Dreaded Data Sufficiency Questions That Will Test Your Knowledge of Number Properties

Here is an often-repeated complaint we hear from test takers – Data Sufficiency questions that deal with number properties are very difficult to handle (even for people who find problem-solving number properties questions manageable)! They feel that such questions are time consuming and often involve too many cases.

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

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

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

Statement 1: b^a is negative
Statement 2: a^b is negative

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

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

Statement 1: b^a is negative

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

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

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

Let’s look at Statement 2:

Statement 2: a^b is negative

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

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

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

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

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

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

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

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

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: Using Ingenuity on GMAT Remainder Questions

We have looked at various types of GMAT remainder questions and discussed how to tackle them in a few previous posts. Specifically, we have examined the concepts of general divisibility, divisibility as applied to GMAT questions, and divisibility specifically applied to remainders. There is one concept, however, that we haven’t discussed yet, and that is using ingenuity on remainder questions.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(z – 3)^2 = z^2 – 6z + 9

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

z^2 – 6z + 9 + 8z = z^2 + 2z + 9

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

Let’s look at the second statement:

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

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

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

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

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!

Which is Worse to Encounter on a GMAT Question: Median or Mean?

Hypothetically speaking, given a choice between a question on median and one on mean, which would you choose? (Not that we are fortunate enough to have a choice on test day, but no harm in dreaming!) I would certainly pick the question testing on median, and here is why:

Median is the value at a point – to be precise, the point which divides the increasing data set into two equal halves. You don’t care what is on the left and what is on the right of this point, so an outlier will do nothing to the median. The mean, however depends on every value in the set. If you increase one element of data, the mean of the set changes – outliers can drastically change the value of the mean. Hence, every element has to be kept in mind! With the median, there is a lot less to worry about.

Let’s illustrate this with an example data sufficiency question:

Question on Median:
At a bakery, cakes are sold every day for a certain number of days. If 6 or more cakes were sold for 20% of the total number of days, is the median number of cakes sold less than 4?

Statement 1: On 75% of the days that less than 6 cakes were sold, the number of cakes sold each day was less than 4.
Statement 2: On 50% of the days that 4 or more cakes were sold, the number of cakes sold each day was 6 or more.

The following is the number of cakes sold on any of the days mentioned in the question:

Say there were 100 days (since all figures are in terms of percentages, we can assume a number to simplify our understanding).

The question stem tells us that 6 or more cakes were sold for 20% of the days, so for 20 days, 6 or more cakes were sold. Then for 80 days, 1/2/3/4/5 cakes were sold.

With this information in mind, is the median number of cakes sold in one day less than 4?

We know how to get the median. When we arrange all figures in increasing order, the median will be the average of the 50th and the 51st terms. We need to know if the average of the 50th and 51st term is less than 4. Let’s tackle the statements one at a time:

Statement 1: On 75% of the days that less than 6 cakes were sold, the number of cakes sold each day was less than 4.

The number of days that less than 6 cakes were sold = 80. 75% of these 80 days will be 60 days. In 60 days, less than 4 cakes were sold. So the 50th and 51st terms will be less than 4 and so will their average. Hence, the median will be less than 4. This statement alone is sufficient.

Statement 2: On 50% of the days that 4 or more cakes were sold, the number of cakes sold each day was 6 or more.

In 20 days, 6 or more cakes were sold. This constitutes 50% of the days during which 4 or more cakes were sold, so in another 20 days, 4 or 5 cakes were sold. Hence, during the leftover 60 days, less than 4 cakes were sold. The 50th and 51st terms will be less than 4 and so will their average. Hence, the median will be less than 4. This statement alone is also sufficient, so our answer is D.

All we needed to worry about here were the 50th and 51st terms, however the whole problem changes when we talk about mean instead of median.

Same Question on Mean:
At a bakery, cakes are sold every day for a certain number of days. If 6 or more cakes were sold for 20% of the total number of days, is the average number of cakes sold less than 4?

Statement 1: On 75% of the days that less than 6 cakes were sold, the number of cakes sold each day was less than 4.
Statement 2: On 50% of the days that 4 or more cakes were sold, the number of cakes sold each day was 6 or more.

Again, the question stem tells us that 6 or more cakes were sold for 20% of the days, so for 20 days, 6 or more cakes were sold. Then for 80 days, 1/2/3/4/5 cakes were sold.

We now need to ask ourselves is the average number of cakes sold in one day less than 4?

This question asks us about the average. – that is far more complicated than the median. Every value matters when we talk about the average. We need to know the number of cakes sold on each of these 100 days to get the average.

6 or more cakes were sold in 20 days. Note that the number of cakes sold during these 20 days could be any number greater than 6, such as 20 or 50 or 120, etc. The minimum number of cakes sold on these 20 days would be 6*20 = 120. There is no limit to the maximum number of cakes sold.

With this in mind, let’s examine the statements:

Statement 1: On 75% of the days that less than 6 cakes were sold, the number of cakes sold each day was less than 4.

In 80 days, less than 6 cakes were sold. Of this number, 75% is 60 days. In 60 days, less than 4 cakes were sold.

So in 60 days, you have a minimum of 1*60 = 60 cakes sold and a maximum of 3*60 = 180 cakes sold. During the leftover 20 days 4 or 5 cakes were sold, so you have a minimum of 4*20 = 80 cakes and a maximum of 5*20 = 100 cakes.

The minimum value of the average is (120 + 60 + 80)/ 100 = 2.6 cakes, but the maximum average could be anything. Therefore, this statement alone is not sufficient.

Statement 2: On 50% of the days that 4 or more cakes were sold, the number of cakes sold each day was 6 or more.

The 20 days when 6 or more cakes were sold make up 50% of the days when 4 or more cakes were sold. So for another 20 days, 4 or 5 cakes were sold. This gives us a minimum of 4*20 = 80 cakes and a maximum of 5*20 = 100 cakes. For 60 days, 1/2/3 cakes were sold. So in 60 days, you have minimum of 1*60 = 60 cakes sold and a maximum of 3*60 = 180 cakes sold.

The minimum value of the average is (120 + 60 + 80)/ 100 = 2.6 cakes, but again, the maximum average could be anything. This statement alone is also not sufficient.

Note that both statements give you the same information, so if they are not sufficient independently, they are not sufficient together. The answer of this modified question would be E.

Here, we had to assume the minimum and maximum value for each data point to get the range of the average – we couldn’t just rely on one or two data points. Finding the mean during a GMAT question requires much more information than finding the median!

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: Evolving Your GMAT Quant Score with Help from The Evolution Of Rap

If it’s March, it must be Hip Hop Month at the GMAT Tip of the Week space, where this year we’ve been transfixed by Vox’s video on the evolution of rhyme schemes in the rap world.

The video below (which is absolutely worth a watch during a designated study break) explores the way that rap has evolved from simple rhyme schemes (yada yada yada Bat, yada yada yada Hat, yada yada yada Rat, yada yada yada Cat…) to the more complex “wait did he just say what I thought he said?” inside-out rhyme schemes that make you rewind an Eminem or Kendrick Lamar track because your ears must be playing tricks on you.

And if you don’t have the study break time right now, we’ll summarize. While a standard rhyme might have a one-syllable rhyme at the end of each bar (do you like green eggs and HAM, yes I like them Sam I AM), rappers have continued to evolve to the point where nowadays each bar can contain multiple rhyme schemes. Consider Eminem’s “Lose Yourself”:

Snap back to reality, oh there goes gravity
Oh there goes Rabbit he choked, he’s so mad but he won’t
Give up that easy, nope, he won’t have it he knows
His whole back’s to these ropes, it don’t matter he’s dope
He knows that but he’s broke, he’s so stagnant he knows…

Where “gravity,” “Rabbit, he,” “mad but he,” “that easy,” “have it he,” “back’s to these,” “matter he’s,” “that but he’s,” and “stagnant, he” all rhyme with one another, the list of goes/goes/choked/so/won’t/knows/whole/ropes/don’t/dope… keeps that hard “O” sound rhyming consistently throughout, too. And that was 15 years ago…since them, Eminem, Kendrick, and others have continued to build elaborate rhyme schemes that reward those listeners who don’t just listen for the simple rhyme at the end of each bar, but pick up the subtle rhyme flows that sometimes don’t come back until a few lines later.

So what does this have to do with your GMAT score?

One of the most common study mistakes that test-takers make is that they study skills as individual, standalone entities, and don’t look for the subtle ways that the GMAT testmaker can layer in those sophisticated Andre-3000-style combinations. Consider an example of an important GMAT skill, the “Difference of Squares” rule that (x + y)(x – y) = x^2 – y^2. A standard (think early 1980s Sugarhill Gang or Grandmaster Flash) GMAT question might test it in a relatively “obvious” way:

What is the value of (x + y)?

(1) x^2 – y^2 = 0
(2) x does not equal y

Here if you factor Statement 1 you’ll get (x + y)(x – y) = 0, and then Statement 2 tells you that it’s not (x – y) that equals zero, so it must be x + y. This Data Sufficiency answer is C, and the test is essentially just rewarding you for knowing the Difference of Squares.

