abstract problem there’s often no “one way” to do the problem, at least not one that you can readily identify from the first instant you start. Very often you have to take a few small steps so that in doing so, you learn what the problem is all about. When all else fails in a “big-number” problem, try testing the relationship with small numbers so that you can either find a pattern or learn more about how you can better attack the bigger numbers. Sometimes your biggest test-day blunder is not allowing yourself to blunder around enough to figure the problem out.

Congratulations: that’s the hardest GMAT problem you’ve solved yet! (And bonus points if you noticed that the answer choices differed by 8, 9, 9, and 8. I still have OCD, and a terrible sense of humor.)

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You Can Do It! How to Work on GMAT Work Problems

Pump UpRate questions, so far as I can remember, have been a staple of almost every standardized test I’ve ever taken. I recall seeing them on proficiency tests in grade school. They showed up on the SAT. They were on the GRE. And, rest assured, dear reader, you will see them on the GMAT. What’s peculiar is that despite the apparent ubiquity of these problems, I never really learned how to do them in school. This is true for many of my students as well, as they come into my class thinking that they’re just not very good at these kinds of questions, when, in actuality, they’ve just never developed a proper approach. This is doubly true of work problems, which are just a kind of rate problem.

When dealing with a complex work question there are typically only two things we need to keep in mind, aside from our standard “rate * time = work” equation. First, we know that rates are  additive. If I can do 1 job in 4 hours, my rate is 1/4. If you can do 1 job in 3 hours, your rate is 1/3. Therefore, our combined rate is 1/4 + 1/3, or 7/12. So we can do 7 jobs in 12 hours.

The second thing we need to bear in mind is that rate and time have a reciprocal relationship. If our rate is 7/12, then the time it would take us to complete a job is 12/7 hours. Not so complex. What’s interesting is that these simple ideas can unlock seemingly complex questions. Take this official question, for example:

Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours; pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank.

A) 1/3

B) 1/2

C) 2/3

D) 5/6

E) 1

So let’s start by assigning some variables. We’ll call the rate for p ump A, Ra. Similarly, we’ll designate the rate for pump B as Rb,and the rate for pump C as Rc.

If the time for A and B together to fill the tank is 6/5 hours, then we know that their combined rate is 5/6, because again, time and rate have a reciprocal relationship. So this first piece of information yields the following equation:

Ra + Rb = 5/6.

If A and C can fill the tank in 3/2 hours, then, employing identical logic, their combined rate will be 2/3, and we’ll get:

Ra + Rc = 2/3.

Last, if B and C can fill tank in 2 hours, then their combined rate will be ½, and we’ll have:

Rb+ Rc = 1/2.

Ultimately, what we want here is the time it would take all three pumps working together to fill the tank. If we can find the combined rate, or Ra + Rb + Rc, then all we need to do is take the reciprocal of that number, and we’ll have our time to full the pump. So now, looking at the above equations, how can we get Ra + Rb + Rc on one side of an equation? First, let’s line our equations up vertically:

 Ra + Rb = 5/6.

Ra + Rc = 2/3.

Rb+ Rc = 1/2.

 Now, if we sum those equations, we’ll get the following:

2Ra + 2Rb + 2Rc = 5/6 + 2/3 + 1/2. This simplifies to:

2Ra + 2Rb + 2Rc = 5/6 + 4/6 + 3/6 = 12/6 or 2Ra + 2Rb + 2Rc  = 2.

Dividing both sides by 2, we’ll get: Ra + Rb + Rc  = 1.

This tells us that the pumps, all working together can do one tank in one hour. Well, if the rate is 1, and the time is the reciprocal of the rate, it’s pretty obvious that the time to complete the task is also 1. The answer, therefore, is E.

Takeaway: the most persistent myth we have about our academic limitations is that we’re simply not good at a certain subset of problems when, in truth, we just never properly learned how to do this type of question. Like every other topic on the GMAT, rate/work questions can be mastered rapidly with a sound framework and a little practice. So file away the notion that rates can be added in work questions and that time and rate have a reciprocal relationship. Then do a few practice questions, move on to the next topic, and know that you’re one step closer to mastering the skills that will lead you to your desired GMAT score.

*GMATPrep question courtesy of the Graduate Management Admissions Council.

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

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

Don’t Panic on the GMAT!

Letter of RecommendationYou’ve made it. After months of study, mountains of flash cards, and enough time spent on our YouTube channel that you’re starting to feel like Brian Galvin is one of your roommates, you’re at the test center and the GMAT — not the essay or something, but the real GMAT, in all its evil glory, complete with exponents and fractions — is about to begin. You’re nervous but excited, and cautiously optimistic for the first question: maybe it’ll be something like “What’s (2²)³?” or a work rate problem about how long it’d take George Jetson to burn down a widget factory. You mostly remember these questions, so you click “Begin”, and this is what you see:

A palindrome is a number that reads the same front-to-back as it does back-to-front (e.g. 202, 575, 1991, etc.) p is the smallest integer greater than 200 that is both a prime and a palindrome. What is the sum of the digits of p?

A) 3

B) 4

C) 5

D) 6

E) 7

Thud.

I don’t know about you, but I’m petrified. I mean, yeah, I know what you’re saying — I’m the bozo who just dreamed up that question — but I don’t know where it came from, and I’m sort of thinking I might need to summon an exorcist, because I must be possessed by a math demon. What does that question even say? How the heck are we going to solve it?

This is such a common GMAT predicament to be in that I’m willing to bet that 99% of test takers experience it: the feeling that you don’t even know what the question is saying, and the sense of creeping terror that maybe you don’t know what any of these questions are saying. This is by design, of course. The test writers love these sort of “gut check” questions that test your ability to calmly unpack and reason out a cruel and unusual prompt. So many students take themselves out of the game by panicking, but like any GMAT question, once we get past the intimidation factor, the problem is simple at heart. Let’s try to model the process.

We’ll start by clarifying our terms. Palindrome, palindrome … what on earth is a palindrome!? Is that some sort of hovercraft where Sarah Palin lives? Where are our flash cards? Maybe we should just go to law school or open a food truck or something, this test is absurd.

Wait, the answer is right in front of us, in the very first line! “A palindrome is a number that reads the same back-to-front as it does front-to-back.” Phew, OK, and there are even some examples. So a palindrome is a number like 101, 111, 121, etc. Alright, got that. And it’s prime … prime, prime … OK, right, that WAS on a flashcard: a prime number is a number with exactly two factors, such as 2, or 3, or 5, or 7. So if we were to make lists of each of these numbers, primes and palindromes, we’d have

Primes: 2, 3, 5, 7, 11, 13, 17, 19, …

Palindromes: 101, 111, 121, 131, …

and we want the first number that’s greater than 200 that appears on both lists. OK!

Now let’s think of where to start. We know our number is greater than 200, so 202 seems promising. But that can’t be prime: it’s even, so it has at least three factors (1, itself, and 2). Great! We can skip everything that begins/ends with 2, and fast forward to 303. That looks prime, but what was it that Brian kept telling us about divisibility by 3 … ah, yes, test the sum of the digits! 3 + 0 + 3 = 6, and 6 divides by 3, so 303 also divides by 3.

Our next candidate is 313. This seems to be our final hurdle: a lot of quick arithmetic. That’s what the question is testing, after all, right? How quickly can you factor 313?

It sure seems that way, but take one last look at the answers. The GMAT tests efficiency as much as anything else, and it has a way of hiding easter eggs for the observant. Our largest answer is 7, and what’s 3+1+3? 7! So this MUST be the answer, and any time spent factoring 313 is wasted time.

We made it! In hindsight, that didn’t really feel like a math problem, did it? It was testing our ability to:

1) Remember a definition (“prime”)

2) Actually read the question stem (“a palindrome is…”)

3) Not panic, and try a few numbers (“202”? “303”?)

4) Realize that heavy calculation is for suckers, and that the answer might be right in front of us (“check the answers”)

So we just had to remember, actually read the directions, have the courage to try something to see where it leads, and look for clues directly around us. I don’t know about you, but if I were running a business, those are exactly the sort of skills I’d want my employees to have; maybe these test writers are on to something after all!

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

Quarter Wit, Quarter Wisdom: Permutation Involving Sum of Digits

Quarter Wit, Quarter WisdomWe have seen in previous posts how to deal with permutation and combination questions on the GMAT. There is a certain variety of questions that involve getting a bunch of numbers using permutation, and then doing some operations on the numbers we get. The questions can get a little overwhelming considering the sheer magnitude of the number of numbers involved! Let’s take a look at that concept today. We will explain it using an example and then take a question as an exercise:

 

What is the sum of all four digit integers formed using the digits 1, 2, 3 and 4 such that each digit is used exactly once in each integer?

First of all, we will use our basic counting principle to find the number of integers that are possible.

The first digit can be chosen in 4 ways. The next one in 3 ways since each digit can be used only once. The next one in 2 ways and there will be only one digit left for the last place.

This gives us a total of 4*3*2*1 = 24 ways of writing such a four digit number. This is what some of the numbers will look like:

1234

1243

1324

1342

2143

4321

Now we need to add these 24 integers to get their sum. Note that since each digit has an equal probability of occupying every place, out of the 24 integers, six integers will have 1 in the units place, six will have 2 in the units place, another six will have 3 in the units place and the rest of the six will have 4 in the units place. The same is true for all places – tens, hundreds and thousands.

Imagine every number written in expanded form such as:

1234 = 1000 + 200 + 30 + 4

2134 = 2000 + 100 + 30 + 4

…etc.

For the 24 numbers, we will get six 1000’s, six 2000’s, six 3000’s and six 4000’s.

In addition, we will get six 100’s, six 200’s, six 300’s and six 400’s.

For the tens place, will get six 10’s, six 20’s, six 30’s and six 40’s.

And finally, in the ones place we will get six 1’s, six 2’s, six 3’s and six 4’s.

Therefore, the total sum will be:

6*1000 + 6*2000 + 6*3000 + 6*4000 + 6*100 + 6*200 + … + 6*3 + 6*4

= 6*1000*(1 + 2 + 3 + 4) + 6*100*(1 + 2 + 3 + 4) + 6*10*(1 + 2 + 3 + 4) + 6*1*(1 + 2 + 3 + 4)

= 6*1000*10 + 6*100*10 + 6*10*10 + 6*10

= 6*10*(1000 + 100 + 10 + 1)

= 1111*6*10

= 66660

Note that finally, there aren’t too many actual calculations, but there is some manipulation involved. Let’s look at a GMAT question using this concept now:

What is the sum of all four digit integers formed using the digits 1, 2, 3 and 4 (repetition is allowed)

A444440

B) 610000

C) 666640

D) 711040

E) 880000

Conceptually, this problem isn’t much different from the previous one.

Using the same basic counting principle to get the number of integers possible, the first digit can be chosen in 4 ways, the next one in 4 ways, the next one in again 4 ways and finally the last digit in 4 ways. This is what some of the numbers will look like:

1111

1112

1121

and so on till 4444.

As such, we will get a total of 4*4*4*4 = 256 different integers.

Now we need to add these 256 integers to get their sum. Since each digit has an equal probability of occupying every place, out of the 256 integers, 64 integers will have 1 in the units place, 64 will have 2 in the units place, another 64 integers will have 3 in the units place and the rest of the 64 integers will have 4 in the units place. The same is true for all places – tens, hundreds and thousands.

Therefore, the total sum will be:

64*1000 + 64*2000 + 64*3000 + 64*4000 + 64*100 + 64*200 + … + 64*3 + 64*4

= 1000*(64*1 + 64*2 + 64*3 + 64*4) + 100*(64*1 + 64*2 + 64*3 + 64*4) + 10*(64*1 + 64*2 + 64*3 + 64*4) + 1*(64*1 + 64*2 + 64*3 + 64*4)

= (64*1 + 64*2 + 64*3 + 64*4) * (1000 + 100 + 10 + 1)

= 64*10*1111

= 711040

So our answer is D.

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

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!

Use Number Lines on the GMAT, Not Memory!

SAT/ACTI’ve written in the past about how the biggest challenge on many GMAT questions is the strain they put on our working memory. Working memory, or our ability to process information that we hold temporarily, is by definition quite limited. It’s why phone numbers only contain seven digits – any more than that and most people wouldn’t be able to recall them. (Yes, there was a time, in the dark and distant past, when we had to remember phone numbers.)

One of the most simple and effective strategies we can deploy to combat our working memory limitations is to simply list out the sample space of scenarios we’re dealing with. If we were told, for example, that x is a prime number less than 20, rather than internalize this information, we can jot down x = 2, 3, 5, 7, 11, 13, 17, or 19. The harder and more abstract the question, the more necessary such a strategy will prove to be.

Take this challenging Data Sufficiency question, for example:

On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line?

1) xyz < 0

2) xy <0

The reader is hereby challenged to attempt this exercise in his or her head without inducing some kind of hemorrhage.

So, rather than try to conceptualize this problem mentally, let’s start by actually writing down all the number line configurations that we might have to deal with before even glancing at the statements. We know that x and z are closer than x and y. So we could get the following:

x____z_______________________y

z____x_______________________y

Or we can swap x and y to generate a kind of mirror image

y______________________x_____z

y______________________z_____x

The above number lines are the only four possibilities given the constraints provided in the question stem. Now we have something concrete and visual that we can use when evaluating the statements.

Statement 1 tells us that the product of the three variables is negative. If you’ve internalized your number properties – and we heartily encourage that you do – you know that a product is negative if there are an odd number of negative elements in said product. In this case, that means that either one of the variables is negative, or all three of them are. So let’s use say one of the variables is negative. By placing a 0 strategically, we can use any of our above number lines:

x__0__z______________________y

z__0__x______________________y

y__0___________________x_____z

y__0___________________z_____x

Each of these scenarios will satisfy that first statement. But we only need two.

In our first number line, z is between x and y, so we get a YES to the question.

In our second number line, z is not between x and y, so we get a NO to the question.

Because we can get a YES or a NO to the original question, Statement 1 alone is not sufficient. Eliminate answer choices A and D.

Statement 2 tells us that the product of x and y is negative. Thus, we know that one of the variables is positive, and one of the variables is negative. Again, we can simply peruse our number lines and select a couple of examples that satisfy this condition.