The GMAT it cares
’bout the Difference of Squares
When there’s squares and subtraction
Put this rule into action

A slightly more sophisticated question (think late 1980s/early 1990s Rob Bass or Run DMC) won’t so obviously show you the Difference of Squares. It might “hide” that behind a square that few people tend to see as a square, the number 1:

If y = 2^(16) – 1, the greatest prime factor of y is:

(A) Less than 6
(B) Between 6 and 10
(C) Between 10 and 14
(D) Between 14 and 18
(E) Greater than 18

Here, many people don’t recognize 1 as a perfect square, so they don’t see that the setup is 2^(16) – 1^(2), which can be factored as:

(2^8 + 1)(2^8 – 1)

And that 2^8 – 1 can be factored again, since 1 remains 1^2:

(2^8 + 1)(2^4 + 1)(2^4 – 1)

And that ultimately you could do it again with 2^4 – 1 if you wanted, but you should know that 2^4 is 16 so you can now get to work on smaller numbers. 2^8 is 256 and 2^4 is 16, so you have:

257 * 17 * 15

And what really happens now is that you have to factor out 257 to see if you can break it into anything smaller than 17 as a factor (since, if not, you can select “greater than 18”). Since you can’t, you know that 257 must have a prime factor greater than 18 (it turns out that it’s prime) and correctly select E.

The lesson here? This problem directly tests the Difference of Squares (you don’t want to try to calculate 2^16, then subtract 1, then try to factor out that massive number) but it does so more subtly, layering it inside the obvious “prime factor” problem like a rapper might embed a secondary rhyme scheme in the middle of each bar.

But in really hard problems, the testmaker goes full-on Greatest of All Time rapper, testing several things at the same time and rewarding only the really astute for recognizing the game being played. Consider:

The size of a television screen is given as the length of the screen’s diagonal. If the screens were flat, then the area of a square 21-inch screen would be how many square inches greater than the area of a square 19-inch screen?

(A) 2
(B) 4
(C) 16
(D) 38
(E) 40

Now here you KNOW you’re dealing with a geometry problem, and it also looks like a word problem given the television backstory. As you start calculating, you’ll know that you have to take the diagonal of each square TV and use that to determine the length of each side, using the 45-45-90 triangle ratio, where the diagonal = x√2. So the length of a side of the smaller TV is 19/√2 and the length of a side of the larger TV is 21/√2.

Then you have to calculate the area, which is the side squared, so the area of the smaller TV is (19/√2)^2 and the area of the larger TV is (21/√2)^2. This is starting to look messy (Who knows the squares for 21 and 19 offhand? And radicals in denominators never look fun…) UNTIL you realize that you have to subtract the two areas. Which means that your calculation is:

(21/√2)^2 – (19/√2)^2

This fits perfectly in the Difference of Squares formula, meaning that you can express x^2 – y^2 as (x + y)(x – y). Doing that, you have:

[(21 + 19)/√2][(21-19)/√2]

Which is really convenient because the math in the numerators is easy and leaves you with:

(40/√2) * (2/√2)

And when you multiply them, the √2 terms in the denominators square out to 2, which factors with the 2 in the numerator of the right-side fraction, and everything simplifies to 40. And then, in classic “oh this guy’s effing GOOD” hip-hop style (like in the Eminem lyric “you’re witnessing a massacre like you’re watching a church gathering take place” and you realize that he’s using “massacre” and “mass occur” – the church gathering taking place – simultaneously), you realize that you should have seen it coming all along. Because when you subtract the area of one square minus the area of another square you’re LITERALLY taking the DIFFERENCE of two SQUARES.

So what’s the point?

Too often people study for the GMAT like they’d listen to 1980s rap. They expect the Difference of Squares to pair nicely at the end of an Algebra-with-Exponents bar, and the Isosceles Right Triangle formula to pair nicely with a Triangle question. They learn skills in distinct silos, memorize their flashcards in nice, tidy sets, and then go into the test and realize that they’re up against an exam that looks a lot more like a 2017 mixtape with layers of rhyme schemes and motives.

You need to be prepared to use skills where they don’t seem to obviously belong, to jot down and rearrange your scratchwork, label your unknowns, etc., looking for how you might reposition the math you’re given to help you bring in a skill or concept that you’ve used countless times, just in totally different contexts. The GMAT testmaker has a much more sophisticated flow than the one you’re likely studying for, so pay attention to that nuance when you study and you’ll have a much better chance of keeping your score 800.

Getting ready to take the GMAT? We have free online GMAT seminars running all the time. And as always, be sure to follow us on Facebook, YouTubeand Twitter!

By Brian Galvin.

Quarter Wit, Quarter Wisdom: When a Little Information is Enough to Solve a GMAT Problem

We have reviewed what standard deviation is in a past post. We know what data is necessary to calculate the standard deviation of a set, but in some cases, we could actually do with a lot less information than the average test-taker may think they need.

Let’s explore this idea through an example GMAT data sufficiency question:

What is the standard deviation of a set of numbers whose mean is 20?

Statement 1: The absolute value of the difference of each number in the set from the mean is equal.
Statement 2: The sum of the squares of the differences from the mean is greater than 100.

We need to determine whether the information we have been given is sufficient to get us the exact value of the standard deviation of a particular set of numbers. To find the standard deviation of a set, we need to know the deviation of each term from the mean so that we can square those deviations, sum the squares, divide them by the number of terms, and then find the square root.

Essentially, to find the standard deviation we either need to know each element of the set, or we need to know the deviation of each element from the mean (which will also give us the number of terms), or we need to know the sum of the square of deviations and the number of terms in the set.

The question stem here tells us that the mean of the set is 20. We have no other information about any of the actual elements of the set or the number of elements. With this in mind, let’s examine each of the statements:

Statement 1: The absolute value of the difference of each number in the set from the mean is equal.

With this statement, we don’t actually know what the absolute value of the difference is. We also don’t know how many elements there are. The set could be something like:

19, 21 (each term is exactly 1 away from the mean 20)
or
18, 18, 22, 22 (each term is exactly 2 away from the mean 20)
etc.

The standard deviation in each case will be different. We don’t know the elements of the set and we don’t know the number of elements in the set. Because of this, there is no way for us to know the value of the standard deviation – this statement alone is not sufficient.

Statement 2: The sum of the squares of the differences from the mean is greater than 100.

“Greater than 100” encompasses a large range of numbers – it could be any value larger than 100. Again, we cannot find the exact standard deviation of the set, so this statement is also not sufficient alone.

Using both statements together, we still do not have any idea of what the elements of the set are or what the sum of the squares of the differences from the mean is. We also still don’t know the number of elements. Hence, both statements together are not sufficient, so the answer is E.

Now, let us add just one more piece of information to the problem in this similar question:

What is the standard deviation of a set of 7 numbers whose mean is 20?

Statement 1: The absolute value of the difference of each number in the set from the mean is equal.
Statement 2: The sum of the squares of the differences from the mean is greater than 100.

What would you expect the answer to be? Still E, right? The sum of the deviations are still unknown and the exact elements of the set are still unknown – all we know is the number of elements. Actually, this information is already too much. All we need to know is that the number of elements is odd and suddenly we can find the standard deviation.

Here is why:

Statement 1 is quite tricky.

If we have an odd number of elements, in which case can the absolute values of the differences of each number in the set from the mean be equal?

Think about it – the mean of the set is 20. What could a possible set look like such that the mean is 20 and the absolute values of the differences of each number in the set from the mean are equal. Try to think of such a set with just 3 elements. Can you come up with one?

19, 19, 21? No, the mean is not 20

19, 20, 21? No, the absolute value of the difference of each number in the set from the mean is not equal. 19 is 1 away from mean but 20 is 0 away from mean.

Note that in this case, the only possible set that could fit the given criteria is one consisting of just an odd number of 20s (all elements in this set must be 20). Only then can each number be equidistant from the mean, i.e. each number would be 0 away from mean. If the numbers of the set all have equal elements, then obviously the standard deviation of the set is 0. It doesn’t matter how many elements it has; it doesn’t matter what the mean is! In this case, Statement 1 alone is sufficient so the answer would be A.

Takeaway:
If a set has an even number of distinct terms, the absolute values of the distances of each term from the mean could be equal. But if a set has an odd number of terms and the absolute values of the distances of each term from the mean are equal, all the terms in the set must be the same and will be equal to the mean.

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: Keep Your GMAT Score Safe from the Bowling Green Massacre

The hashtag of the day is #bowlinggreenmassacre, inspired by an event that never happened. Whether intentionally or accidentally (we’ll let you and your news agency of choice decide which), White House staffer Kellyanne Conway referenced the “event” in an interview, inspiring an array of memes and references along the way.

Whatever Ms. Conway’s intentions (or lack thereof; again we’ll let you decide) with the quote, she is certainly guilty of inadvertently doing one thing: she didn’t likely intend to help you avoid a disaster on the GMAT, but if you’re paying attention she did.

Your GMAT test day does not have to be a Bowling Green Massacre!

Here’s the thing about the Bowling Green Massacre: it never happened. But by now, it’s lodged deeply enough in the psyche of millions of Americans that, to them, it did. And the same thing happens to GMAT test-takers all the time. They think they’ve seen something on the test that isn’t there, and then they act on something that never happened in the first place. And then, sadly, their GMAT hopes and dreams suffer the same fate as those poor souls at Bowling Green (#thoughtsandprayers).