In our first number line, z is between x and y, so we get a YES to the question.

In our third number line, z is not between x and y, so we get a NO to the question.

Like with Statement 1, because we can get a YES or NO to the original question, Statement 2 alone is also not sufficient. Eliminate answer choice B.

When testing the statements together, we know two pieces of information. Statement 1 tells us that either one variable is negative or all three are. Statement 2 tells us that, between x and y, we have one negative and one positive. Therefore, together, we know that either x or y is negative, and the remaining variables are all positive. Now all we have to do is peruse our sample space and locate these scenarios. It turns out that we can use the same two number lines we used when testing Statement 2:

In our first number line, z is between x and y, so we get a YES to the question.

In our third number line, z is not between x and y, so we get a NO to the question.

So even together, the statements are not sufficient to answer the question – the correct answer is E.

Takeaway: on the GMAT there’s no reason to strain your brain any more than is necessary. The more concrete you can make the information you’re provided on a given question, the more likely it is that you’ll be able to properly execute whatever math or logic maneuvers you’re asked to perform.

*GMATPrep question courtesy of the Graduate Management Admissions Council.

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

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

Burn the Which! On the GMAT, That Is…

holy-grail-witchIt’s Halloween as I write this, but by the time you read it, November will be upon us, and you’ll be several days into a serious candy hangover. With that in mind, you’re probably in the mood for something boring and self-disciplined — or just to throw up — and I couldn’t think of anything that better accomplishes both than a little bit of sentence correction.

Unless you were a big reader in high school (or you’re a incorrigible grammar nerd, the type who brightens up when I use the word “incorrigible”), sentence correction is probably your least favorite part of the GMAT. You should know English, you do know English, but the GMAT wants you to feel like you don’t, and it’s amazing the rest of us can even find the will to live while we listen to you talk. It wasn’t bad enough for the test writers to undo years of hard work with your therapist to forget your high school math; now they’re making you feel inadequate about your own language (or in some cases, a language you busted your butt to learn as an adult).

Like it or not, however, that’s the game here: testing the subtle differences between everyday English, the sort you speak and type and are reading from me right now (“This is him! Who were you looking for?”), and black-tie, formal English, the kind you use to lose friends and alienate people (“This is he! For whom were you looking?”). And no word — see how I started that sentence with “and”? I’m on your side! — better stands for this distinction than “which”, a seemingly simple, everyday word that you and the GMAT test writers will be fighting a brutal war to control.

In your world, after all, “which” is used to describe anything. In your world, “He was being such a jerk, which was totally uncalled for,” or, “I took the GMAT this morning, which was the worst thing I’ve done since I felt off the stage in the first act of that school play,” are perfectly grammatical. In the GMAT’s world, however, they aren’t, and it’s all because of that “which”. In your world, “which” can describe the gist of the sentence, but in the GMAT’s world, it describes the noun that precedes it. Luckily, there’s an easy, 99% accurate way to test GMAT-approved usage of “which”:

CORRECT: (non-human noun), which (phrase describing that noun)

INCORRECT: almost anything else

So these are correct:

“The sun, which is actually a star, was once considered a god.”

“My car, which has multiple dents, two differently colored front doors, and a dog sleeping on it, is a bit of a fixer-upper.”

“I finally saw Wayne’s World 2, which I’ve been hearing about for years.”

In each case, “which” directly connects a noun to a phrase that describes that noun. The sun IS actually a star, my car DOES have multiple dents, and Wayne’s World 2 WAS what I’d been hearing about (I’ve been living under a rock since 1992).

As obnoxious as this rule is — and by no means do I encourage you to follow it in your own writing or speech — it’s easy to remember on test day. If you see “which” begin a modifier, make sure that it’s next to the noun it describes. If it is, lovely! If it isn’t, burn that which!

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

Quarter Wit, Quarter Wisdom: The Tricky Critical Reasoning Conclusion

Quarter Wit, Quarter WisdomAs discussed previously, the most important aspect of a strengthen/weaken question on the GMAT is “identifying the conclusion,” but sometimes, that may not be enough. Even after you identify the conclusion, you must ensure that you have understood it well. Today, we will discuss the “tricky conclusions.”

First let’s take a look at some simple examples:

Conclusion 1: A Causes B.

We can strengthen the conclusion by saying that when A happens, B happens.

We can weaken the conclusion by saying that A happened but B did not happen.

How about a statement which suggests that “C causes B,” or, “B happened but A did not happen”?

Do these affect the conclusion? No, they don’t. The relationship here is that A causes B. Whether there are other factors that cause B too is not our concern, so whether B can happen without A is none of our business.

Conclusion 2: Only A Causes B.

This is an altogether different conclusion. It is apparent that A causes B but the point of contention is whether A is the only cause of B.

Now here, a statement suggesting, “C causes B,” or, “B happened but A did not happen,” does affect our conclusion. These weaken our conclusion – they suggest that A is not the only cause of B.

This distinction can be critical in solving the question. We will now illustrate this point with one of our own GMAT practice questions:

Two types of earthworm, one black and one red-brown, inhabit the woods near the town of Millerton. Because the red-brown worm’s coloring affords it better camouflage from predatory birds, its population in 1980 was approximately five times that of the black worm. In 1990, a factory was built in Millerton and emissions from the factory blackened much of the woods. The population of black earthworms is now almost equal to that of the red-brown earthworm, a result, say local ecologists, solely stemming from the blackening of the woods.

Which of the following, if true, would most strengthen the conclusion of the local ecologists?

(A) The number of red-brown earthworms in the Millerton woods has steadily dropped since the factory began operations.

(B) The birds that prey on earthworms prefer black worms to red-brown worms.

(C) Climate conditions since 1990 have been more favorable to the survival of the red-brown worm than to the black worm.

(D) The average life span of the earthworms has remained the same since the factory began operations.

(E) Since the factory took steps to reduce emissions six months ago, there has been a slight increase in the earthworm population.

Let’s look at the argument.

Premises:

  • There are two types of worms – Red and Black.
  • Red has better camouflage from predatory birds, hence its population was five times that of black.
  • The factory has blackened the woods and now the population of both worms is the same.

Conclusion:

From our premises, we can determine that the blackening of the woods is solely responsible for equalization of the population of the two earthworms.

We need to strengthen this conclusion. Note that there is no doubt that the blackening of the woods is responsible for equalization of populations; the question is whether it is solely responsible.

(A) The number of red-brown earthworms in the Millerton woods has steadily dropped since the factory began operations.

Our conclusion is that only the blackening of the woods caused the numbers to equalize (either black worms are able to hide better or red worms are not able to hide or both), therefore, we need to look for the option that strengthens that there is no other reason. Option A only tells us what the argument does anyway – the population of red worms is decreasing (or black worm population is increasing or both) due to the blackening of the woods. It doesn’t strengthen the claim that only blackening of the woods is responsible.

(B) The birds that prey on earthworms prefer black worms to red-brown worms.

The fact that birds prefer black worms doesn’t necessarily mean that they get to actually eat black worms. Even if we do assume that they do eat black worms over red worms when they can, this strengthens the idea that “the blackening of the woods is responsible for equalization of population,” but does not strengthen the idea that “the blackening of the woods is solely responsible for equalization,” hence, this is not our answer.

(C) Climate conditions since 1990 have been more favorable to the survival of the red-brown worm than to the black worm.

Option C tells us that another factor that could have had an effect on equalization (i.e. climate) is not responsible. This strengthens the conclusion that better camouflage is solely responsible – it doesn’t prove the conclusion beyond doubt, since there could be still another factor that could be responsible, but it does discard one of the other factors. Therefore, it does improve the probability that the conclusion is true.

(D) The average life span of the earthworms has remained the same since the factory began operations.

This option does not distinguish between the two types of earthworms. It just tells us that as a group, the average lifespan of the earthworms has remained the same. Hence, it doesn’t affect our conclusion, which is based on the population of two different earthworms.

(E) Since the factory took steps to reduce emissions six months ago, there has been a slight increase in the earthworm population.

Again, this option does not distinguish between the two types of earthworms. It just tells us that as a group, the earthworm population has increased, so it also does not affect our conclusion, which is based on the population of two different earthworms.

Therefore, our answer is C.

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

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!

GMAT Tip of the Week: Movember and Moving Your GMAT Score Higher

GMAT Tip of the WeekOn this first Friday of November, you may start seeing some peach fuzz sprouts on the upper lips of some of your friends and colleagues. For many around the world, November means Movember, a month dedicated to the hopefully-overlapping Venn Diagram of mustaches and men’s health. Why – other than the fact that this is a GMAT blog – do we mention the Venn Diagram?

Because while the Movember Foundation is committed to using mustaches as a way to increase both awareness of and funding for men’s health issues (in particular prostate and testicular cancer), many young men focus solely on the mustache-growth facet of the month. And “I’m growing a mustache for Movember” without the fundraising follow-through is akin to the following quotes:

“I’m growing a mustache for Movember.”

“I’m running a marathon for lymphoma research.”

“I’m dumping a bucket of ice water over my head on Facebook.”

“I’m taking a GMAT practice test this weekend.”/”I’m going to the library to study for the GMAT.”

Now, those are all noble sentiments expressed with great intentions. But another thing they all have in common is that they’re each missing a critical action step in their mission to reach their desired outcome. Growing a mustache does very little to prevent or treat prostate cancer. Running a marathon isn’t what furthers scientists’ knowledge of lymphoma. Dumping an ice bucket over your head is more likely to cause pneumonia than to cure ALS. And taking a practice test won’t do very much for your GMAT score.

Each of those actions requires a much more thorough and meaningful component. It’s the fundraising behind Movember, Team in Training, and the Ice Bucket Challenge that advances those causes. It’s your effort to use your mustache, sore knees, and Facebook video to encourage friends and family to seek out early diagnosis or to donate to the cause. And it’s the follow-up to your GMAT practice test or homework session that helps you increase your score.

This weekend, well over a thousand practice tests will be taken in the Veritas Prep system, many by young men a week into their mustache growth. But the practice tests that are truly valuable will be taken by those who follow up on their performance, adding that extra step of action that’s all so critical. They’ll ask themselves:

Which mistakes can I keep top-of-mind so that I never make them again?

How could I have budgeted my time better? Which types of problems take the most time with the least probability of a right answer, and which types would I always get right if I just took the extra few seconds to double check and really focus?

Based on this test, which are the 2-3 content areas/question types that I can markedly improve upon between now and my next practice test?

How will I structure this week’s study sessions to directly attack those areas?

And then they’ll follow up on what they’ve learned, following the new week’s plan of attack until it’s time to again take the first step (a practice test) with the commitment to take the substantially-more-important follow-up steps that really move the needle toward success.

Taking a practice test and growing a Movember mustache are great first steps toward accomplishing noble goals, but in classic Critical Reasoning form, premise alone doesn’t guarantee the conclusion. So make sure you don’t leave the GMAT test center this November with an ineffective mustache and a dismal score – put in the hard work that has to accompany that first step, and this can be a Movember to Remember.

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!

By Brian Galvin.

The Critical Role of Reading in GMAT Critical Reasoning Questions!

Quarter Wit, Quarter WisdomMost non-native English users have one question: How do I improve my Verbal GMAT score?  There are lots of strategies and techniques we discuss in our books, in our class and on our blog. But one thing that we seriously encourage our students to do (that they need to do on their own) is read more – fiction, non fiction, magazines (mind you, good quality), national dailies, etc. Reading high quality material helps one develop an ear for correct English. It is also important to understand the idiomatic usage of English, which no one can teach in the class. At some time, most of us have thought how silly some things are in the English language, haven’t we?

For example:

Fat chance” and “slim chance’”mean the same thing – Really? Shouldn’t they mean opposite things?

But “wise man” and “wise guy” are opposites – Come on now!

A house burns up as it burns down and you fill in a form by filling it out?

And let’s not even get started on the multiple unrelated meanings many words have – The word on the top of the page, “critical,” could mean “serious” or “important” or “inclined to find fault” depending on the context!

Well, you really must read to understand these nuances or eccentricities, if you may, of the English language. Let’s look at an official question today which many people get wrong just because of the lack of familiarity with the common usage of phrases in English. But before we do that, some quick statistics on this question – 95% students find this question hard and more than half answer it incorrectly. And, on top of that, it is quite hard to convince test takers of the right answer.

Some species of Arctic birds are threatened by recent sharp increases in the population of snow geese, which breed in the Arctic and are displacing birds of less vigorous species. Although snow geese are a popular quarry for hunters in the southern regions where they winter, the hunting season ends if and when hunting has reduced the population by five percent, according to official estimates. Clearly, dropping this restriction would allow the other species to recover.

Which of the following, if true, most seriously undermines the argument?

(A) Hunting limits for snow geese were imposed many years ago in response to a sharp decline in the population of snow geese.

(B) It has been many years since the restriction led to the hunting season for snow geese being closed earlier than the scheduled date.

(C) The number of snow geese taken by hunters each year has grown every year for several years.

(D) As their population has increased, snow geese have recolonized wintering grounds that they had not used for several seasons.

(E) In the snow goose’s winter habitats, the goose faces no significant natural predation.

As usual, let’s start with the question stem – “… most seriously undermines the argument”

This is a weaken question. The golden rule is to focus on the conclusion and try to weaken it.

Let’s first understand the argument:

Snow geese breed in the Arctic and fly south for the winter. They are proliferating, and that is bad for other birds. Southern hunters reduce the number of geese when they fly south. There is a restriction in place that if the population of the geese that came in reduces by 5%, hunting will stop. So if 1000 birds flew south and 50 were hunted, hunting season will be stopped. The argument says that we should drop this restriction to help other Arctic birds flourish (conclusion), then hunters will hunt many more geese and reduce their numbers.

What is the conclusion here? It is: “Clearly, dropping this restriction would allow the other species to recover.”

You have to try to weaken it, i.e. give reasons why even after dropping this restriction, it is unlikely that other species will recover. Even if this restriction of “not hunting after 5%” is dropped and hunters are allowed to hunt as much as they want, the population of geese will still not reduce.

Now, first look at option (B);

(B) It has been many years since the restriction led to the hunting season for snow geese being closed earlier than the scheduled date.

What does this option really mean?

Does it mean the hunting season has been closing earlier than the scheduled date for many years? Or does it mean the exact opposite, that the restriction came into effect many years ago and since then, it has not come into effect.

It might be obvious to the native speakers and to the avid readers, but many non-native test takers actually fumble here and totally ignore option (B) – which, I am sure you have guessed by now, is the correct answer.

The correct meaning is the second one – the restriction has not come into effect for many years now. This means the restriction doesn’t really mean much. For many years, the restriction has not caused the hunting season to close down early because the population of geese hunted is less than 5% of the population flying in. So if the hunting season is from January to June, it has been closing in June, only, so even if hunters hunt for the entire hunting season, they still do not reach the 5% of the population limit (Southern hunters hunt less than 50 birds when 1000 birds fly down South).

Whether you have the restriction or not, the number of geese hunted is the same. So even if you drop the restriction and tell hunters that they can hunt as much as they want, it will not help as they will not want to hunt geese much anyway. This implies that even if the restriction is removed, it is likely that there will be no change in the situation. This definitely weakens our conclusion that dropping the restriction will help other species to recover.

So when people ignore (B), on which option do they zero in? Some fall for (C) but many fall for (D). Let’s look at all other options now:

(A) Hunting limits for snow geese were imposed many years ago in response to a sharp decline in the population of snow geese.

This is out of scope to our argument. It doesn’t really matter when and why the limits were imposed.

(C) The number of snow geese taken by hunters each year has grown every year for several years.

This doesn’t tell us how dropping the restriction would impact the population of geese, it just tells us what has happened in the past – the number of geese hunted has been increasing. If anything, it might strengthen our conclusion if the number of geese hunted is close to 5% of the population. When the population decreases by 5%, if the restriction is dropped, chances are that more geese will be hunted and other species will recover. We have to show that even after dropping the restriction, the other species may not recover.

(D) As their population has increased, snow geese have recolonised wintering grounds that they had not used for several seasons.

With this answer choice, “wintering grounds” implies the southern region (where they fly for winter). In the South, they have recolonised regions they had not occupied for a while now, which just tells you that the population has increased a lot and the geese are spreading. It doesn’t say that removing the restrictions and letting hunters hunt as much as they want will not help. In fact, if anything, it may make the argument a little stronger. If the geese are occupying more southern areas, hunting grounds may become easily accessible to more hunters and dropping hunting restrictions may actually help more!

(E) In the snow goose’s winter habitats, the goose faces no significant natural predation.

We are concerned about the effect of hunting, thus natural predation is out of scope.

Therefore, our answer is (B).

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 and Google+, and Twitter!

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!

GMAT Tip of the Week: Trick or Treat

ZombieinsteinOne of the most dreaded things about the GMAT is the time-honored “Testmaker Trick” – the device that the GMAT question author uses to sucker you into a trap answer on a question. You’ve done all the math right, but forgot to consider negative numbers or submitted the answer for x when the question really asks for y. The “Testmaker Tricks” are enough to make you resent the test and to see it in a derogatory light. This is a grad school test, not Simon Says! Why should it matter that Simon didn’t say “positive”?

But as we head into Halloween weekend, it’s an appropriate time for you to think back to the phrase that earned you pounds and pounds of candy (and maybe tons if you followed Jim Harbaugh’s double-costume strategy): Trick or Treat.

In a GMAT context, that means that on these challenging questions, what tricks one examinee is the “treat” or reward for those who buy into the critical thinking mindset that the GMAT is set up to reward. The GMAT testmakers themselves are defensive about the idea of the “trap” answer, preferring to see it as a reward system; the intent isn’t to “trick” people as much as it is to “treat” higher-order thinking and critical reasoning. Consider the Data Sufficiency example:

Is x > 3z?

(1) x/z > 3

(2) z > 0

Here the “trick” that the testmaker employs is that of negative numbers. Many people will say that Statement 1 is sufficient (just multiply both sides by z and Statement 1 directly answers the questions, x > 3z), but it’s important to remember that z could be negative, and if it were negative you’d have to flip the sign, as you do in an inequality problem when you multiply or divide by a negative. In that case x < 3z and the answer is an emphatic no.

Now, those test takers who lament the trick after getting it wrong are somewhat justified in their complaint that “you forgot about negatives!” is a pretty cheap trick. But that’s not the entire question: Statement 2 exists, too, and it’s a total throwaway when you consider it alone. Why is it there? It’s there to “treat” those who are able to leverage that hint: why would it matter if z is greater than 0? That statement provides a very important clue as to how you should have been thinking when you looked at Statement 1.

If your initial read of Statement 1 – under timed pressure in the middle of a test, mind you – had you doing that quick algebra and making the mistake of saying that it’s sufficient, that’s understandable. But if you blew right past the clear hint in the second statement, you missed a very important opportunity to seize the treat. To some degree this problem is about the math, but the GMAT often adds that larger degree of leveraging hints – after all, much of business success comes down to your ability to find an asset that others have overlooked, or to get more value out of an asset than anyone else could.

So as you study for the GMAT, keep that Halloween spirit close by. When you miss a problem because of a dirty “trick,” take a second to also go back and see if you missed a potential treat – a reward that the GMAT was dangling just out of reach so that only the most critical thinkers could find it and take advantage. GMAT problems aren’t all ghosts, goblins, and ghouls out to frighten and trick you; often they include very friendly pieces of information just disguised or camouflaged enough that you have to train yourself to spot the treat.

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 and Google+, and Twitter!

By Brian Galvin.

Does the GMAT Even Really Measure Anything?

SAT/ACTAt some point, in pretty much every class I teach, a student will ask me what the GMAT really measures. The tone of the question invariably suggests that the student doesn’t believe that the test accurately assesses anything of real significance, that the frustrations and anxieties we endure when preparing for the exam are little more than a form of admissions sadism.

When it comes to standardized testing, a certain amount of cynicism is understandable – if person A has a better grasp on the fundamentals of geometry and algebra than person B, why on earth would we conclude on that basis that person A will be more likely to have a successful career in a field totally unrelated to geometry and algebra?

Of course, I have my stock answer: the test is designed to reward flexible thinking, to provide feedback on our ability to make good decisions under pressure. And though I do believe this, I’m also well aware that tests have their limitations. There are many incredibly talented and intelligent people who struggle in the artificial conditions of a testing environment, and no 3.5 hour exam will be able to fully capture an individual’s potential. At some level, we all know this. It’s why the admissions process is holistic. Still, your GMAT score is important, so I thought it worthwhile to do a bit of research about what the data says regarding how well the test predicts future success.

In 2005, GMAC issued a report in which it examined data from 1997-2004 about the correlation between GMAT scores and graduate school grades. The report summarizes a regression analysis in which researchers generated what they term a “validity coefficient.” A coefficient of “1” would mean that the correlation between the GMAT and graduate school grades was perfect – the two variables would move in lockstep. According to this report, any coefficient between .3 and .4 is considered useful for admissions.

The GMAT’s validity coefficient came out to .459, suggesting that the test does, in fact, have some predictive value, and this predictive value seems to be superior to other variables that admissions committees consider. The validity coefficient for undergraduate grades, for example, was .283. (And when the variables are combined, the validity coefficient is higher any individual coefficient.) So is that the end of the story? Can I rebuff my students’ complaints about standardized testing by sending them an abstract of this report? It’s not quite that simple.

In the conclusion section of the paper, we’re offered the following: “When examining the validity data in this study, one should recognize that there is a great deal of variability across programs and that the relative importance for each of the investigated variables differs for each program. This is to be expected.”

So one interpretation of the data is that the GMAT does a pretty good job of predicting how well students will do in their MBA programs. But if you’ve been studying for the GMAT for any length of time, hopefully your “correlation is not causation” reflex was triggered. What if students with higher GMAT scores attend more selective schools and then it turns out that those selective schools have more lenient grading policies because they figure that the necessary vetting has already been performed? In this case, the correlation between GMAT score and grades wouldn’t be shedding much light on how well the test-takers would perform academically, but rather, would be providing information about what kinds of programs test-takers would eventually attend.

Moreover, one could argue that looking at the correlation between GMAT scores and grad school grades is of limited usefulness. Schools no doubt hope their students do well in their classes, but it stands to reason that admissions decisions are also informed by predictions about what prospective students can contribute to the school’s community, as well as what kind of future career success these students can expect after they graduate. What, then, is the correlation between graduate grades and career success beyond the classroom? And how would we even begin to measure or define “success”? These are complex questions with no good answer.

Furthermore, while the paper appeared statistically rigorous to me, amateur that I am, we still have to consider that it was commissioned by GMAC, the company that administers the test, so there is a conflict of interest to bear in mind.  A recent article by the Journal of Education for Business questioned the results of the earlier research and insisted that the section of the GMAT that best predicted conventional managerial qualities, such as leadership initiative and communication skill, was the Analytical Writing section, the component of the test that admissions committees care about least and that had the lowest validity coefficient, according to the earlier paper.

Needless to say, though I found these papers interesting, they provided me with no definitive answers to offer my students when they ask about what the GMAT really measures. And, paradoxically enough, this is something we should find encouraging. If the GMAT were measuring any kind of fixed inherent quality, there’d be little point in prepping for the test. But if the test requires a unique skillset, that skillset can be mastered, irrespective of how directly applicable that skillset will be to future endeavors. Pragmatically speaking, the thing that matters most is that admissions committees do care about the GMAT score. So my ultimate message to my students is this: stop worrying about what the GMAT measures, and instead, harness that energy to focus on what you need to do to maximize your score.

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

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

How to Solve Tough GMAT Quant Problems by Blending Strategies

victorinox_mountaineer_lgI wrote a post a few weeks ago in which I discussed the importance of blending strategies on certain questions. It’s a mistake to pigeonhole a complex problem as one in which a single tool will be most effective. By test day you will have cultivated a veritable Swiss army knife of strategies and you want to be able to switch from one to another seamlessly.

This philosophy came to mind the other day when a student sent me the following official problem:

For a certain art exhibit, a museum sold admission tickets to a group of 30 people every 5 minutes from 9:00 in the morning to 5:55 in the afternoon, inclusive. The price of a regular admission ticket was $10 and the price of a student ticket was $6. If on one day 3 times as man regular admissions tickets were sold as student tickets, was the total revenue from ticket sales that day?

A) 24,960

B) 25,920

C) 28,080

D) 28,500

E) 29,160

Oh boy. There’s a lot going on here. So let’s start by simply finding the total number of tickets sold. We know that every 5 minutes, 30 tickets are sold. We know that there are twelve 5-minute increments each hour, so 12*30 = 360 tickets are sold each hour. We see that the museum will be open for a total of 9 hours, so a total of 9*360 = 3240 tickets are sold during that time.

We’ve got two different kinds of tickets – general and student. The general tickets were $10 and the student tickets were $6. And we know that 3 times as many general tickets were sold as student tickets. So the tickets were overwhelmingly for general admission. If they were all for general admissions tickets, we know that the revenue would have been 3240*10 = 32,400. Because 25% of the tickets were sold for $6, we know that the correct answer will be a bit below this value. If we were short on time, E would be a pretty reasonable guess.

But say we’ve achieved a level of mastery where we don’t need to guess. Hopefully, you recognized that if the ratio of general tickets to student tickets is 3:1, we’re dealing with a kind of weighted average, meaning we can use a number line to find the average overall ticket price, which will be much closer to $10 than to $6. So we know the average price is greater than $8, as this would be the average price if the same number of both kinds of tickets were sold. What about $9? On the number line, we’ll have the following: 6——–9—-10.

9 is three units away from 6 and one unit away from 10, thus yielding our desired 3:1 ratio. Now we know that the average price is $9 per ticket.

So all we have to do is calculate 3240 * 9, as 3240 tickets were sold for an average of $9 each, and we have our answer. That math isn’t too bad, but we can incorporate a couple more useful strategies to save some time. We know that 3000*9 = 27,000, so clearly 3240*9 is greater than 27,000. Now we can eliminate A and B from contention.  Next, we can see that going from right to left, the first non-zero digit of 3240*9 will be 6, as 4*9 = 36. Among C, D, and E, the only answer choice that has a 6 in the tens place is E, which is our answer.

Takeaway: In a single question, we ended up doing a bit of estimation, using the answer choices, employing some rudimentary logic, and using the number line to simplify a weighted average. Just as important as what we did do, is what we avoided doing – a lot of grinding calculation.

We cannot emphasize this enough: the Quant section is not a math test. It’s an opportunity to demonstrate fluid thinking under pressure. So when you’re doing practice questions, work on employing every tool in your Swiss army knife of strategies. By the day of the test, the more fluidly you can switch from one tool to another, the better you’ll be able to handle even the most challenging problems.

*GMATPrep question courtesy of the Graduate Management Admissions Council.

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

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

Complicated GMAT Work-Rate Questions Made Easy!

Quarter Wit, Quarter WisdomToday, we will take up a gem of a work-rate question from our own curriculum. Its basics lie in a post on joint variation that we discussed many weeks ago. Here is a quick recap of the actual methodology:

If 10 workers complete a work in 5 days working 8 hours a day, how much work will be done by 6 workers in 10 days working 2 hours a day?

Here is what it looks like:

 

10 workers……………..5 days …………….. 8 hours ……………1 work

6 workers………………10 days …………… 2 hours …………… ? work

We need to find the amount of work done, so we start with the work done in the first case and then multiply it by the respective ratios:

Work done = 1 * (6/10) * (10/5) * (2/8) = 3/10

We multiply by 6/10 because number of men decreases from 10 to 6. The work done will reduce, so we multiply by 6/10 (the fraction less than 1).

We also multiply by 10/5 because number of days increases from 5 to 10. Because of this, the work done will increase, so we multiply by 10/5 (the fraction more than 1).

We also multiply by 2/8 because number of hours decreases from 8 to 2. Because of this, the work done will decrease, hence, we multiply by 2/8 (the fraction less than 1).

So the process is super simple – start with what you need to find out, say x, and multiply it by the ratio of each thing that changes from A to B. Whether you multiply by A/B or B/A depends on whether with this change increases or reduces x. If x increases, you will multiply by the fraction that is greater than 1, however if x decreases, you will multiply by the fraction that is less than 1.