Here’s how it works:

The Quant Section’s Bowling Green Massacre
On the Quant section, particularly with Data Sufficiency, your mind will quickly leap to conclusions or jump to use a rule that seems relevant. Consider the example:

What is the perimeter of isosceles triangle LMN?

(1) Side LM = 4
(2) Side LN = 4√2

A. Statement (1) ALONE is sufficient, but statement (2) alone is insufficient
B. Statement (2) ALONE is sufficient, but statement (1) alone is insufficient
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D. EACH statement ALONE is sufficient
E. Statements (1) and (2) TOGETHER are NOT sufficient

When people see that square root of 2, their minds quickly drift back to all those flash cards they studied – flash cards that include the side ratio for an isosceles right triangle: x, x, x√2. And so they then leap to use that rule, inferring that if one side is 4 and the other is 4√2, the other side must also be 4 to fit the ratio and they can then calculate the perimeter. With both statements together, they figure, they can derive that perimeter and select choice C.

But think about where that side ratio comes from: an isosceles right triangle. You’re told in the given information that this triangle is, indeed, isosceles. But you’re never told that it’s a right triangle. Much like the Bowling Green Massacre, “right” never happened. But the mere suggestion of it – the appearance of the √2 term that is directly associated with an isosceles, right triangle – baits approximately half of all test-takers to choose C here instead of the correct E (explanation: “isosceles” means only that two sides match, so the third side could be either 4, matching side LM, or 4√2, matching side LN).

Your mind does this to you often on Data Sufficiency problems: you’ll limit the realm of possible numbers to integers, when that wasn’t defined, or to positive numbers, when that wasn’t defined either. You’ll see symptoms of a rule or concept (like √2 leads to the isosceles right triangle side ratio) and assume that the entire rule is in play. The GMAT preys on your mind’s propensity for creating its own story when in reality, only part of that story really exists.

The Verbal Section’s Bowling Green Massacre
This same phenomenon appears on the Verbal section, too – most notably in Critical Reasoning. Much like what many allege that Kellyanne Conway did, your mind wants to ascribe particular significance to events or declarations, and it will often exaggerate on you. Consider the example:

About two million years ago, lava dammed up a river in western Asia and caused a small lake to form. The lake existed for about half a million years. Bones of an early human ancestor were recently found in the ancient lake-bottom sediments that lie on top of the layer of lava. Therefore, ancestors of modern humans lived in Western Asia between two million and one-and-a-half million years ago.

Which one of the following is an assumption required by the argument?

A. There were not other lakes in the immediate area before the lava dammed up the river.
B. The lake contained fish that the human ancestors could have used for food.
C. The lava that lay under the lake-bottom sediments did not contain any human fossil remains.
D. The lake was deep enough that a person could drown in it.
E. The bones were already in the sediments by the time the lake disappeared.

The key to most Critical Reasoning problems is finding the conclusion and knowing EXACTLY what the conclusion says – nothing more and nothing less. Here the conclusion is the last sentence, that “ancestors of modern humans lived” in this region at this time. When people answer this problem incorrectly, however, it’s almost always for the same reason. They read the conclusion as “the FIRST/EARLIEST ancestors of modern humans lived…” And in doing so, they choose choice C, which protects against humans having come before the ones related to the bones we have.

“First/earliest” is a classic Bowling Green Massacre – it’s a much more noteworthy event (“scientists have discovered human ancestors” is pretty tame, but “scientists have discovered the FIRST human ancestors” is a big deal) that your brain wants to see. But it’s not actually there! It’s just that, in day to day life, you’d rarely ever read about a run-of-the-mill archaeological discovery; it would only pop up in your social media stream if it were particularly noteworthy, so your mind may very well assume that that notoriety is present even when it’s not.

In order to succeed on the GMAT, you need to become aware of those leaps that your mind likes to take. We’re all susceptible to:

• Assuming that variables represent integers, and that they represent positive numbers
• Seeing the symptoms of a rule and then jumping to apply it
• Applying our own extra superlatives or limits to conclusions

So when you make these mistakes, commit them to memory – they’re not one-off, silly mistakes. Our minds are vulnerable to Bowling Green Massacres, so on test day #staywoke so that your score isn’t among those that are, sadly, massacred.

By Brian Galvin.

Advanced Number Properties on the GMAT – Part VII

We have seen a number of posts on divisibility, odd-even concepts and perfect squares. Individually, each topic has very simple concepts but when they all come together in one GMAT question, it can be difficult to wrap one’s head around so many ideas. The GMAT excels at giving questions where multiple concepts are tested. Let’s take a look at one such Data Sufficiency question today:

If p, x, and y are positive integers, y is odd, and p = x^2 + y^2, is x divisible by 4?

1) When p is divided by 8, the remainder is 5.
2) x – y = 3

This Data Sufficiency question has a lot of information in the question stem.  First, we need to sort through this information before we move on to the statements.

We know that p, x and y are positive integers. y is an unknown odd number, so it can be written in the form 2n + 1. We also know that p = x^2 + y^2.

Because y is written in the form 2n + 1, y^2 can be written as:

y^2 =(2n + 1)^2

y^2 = 4n^2 + 4n + 1

y^2 = 4n(n + 1) + 1

An interesting thing to note here is that one case of n and (n+1) will be odd and the other will be even. In every case, n(n + 1) is even. Therefore, y^2 is 1 more than a multiple of 8. In other words, we can write it as y^2 = 8m + 1.

Now we can say p = x^2 + 8m + 1.

With this in mind, is x divisible by 4? Let’s examine the statements to find out:

Statement 1: When p is divided by 8, the remainder is 5.

Because y^2 = 8m + 1, we can see that when y^2 is divided by 8, the remainder will be 1. Therefore, to get a remainder of 5 when p is divided by 8, when x^2 is divided by 8, we should get a remainder of 4.

Now we know that x^2 can be written in the form 8a + 4 (i.e. we can make “a’” groups of 8 each and have 4 leftover).

x^2 = 4*(2a + 1)

So x = 2 * √(an odd number)

Note that square root of an odd number will be an odd number only. If there is no 2 in the perfect square, obviously there was no 2 in the number, too.

So, x = 2 * some other odd number, which means x will be a multiple of 2, but not of 4 definitely. This statement alone is sufficient.

Now let’s look at the next statement:

Statement 2: x – y = 3

Since y is odd, we can say that x will be even (an even – an odd = an odd). But whether x is divisible by 2 only or by 4 as well, we cannot say since we have no constraints on p.

This statement alone is not sufficient to answer the question. Therefore, our answer is A.

Test takers might feel that not every step in this solution is instinctive. For example, how do we know that we should put y^2 in the form 4n(n+1) + 1? Keep the target in mind – we know that we need to find whether x is divisible by 4. Hence, try to get everything in terms of multiples of 4 + a remainder.

See you next week!

(For more advanced number properties on the GMAT, check out Parts IIIIIIIV, V and VI of this series.)

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!

The Patterns to Solve GMAT Questions with Reversed-Digit Numbers – Part II

, I wrote about the GMAT’s tendency to ask questions regarding the number properties of two two-digit numbers whose tens and units digits have been reversed.

The biggest takeaways from that post were:

1. Anytime we add two two-digit numbers whose tens and units digits have been reversed, we will get a multiple of 11.
2. Anytime we take the difference of two two-digit numbers whose tens and units digits have been reversed, we will get a multiple of 9.

For the hardest GMAT questions, we’re typically mixing and matching different types of number properties and strategies, so it can be instructive to see how the above axioms might be incorporated into such problems.

Take this challenging Data Sufficiency question, for instance:

When the digits of two-digit, positive integer M are reversed, the result is the two-digit, positive integer N. If M > N, what is the value of M?

(1) The integer (M –N) has 12 unique factors.

(2) The integer (M –N) is a multiple of 9.

The average test-taker looks at Statement 1, sees that it will be very difficult to simply pick numbers that satisfy this condition, and concludes that this can’t possibly be enough information. Well, the average test-taker also scores in the mid-500’s, so that’s not how we want to think.

First, let’s concede that Statement 1 is a challenging one to evaluate and look at Statement 2 first. Notice that Statement 2 tells us something we already know – as we saw above, anytime you have two two-digit numbers whose tens and units digits are reversed, the difference will be a multiple of 9. If Statement 2 is useless, we can immediately prune our decision tree of possible correct answers. Either Statement 1 alone is sufficient, or the statements together are not sufficient, as Statement 2 will contribute nothing. So right off the bat, the only possible correct answers are A and E.

If we had to guess, and we recognize that the average test-taker would likely conclude that Statement 1 couldn’t be sufficient, we’d want to go in the opposite direction – this question is significantly more difficult (and interesting) if it turns out that Statement 1 gives us considerably more information than it initially seems.

In order to evaluate Statement 1, it’s helpful to understand the following shortcut for how to determine the total number of factors for a given number. Say, for example, that we wished to determine how many factors 1000 has. We could, if we were sufficiently masochistic, simply list them out (1 and 1000, 2 and 500, etc.). But you can see that this process would be very difficult and time-consuming.

Alternatively, we could do the following. First, take the prime factorization of 1000. 1000 = 10^3, so the prime factorization is 2^3 * 5^3. Next, we take the exponent of each prime base and add one to it. Last, we multiply the results. (3+1)*(3+1) = 16, so 1000 has 16 total factors. More abstractly, if your number is x^a * y^b, where x and y are prime numbers, you can find the total number of factors by multiplying (a+1)(b+1).