On this same concept, let’s look at the question:

16 horses can haul a load of lumber in 24 minutes. 12 horses started hauling a load and after 14 minutes, 12 mules joined the horses. Will it take less than a quarter-hour for all of them together to finish hauling the load?

Statement 1: Mules work more slowly than horses.

Statement 2: 48 mules can haul the same load of lumber in 16 minutes.

Let’s see what data we have in the question stem:

16 horses …….. 24 mins ………. 1 work

12 horses …….. 14 mins ………. ? work

Work done = 1*(14/24)*(12/16) = (7/16)th of the work

We multiply by 14/24 because if the time taken to do the work decreases, the work done will also decrease. 14/24 is less than 1 so it will decrease the work done.

We also multiply by 12/16 because if the number of horses decreases, the work done will also decrease. 12/16 is less than 1 so it will decrease the work done.

All in all, we now know that 12 horses complete 7/16th of the work in 14 mins. So there is still 1 – 7/16 = 9/16 of the work left to do.

Now let’s review the two statements.

Statement 1: Mules work more slowly than horses.

This statement doesn’t give us any figures, so how can we analyse it mathematically? What we can do is find the range in which the time taken by all the horses and mules together will lie according to this statement.

Case 1: When mules work at a rate that is infinitesimally smaller than the rate of horses.

In this case, 12 mules are equivalent to 12 horses. So we have a total of 12 + 12 = 24 horses working together to complete (9/16)th of the work.

16 horses …….. 24 mins ………. 1 work

24 horses ……… ? mins ………. 9/16 work

Time taken = 24*(16/24)*(9/16) = 9 mins

Since the mules are slower than the horses, the time taken to complete the work will be more than 9 minutes. How much more than 9 minutes, we do not know. Now look at the flip side:

Case 2: When the mules work at a rate close to 0.

If the mules work slower, time taken will be more till the point when mules work so slowly that they do almost no work.

16 horses …….. 24 mins ………. 1 work

12 horses ……… ? mins ………. 9/16 work

Time taken = 24*(16/12)*(9/16) = 18 minutes

Therefore, depending on how fast/slow the mules are, the time taken to do the rest of the work could be anywhere from 9 minutes to 18 minutes. Therefore the time taken could be either less or more than 15 minutes – this statement alone is not sufficient.

Statement 2: 48 mules can haul the same load of lumber in 16 minutes.

We now know exactly how fast the mules are, so this must be sufficient to say whether the time taken to do the rest of the work was less or more than 15 minutes – we don’t need to actually find the time taken here – therefore, the answer is B, Statement 2 alone is sufficient.

However, if you would like to find out for practice, just find the equivalence between the horses and the mules first.

To haul the load in 16 minutes, we need 48 mules

To haul the load in 24 minutes, we need 48 * (16/24) = 32 mules

So 32 mules are equivalent to 16 horses (because 16 horses haul the load in 24 minutes). This means that 2 mules are equivalent to 1 horse, and 12 mules are, therefore, equivalent to 6 horses.

So now, in effect we have a total of 12 + 6 = 18 horses, and the situation now becomes this:

16 horses …….. 24 mins ………. 1 work

18 horses ……… ? mins ………. 9/16 work

Time taken = 24*(16/18)*(9/16) = 12 minute – less than a quarter-hour to finish the work.

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 and Google+, and Twitter!

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!

GMAT Tip of the Week: Percents Are Easy, Words Are Hard

GMAT Tip of the WeekPop quiz: 1) Your restaurant bill came to exactly $64.00 and you want to leave a 20% tip. How much do you leave? 2) You’re running a charity half-marathon and your fundraising goal is $6000. You’ve raised $3300. What percent of your goal have you reached? 3) Your $20,000 investment is now worth $35,000. By what percent has your investment increased in value?

[Answers: $12.80; 55%; 75%]

 

If that was easy for you, good. It better have been. After all, you’re applying to graduate school and that’s maybe 6th grade math in three real-life contexts. Percents are not hard! But percent problems can be. And that’s what savvy GMAT test-takers need to learn:

On the GMAT, percent problems aren’t hard because of the numbers. They’re hard because of the words.

Consider two situations:

1) A band sells concert t-shirts online for $20 each, and in California, web-based sales are subject to a 10% sales tax. How much does a California-based purchaser pay in sales tax after buying a t-shirt?

2) At a concert in California, a band wants to sell t-shirts for $20. For simplicity’s sake at a cash-only kiosk, the band wants patrons to be able to pay $20 even – hopefully paying with a single $20 bill – rather than having to pay sales tax on top. If t-shirts are subject to a 10% tax on the sale price, and the shirts are priced so that the after-tax price comes to $20, how much will a patron pay in sales tax after buying a t-shirt?

So what are the answers?

The first, quite clearly, should be $2. Take 10% of the $20 price and there’s your answer. And taking 10% is easy – just divide by 10, which functionally means moving the decimal point one place to the left and keeping the digits the same.

The second is not $2, however, and the reason is critical to your preparation for percent questions above the 600 level on the GMAT: the percent has to be taken OF the proper value. Patrons will pay 10% OF the before-tax price, not 10% of the after-tax price. $20 is the after-tax price (just as $22 is the after-tax price in the first example…note that there you definitely did not take the 10% of the $22 after-tax price!). So the proper calculation is:

Price + 10% of the Price = $20

1.1(P) = 20

P = 20/1.1 = 18.18

So the price comes out to $18.18, meaning that $1.82 is the amount paid in tax.

While the calculation of 20/1.1 may have been annoying, it’s not “clever” or “hard” – the reason that many people will just say $2.00 to both isn’t that they screwed up dividing $20 by 1.1, but instead because they saw a percent problem with two numbers (10% and $20) and just “calculated a percent.” That’s what makes the majority of GMAT percent problems tricky – they require an attention to detail, to precision in wording, for examinees to ensure that the (generally pretty darned easy) percent calculations are taking the percent of the proper value.

They’re logic puzzles that require a bit of of arithmetic, not simple arithmetic problems that just test your ability to divide by 10 absent critical thought. So as you approach GMAT percent problems, remember that the math should be the easy part. GMAT percent problems are often more about reading comprehension and logic than they are about multiplication and division.

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 and Google+, and Twitter!

By Brian Galvin.

How “Back to the Future” Can Help Your GMAT Score!

Back to the FutureAs a social media user, you’re probably very away that today – October 21, 2015 – is “The Future,” the day from Back to the Future II when Marty McFly and Doc Brown (along with a sleeping Jennifer) visit Hill Valley 30 years in the future. And while we don’t yet have hoverboards and while the bold prediction that the Cubs will have just won the World Series seems to be slipping away, the Back to the Future trilogy does offer some incredibly valuable GMAT lessons. How can Marty McFly help you better understand the GMAT and increase your score?

1) The Space-Time Continuum

Throughout the Back to the Future series, Doc Brown was keenly aware of the impacts that any slight alteration to the past would have on the future (as it turns out, stopping your parents from meeting or allowing the scores of all future sporting events to fall into the hands of your family’s mortal enemy could have disastrous results!). The GMAT works on a similar premise: because the GMAT is adaptive, each question impacts the future questions you will see – the events are connected and sequential. Which means:

A) You can’t go back and change your answers. That would violate the “Space-Time Continuum” nature of the GMAT (changing #5 would mean that questions 6-37 would all be different, so it’s just not an option). And THAT means that you have to make good decisions in real-time – you need to double-check for careless errors before you submit, because if you realize later that you blew it, that question is gone.

B) You can’t afford a disastrous start. It’s not that the first 10 questions matter exponentially more (as the old myth goes), but they are slightly more important if only for this reason: a strong early performance means that you’re seeing harder questions once you’re in your groove, and a poor early performance means that you’re seeing easier questions and have a much lower margin for error. Throughout each section you’ll make a few mistakes and you’ll hit a lucky guess or two.

If you’ve done well and avoided careless mistakes early, then your mistakes and lucky guesses will be on harder questions. If you haven’t, then those mistakes come on easier questions and pull down your score all the more. It *is* possible to recover from a poor start…it just requires you to be a lot closer to perfect and that can be hard to do on test day. Please note: you don’t need to get all 10 right to consider it a good start!  6 or 7 will probably put you on a track you’re happy with; the key is to just make sure you’re not making too many silly mistakes early and missing the questions that you should get right.

2) Save the Clock Tower! 

Back to the Future taught a generation the importance of timeline, and that’s critical on the GMAT. You need to be mindful of time and ensure that you have enough to finish each section. Just like in the movies, where mismanagement of time and unforeseen events created precarious situations (would Doc get the wire connected before lightning struck? Would Marty get to that point at the proper time? Would Doc reach Clara before the train tumbled off the cliff?), the GMAT offers you plenty of opportunities to waste time and get off schedule (and maybe your score falls off a cliff, or you’re the one stuck in the past…an era when master’s degrees were far from the norm).

You need to conserve time on the test so that you don’t find a catastrophe waiting at the end. Which means that sometimes you have to let a hard problem go so that it doesn’t suck up several minutes of your time (even if the hard problem seems to be calling you “chicken”!). Like Marty should have done in most time-travel situations, have a plan for how you’ll address events in a timely fashion and stick to it. If you want to have 53 minutes left after 10 questions and you have 51, know that you’ll probably have to guess soon to get back on track.

3) Find Your Skateboard

1985 was easy for Marty, like a 400-500 level GMAT problem. If he needed to quickly get from one place to another, he’d hop on his skateboard and grab the back of a truck. But 1955 and 2015 were quite different – there weren’t conventional skateboards for him to use, so he had to improvise either by breaking a scooter in two or learning how to handle a hoverboard.

The GMAT is similar: the tools you’ll use to solve problems (find skateboard, let a Tannen chase you, veer off at the last second leaving him to crash into a pile of manure) are extremely similar, but just different enough that it may not be obvious what to do at first. Your job as you study is to learn how to look for that “skateboard.”

On exponent problems, for example, the key is almost always getting the given information to a point where you can perform the rules you know. And since those rules are almost always requiring you to deal with exponents with the same base and that the terms are being multiplied or divided, your “finding the skateboard” process usually involves factoring non-prime bases into prime factors and factoring addition and subtraction into multiplication. Much like Marty McFly in a new decade, you’ll find yourself seeing slightly-familiar, but yet totally different situations on the test – your job is to focus more on the similarity and seek out a couple steps to get it to where the rest is rote.

4) Be a Man (or Woman) of Action

In the original Back to the Future, you saw how the entire future changed with just one action: the ever analytical and incredibly intelligent George McFly just wasn’t a confident or action-oriented man, and so despite Marty’s best efforts to talk him up to Lorraine and to get him to be a bit more debonair, the McFly family future was fading quickly. Until…George had the opportunity to stop analyzing and just “do,” telling Biff to “get your damn hands off” Lorraine and ultimately punching Biff in the mouth. From that point on, the George-and-Lorraine romance was on (again?) and the future was just a matter of density. I mean…destiny.

If you’re reading a blog post about the GMAT you’re certainly not the type that Principal Strickland would call a slacker, but there’s a good likelihood that you’ll perform on test day like the “old” George McFly: intelligent and capable, but timid and over-analytical. Particularly with the timed nature of the GMAT, you often just have to go with an instinct and try it out, whether that means writing down an equation and then double checking that you like your math (as opposed to reading the question again and again) or testing your theory that you’re allowed to cross-multiply there (test it with small numbers and see if you get the answer you should).

The biggest mistake that the truly-capable make on the GMAT is one of paralysis by analysis; they’re afraid to put pen to paper to “try something” and then they become acutely aware of the time ticking past them and panic all the more. Avoid that trap! Be willing to try, to take action, and you’ll find that – like the owner of  DeLorean time machine – you have plenty of time.

On this 30th anniversary of Marty’s journey to the future, plan for your future 30 years down the road. The way you study for the GMAT, the way you manage your time and confidence on the test – they could have a major impact on what your future looks like. Heed the lessons that Doc and Marty taught you, and you could leave the test center saying, “Roads? Where I’m going, we don’t need roads,” of course because most elite b-school campuses are all about sidewalks.

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 and Google+, and Twitter!

By Brian Galvin.

7 Formulas for Tackling Three Overlapping Sets on the GMAT

Quarter Wit, Quarter WisdomIn a previous post, we saw how to solve three overlapping sets questions using venn diagrams. Today, we will look at all of the various formulas floating around on three overlapping sets. Most of these are self explanatory but we will look into the details of some of them.

 

 

 

There are two basic formulas that we already know:

1) Total = n(No Set) + n(Exactly one set) + n(Exactly two sets) + n(Exactly three sets)

2) Total = n(A) + n(B) + n(C) – n(A and B) – n(B and C) – n(C and A) + n(A and B and C) + n(No Set)

From these two formulas, we can derive all others.

n(Exactly one set) + n(Exactly two sets) + n(Exactly three sets) gives us n(At least one set). So we get:

3) Total = n(No Set) + n(At least one set)

From (3), we get n(At least one set) = Total – n(No Set)

Plugging this into (2), we then get:

4) n(At least one set) = n(A) + n(B) + n(C) – n(A and B) – n(B and C) – n(C and A) + n(A and B and C)

Now let’s see how we can calculate the number of people in exactly two sets. There is a reason we jumped to n(Exactly two sets) instead of following the more logical next step of figuring out n(At least two sets) – it will be more intuitive to get n(At least two sets) after we find n(Exactly two sets).

n(A and B) includes people who are in both A and B and it also includes people who are in A, B and C. Because of this, we should remove n(A and B and C) from n(A and B) to get n(A and B only). Similarly, you get n(B and C only) and n(C and A only), so adding all these three will give us number of people in exactly 2 sets.

n(Exactly two sets) = n(A and B) – n(A and B and C) + n(B and C) – n(A and B and C) + n(C and A) – n(A and B and C). Therefore:

5) n(Exactly two sets) = n(A and B) + n(B and C) + n(C and A) – 3*n(A and B and C)

Now we can easily get n(At least two sets):

6) n(At least two sets) = n(A and B) + n(B and C) + n(C and A) – 2*n(A and B and C)

This is just n(A and B and C) more than n(Exactly two sets). That makes sense, doesn’t it? Here, you include the people who are in all three sets once and n(Exactly two sets) converts to n(At least two sets)!