Now let’s apply this process to Statement 1. Imagine that the difference of M and N comes out to some two-digit number that can be expressed as x^a * y^b. If we have a total of 12 factors, then we know that (a+1)(b+1) = 12. So, for example, it would work if a = 3 and b = 2, as a + 1 = 4 and b + 1 = 3, and 4*3 =12. But it would also work if, say, a = 5 and b = 1, as a + 1 = 6 and b + 1 = 2, and 6*2 = 12. So, let’s list out some numbers that have 12 factors:

1. 2^3 * 3^2 (3+1)(2+1) = 12
2. 2^5 * 3^1 (5+1)(1+1) = 12
3. 2^2 * 3^3 (2+1)(3+1) = 12

Now remember that M – N, by definition, is a multiple of 9, which will have at least 3^2 in its prime factorization. So the second option is no longer a candidate, as its prime factorization contains only one 3. Also recall that we’re talking about the difference of two two-digit numbers. 2^2 * 3^3 is 4*27 or 108. But the difference between two positive two-digit numbers can’t possibly be a three-digit number! So the third option is also out.

The only possibility is the first option. If we know that the difference of the two numbers is 2^3 * 3^2, or 8*9 = 72, then only 91 and 19 will work. So Statement 1 alone is sufficient to answer this question, and the answer is A.

Algebraically, if M = 10x + y, then N = 10y + x.

M – N = (10x + y) – (10y + x) = 9x – 9y = 9(x – y).

If 9(x – y) = 72, then x – y = 8. If the difference between the tens and units digits is 8, the numbers must be 91 and 19.

Takeaway: the hardest GMAT questions will require a balance of strategy and knowledge. In this case, we want to remember the following:

• Anytime we take the difference of two two-digit numbers whose tens and units digits have been reversed, we will get a multiple of 9.
• If one statement is easier to evaluate than the other, tackle the easier one first. If it’s the case that one statement gives you absolutely nothing, and the other is complex, there is a general tendency for the complex statement alone to be sufficient.
• For the number x^a * y^b, where x and y are prime numbers, you can find the total number of factors by multiplying (a+1)(b+1).

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

Data Sufficiency Questions: How to Know When Both Statements Together Are Not Sufficient

Today we will discuss a problem we sometimes face while attempting to solve Data Sufficiency questions for which the answer is actually E (when both statements together are not sufficient to answer the question). Ideally, we would like to find two possible answers to the question asked so that we know that the data of both statements is not sufficient to give us a unique answer. But what happens when it is not very intuitive or easy to get these two distinct cases?

Let’s try to answer these questions in today’s post using using one of our own Data Sufficiency questions.

A certain car rental agency rented 25 vehicles yesterday, each of which was either a compact car or a luxury car. How many compact cars did the agency rent yesterday?

(1) The daily rental rate for a luxury car was \$15 higher than the rate for a compact car.
(2) The total rental rates for luxury cars was \$105 higher than the total rental rates for compact cars yesterday

We know from the question stem that the total number of cars rented is 25. Now we must find how many compact cars were rented.

There are four variables to consider here:

1. Number of compact cars rented (this is what we need to find)
2. Number of luxury cars rented
3. Daily rental rate of compact cars
4. Daily rental rate of luxury cars

Let’s examine the information given to us by the statements:

Statement 1: The daily rental rate for a luxury car was \$15 higher than the rate for a compact car.

This statement gives us the difference in the daily rental rates of a luxury car vs. a compact car. Other than that, we still only know that a total of 25 cars were rented. We have no data points to calculate the number of compact cars rented, thus, this statement alone is not sufficient. Let’s look at Statement 2:

Statement 2: The total rental rates for luxury cars was \$105 higher than the total rental rates for compact cars yesterday.

This statement gives us the difference in the total rental rates of luxury cars vs. compact cars (we do not know the daily rental rates). Again, we have no data points to calculate the number of compact cars rented, thus, this statement alone is also not sufficient.

Now, let’s try to tackle both statements together:

The daily rate for luxury cars is \$15 higher than it is for compact cars, and the total rental rates for luxury cars is \$105 higher than it is for compact cars. What constitutes this \$105? It is the higher rental cost of each luxury car (the extra \$15) plus adjustments for the rent of extra/fewer luxury cars hired. That is, if n compact cars were rented and n luxury cars were rented, the extra total rental will be 15n. But if more  luxury cars were rented, 105 would account for the \$15 higher rent of each luxury car and also for the rent of the extra luxury cars.

Event with this information, we still should not be able to find the number of compact cars rented. Let’s find 2 cases to ensure that answer to this question is indeed E – the first one is quite easy.

The total extra money collected by renting luxury cars is \$105.

105/15 = 7

Say out of 25 cars, 7 are luxury cars and 18 are compact cars. If the rent of compact cars is \$0 (theoretically), the rent of luxury cars is \$15 and the extra rent charged will be \$105 (7*15 = 105) – this is a valid case.

Now how do we get the second case? Think about it before you read on – it will help you realize why the second case is more of a challenge.

Let’s make a slight change to our current numbers to see if they still fit:

Say out of 25 cars, 8 are luxury cars and 17 are compact cars. If the rent of compact cars is \$0 and the rent of luxury cars is \$15, the extra rent charged should be \$15*8 = \$120, but notice, 9 morecompact cars were rented than luxury cars. In reality, the extra total rent collected is \$105 – the \$15 reduction is because of the 9 additional compact cars. Hence, the daily rental rate of each compact car would be \$15/9 = \$5/3.

This would mean that the daily rental rate of each luxury car is \$5/3 + \$15 = \$50/3

The total rental cost of luxury cars in this case would be 8 * \$50/3 = \$400/3

The total rental cost of compact cars in this case would be 17 * \$5/3 = \$85/3

The difference between the two total rental costs is \$400/3 – \$85/3 = 315/3 = \$105

Everything checks out, so we know that there is no unique answer to this question – for any number of compact cars you use, you will come up with the same answer. Thus, Statements 1 and 2 together are not sufficient.

The strategy we used to find this second case to test is that we tweaked the numbers we were given a little and then looked for a solution. Another strategy is to try plugging in some easy numbers. For example:

Instead of using such difficult numbers, we could have tried an easier split of the cars. Say out of 25 cars, 10 are luxury and 15 are compact. If the rent of compact cars is \$0 and the rent of luxury cars is \$15, the extra rent charged should be 10*\$15 = \$150 extra, but it is actually only \$105 extra, a difference of \$45, due to the 5 additional compact cars. The daily rental rent of 5 extra compact cars would be \$45/5 = \$9. Using these numbers in the calculations above, you will see that the difference between the rental costs is, again, \$105. This is a valid case, too.

Hence, there are two strategies we saw in action today:

• Tweak the numbers slightly to see if you will get the same results
• Go for the easy split when choosing numbers to plug in

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: Taking The GMAT Like It’s Nintendo Switch

The non-election trending story of the day is the announcement of the forthcoming Nintendo Switch gaming system, a system that promises to help you take the utmost advantage of your leisure time…but that may help you maximize the value of your GMAT experience, too.

How?

The main feature of the Switch (and the driving factor behind its name) is its flexibility. It can be an in-home gaming system attached to a fixed TV set, but then immediately Switch to a hand-held portable system that allows you to continue your game on the go. Nintendo’s business plan is primarily based on offering flexibility…and on the GMAT, your plan should be to prove to business schools that you can offer the same.

The GMAT, of course, tests algebra skills and critical thinking skills and grammar skills, but beneath the surface it also has a preference for testing flexibility. Many problems will punish those with pure tunnel vision, but reward those who can identify that their first course of action isn’t working and who can then Switch to another plan. This often manifests itself in:

• Math problems that seem to require algebra…but halfway through beg to be back-solved using answer choices.
• Sentence Correction problems that seem to ask you to make a decision about one major difference…but for which the natural choices leave you with clearer-cut errors elsewhere.
• Critical Reasoning answer choices that seem out of scope at first, but reward those who read farther and then see their relevance.
• Data Sufficiency problems for which you’ve made a clear, confident decision on one statement…but then the other statement shows you something you hadn’t considered before and forces you to reconsider.
• The overall concept that if you’re a one-trick pony – you’re a master of plugging in answer choices, for example – you’ll find questions that just won’t reward that strategy and will force you to do something else.

Flexibility matters on the GMAT! As an example, consider the following Data Sufficiency question:

Is x/y > 3?

1) 3x > 9y
2) y > 3y

If you’re like many, you’ll confidently address the algebra in Statement 1, divide both sides by 3 to get x > 3y, and then see that if you divide both sides by y, you can make it look exactly like the question stem: x/y > 3. And you may very well say, “Statement 1 is sufficient!” and confidently move on to Statement 2.

But when you look at Statement 2 – either conceptually or algebraically – something should stand out. For one, there’s no way that it’s sufficient because it doesn’t help you determine anything about x. And secondly, it brings up the point that “y is negative” (algebraically you’d subtract y from both sides to get 0 > 2y, then divide by 2 to get 0 > y). And here’s where, if it hasn’t already, your mind should Switch to “positive/negative number properties” mode. If you weren’t thinking about positive vs. negative properties when you considered Statement 1, this one gives you a chance to Switch your thinking and reconsider – what if y were negative? Algebraically, you’d then have to flip the sign when you divide both sides by y:

3x > 9y : Divide both sides by 3

x > 3y : Now divide both sides by y, but remember that if y is positive you keep the sign (x/y > 3), and if y is negative you flip the sign (x/y < 3).

With this in mind, Statement 1 doesn’t really tell you anything. x/y can be greater than 3 or less than 3, so all Statement 1 does is eliminate that x/y could be exactly 3. Now you have the evidence to Switch your answer. If you initially thought Statement 1 was sufficient, Statement 2 has given you a chance to reassess (thereby demonstrating flexibility in thinking) and realize that it’s not, until you know whether y is negative or positive.

Statement 2 supplies that missing piece, and the answer is thus C. But more important is the lesson – because the GMAT so values mental flexibility, it will often provide you with clues that can help you change your mind if you’re paying attention. So on the GMAT, take a lesson from Nintendo Switch: flexibility is an incredibly marketable skill, so look for clues and opportunities to Switch your line of thinking and save yourself from trap answers.

By Brian Galvin.

How to Solve “Unsolvable” Equations on the GMAT

The moment we see an equation involving the variable x, we have a habit of jumping right into attempting to solve it. But what happens when we are not able to solve it? Let’s say, for example, we have an equation such as x^2 + 1 = 0. How would we solve for x here? We can’t because x has no real value. Note that x^2 is non-negative – it would be either 0 or positive. 1, we know, is positive. So together, a positive number and a non-negative number cannot add up to 0.

In this example, it relatively easy to see that the equation has no real solution. In others, it may not be so obvious, so we will need to use other strategies.

We know how to solve third degree equations. The first solution is found by trial and error – we try simple values such as -2, -1, 0, 1, 2, etc. and are usually able to find the first solution. Then the equation of third degree is split into two factors, including a quadratic. We know how to solve a quadratic, and that is how we get all three solutions, if it has any.

But what if we are unable to find the first solution to a third degree equation by trial and error? Then we should force ourselves to wonder if we even need to solve the equation at all. Let’s take a look at a sample question to better understand this idea:

Is x < 0?
(1) x^3 + x^2 + x + 2 = 0
(2) x^2 – x – 2 < 0

In this problem, x can be any real number – we have no constraints on it. Now, is x negative?

Statement 1: x^3 + x^2 + x + 2 = 0

If we try to solve this equation as we are used to doing, look at what happens:

If you plug in x = 2, you get 16 = 0
If you plug in x = 1, you get 5 = 0
If you plug in x = 0, you get 2 = 0
If you plug in x = -1, you get 1 = 0
If you plug in x = -2, you get -4 = 0

We did not find any root for the equation. What should we do now? Note that when x goes from -1 to -2, the value on the left hand side changes from 1 to -4, i.e. from a positive to a negative. So, in between -1 and -2 there will be some value of x for which the left hand side will become 0. That value of x will not be an integer, but some decimal value such as -1.3 or -1.4, etc.

Even after we find the first root, making the quadratic will be very tricky and then solving it will be another uphill task. So we should ask ourselves whether we even need to solve this equation.

Think about it – can x be positive? If x is indeed positive, x^3, x^2 and x all will be positive. Then, if we add four positive numbers (x^3, x^2, x and 2) we will get a positive sum – we cannot get 0. Obviously x cannot be 0 since that will give us 2 = 0.

This means the value of x must be negative, but what it is exactly doesn’t matter. We know that x has to be negative, and that is sufficient to answer the question.

Statement 2: x^2 – x – 2 < 0

This, we can easily solve:

x^2 – 2x + x – 2 < 0
(x – 2)*(x + 1) < 0

We know how to solve this inequality using the method discussed here.

This this will give us -1 < x < 2.

Since x can be a non-integer value too, x can be negative, 0, or positive. This statement alone is not sufficient,and therefore, the answer is A.

To evaluate Statement 1, we didn’t need to solve the equation at all. We figured out everything we wanted to know by simply using some logic.

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: As You Debate Over Answer Choices… Just Answer The Freaking Question!

If you’re like many – to the dismay of the NFL and the advertising industry – you’re planning to watch another presidential debate this coming Sunday. And just like Trump-Clinton I and Pence-Kaine earlier this week, this debate will provide plenty of opportunities to be annoyed, frustrated, and disappointed…but it will also provide an ever-important lesson about the GMAT.

It’s no surprise that candidate approval ratings are low for the same reason that far too many GMAT scores are lower than candidates would hope. Why?

People don’t directly answer the question.

This is incredibly common in the debates, where the poor moderators are helpless against the talking points and stump speeches of the candidates. The public then suffers because people cannot get direct answers to the questions that matter. This is also very common on the GMAT, where students will invest the time in critical thought and calculation, and then levy an answer that just doesn’t hit the mark. Consider the example:

Donald has \$520,000 in campaign money available to spend on advertising for the month of October, and his advisers are telling him that he should spend a minimum of \$360,000 in the battleground states of Ohio, Florida, Virginia, and North Carolina. If he plans to spend the minimum amount in battleground states to appease his advisers, plus impress his friends by a big ad spend specific to New York City (and then he will skip advertising in the rest of the country), how much money will he have remaining if he wants 20% of his ad spend to take place in New York City?

(A) \$45,000
(B) \$52,000
(C) \$70,000
(D) \$90,000
(E) \$104,000

As people begin to calculate, it’s common to try to determine all of the facets of Donald’s ad spend. If he’s spending only the \$360,000 in battleground states plus the 20% he’ll spend in New York City, then \$360,000 will represent 80% of his total ad spend. If \$360,000 = 0.8(Total), then the total will be \$450,000. That means that he’ll spend \$90,000 in New York City. Which is answer choice D…but that’s not the question!

The question asked for how much of his campaign money would be left over, so the calculation you need to focus on is the \$520,000 he started with minus the \$450,000 he spent for a total of \$70,000, answer choice C. And in a larger context, you can learn a major lesson from Wharton’s most famous alumnus: it’s not enough for your answer to be related to the question. On the GMAT, you must answer the question directly! So make sure that you:

1. Double check which portion of a word problem the question asked for. Don’t be relieved when your algebra spits out “a” number. Make sure it’s “the” number.
2. Be careful with Strengthen/Weaken Critical Reasoning problems. A well-written Strengthen problem will likely have a good Weaken answer choice, and vice-versa.
3. In algebra problems, make sure to identify the proper variable (or combination of variables if they ask, for example “What is 6x – y?”).
4. With Data Sufficiency problems, pay attention to the exact values being asked for. One of the most common mistakes that people make is saying that a statement is insufficient because they’re looking to fill in all variables, when actually it is sufficient to answer the exact combination that the test asked for.

As you watch the debate this weekend, notice (How could you not?) how absurd it is that the candidates just about never directly answer the question…and then vow to not make the same mistake on your GMAT exam.

By Brian Galvin.

How to Use the Pythagorean Theorem With a Circle

It does not surprise anyone when they learn that the properties of circles are tested on the GMAT. Most test-takers will nod and rattle off the relevant equations by rote: Area = Π*radius ^2; Circumference  = 2Π* radius; etc. However, many of my students are caught off guard to learn that the equation for a circle on the coordinate plane is our good friend the Pythagorean theorem. Why on earth would an equation for a right triangle describe a circle?

Take a look at the following diagram in which a circle is centered on the origin (0,0) in the coordinate plane:

Designate a random point on the circle (x,y.) If we draw a line from the center of the circle to x,y, that line is a radius of the circle. Call it r. If we drop a line down from (x,y) to the x-axis, we’ll have a right triangle:

Note that the base of the triangle is x, and the height of the triangle is y. So now we have our Pythagorean theorem: x^2 + y^2 = r^2. This is also the equation for a circle centered on the origin on the coordinate plane. [The more general equation for a circle with a center (a,b) is (x-a)^2 + (y-b)^2 = r^2. When a circle is centered on the origin, (a,b) is simply (0,0.)]

This ends up being an immensely useful tool to use on the GMAT. Take the following question, for example:

A certain circle in the xy-plane has its center at the origin. If P is a point on the circle, what is the sum of the squares of the coordinates of P?

(1) The radius of the circle is 4
(2) The sum of the coordinates of P is 0

So let’s draw this, designating P as (x,y):

Now we draw our trust right triangle by dropping a line down from P to the x-axis, which will give us this:

We’re looking for x^2 + y^2. Hopefully, at this point, you notice what the question is going for – because we have a right triangle, x^2 + y^2 = r^2, meaning that all we need is the radius!

Statement 1 is pretty straightforward – if r = 4, we can insert this into our equation of x^2 + y^2 = r^2 to get x^2 + y^2 = 4^2. So x^2 + y^2 = 16. Clearly, this is sufficient.

Now look at Statement 2. If the sum of x and y is 0, we can say x = 1 and y = -1 or x = 2 and y = -2 or x = 100 and y = -100, etc. Each of these will yield a different value for x^2 + y^2, so this statement alone is clearly not sufficient. Our answer is A.

Takeaway: any shape can appear on the coordinate plane. If the shape in question is a circle, remember to use the Pythagorean theorem as your equation for the circle, and what would have been a challenging question becomes a tasty piece of baklava. (We are talking about principles elucidated by the ancient Greeks, after all.)