Now, we go on to find n(Exactly one set). From n(At least one set), let’s subtract n(At least two sets); i.e. we subtract (6) from (4)

n(Exactly one set) = n(At least one set) – n(At least two sets), therefore:

7) n(Exactly one set) = n(A) + n(B) + n(C) – 2*n(A and B) – 2*n(B and C) – 2*n(C and A) + 3*n(A and B and C)

You don’t need to learn all these formulas. Just focus on first two and know how you can arrive at the others if required. Let’s try this in an example problem:

Among 250 viewers interviewed who watch at least one of the three TV channels namely A, B &C. 116 watch A, 127 watch C, while 107 watch B. If 50 watch exactly two channels. How many watch exactly one channel?

(A) 185

(B) 180

(C) 175

(D) 190

(E) 195

You are given that:

n(At least one channel) = 250

n(Exactly two channels) = 50

So we know that n(At least one channel) = n(Exactly 1 channel) + n(Exactly 2 channels) + n(Exactly 3 channels) = 250

250 = n(Exactly 1 channel) + 50 + n(Exactly 3 channels)

Let’s find the value of n(Exactly 3 channels) = x

We also know that n(At least one channel) = n(A) + n(B) + n(C) – n(A and B) – n(B and C) – n(C and A) + n(A and B and C) = 250

Also, n(Exactly two channels) = n(A and B) + n(B and C) + n(C and A) – 3*n(A and B and C)

So n(A and B) + n(B and C) + n(C and A) = n(Exactly two channels) + 3*n(A and B and C)

Plugging this into the equation above:

250 = n(A) + n(B) + n(C) – n(Exactly two channels) – 3*x + x

250 = 116 + 127 + 107 – 50 – 2x

x = 25

250 = n(Exactly 1 channel) + 50 + 25

n(Exactly 1 channel) = 175, so your answer is C.

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 and Google+, and Twitter!

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!

GMAT Tip of the Week: Your GMAT Verbal (Donald) Trump Card

GMAT Tip of the WeekThe general consensus coming out of this week’s Democratic debate for the 2016 U.S. Presidency was this: the Democrats were quick to defend and agree with each other, particularly in contrast to the recent Republican debates in which the candidates were much more apt to attack each other.

The Democrats discussed, but the Republicans DEBATED, fiercely and critically. And – putting politics aside – one of the main issues on which those Republican candidates have attacked each other is “who is the more successful CEO/entrepreneur?” (And the answer to that? Likely Wharton’s finest: Donald “You’re Fired” Trump.)

So as you watch the political debates in between GMAT study sessions, keep this in mind: on the GMAT verbal section, you want to think more like a Republican candidate, and if possible you want to think like The Donald. Trump thinking is your Trump card: on GMAT verbal, you should attack, not defend.

Why?

Because incorrect answers are very easy to defend if that’s your mindset. They’re wrong because of a small (but significant) technicality, but to the “I see the good in all answer choices” eye, they’ll often look correct. You want to be in attack mode, critically eliminating answer choices and enjoying the process of doing so. Consider an example:

From 1998 to 2008, the amount of oil exported from the nation of Livonia increased by nearly 20% as the world’s demand soared. Yet over the same period, Livonia lost over 8,000 jobs in oil drilling and refinement, representing a 25% increase in the nation’s unemployment rate.

Which of the following, if true, would best explain the discrepancy outlined above?

A) Because of a slumping local economy, Livonia also lost 5,000 service jobs and 7,500 manufacturing jobs.

B) Several other countries in the region reported similar percentages of jobs lost in the oil industry over the same period.

C) Because of Livonia’s overvalued currency, most of the nation’s crude oil is now being refined after it has been exported.

D) Technological advancements in oil drilling techniques have allowed for a greater percentage of the world’s oil to be obtained from underneath the ocean floor.

E) Many former oil employees have found more lucrative work in the Livonia’s burgeoning precious metals mining industry.

The paradox/discrepancy here is that oil exports are up, but that jobs in oil drilling and refinement are down. What’s a Wharton-bound Trump to do here? Donald certainly wouldn’t overlook the word “Critical” in “Critical Reasoning.” Almost immediately, he’d be attacking the two-part job loss – it’s not that “oil jobs” are down, it’s that oil jobs in “drilling AND refinement” are down. Divide and conquer, he’d think, one of those items (either drilling or refinement) is bound to be a “lightweight” ready to be attacked.

Choice A is something that you could talk yourself into. “Hey, the economy overall is down, so it only makes sense that oil jobs would be down, too.” But think critically – you ALREADY know that the oil sector is not down. Oil exports are up 20% and global demand is soaring, so these oil jobs should be different. Critical thinking shows you that the general economy and this particular segment are on different tracks. Choice A does not explain the discrepancy.

Choice B is similar: if you’re looking for a reason to make it right, you might think, “See, it’s just part of what’s going on in the world.” But again, be critical. This is a bad answer, because it overlooks information you already have. Livonia’s oil exports are up, so absent a major reason that those exports are occurring without human labor, we don’t have a sound explanation.

Choice C hits on Trump’s “divide and conquer” attack strategy outlined above: if a conclusion to a Critical Reasoning problem includes the word “AND” there’s a very high likelihood that one of the two portions is the weak link. So fixate on that “and” and try to find which is the lightweight. Here you see that the oil is being exported from Livonia, but no longer being REFINED there. Those are the jobs that are leaving the country, and that explains why exports could be up with employment going down.

Choice D is tempting (statistically the most popular incorrect answer choice to this problem, with Trump-like polling numbers in the ~25% range). Why? Because you’re conditioned to think, “Oh, they’re losing jobs to technology.” So if you’re looking to find a correct answer without much critical thought and effort, this one shines like a beacon. But get more critical on the second half of the sentence: it’s not that technology makes it easier to obtain oil without human labor, it’s that technology is allowing for more drilling from the ocean. But that’s irrelevant, because, again, Livonia’s exports are up! So whether it’s Livonia getting that seafloor oil or other countries doing so, the fact remains that with oil exports up, you’d think that Livonia would have more jobs in oil, and this answer doesn’t explain why that’s not the case.

Here it pays to be critical all the way through the sentence: just because the first few words match what you think you might want to hear, that doesn’t mean that the entire statement is true. Think of this in Trump terms: Megyn Kelly might start a sentence with, “Mr. Trump, you’re arguably the most successful businessman of your generation,” (and you know Trump will love that) but if she follows that with, “But many would argue that your success was largely a result of your father’s money and that your manipulation of bankruptcy laws is unbefitting of an American president,” you know he’d be in attack mode immediately thereafter. Don’t fall in love with the first few words of an answer choice – stay ready to attack at a moment’s notice!

And choice E is similarly vulnerable to attack: yes some oil employees may have taken other jobs, but someone has to be doing the oil work. And if unemployment is up overall (as you know from the stimulus) then people are waiting to take those jobs, so the fact that some employees have left doesn’t explain why no one has filled those spots. When Donald Trump had to surrender his post as the star of The Apprentice, Arnold Schwarzenegger was ready to take his place; so, too, should unemployed members of the labor pool in Livonia be ready to take those oil jobs, absent a major reason why they wouldn’t, and choice E fails to present one.

Overall, your job on GMAT Verbal is to be as critical as possible. You’re there to debate the answer choices, not to defend or discuss them. As you read the conclusion of a Critical Reasoning problem, you want to be scanning for a “lightweight” word or phrase that makes it all the more vulnerable to attack. And as you read each answer choice, you shouldn’t be quick to see the good in the sentence, but instead you should be probing it to see where it’s weak and vulnerable to attack.

Let the answer choices view you as a bully – you’re not at the GMAT test center to make friends. Always be attacking, always be looking for words, phrases, or ideas that are an answer choice’s undoing. Trump logic is your Trump card, take joy from telling four of five answer choices “You’re Fired.”

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 and Google+, and Twitter!

By Brian Galvin.

The Importance of Catching Details on the GMAT

Magnifying GlassIn our everyday lives, we all understand that attention to linguistic detail is important. When my wife tells me I need to pick up my daughter, I don’t unconsciously filter out minor elements like where my daughter is or what time I’m supposed to get her. Similarly, if you were on the phone making plans with a friend, you’d never hang up before knowing what you’d made plans to do.

Details aren’t just important – almost every conversation we have would be totally incoherent if we didn’t pay attention to them. And yet, for whatever reason, on the GMAT, we have a tendency to skim over these very same details without absorbing them. This tendency, I find, is particularly pronounced on Critical Reasoning questions. Take this question, which I reviewed with a student the other day:

Citizens of Parktown are worried by the increased frequency of serious crimes committed by local teenagers. In response the city government has instituted a series of measures designed to keep teenagers at home in the late evening. Even if the measures succeeded in keeping teenagers at home, however, they are unlikely to affect the problem that concerns citizens, since more crimes committed by local teenagers take place between 3p.m. and 6p.m. 

Which of the following, if true, most substantially weakens the argument? 

A) Similar measures adopted in other places have failed to reduce the number of teenagers on the streets in the late evening. 

B) The crimes committed by teenagers in the afternoon are mostly small thefts and inconsequential vandalism 

C) Teenagers are much less likely to commit serious crimes when they are at home than when they are not at home 

D) Any decrease in the need for police patrols in the late evening would not mean that there could be more intensive patrolling in the afternoon 

E) The schools in Parktown have introduced a number of after-school programs that will be available to teenagers until 6 p.m. on weekday afternoons.

My student broke down the argument quickly. He saw that the conclusion was that the city’s plan to keep teenagers at home in the late evening was unlikely to be successful because most teenage crimes were committed earlier in the day.

When I asked him to reiterate what the fine citizens of Parktown were concerned about, he shrugged and said ‘crime.’ Of course, this wasn’t wrong, per se, but it was incomplete. When I followed up and asked what kind of crime they were worried about, he was puzzled at first. It wasn’t until I asked him to reread the first sentence of the argument and to pay very close attention to adjectives that it clicked.

The citizens were worried about serious crime. And this makes sense. If someone told you that the neighborhood you were about to move in to had a very high crime rate, your reaction would not be the same if those crimes consisted largely of jay-walking as it would if you discovered that those crimes were more serious offenses. I then asked him to reread the sentence at the very end of the passage. This time, he got it.

While it’s true that the majority of crimes were committed between 3 and 6 pm, the argument doesn’t specify what kinds of crimes were committed during these hours. It’s this gap between the crimes that the citizens were concerned about – serious ones – and the crimes we’re given evidence about – all crimes – that is the key to this question. If the teenagers are jay-walking in the early afternoon, but engaging in far more damaging behavior in the evening, the plan to impose the curfew still makes sense, even if, technically, those jay-walking offenses constitute a majority of the crimes committed.

Now let’s go to the answer choices:

A: We’re trying to weaken the idea that the plan won’t work. If the plan didn’t work in other places, that certainly doesn’t weaken the idea that the plan won’t work in Parktown. A is out.

B: This looks good. Even though the crimes committed between 3 and 6 constitute a majority of the total crimes, these crimes are trivial. The citizens of Parktown are worried about serious crimes, which, if they’re committed at night, the curfew would help prevent. B is the correct answer.

C: This does nothing to address the core issue of the argument, which is that the plan won’t work because most crimes are committed before the curfew takes effect.

D: While decreasing the need for police patrols is a laudable objective, this isn’t relevant to the argument. Moreover, if the police patrols weren’t more available in the afternoon, when most crimes are committed, there’s certainly no reason to have more confidence that the curfew would be effective.

E: I am a fan of after-school programs, but the availability of such activities sheds little light on whether the curfew will work. After all, if teenagers are determined to commit crimes in the afternoon, the fact that they could join the Glee Club if they want to is unlikely to serve as an effective deterrent to whatever mischief they had planned.

Takeaway: Typically, when we talk about modifiers, we’re doing so in the context of Sentence Correction, but modifiers are no less important in Critical Reasoning. Information about “what kind,” “where,” and “when,” will be absolutely crucial to assessing any argument we encounter. If a modifier is present in the argument’s conclusion, but not in the argument’s premises, that is something we want to note. We make the effort to pay attention to these details when dealing with the mundane activities of our everyday lives, so let’s not neglect those same details on the GMAT.

*GMATPrep question courtesy of the Graduate Management Admissions Council.

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

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

Manipulating Standard Formulas on the GMAT

Quarter Wit, Quarter WisdomWe know the formula we need to use to find the sum of n consecutive positive integers starting from 1. The formula is given as n(n+1)/2.

So the sum of first four positive integers is 4 * (4 + 1)/2 = 10.

This might seem a bit cumbersome, since it is easy to see that 1 + 2 + 3 + 4 = 10, but we know that the formula comes in very handy when n is a large number. For example, the sum of first 50 positive integers = 50 * 51/2 = 1275. Obviously, this will be a lot harder when done the “1 + 2 + 3 + 4 … + 49 + 50” way.

Now the question is, how do we adjust the same formula to find the sum of consecutive integers which do not start from 1?

Say, how do we find the sum of all positive integers from 8 to 20? The formula assumes a starting point of 1, so then we insert only the last number, n. How do we manage the 8? Let’s try to figure it out

Say the sum of first 20 positive integers = 1 + 2 + 3 + 4 + …. + 19 + 20 = 20 * 21/2

(1 + 2 + 3 +… + 7) + (8 + 9 +10 + … + 19 + 20) = 20 * 21/2

We need the value of the part in red, let’s call it the required sum.

(1 + 2 + 3 +… + 7) + The Required Sum = 20 * 21/2

Note here that we know the sum of 1 + 2 + 3 + … + 7 = 7 * 8/2

So, 7*8/2 + The Required Sum = 20 * 21/2, therefore the Required Sum = 20*21/2 – 7*8/2

To get the sum of consecutive integers from 8 to 20, we find the sum of all integers from 1 to 20 (using the formula we know) and subtract the sum of integers from 1 to 7 out of it (using the same formula).

To generalize, the sum of all positive integers from m to n is given as:

n(n+1)/2 – (m-1)*m/2

Let’s look at a question based on this concept:

If the sum of the consecutive integers from –40 to n inclusive is 356, what is the value of n?

(A) 47

(B) 48

(C) 49

(D) 50

(E) 51

If you are thinking that we haven’t gone over how to adjust the formula for negative numbers, you are right, but what we have discussed is enough to solve this question.

Numbers around 0 are symmetrical. So 1 and -1 add up to equal 0. Similarly, 2 and -2 add up to equal 0, and so on…

-40, -39 … 0 … 39, 40, 41, 42, 43, 44, 45 …

The sum of all numbers from -40 to 40 will be 0. Or another way to look at it is that 0 is the mean of all numbers from -40 to 40. So the total sum of these numbers will also be 0.

The given sum is actually the sum of numbers from 41 to n only.

We know how to calculate that:

n(n+1)/2 – 40*41/2 = 356

n(n+1) = 2352

From the options, we see that n cannot be 49 or 50 because the product of 49*50 or 50*51 will end in 0, so plug in n = 48 to check whether 48*49 is equal to 2352. It is, therefore our answer is B

(Had we obtained a lower product than required, we could have said that n must be 51. Had we obtained a higher product than was required, we could have said that n is 47.)

Another method:

Use the concept of arithmetic mean and ballpark. The mean of numbers from 41 to 47 or 48 or 49… will be somewhere between 44 and 46.

Let’s estimate the number of integers we need to get the sum of about 356. Each additional integer is quite large (more than 45) therefore, a difference of about 10-15 in the sum due to the various possible values of the mean will be immaterial.

45*7 = 315

45*8 = 360

This brings us very close to the value of 356.

Assuming there are 8 integers, their values will be from 41 to 48. The average of these 8 numbers will be 44.5. The total sum will be 44.5 * 8 = 356. It matches, so our answer is still B.

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

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!

GMAT Tip of the Week: What To Do When The GMAT Gets All Netflix On You

GMAT Tip of the WeekPicture this: a friend texts you and asks, “Do you want to get a pizza and watch a movie after work?”. Do you find that odd at all?

But now picture this: that same friend asks, instead, “Do you want to get a pepperoni, mushroom and olive pizza with white sauce on thin crust from Domino’s and watch a Critically-Acclaimed Inspiring Underdog movie on Neflix after work?”. That’s strange, right? And why is that? Because it’s so specific.

Well, on the GMAT you’ll often see questions that ask for something oddly specific; “What is the value of x?” is pretty normal, but “What is the value of 6x – y?” is the equivalent of the specific pizza and odd Netflix category question. Why did they ask that? Often that’s a clue, and if you notice that clue it will help you better set up the problem. Consider this example:

Specific Question

 

 

 

 

Reflect on what this question is asking about. Not x. Not y. But to paraphrase Netflix, “a partially coefficiented combination of additive variables with a strong horizontal lead.” 6x – y. That’s oddly specific, so your first inclination should be, “Is there an easy way to get 6x – y?” as opposed to, “Let’s start solving for x” (which of course you can’t do here…that’s why E is a trap answer choice).

With that in mind, even if you’ve forgotten (or temporarily blanked on) some exponent rules, you should immediately be thinking, “I have 2x – how does that become 6x,” and, “Where does the subtraction come from?”.

The 6x, of course, comes from breaking 27 down into 3^3, so that you have (3^3)^2x, which then becomes 3^6x. And then with that, you have a fraction:

Exponent

 

 

 

And that’s where the subtraction comes from. When you divide two exponents of the same base, you subtract the exponents, so now you have your 6x – y ready to go. Of course, from there, you need to get a base of 3 on the other side of the equation, so you can express 81 as 3^4, and now you know that 6x – y = 4, answer choice B.

Most importantly here, when the GMAT asks you an oddly-specific question in the vein of the oddly-specific Netflix category, you should seize on that specificity. Very frequently on the GMAT, it’s easier to solve for that oddly-specific combination of variables than it is to solve for any of the individual variables themselves!

On Problem Solving questions this can save you plenty of time, taking that extra few seconds to ask yourself how you’d arrive at that specific combination. On Data Sufficiency, this practice can be even more a matter of correct or incorrect. Data Sufficiency problems often give you sufficient information to arrive at the oddly-specific combination from the question stem, but insufficient information to determine any of the individual components. Imagine this problem as a Data Sufficiency problem:

Data Sufficiency 1

 

 

 

 

Here, as you know from above, Statement 1 is sufficient, but if you go into the problem trying to solve for the variables individually, you’ll likely think that you need Statement 2 so that you can plug the value of y back into Statement 1 to supply the value of x. That way you’ll have the entire picture filled in: x = 1, y = 2, and 6x – y = 4.

But you don’t NEED Statement 2, so on a question like this the GMAT will punish you for not seeing that Statement 1 alone is sufficient. And it’s only sufficient because of that oddly-specific question stem. Check out this follow-up question (with a similar setup, but variables changed to a and b since the actual numbers will change):

Data Sufficiency 2

 

 

 

 

Here you cannot use Statement 1 to get directly to the oddly-specific question stem. You can get to 4a – b = 4, but that doesn’t tell you about 6a – b. So here, the answer is C because you need Statement 2 so that you can solve for each variable individually.

More often than not, when the GMAT asks for an oddly-specific combination of variables it provides a way to arrive at it. So pay attention to the question itself: if it’s asking for something out of the ordinary or oddly specific, see that as a thinly-veiled clue that allows you to be the Confident GMAT Problem Solver With Excellent Think Like The Testmaker Skills En Route To A 700+ that you know you can be.

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

By Brian Galvin.

Read the Last Piece First on the GMAT!

Make Studying FunWhen I was in grad school, I had a writing teacher who insisted on reading the last page of a novel before she read the first. Her reasoning was that she was starting a kind of journey, and she was curious to know where she’d be going before she could decide whether she wished to embark. Now, as a devoted reader, I couldn’t find this strategy more abhorrent. Uncertainty and mystery are integral parts of the pleasure of reading fiction. Why ruin it?

However, when it comes to the GMAT, I am quite content to ruin the suspense of a question in favor of deriving a more convenient and efficient means of solving it. Interestingly, it turns out that when a question offers multiple bits of information, starting with the last piece can often be a way of dramatically simplifying the problem.

Take the following problem that a tutoring student of mine encountered on her GMATPrep test:

Mary’s income is 60 percent more than Tim’s income, and Tim’s income is 40% less than Juan’s income. What percent of Juan’s income is Mary’s income?

A) 124%

B) 120%

C) 96%

D) 80%

E) 64%

She approached the question like many test-takers would: she started with the first piece of information, and called Mary’s income $100. And then she got stuck. She realized that Tim’s income isn’t $40 here, as $100 is more than double $40, so clearly Mary’s income would not then be 60% greater than Tim’s (though Tim’s would have been 60% less than Mary’s.) So then, I suggested, why not start at the end?

The last person mentioned here is Juan, so let’s call Juan’s income $100. She then knocked out the remaining calculations in about 30 seconds. If Juan’s income is $100, and Tim’s income is 40% less than Juan’s, than Tim’s income would be $60. And if Tim’s income is $60, and Mary’s income is 60% more than Tim’s, Mary’s income would be 60 + 60% of 60 = 60 + 36 = 96. (Or 1.6 * 60 = 96.) If Mary’s income is $96 and Juan’s is $100, then clearly, Mary’s income is 96% of Juan’s, and the answer is C. Not bad.

Let’s try it again on another question:

In a certain region, the number of children who have been vaccinated against rubella is twice the number who have been vaccinated against mumps. The number who have been vaccinated against both is twice the number who have been vaccinated only against mumps. If 5000 have been vaccinated against both, how many have been vaccinated only against rubella?

A) 2500

B) 7500

C) 10000

D) 15000

E) 17500

First, note that this is a classic overlapping sets questions, so let’s set up a simple matrix:

DG Graph 1

 

 

 

 

 

But now, let’s start by inserting the last piece of information we’re given. 5000 have been vaccinated against both, so that goes in the Mumps/Rubella Vaccine cell. Now we’ve got:

DG Graph 2

 

 

 

 

 

Next, we’ll work backwards. We’re told that the number that have been vaccinated against both (5000) is twice the number that have been vaccinated against only mumps. So the number that have been vaccinated against only mumps must be 2500. Now our table looks like this:

DG Graph 3

 

 

 

 

 

Now we know that 7500 people have been vaccinated against Mumps. Last, we’re told that the number vaccinated against Rubella is twice the number that have been vaccinated against Mumps, which means that 15,000 people have been vaccinated against Rubella. If 15,000 total have been vaccinated against Rubella, and 5000 of those have been vaccinated against both, then, according to our table, 10,000 have been vaccinated against only Rubella. So C is our answer.


DG Graph 4

 

 

 

 

 

Takeaway: The GMAT question writer is going to provide information to you in a very strategic way. If the most useful piece of info comes at the end of a lengthier question, the question will be harder if you start at the beginning. So be like my zany grad school teacher and start at the end. It may ruin the suspense, but as a consolation, you’re more likely to get the question right, and I’m guessing that’s a trade-off most of us are more than happy to make.

*GMATPrep questions courtesy of the Graduate Management Admissions Council.

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

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

How Understanding Sampling Can Help You Conquer the GMAT

Quarter Wit, Quarter WisdomToday, we will discuss the concept of sampling. People with a statistics background will be very comfortable with it, but if you have not studied statistics, a little bit of knowledge will be helpful. You are not required to know this for the GMAT, however there could be questions framed on the sampling premise, and you will be far more comfortable solving them with some understanding in place. A sample is a selection made from a larger group (the “population”) which helps you examine certain characteristics of the larger group using limited resources.

For example:

In a large population, say all the people in a state, it is difficult to find the number of people with a certain trait, such as red hair. So you pick up 100 people at random (from different families, different areas, different backgrounds) and find the number of people who have red hair in this selection of 100.

Let’s say 12 have red hair. You can then generalize that approximately 12% of the whole population has red hair. The more unbiased your sample, the better the approximation.

In this example, you found something about the entire population (12% has red hair) based on a small sample and hence, using few resources. To find the actual percentage of people who have red hair in the entire population, you would need far more effort, time and money. Usually the use of fewer resources justifies the use of sampling even though it comes with some error.

So that is a bit of background on sampling. It will help you make sense of the  official question given below:

In a certain pond, 50 fish were caught, tagged, and returned to the pond. A few days later, 50 fish were caught again, of which 2 were found to have been tagged. If the percent of tagged fish in the second catch approximates the percent of tagged fish in the pond, what is the approximate number of fish in the pond?

A) 400

B) 625

C) 1,250

D) 2,500

E) 10,000

This is what took place: From a pond, 50 fish were caught, tagged and returned to the pond. Then 50 were caught again and 2 of those were found to be tagged.

Why was this done?

The total number of fish in the pond is the population of the pond. It is unknown. Since counting the total number of fish in the pond was hard, they tagged 50 of them and let them disperse evenly in the population. This means they gave a certain trait to a known number of fish in the pond – they tagged 50 fish.

Then they caught 50 fish again and these fish became the sample. Out of these 50, 2 were found to be tagged. So 2 of the 50 fish caught were found to have the trait given (tagged) – 4% of our sample was tagged.

The question tells us that “… the percent of tagged fish in the second catch approximates the percent of tagged fish in the pond …” that is, the question tells us that the sample is representative of the population. This implies that 50 (the number of fish we tagged) is 4% of the entire fish population of the pond.

50 = 4% of Total Fish Population, therefore, we can calculate that the Total Fish Population = 50 * 100/4 = 1250. Our answer is then C.

Using sampling, we were able to calculate the total population of the pond without actually counting each fish. For increased accuracy, often the exercise of taking samples is repeated many times and then some kind of average is used to get the best approximation.

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

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!

How to Solve Probability Problems on the GMAT

Roll the DiceAt some point in every class I teach, I’ll take a poll from my students about which topics they struggle with most. The answers will vary, but one topic that comes up over and over again is probability. Though I do recall finding probability somewhat vexing when I studied it as an undergraduate, I’ve always found it surprising that this is such an area of concern for GMAT test-takers, for the simple reason that probability questions, historically, have tended not to be very common on the test.

I suspect there are two reasons for the concern. First, is simply that the human brain isn’t naturally wired to do probability very well. Whereas many branches of mathematics are thousands of years old and have their roots in ancient civilizations, there was no working theory of probability until the 16th century.

This is pretty surprising. The ancient Greeks, for example, possessed the rudiments of integral calculus, but when it came to probability, they were clueless. Moreover, there is plenty of research demonstrating that, even now, well-educated adults struggle with probability even when the question touches on material within their field of expertise.

Secondly, GMAC seems to be realizing that probability is such an elastic concept that other question types can be incorporated into a probability question. Consequently, probability questions have been showing up a bit more frequently on some of the newer material released by GMAC. If we’re not wired to do probability very well, and these questions are showing up more frequently, some anxiety about the topic is inevitable.

The reason that probability can encompass other categories so easily is that the probability of an event occurring is, at heart, a simple ratio: the number of desired outcomes/the number of total possible outcomes. To simplify matters, it can be helpful to break this ratio into its component parts. First find the total possible number of outcomes. Then find the number of desired outcomes. When we think about the issue this way, it seems much more manageable. Take this newer official question, for example:

If an integer n to be chosen randomly between 1 and 96 inclusive, what is the probability that n(n+1)(n+2) is divisible by 8 ? 

A) 1/4 

B) 3/8

C) 1/2

D) 5/8

E) 3/4 

On the surface, this is a probability question, but because we’re talking about divisibility, it’s also testing our knowledge of number properties. So let’s start by thinking about our total possible outcomes. There are 96 numbers between 1 and 96 inclusive, so clearly, there are 96 total possible outcomes when we select a number at random. We have the denominator of our fraction.

Now we just have to figure out how many ways we can multiply three consecutive numbers, n(n+1)(n+2), to get a multiple of 8. Put another way, any multiple of 8, or 2^3, must contain three 2’s. One way this can happen is if the middle number, n+1, is odd, because every odd number must be sandwiched between a multiple of 2 and a multiple of 4.

If n+1 is 3, for example, you’d have 2*3*4, which is a multiple of 8. (We need three 2’s in all. The 2 gives us one, and the 4 donates the other 2’s.) If n+1 is 5, you’d have 4*5*6, which is also a multiple of 8. (The 4 donates two 2’s and the 6 donates one. So long as we have three 2’s, we have a multiple of 8.) Between 1 and 96, we’ve got 48 odd numbers.