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

GMAT Tip of the Week: 6 Reasons That Your Test Day Won’t Be A Labor Day

As the northern hemisphere drifts toward autumn, two events have become just about synonymous: Labor Day and Back to School. If you’re spending this Labor Day weekend getting yourself ready to go back to graduate school, you may well labor over GMAT study materials in between barbecues and college football games. And if you do, make sure you heed this wisdom: GMAT test day should not be Labor Day!

What does that mean?

On a timed test like the GMAT, one of the biggest drains on your score can be a combination of undue time and undue energy spent on problems that could be done much simpler. “The long way is the wrong way” as a famous GMAT instructor puts it – those seconds you waste, those extra steps that could lead to error or distraction, they’ll add up over the test and pull your score much lower than you’d like it to be. With that in mind, here are six ways to help you avoid too much labor on test day:

QUANTITATIVE SECTION
1) Do the math in your order, only when necessary.
Because the GMAT doesn’t allow a calculator, it heavily rewards candidates who can find efficient ways to avoid the kind of math for which you’d need a calculator. Very frequently this means that the GMAT will tempt you with calculations that you’d ordinarily just plug-and-chug with a calculator, but that can be horribly time-consuming once you start.

For example, a question might require you to take an initial number like 15, then multiply by 51, then divide by 17. On a calculator or in Excel, you’d do exactly that. But on the GMAT, that calculation gets messy. 15*51 = 765 – a calculation that isn’t awful but that will take most people a few steps and maybe 20 seconds. But then you have to do some long division with 17 going into 765. Or do you? If you’re comfortable using factors, multiples, and reducing fractions, you can see those two steps (multiply by 51, divide by 17) as one: multiply by 51/17, and since 51/17 reduces to 3, then you’re really just doing the calculation 15*3, which is easily 45.

The lesson? For one, don’t start doing ugly math until you absolutely know you have to perform that step. Save ugly math for later, because the GMAT is notorious for “rescuing” those who are patient enough to wait for future steps that will simplify the process. And, secondly, get really, really comfortable with factors and divisibility. Quickly recognizing how to break a number into its factors (51 = 3*17; 65 = 5*13; etc.) allows you to streamline calculations and do much of the GMAT math in your head. Getting to that level of comfort may take some labor, but it will save you plenty of workload on test day.

2) Recognize that “Answers Are Assets.”
Another way to avoid or shortcut messy math is to look at the answer choices first. Some problems might look like they involve messy algebra, but can be made much easier by plugging in answer choices and doing the simpler arithmetic. Other times, the answer choices will lead themselves to process of elimination, whether because some choices do not have the proper units digit, or are clearly too small.

Still others will provide you with clues as to how you have to attack the math. For example, if the answer choices are something like: A) 0.0024; B) 0.0246; C) 0.246; D) 2.46; E) 24.6, they’re not really testing you on your ability to arrive at the digits 246, but rather on where the decimal point should go (how many times should that number be multiplied/divided by 10). You can then set your sights on the number of decimal places while not stressing other details of the calculation.

Whatever you do, always scan the answer choices first to see if there are easier ways to do the problem than to simply slog through the math. The answers are assets – they’re there for a reason, and often, they’ll provide you with clues that will help you save valuable time.

3) Question the Question – Know where the game is being played.
Very often, particularly in Data Sufficiency, the GMAT Testmaker will subtly provide a clue as to what’s really being tested. And those who recognize that can very quickly focus on what matters and not get lost in other elements of the problem.

For example, if the question stem includes an inequality with zero (x > 0 or xy < 0), there’s a very high likelihood that you’re being tested on positive/negative number properties. So, when a statement then says something like “1) x^3 = 1331”, you can hold off on trying to take the cube root of 1331 and simply say, “Odd exponent = positive value, so I know that x is positive,” and see if that helps you answer the question without much calculation. Or if the problem asks for the value of 6x – y, you can say to yourself, “I may not be able to solve for x and y individually, but if not, let’s try to isolate exactly that 6x – y term,” and set up your algebra accordingly so that you’re efficiently working toward that specific goal.

Good test-takers tend to see “where the game is being played” by recognizing what the Testmaker is testing. When you can see that a question is about number properties (and not exact values) or a combination of values (and not the individual values themselves) or a comparison of values (again, not the actual values themselves), you can structure your work to directly attack the question and not fall victim to a slog of unnecessary calculations.

VERBAL SECTION
4) Focus on keywords in Critical Reasoning conclusions.
The Verbal section simply looks time-consuming because there’s so much to read, so it pays to know where to spend your time and focus. The single most efficient place to spend time (and the most disastrous if you don’t) is in the conclusion of a Strengthen or Weaken question. To your advantage, noticing a crucial detail in a conclusion can tell you exactly “where the game is being played” (Oh, it’s not how much iron, it’s iron PER CALORIE; it’s not that Company X needs to reduce costs overall, it’s that it needs to reduce SHIPPING costs; etc.) and help you quickly search for the answer choices that deal with that particular gap in logic.

On the downside, if you don’t spend time emphasizing the conclusion, you’re in trouble – burying a conclusion-limiting word or phrase (like “per calorie” or “shipping”) in a long paragraph can be like hiding a needle in a haystack. The Testmaker knows that the untrained are likely to miss these details, and have created trap answers (and just the opportunity to waste time re-reading things that don’t really matter) for those who fall in that group.

5) Scan the Sentence Correction answer choices before you dive into the sentence.
Much like “Answers are Assets” above, a huge help on Sentence Correction problems is to scan the answer choices quickly to see if you can determine where the game is being played (Are they testing pronouns? Verb tenses?). Simply reading a sentence about a strange topic (old excavation sites, a kind of tree that only grows on the leeward slopes of certain mountains…) and looking for anything that strikes you as odd or ungrammatical, that takes time and saps your focus and energy.

However, the GMAT primarily tests a handful of concepts over and over, so if you recognize what is being tested, you can read proactively and look for the words/phrases that directly control that decision you’re being asked to make. Do different answers have different verb tenses? Look for words that signal time (before, since, etc.). Do they involve different pronouns? Read to identify the noun in question and determine which pronoun it needs. You’re not really being tasked with “editing the sentence” as much as your job is to make the proper decision with the choices they’ve already given you. They’ve already narrowed the scope of items you can edit, so identify that scope before you take out the red marking pen across the whole sentence.

As the Veritas Prep Reading Comprehension lesson teaches, stop at the end of each paragraph of a reading passage to ask yourself whether you understand Scope, Tone, Organization, and Purpose. The top two time-killers on Reading Comprehension passages/problems are re-reading (you get to the end and realize you don’t really know what you just read) and over-reading (you took several minutes absorbing a lot of details, but now the clock is ticking louder and you haven’t looked at the questions yet).

STOP will help you avoid re-reading (if you weren’t locked in on the first paragraph, you can reread that in 30 seconds and not wait to the end to realize you need to reread the whole thing) and will give you a quick checklist of, “Do I understand just enough to move on?” Details are only important if you’re asked about them, so focus on the major themes (Do you know what the paragraph was about – a quick 5-7 word synopsis is perfect – and why it was written? Good.) and save the details for later.

It may seem ironic that the GMAT is set up to punish hard-workers, but in business, efficiency is everything – the test needs to reward those who work smarter and not just harder, so an effective test day simply cannot be a Labor Day. Use this Labor Day weekend to study effectively so that test day is one on which you prioritize efficiency, not labor.

Getting ready to take the GMAT? We have free online GMAT seminars running all the time. And as always, be sure to follow us on Facebook, YouTubeand Twitter!

By Brian Galvin.

How to Solve “Hidden” Factor Problems on the GMAT

One of the interesting things to note about newer GMAC Quant questions is that, while many of these questions test our knowledge of multiples and factors, the phrasing of these questions is often more subtle than earlier versions you might have seen. For example, if I ask you to find the least common multiple of 6 and 9, I’m not being terribly artful about what topic I’m testing you on – the word “multiple” is in the question itself.

But if tell you that I have a certain number of cupcakes and, were I so inclined, I could distribute the same number of cupcakes to each of 6 students with none left over or to each of 9 students with none left over, it’s the same concept, but I’m not telegraphing the subject in the same conspicuous manner as the previous question.

This kind of recognition comes in handy for questions like this one:

All boxes in a certain warehouse were arranged in stacks of 12 boxes each, with no boxes left over. After 60 additional boxes arrived and no boxes were removed, all the boxes in the warehouse were arranged in stacks of 14 boxes each, with no boxes left over. How many boxes were in the warehouse before the 60 additional boxes arrived?

(1) There were fewer than 110 boxes in the warehouse before the 60 additional arrived.
(2) There were fewer than 120 boxes in the warehouse after the 60 additional arrived.

Initially, we have stacks of 12 boxes with no boxes left over, meaning we could have 12 boxes or 24 boxes or 36 boxes, etc. This is when you want to recognize that we’re dealing with a multiple/factor question. That first sentence tells you that the number of boxes is a multiple of 12. After 60 more boxes were added, the boxes were arranged in stacks of 14 with none left over – after this change, the number of boxes is a multiple of 14.