The other way we can get a multiple of 8, when we multiply n(n+1)(n+2)  is if n + 1 is itself a multiple of 8. Clearly 7*8*9 will be a multiple of 8. As will 15*16*17. We can either count the multiples of 8 between 1 and 96, or we can use the trusty formula: [(High-Low)/Interval] + 1. The first multiple of 8 between 1 and 96 is 8. The largest is 96. And the interval will be 8. So we get [(96-8)/8] + 1 = 11 + 1 = 12 multiples of 8.

So we have two categories of desired outcomes: there are 48 ways that n+1 can be odd, and there are 12 ways that n+1 can be a multiple of 8, giving us a total of 48 + 12 = 60 desired outcomes.

We’re done! The number of desired outcomes/number of total possible outcomes is 60/96, which will reduce to 5/8. The correct answer is D.

Takeaway: There’s no reason to be intimidated by probability questions, particularly when we remember that a probability calculation can be viewed as a ratio of two numbers. If we break the problem into its constituent parts, the question is often revealed to be quite a bit easier than it seems at first glance, a realization that proves true for almost any challenging GMAT problem.

*GMATPrep question courtesy of the Graduate Management Admissions Council.

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

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

Advanced Exponent Properties for the GMAT

Quarter Wit, Quarter WisdomToday, let’s discuss the relative placements of exponents on the number line.

We know what the graph of 2^x looks like:

 

 

 

 

Graph of 2^x

 

 

 

 

 

 

 

It shows that when x is positive, with increasing value of x, 2^x increases very quickly (look at the first quadrant), but we don’t know exactly how it increases.

It also shows that when x is negative, 2^x stays very close to 0. As x decreases, the value of 2^x decreases by a very small amount.

Now note the spacing of the powers of 2 on the number line:

Number line jpg

2^0 = 1

2^1 = 2

2^2 = 4

2^3 = 8

and so on…

2^1 = 2 * 2^0 = 2^0 + 2^0

2^2 = 2 * 2^1 = 2^1 + 2^1

2^3 = 2 * 2^2 = 2^2 + 2^2

2^4 = 2 * 2^3 = 2^3 + 2^3

So every power of 2 is equidistant from 0 and the next power. This means that a power of 2 would be much closer to 0 than the next higher powers. For example, 2^2 is at the same distance from 0 as it is from 2^3.

But 2^2 is much closer to 0 than it is to 2^4, 2^5 etc.

Let’s look at a question based on this concept. Most people find it a bit tough if they do not understand this concept:

Given that x = 2^b – (8^30 + 16^5), which of the following values for b yields the lowest value for |x|?

A) 35

B) 90

C) 91

D) 95

E) 105

We need the lowest value of |x|. We know that the smallest value any absolute value function can take is 0. So 2^b should be as close as possible to (8^30 + 16^5) to get the lowest value of |x|.

Let’s try to simplify:

(8^30 + 16^5)

= (2^3)^30 + (2^4)^5

= 2^90 + 2^20

Which value should b take such that 2^b is as close as possible to 2^90 + 2^20?

2^90 + 2^20 is obviously larger than 2^90. But is it closer to 2^90 or 2^91 or higher powers of 2?

Let’s use the concept we have learned today – let’s compare 2^90 + 2^20 with 2^90 and 2^91.

2^90 = 2^90 + 0

2^91 = 2^90 + 2^90

So now if we compare these two with 2^90 + 2^20, we need to know whether 2^20 is closer to 0 or closer to 2^90.

We already know that 2^20 is equidistant from 0 and 2^21, so obviously it will be much closer to 0 than it will be to 2^90.

Hence, 2^90 + 2^20 is much closer to 2^90 than it is to 2^91 or any other higher powers.

We should take the value 90 to minimize |x|, therefore the answer is B.

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

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!

5 Tips for Being Efficient with Data Sufficiency Problems

GoalsGMAT Data Sufficiency problems present test takers with several unique issues.  First, this type of problem has likely not been encountered before and thus, there is a learning curve for not only what the “rules” regarding such problems are, but also how to approach the various questions. Additionally, Data Sufficiency problems do not lend themselves to being solved through brute force – with Data Sufficiency questions you do not really need to answer the question in order to solve the problem.

Data Sufficiency questions are inherently more difficult than Problem Solving questions because they are more conceptual in nature.  Take for example the following problem:

Is xy > 0?

1) x < 6

2) 0 ≤ y < x

Right off the bat, we see that there are 2 variables, so to answer the question we need to know the values of x and y.  However, this problem is better viewed conceptually – instead of  determining the actual values of x and y, if we recognize that this problem is really testing us on the Properties of Numbers, we realize that what is actually being asked is if x and y are either both positive or both negative. Once we re-phrase the question this way, the problem is much easier to deal with.  Statement 1 says that x is less than 6, but this does not tell us definitively whether x is positive or negative. Nor, does Statement 1 give us any information about y.  Thus, Statement 1 is not sufficient.

Statement 2 gives us information about both x and y.  We now know that y is less than or equal to 0, and x is greater than y. This looks promising.  But, since y could be 0 (or greater than 0), we cannot say that xy is greater than 0.  Statement 2 is not sufficient.  Taking both Statements together provides no more information about y, so we still cannot answer the question (although some might be tempted to overlook the less than or equal to portion).

Here are some tips to efficiently and strategically approach these unique problems:

  • Memorize the answer choices! They are the same for every Data Sufficiency question on the GMAT, so you can save valuable time by knowing them and knowing that if Statement 1 is sufficient, your answer choices are either A or D.
  • Before reading the statements, try to verbalize what information you need to answer the question. This will help you to determine whether the statements provide the information you need.
  • Leverage as much information as you can from the prompt. Often times, important information is included in the prompt but not readily apparent.
  • Be very wary of statements that provide information that blatantly and obviously answers the question. If a question asks what the value of x is and one statement tells you x = 6, take a very close look at the other statement. Many times, the other statement will contain information that is difficult to decipher and the test makers are baiting you to select the obvious answer and move on.
  • Be on the lookout for statements that give no new information. The circumference of a circle, for instance, contains just as much information as the length of the radius. If you know the circumference, you can find the radius; conversely, if you know the radius, you can find the circumference. Often on Data Sufficiency questions, Statement 2 will just be a repackaging of the same information provided by Statement 1.

Even though GMAT Data Sufficiency problems require some different thinking, with some strategic practice, you will master them. Start with becoming familiar with the structure of the questions and the concepts they most commonly test.

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

By Dennis Cashion, a Veritas Prep instructor based in Denver.

The GMAT Quant Decimal Trend You NEED to Know

Pi to the 36th digitWhenever GMAC releases new material, I’m always on the lookout for conspicuous trends – esoteric or little known rules that end up being applicable in multiple questions. One type of question that has recently shown up with greater frequency involves terminating decimals. The concept isn’t, in the abstract, a terribly hard one. ½, for example, is .5, and so this is a terminating decimal. It ends. 1/3, on the other hand is .33333…, and continues indefinitely, so it’s not a terminating decimal. That’s not so hard. So, you say to yourself: all I have to do is perform a little division, and then I can see for myself if the decimal terminates or not, right?

But then you see a question like this:

Which of the following fractions has a decimal equivalent that is a terminating decimal?

A) 10/189

B) 15/196

C) 16/225

D) 25/144

E) 39/128

Once you spend a little time trying to divide 10 by 189, you realize that the question is going to be incredibly painful and time-consuming if you have to keep applying this approach until you find a fraction that results in a terminating decimal. So let’s be mindful of the fact that the purpose of the GMAT is not to test one’s facility for engaging in tedious arithmetic, but rather to assess our ability to recognize patterns under pressure.

Generally speaking, the best way to uncover a pattern is to use simple numbers first and then extrapolate our results to the more complex scenario we’re tasked with evaluating. We already established above that ½ is a terminating decimal and 1/3 is not. Let’s continue in that vein and see what we find (terminating decimals are in bold):

½ = .5

1/3 = .3333…

¼ = .25

1/5 = .2

1/6 = .166666…

1/7= .142857…

1/8 = .125

1/9 = .1111

1/10 = .1

Next, let’s examine our terminating decimal expressions and see if these numbers have any elements in common. Each of these fractions, it turns out, has a denominator whose prime factorization is composed solely of two prime bases, 2 or 5 or both. This turns out to be a general principle: if a fraction has been simplified, and the prime factorization of the denominator can be expressed in the form of 2^x * 5^y where x and y are non-negative integers, the fraction can be expressed as a terminating decimal.

Now back to our question. We can rephrase the question to be, “Which of the following denominators has a prime factorization that consists solely of 2’s or 5’s or both?”

Not bad. That certainly makes life a little easier. But before we dive in and begin taking prime factorizations with reckless abandon, let’s think like the test-maker. There is no way to do this question without working with the answer choices. Most test-takers will begin with A and work their way down. If you’re trying to create a difficult time-consuming question, where would you bury the correct answer? Probably towards D or E. So when we encounter this kind of scenario, we’re better off if we start at the bottom and work our way up.

E) 39/128. The denominator is 128, which has a prime factorization of 2^7. Because the denominator consists solely of 2’s, this fraction, when expressed as a decimal, must terminate. We’re done. E is the answer. (Intuitively, this makes sense, as all we’re really doing is cutting our numerator in half seven times.) Much easier than doing long division.

Before we commit this principle to memory, let’s make sure that it will be helpful in other contexts. After all, the rule that unlocks a single question won’t be terribly useful to us. So here is the same concept utilized in a Data Sufficiency question:

Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 24, 0.82, and 5.096 are three terminating decimals. If r and s are positive integers and the ratio r/s is expressed as a decimal, is r/s a terminating decimal? 

(1) 90 < r < 100 

(2) s = 4 

Notice how much easier this question is if we rephrase it as “if r/s is in its most simplified form, does the prime factorization of the denominator consist entirely of 2’s or 5’s?”

Statement 1 can’t be sufficient on its own, as it tells us nothing about the denominator. 91/2 is a terminating decimal, for example, but 91/3 is not.

Statement 2 tells us that the denominator is 4, or 2^2. If we’ve internalized our terminating decimal rule, we see right away that this must be sufficient, as anything dividing by 4 will result in a terminating decimal. The answer is B, Statement 2 alone is sufficient to answer the question.

Takeaway: When studying for the GMAT, it can feel as though there are an infinite number of rules, axioms, and formulas to memorize. Our job, when preparing, is to find the rules that are applicable in multiple contexts and internalize those. If we encounter a problem that seems unusually time-consuming, and no rule springs to mind, we can derive the necessary pattern on the spot by working with simple numbers.

*GMATPrep question courtesy of the Graduate Management Admissions Council.

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

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

GMAT Tip of the Week: Yogi Berra Teaches GMAT Sentence Correction

GMAT Tip of the WeekThe world lost a legend this week with the passing of Yogi Berra, a New York Yankee and World War II hero. Yogi was universally famous – his name was, of course, the inspiration for beloved cartoon character Yogi Bear’s – but to paraphrase the man himself, those who knew him didn’t really know him.

As news of his passing turned into news reports summarizing his life, many were stunned by just how illustrious his career was: 18 All-Star game appearances (in 19 pro seasons), 10 World Series championships as a player, 3 American League MVP awards, part of the Normandy campaign on D-Day… To much of the world, he was “the quote guy” who also had been a really good baseball player. His wordsmithery is what we all remembered:

  • Never answer an anonymous letter.
  • It ain’t over ’til it’s over.
  • It gets late early out here.
  • Pair up in threes.

And his command (or butchering) of the English language is what you should remember as you take the GMAT. Yogi Berra famously “didn’t say some of the things I said” but he did, however inadvertently, have a lot to say about GMAT Sentence Correction:

Pronouns Matter

What’s funny about his quote, “Always go to other people’s funerals, otherwise they won’t come to yours”?

It’s the pronoun “they.” You know what Yogi means – go to other people’s funerals so that other people will come to yours. But in that sentence, the logical referent for “they” is “other people(‘s)”, and those other people have already been designated in the sentence as people who have already died. So the meaning is illogical: those same people cannot logically attend a funeral in the future. When you use a pronoun, it has to refer back to a specific noun. If that noun cannot logically do what the pronoun is said to be doing, that’s a Sentence Correction, illogical meaning problem.

What’s funny about his quote, “When you come to a fork in the road, take it”?

Again, it’s the pronoun, this time “it.” Since a fork in the road is a place where the road diverges into two paths, you can’t take “it” – you have to pick one path. And this is a good example of another sentence correction theme. In order to fix this thought (and the one above), there’s really not a pronoun that will work. “Them” has no logical referent (there’s only one fork) so the meaning is extremely important.

The only way to fix it is to change something prior in the sentence. Perhaps, “When you come to a turnoff on the road, take it,” or, “when the road presents a turn, take it.” On the GMAT, a pronoun error isn’t always fixed by fixing the pronoun – often the correct answer will change the logic that precedes the pronoun so that in the correct answer the previously-incorrect pronoun is correct.

Modifiers Matter

What’s funny about his quote, “Congratulations. I knew the record would stand until it was broken”?

Of course records stand until they’re broken, but in a grammatical sense Yogi’s primary mistake was his placement of the modifier “until it was broken.” What he likely meant to say is, “Until the record was broken, I thought it might stand forever.” That’s a perfectly logical thought, but we all laugh at the statement he actually made because the placement of the modifier creates a laughable meaning. So learn to spot similarly-misplaced modifiers by checking to make sure the language means exactly what it should.

Redundancy Is Funny (but sometimes has its place)

What’s funny about, “We made too many wrong mistakes,” and “It’s like déjà vu all over again”?

They’re redundant. A mistake is, by nature, something that went wrong. And déjà vu is the feeling that something happened before, so of course it’s “all over again.” Redundancy does come up on the GMAT, but as Yogi himself would point out, there’s a fine line between “redundant (and wrong)” and “a useful literary device”.

Take, for example, his famed, “It ain’t over ’til it’s over” quote. In a sports context, even though the word “over” is repeated, that sentence carries a lot of useful meaning: “when someone might say that the game is over, if there is still time (or outs) remaining there’s always a chance to change the result.” The world chuckles at this particular Yogi quote, but in actuality it’s arguably his most famous because, in its own way, it’s quite poignant.