Because 60 is, itself, a multiple of 12, the new number must remain a multiple of 12, as well. [If we called the old number of boxes 12x, the new number would be 12x + 60. We could then factor out a 12 and call this number 12(x + 5.) This number is clearly a multiple of 12.] Therefore the new number, after 60 boxes are added, is a multiple of both 12 and 14. Now we can find the least common multiple of 12 and 14 to ensure that we don’t miss any possibilities.

The prime factorization of 12: 2^2 * 3

The prime factorization of 14: 2 * 7

The least common multiple of 12 and 14: 2^2 * 3 * 7 = 84.

We now know that, after 60 boxes were added, the total number of boxes was a multiple of 84. There could have been 84 boxes or 168 boxes, etc. And before the 60 boxes were added, there could have been 84-60 = 24 boxes or 168-60 = 108 boxes, etc.

A brief summary:

After 60 boxes were added: 84, 168, 252….

Before 60 boxes were added: 24, 108, 192….

That feels like a lot of work to do before even glancing at the statements, but now look at how much easier they are to evaluate!

Statement 1 tells us that there were fewer than 110 boxes before the 60 boxes were added, meaning there could have been 24 boxes to start (and 84 once 60 were added), or there could have been 108 boxes to start (and 168 once 60 were added). Because there are multiple potential solutions here, Statement 1 alone is not sufficient to answer the question.

Statement 2 tells us that there were fewer than 120 boxes after 60 boxes were added. This means there could have been 84 boxes – that’s the only possibility, as the next number, 168, already exceeds 120. So we know for a fact that there are 84 boxes after 60 were added, and 24 boxes before they were added. Statement 2 alone is sufficient, and the answer is B.

Takeaway: questions that look strange or funky are always testing concepts that have been tested in the past – otherwise, the exam wouldn’t be standardized. By making these connections, and recognizing that a verbal clue such as “none left over” really means that we’re talking about multiples and factors, we can recognize even the most abstract patterns on the toughest of GMAT questions.

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

Quarter Wit, Quarter Wisdom: The Power of Deduction on GMAT Data Sufficiency Questions

In a previous post, we have discussed how to find the total number of factors of a number. What does the total number of factors a number has tell us about that number? One might guess, “Not a lot,” but it actually does tell us quite a bit! If the total number of factors is odd, you know the number must be a perfect square. If the total number of factors is even, you know the number is not a perfect square.

We know that the total number of factors of a number A (prime factorised as X^p * Y^q *…) is given by (p+1)*(q+1)… etc.

So, if we know that a number has, say, 6 total factors, what can we say about the number?

6 = (p+1)*(q+1) = 2*3, so p = 1 and q = 2 or vice versa.

A = X^1 * Y^2 where X and Y are distinct prime numbers.

Today, we will look at a data sufficiency question in which we can use factors to deduce much more information than what we might first guess:

When the digits of a two-digit, positive integer M are reversed, the result is the two-digit, positive integer N. If M > N, what is the value of M?

Statement 1: The integer (M – N) has 12 unique factors.
Statement 2: The integer (M – N) is a multiple of 9.

With this question, we are told that M is a two-digit integer and N is obtained by reversing it. So if M = 21, then N = 12; if M = 83, then N = 38 (keeping in mind that M must be greater than N). In the generic form:

M = 10a + b and N =10b + a (where a and b are single-digit numbers from 1 to 9. Neither can be 0 or greater than 9 since both M and N are two-digit numbers.)

We also know that no matter what M and N are, M > N. Therefore:

10a + b > 10b + a
9a > 9b
a > b

Let’s examine both of our given statements:

Statement 1: The integer (M – N) has 12 unique factors.

First, let’s figure out what M – N is:

M – N = (10a + b) – (10b + a) = 9a – 9b

Say M – N = A. This would mean A = 9(a-b) = 3^2 * (a-b)

The total number of factors of A where A = X^p * Y^q *… can be calculated using the formula (p+1)*(q+1)* …

We know that A has 3^2 as a factor, so X = 3 and p = 2. Therefore, the total number of factors would be (2+1)*(q+1)*… = 3*(q+1)*… = 12, so (q+1)*… must be 4.

Case 1:
This means q may be 3 so that (q+1) is 4. Since a-b must be less than or equal to 9 and must also be the cube of a number, (a-b) must be 8. (Note that a-b cannot be 1 because then the total number of factors of A would only be 3.)

So, a must be 9 and b must be 1 in this case (since a > b). The integers will be 91 and 19, and since M > N, M = 91.

Case 2:
Another possibility is that (a-b) is a product of two prime factors (other than 3), both with the power of 1. In that case, the total number of factors = (2+1)*(1+1)*(1+1) = 12

Note, however, that the two prime factors (other than 3) with the smallest product is 2*5 = 10, but the difference of two single-digit positive integers cannot be 10. This means that only Case 1 can be true, therefore, Statement 1 alone is sufficient. This is certainly not what we expected to find from just the total number of factors!

Statement 2: The integer (M – N) is a multiple of 9.

M – N = (10a + b) – (10b + a) = 9a – 9b, so M – N = 9 (a-b) . This is already a multiple of 9.

We get no new information with this statement; (a-b) can be any integer, such as 2 (a = 5, b = 3 or a = 7, b = 5), etc. This statement alone is insufficient, therefore our answer is A.

Don’t take the given data of a GMAT question at face value, especially if you are expecting questions from the 700+ range. Ensure that you have deduced everything that you can from it before coming to a conclusion.

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: Making Your GMAT Score SupeRIOr to Ryan Lochte’s

What’s the worst thing that can happen on your GMAT exam? Is it running out of time well before you’re done? Or blanking on nearly every math formula you’ve studied?

Whatever it is, it can’t be nearly as bad as being pulled over by fake cops – no lights or nothing, just a badge – then being told to get on the ground and having a gun placed on your forehead and being like, “whatever.” So your big event of 2016 will already go a lot better than Ryan Lochte’s did; you have that going for you.

What else do you have going for you on the GMAT? The ability to learn from the most recent few days of Lochte’s life. Lochte’s biggest mistake wasn’t vandalizing a gas station bathroom at 4am, but rather making up his own story and creating an even larger mess. And that’s a huge lesson that you need to keep in mind for the GMAT:

Don’t make up your own story.

Here’s what that means, on three major question types:

DATA SUFFICIENCY
People make up their own story on Data Sufficiency all the time. And like a prevailing theory about Lochte (he didn’t connect the vandalism of the bathroom to the men coming after him for restitution; he really did think that he had been robbed for no reason), it’s not that they’re intentionally lying. They’re just “conveniently” misremembering what they’ve read or connecting dots that weren’t actually connected in real life. Consider the question:

The product of consecutive integers a, b, c, and d is 5040. What is the value of integer d?

(1) d is prime
(2) d < c < b < a

Once people have factored 5040 into 7*8*9*10, they can then quickly recognize that Statement 1 is sufficient: the only prime number in that bunch is 7, so d must be 7. But then when it comes to Statement 2, they’ve often made up their own story. By saying “d is the smallest, and, yep, that’s 7!” they’re making up the fact that these consecutive integers are positive. That was not specifically stated! So it could be 7, 8, 9, and 10 or it could be -7, -8, -9, and -10, making d either -10 or 7. And the GMAT (maybe like an NBC interviewer?) makes it easy for you to make up your own story.

With Statement 1, prime numbers must be positive, so if you weren’t already thinking only about positives, the question format nudges you further in that direction. The answer is A when people often mistakenly choose D, and the reason is that the question makes it easy for you to make up your own story when looking at Statement 2. So before you submit an answer, always ask yourself, “Am I only using the facts explicitly provided to me, or am I somehow making up my own story?”

CRITICAL REASONING
Think of your friends who are good storytellers. We hate to break it to you, but they’re probably making at least 10-20% of those stories up. Which makes sense. “It was a pretty big fish,” is a lot less compelling than, “It was the biggest fish any of us had ever seen!” Case in point, the Olympics themselves.

No commentator this week has said that Michael Phelps, Lochte’s teammate, is “a really good swimmer.” They’re posing, “Is he the greatest athlete of all time?” because words that end in -st capture attention (and pageviews). Even Lochte was guilty of going overly-specific for dramatic effect: there was, indeed, a gun pointed at his taxi, but not resting on his forehead. His version just makes the story more exciting and dramatic…and you may very well be guilty of such a mistake on the GMAT. Consider:

About two million years ago, lava dammed up a river in western Asia and caused a small lake to form. The lake existed for about half a million years. Bones of an early human ancestor were recently found in the ancient lake bottom sediments on top of the layer of lava. Therefore, ancestors of modern humans lived in Western Asia between 2 million and 1.5 million years ago.

Which one of the following is an assumption required by the argument?

(A) There were not other lakes in the immediate area before the lava dammed up the river.
(B) The lake contained fish that the human ancestors could have used for food.
(C) The lava under the lake-bottom sediments did not contain any human fossil remains.
(D) The lake was deep enough that a person could drown in it.
(E) The bones were already in the sediments by the time the lake disappeared.

The correct answer here is E (if the bones were not already there, then they’re not good evidence that people were there during that time), but the popular trap answer is C. Consider what would happen if C were untrue: that means that there were human fossil remains that pre-date the time period in question.