What does that mean for you on the GMAT? Don’t prioritize redundancy as a primary decision point! GMAT Sentence Correction, by nature, involves plenty of different literary devices and sentence structures, and it’s extremely unlikely that you’ll feel like an expert on all of them.

Students often eliminate correct answers because they perceive redundancy, but a phrase like “not unlike” (a “not” next to an “un-“? That’s a redundant double-negative!) actually has a logical and important meaning (“not unlike” means “it’s not totally different from…there are at least some similarities,” whereas “like” conveys significantly more similarity). Rules for modifiers and pronouns are much more absolute, and you can get plenty of practice with those. Be careful with redundancy because, as Yogi might say, sometimes saying it twice is twice as good as saying it once.

It’s all in your head.

“Baseball is ninety percent mental and the other half is physical.”

To paraphrase the great Yogi Berra, 90% of Sentence Correction is mental and the other half is grammatical. When he talked about baseball, he was talking about the physical tools – the ability to hit, run, throw, catch –  as meaning substantially less than people thought, but the mental part of the game – strategy, mental toughness, stamina, etc. – being more important than people thought. The exact percentages, as his quote so ineloquently suggests, are harder to pin down and less important than the takeaway.

So heed Yogi’s advice as it pertains to Sentence Correction. Memorizing and knowing hundreds of grammar rules is “the other half” (or maybe 10%) of the game – employing good strategy (prioritizing primary Decision Points, paying attention to logical meaning, etc.) is the more-important-but-often-overlooked part of success. However eloquently or inelegantly Yogi Berra may have articulated his lessons, at least he made them memorable.

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

By Brian Galvin.

6 Simple Steps to Attack Critical Reasoning Questions on the GMAT

GMAT ReasoningThe first step in attacking any Critical Reasoning question on the GMAT is to identify the premises and conclusions of the argument being presented. While Strengthen, Weaken, Assumption and Resolve the Paradox questions include a conclusion in the stimulus, Inference questions require you to select the conclusion (answer choice) that directly follows from the information presented in the stimulus.

This can be difficult because several of the answers can appear attractive. Keep in mind, however, that for Inference questions, the correct answer must be true. Answers that are “likely to be true” or “could be true” based on the information provided in the stimulus seem attractive at first, but if they are not true 100% of the time, in every situation, then they are not the correct answer.

Another difficulty in approaching Inference questions is that with the many of the other question types (Strengthen, Weaken, etc.), your job is to select the answer that includes new information that either undermines or supports the conclusion. For Inference questions, you do not want to bring in information that is not in the stimulus. All of the information required to answer the question will be included in the stimulus.

Here is a 6-step approach that can help you to efficiently attack GMAT Critical Reasoning Inference questions:

1) Read the question stem first.

This will allow you quickly categorize the type of Critical Reasoning question (Strengthen, Weaken, Inference, etc.) and let you focus on identifying the premises in the stimulus. Questions such as, “Which of the following can be correctly inferred from the statements above?” and, “If the statements above are true, which of the following must also be true?” signify that you are dealing with an Inference question.

2) Speculate what you think the correct conclusion is.

Sometimes this may be difficult to verbalize, but having an outline or framework of what the “must be true” answer should include will help to eliminate some answer choices.

3) Evaluate the answer choices using your speculated answer.

You want to carefully read all 5 answer choices. As you read the answers, compare them to the answer, or the outline of the answer, you speculated. Some answers are obviously incorrect – either they are too narrow in scope, too extreme to be always be true, or do not follow the criteria laid out in the stimulus. Eliminate these answers. For other answer choices that seem attractive, keep them as possibilities. Once you have read all of the answer choices, you can then compare your list of possible answers using the criteria that the correct answer must be always be true.

4) Become a Defense Lawyer.

When comparing your list of possible answers, try to come up with plausible scenarios that would prove the answer being considered not true. Just because the stimulus says that “everyone sitting in the dentist’s office waiting room at 9:00 a.m. was a patient” does not necessarily mean that they were waiting for an appointment. Some could have already finished their appointment, and some could have been there dropping off another patient. Like a defense lawyer, you need to find every every scenario in which an answer choice might not be true in order to eliminate it from your options.

5) Be aware of exaggerated or extreme answers.

Because the correct answer must always be true, modifiers that exaggerate an element of the premise or make an extreme claim usually signify an incorrect answer. If the stimulus says, “Some of the widgets produced by Company X were defective,” an attractive, yet incorrect answer choice may exaggerate this statement with a modifier such as “most” by claiming, “Most of Company X’s widgets were found to be defective.” Furthermore, answers that include the terms “always”, “never”, “none” and the like are good indicators that the answer will not be true 100% of the time.

6) Be aware of answers that change the scope of the stimulus.

On more difficult Inference questions (as if they were not difficult enough), the test makers will tempt you to select an answer choice that slightly changes an element of the facts laid out in the stimulus. For example, the stimulus might discuss the decrease in the violent crime rate in City A over a certain time period.

The attractive answer that follows all of the elements of having to be true 100% of the time, but is still incorrect might discuss decrease in the murder rate of City A over that time period. While the answer would seem to fit the bill, the murder rate is not the same as the rate of violent crime – this changes the scope of the initial stimulus and we can therefore rule that answer out.

The correct inference or conclusion on Critical Reasoning Inference questions is very close to what is stated explicitly in the stimulus. Remember, the right answer choice on these question types must be true 100% of the time.

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

By Dennis Cashion, a Veritas Prep instructor based in Denver.

How to Use the Answer Choices to Solve GMAT Quant Problems

EssayIn your approach to solving Quantitative problems on the GMAT, do not forget that the answers are part of the problem and often provide valuable information.

Take for example, the following question:

 

 

 

If 3x4y = 177,147 and x – y = 11, then x =?

A) Undefined

B) 0

C) 11

D) 177,136

E) 177,158

Where do we begin here? 177,147 is a large (not familiar) number and there are not one, but two exponents in the equation.  Looking at the answer choices, we can see that D and E cannot be the answer as they are too large, so at least now we have a starting point.  Additionally, we can see that our choices come down to some mixture of x and y, all y, or all x.

If x = 0, then we can say that 177, 147 is not divisible by 3 and is divisible by 4, so checking the divisibility rule is the ticket! Knowing that to be divisible by 4, the last two digits must be divisible by 4, we can see that 177,147 is not divisible by 4, so 4y becomes irrelevant and we realize y must equal 0.  The sum of the digits of 177, 147 is 27, which is divisible by 3, so we can see that the 3portion of the equation is relevant. We can now (correctly) conclude that the correct answer is answer choice C, x = 11.

Answer choices are little used resources by GMAT test takers.  In the heat of battle, we become so focused on solving the problems in front of us that we forget to utilize all of the information at our disposal.  Another way the answer choices can help you is by plugging them back into the problem to see if they work.

This “back-plugging” is useful when the problem to be solved is algebraic in nature and the answer choices are numbers (not variables). You may find it is easier on a certain problem to arithmetically calculate 2, 3 or even 5 answers by plugging in the answer choices, than in creating and manipulating a complex algebraic equation.  In these cases, plugging in answer choice C first will help you to eliminate up to 60% of the answers on the first calculation.

Many times, just understanding what the correct answer should “look like” by employing some reasoning on the front end will allow you to eliminate some, if not all of the incorrect answers.  Consider this problem:

((-1.9)(o.6) – (2.6)(1.2))/6.0 = ?

A) -0.71

B) 1.00

C) 1.07

D) 1.71

E) 2.71

This is not a difficult problem by any measure, and some test takers will not hesitate to jump in and begin multiplying and dividing decimals.  However, by spending a little bit of time looking at the big picture of this problem, an astute test taker would see that the answer must be negative.  The first term is negative and we are subtracting a larger number from it.  Therefore, the correct answer must be A.

So, instead of jumping in and crunching numbers on the GMAT, you can save yourself some time and brain power by using the answer choices to assist you in reasoning your way to the correct answer – or at least in eliminating several of the incorrect answers.

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

By Dennis Cashion, a Veritas Prep instructor based in Denver.

99th GMAT Score or Bust! Lesson 9: Talk Like a Lawyer

raviVeritas Prep’s Ravi Sreerama is the #1-ranked GMAT instructor in the world (by GMATClub) and a fixture in the new Veritas Prep Live Online format as well as in Los Angeles-area classrooms. He’s beloved by his students for the philosophy “99th percentile or bust!”, a signal that all students can score in the elusive 99thpercentile with the proper techniques and preparation. In this “9 for 99th” video seriesRavi shares some of his favorite strategies to efficiently conquer the GMAT and enter that 99th percentile.

First, take a look at the previous lessons in this series: 1, 2, 3, 4, 5, 6, 7 and 8!

Lesson Nine: 

Talk Like a Lawyer. When you click “Agree” on a user contract (think iTunes) or read through a GMAT question, you may just see an overkill of words. But thanks to lawyers, every word on that user agreement is carefully chosen – and that GMAT question is written the same exact way. In this final “9 for 99th” video, Ravi (a member of the bar himself) shows you how to talk and read like a lawyer, noticing those subtle word choices that can make or break your answer to those carefully-written GMAT problems you see on test day.​

Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook, YouTube and Google+, and follow us on Twitter!

Want to learn more from Ravi? He’s taking his show on the road for a one-week Immersion Course in New York this summer, and he teaches frequently in our new Live Online classroom.

By Brian Galvin

Beat the GMAT Verbal Section by Personalizing Questions

Integrated Reasoning GMATMy students often ask why the verbal section has to come at the very end of the GMAT. When they’re fresh, they complain, they’re able to answer a much higher percentage of questions correctly. Of course, this is precisely the point. Part of what the GMAT is assessing is your stamina and focus, both of which will certainly be flagging by the time you’ve been in the testing facility for over three hours.

Moreover, the questions themselves aren’t exactly known for their dazzling wit and soaring narrative verve. They’re boring. Reading Comp. passages are often tedious and technical, while Critical Reasoning arguments can feel so abstract as to be ungraspable. So how do we, as test-takers, combat this?

One answer, when it comes to those abstract Critical Reasoning questions, is to personalize the argument. I’ve blogged in the past about how our reading comprehension improves dramatically when we’re emotionally invested in what we’re reading, so why not attempt to trick ourselves into this state of heightened concentration?

If the CR question is about the impact of pesticide use on crop yields, I imagine I’m the farmer, and the well-being of my family is at stake. If the question is about how overtime pay will impact employee incentives, I imagine I own the business and that the consequence of my company’s compensation structure will impact not only me, but dozens of workers whose livelihood I’m responsible for. By creating these artificial stakes, I find that my brain is able to lock in on the minutia of the question in a way it can’t if the question is about some airy fictional farmer, whom I know exists only in the mind of some bureaucratic question writer.

Take an official question, for example:

In the past the country of Malvernia has relied heavily on imported oil. Malvernia recently implemented a program to convert heating systems from oil to natural gas. Malvernia currently produces more natural gas each year than it uses, and oil production in Malvernian oil fields is increasing at a steady pace. If these trends in fuel production and usage continue, therefore, Malvernian reliance on foreign sources for fuel is likely to decline soon. 

Which of the following would it be most useful to establish in evaluating the argument?

(A) When, if ever, will production of oil in Malvernia outstrip production of natural gas?

(B) Is Malvernia among the countries that rely most on imported oil?

(C) What proportion of Malvernia’s total energy needs is met by hydroelectric, solar, and nuclear power?

(D) Is the amount of oil used each year in Malvernia for generating electricity and fuel for transportation increasing?

(E) Have any existing oil-burning heating systems in Malvernia already been converted to natural-gas-burning heating systems?

If you’re anything like most test-takers, your eyes glaze over a bit. You know that Malvernia is not a real country, that it’s been invented for the sake of the problem. Consequently, the details of energy consumption in this non-existent country are not going to be terribly compelling to, well, anyone. This is by design. So let’s create some artificial stakes. Let’s say you’re the President of Malvernia. The economic well-being of your country, and, therefore, the prospects of your reelection, are going to be impacted by your country’s energy policy. Now let’s break down the facts:

  • Historically, you’ve relied on oil imports.
  • A new program converts heating systems from oil to gas.
  • You produce more gas than you use.
  • Oil production is increasing.

Based on this, you’ve concluded that your reliance on foreign oil will soon decrease. The question is what do you, as President, need to know to determine whether this prediction is valid?

Let’s break down each answer choice:

(A) The question of when production of oil will outstrip production of gas isn’t really relevant. In fact, if you’re using less oil as a result of the change in heating systems, and oil production is up, it’s possible that you can reduce your dependence on foreign oil without having to produce more oil than gas. A is out.

(B) Whether you are among the most dependent countries on foreign oil doesn’t matter. You are now, and we’re trying to determine if you will be in the future. This doesn’t help. Eliminate B.

(C) Hydroelectric, solar, and nuclear power aren’t relevant for this argument. We know that you’re dependent on foreign oil now, irrespective of other energy sources. It’s increased oil production and switching to gas that will, according to the argument, reduce this dependence. C is out of scope.

(D) Let’s say your oil consumption for electricity and transportation is increasing. Suddenly, the fact that you’re switching heating systems from oil to gas might not help – if your oil needs are going up in other areas, you may remain dependent on foreign oil. But if your oil consumption in these other areas is not increasing, that would reduce your dependence on foreign oil because your heating systems are switching to gas. D looks good.

(E) This doesn’t matter at all. We know that the systems are going to switch from oil to gas, so the question of whether some systems have already made the switch sheds no light on whether you will remain dependent on foreign oil.

D is the answer. Once you have the answer to whether your oil consumption for electricity and transportation is increasing, you’ll be better able to assess whether you will remain dependent on foreign oil, and, consequently, whether your reign as supreme ruler of Malvernia will continue.

Takeaway: There is plenty of research indicating that our comprehension improves drastically when we’re reading something we care about. When we put ourselves into the position of the agents having to make decisions in these arguments, we can transform a tedious abstraction into something that has a bit of emotional resonance, which will, in turn, result in a higher GMAT score.

*Official Guide question courtesy of the Graduate Management Admissions Council.

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