But here’s where Lochte Logic is dangerous: you’re not trying to prove that the FIRST humans lived in this period at this time; you’re just trying to prove that humans lived here during that time. And whether or not there were fossils from 2.5 million or 4 million years ago doesn’t change that you still have this evidence of people in that 2 million-1.5 million years ago timeframe.

When people choose C, it’s almost always because they made up their own story about the argument – they read it as, “The earliest human ancestors lived in this place and time,” and that’s just not what’s given. Why do they do that? For Lochte’s very own reasons: it makes the story a little more interesting and a little more favorable.

After all, the average pre-MBA doesn’t spend much time reading about archaeology, but if some discovery is that level of exciting (We’ve discovered the first human! We’ve discovered evidence of aliens!) then it crosses your Facebook/Twitter feeds. You’re used to reading stories about the first/fastest/greatest/last, and so when you get dry subject matter your mind has a tendency to put those words in there subconsciously. Be careful – do not make up your own story about the conclusion!

A similar phenomenon occurs with Reading Comprehension. When you read a long passage, your mind tends to connect dots that aren’t there as it fills in the rest of the story for you. Just like Lochte, who had to fill in the gap of, “Hey what would I have said if someone pointed a gun at me and told me to get on the ground? Oh right…’whatever’ is my default answer for most things,” your mind will start to fill in details that make logical sense.

The problem then comes when you’re asked an Inference question, for which the correct answer must be true based on the passage. For example, if two details in a passage are:

1. Michael swam the fastest race of his life.
2. Ryan’s race was one of the slowest he’s ever swam.

You might answer the question, “Which of the following is a conclusion that can be drawn from the passage?” with:

(A) Michael swam faster than Ryan.

Your mind – particularly amidst a lot of other text between those two facts – wants to logically arrange those two swims together, and with “fastest” for Michael and “slowest” for Ryan, it kind of seems logical that Michael was faster. But those two races are never compared directly to each other. Consider that if Michael and Ryan aren’t Phelps and Lochte, but rather filmmaker Michael Moore and Olympic champion Ryan Lochte, then of course Lochte’s slowest swim would still be way, way faster than Moore’s fastest.

Importantly, Reading Comprehension questions love to bait unwitting test-takers with comparisons as answer choices, knowing that your mind is primed to create your own story and draw comparisons that are probably true, but just not proven. So again, any time you’re faced with an answer that seems obvious, go back and ask yourself if the details you’re using were provided to you, or if instead, you’re making up your own story.

So learn a valuable lesson from Ryan Lochte and avoid making up your own story, sticking only to the clean facts of the matter. Stay true to the truth, and you’ll walk out of the test center saying “Jeah!”

By Brian Galvin.

GMAT Tip of the Week: How to Avoid GMAT (and Pokemon Go) Traps

In seemingly the most important development in world history since humans learned to create fire, Pokemon Go has arrived and is taking the world by storm. Rivaling Twitter and Facebook for mobile phone attention and battling the omnipresent selfie as a means of death-by-mobile-phone, Pokemon Go is everywhere you want to be…and often in places you don’t.

And that is why Pokemon Go is responsible for an ever-important GMAT lesson.

Perhaps most newsworthy about Pokemon Go these days is the dangerous and improper places that it has led its avid users. On the improper side,  such solemn and dignified places as the national Holocaust Museum and Arlington National Cemetery have had to actively prohibit gamers from descending upon mourners/commemorators while playing the game. And as for danger, there have been several instances of thieves luring gamers into traps and therefore robbing them of valuable (if you’re playing the game, you definitely have a smartphone) items.

And the GMAT can and will do the same thing.

How?

If you’re reading this on our GMAT blog, you’ve undoubtedly already learned that, on Data Sufficiency problems, you cannot assume that a variable is positive, or that it is an integer. But think about what makes Pokemon Go users so vulnerable to being lured into a robbery or to losing track of basic human decency. They’re so invested in the game that they lose track of the situations they’re being lured into.

Similarly, the most dangerous GMAT traps are those for which you should absolutely know better, but the testmaker has gotten your mind so invested in another “game” that you lose track of something basic. Consider the example:

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

(1) x is a prime number
(2) y is a 3 digit number

Statement 1 is clearly sufficient. Since y is odd, and an integer, and the product of integers x and y is an even integer, that means that x must be even. And since x also has to be prime (which is how you know it’s an integer, too), the only even prime is 2, making x = 2.

From there your mind is fixated on the game. You can quickly see that in that case y = 111 and x = 2. Which you then have to forget about as you attack Statement 2. But here’s the reason that less than 25% of users in the Veritas Prep Question Bank get this right, while nearly half incorrectly choose D. Statement 1 has gotten your mind fixated on the even/odd/prime game, meaning that you may only be thinking about integers (and positive integers at that) at this point.

That y is a 3-digit number DOES NOT mean that it has to be 111. It could be -111 (making x = -2) or 333 (making x = 2/3). So only Statement 1 alone is sufficient, but the larger lesson is more important. Just like Pokemon Go has the potential to pollute your mind and have you see the real world through its “enhanced reality” lens, so does a statement that satisfies your intellect (“Ah, 2 is the only even prime number!”) give you just enough tunnel vision that you make poor decisions and fall for traps.

The secret here is that almost no one scoring above a 500 carries over all of Statement 1 (“Oh, well I already know that x = 2!”) – a total rookie mistake. It’s that Statement 1 got you fixated on definitions of types of integers (prime, even, odd) and therefore got your mind looking through the “enhanced reality” of integers-only.

The lesson? Much like Pokemon Go, the GMAT has tools to get you so invested in a particular facet of a game that you lose your universal awareness of your surroundings. Know that going in – that you have to consciously step back from that enhanced reality you’ve gained after Statement 1 and look at the whole picture. So take a lesson from Pokemon Go and know when to stop and step back.

Getting ready to take the GMAT? We have free online GMAT seminars running all the time. And as always, be sure to follow us on Facebook, YouTubeand Twitter!

By Brian Galvin.

How to Go From a 48 to 51 in GMAT Quant – Part VII

Both a test-taker at the 48 level and one at the 51 level in the GMAT Quant section, are conceptually strong – given an unlimited time frame, both will be able to solve most GMAT questions correctly. The difference lies in the two things a test-taker at the 51 level does skillfully:

1. Uses holistic, big-picture methods to solve Quant questions.
2. Handles questions he or she finds difficult in a timely manner.

We have been discussing holistic methods on this blog for a long time now and will continue discussing them. (Before you continue reading, be sure to check out parts I, II, III, IVV and VI of this series.)

Today we will focus on “handling the hard questions in a timely manner.” Note that we do not say “solving the hard questions in a timely manner.” Occasionally, one might be required to make a quick call and choose to guess and move on – but again, that is not the focus of this post. We are actually going to talk about the “lightbulb” moment that helps us save on time. There are many such moments for the 51 level test-taker – in fact, the 51 scorers often have time left over after attempting all these questions.

Test takers at the 48 level will also eventually reach the same conclusions but might need much more time. That will put pressure on them the next time they look at the ticking clock, and once their cool is lost, “silly errors” will start creeping in. So it isn’t about just that one question – one can end up botching many other questions too.

There are many steps that can be easily avoided by a lightbulb moment early on. This is especially true for Data Sufficiency questions.

Let’s take an official example:

Pam owns an inventory of unopened packages of corn and rice, which she has purchased for \$17 and \$13 per package, respectively. How many packages of corn does she have ?

Statement 1: She has \$282 worth of packages.

Statement 2: She has twice as many packages of corn as of rice.

A high scorer will easily recognize that this question is based on the concept of “integral solutions to an equation in two variables.” Since, in such real world examples, x and y cannot be negative or fractional, these equations usually have a finite number of solutions.

After we find one solution, we will quickly know how many solutions the equation has, but getting the first set of values that satisfy the equation requires a little bit of brute force.

The good thing here is that this is a Data Sufficiency question – you don’t need to find the actual solution. The only thing we need is to establish that there is a single solution only. (Obviously, there has to be a solution since Pam does own \$282 worth of packages.)

So, the test-taker will start working on finding the first solution (using the method discussed in this post). We are told:

Price of a packet of corn = \$17
Price of a packet of rice = \$13

Say Pam has “x” packets of corn and “y” packets of rice.

Statement 1: She has \$282 worth of packages

Using Statement 1, we know that 17x + 13y = 282.

We are looking for the integer values of x and y.

If x = 0, y will be 21.something (not an integer)
If x = 1, y = 20.something
If x = 2, y = 19.something
If x = 3, y = 17.something

This is where the 51 level scorer stops because they never lose sight of the big picture. The “lightbulb” switches on, and now he or she knows that there will be only one set of values that can satisfy this equation. Why? Because y will be less than 17 in the first set of values that satisfies this equation. So if we want to get the next set that satisfies, we will need to subtract y by 17 (and add 13 to x), which will make y negative.

So in any case, there will be a unique solution to this equation. We don’t actually need to find the solution and hence, nothing will be gained by continuing these calculations. Statement 1 is sufficient.

Statement 2: She has twice as many packages of corn as of rice.

Statement 2 gives us no information on the total number of packages or the total amount spent. Hence, we cannot find the total number of packages of corn using this information alone. Therefore, our answer is A.

I hope you see how you can be alert to what you want to handle these Quant questions in a timely manner.

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 Overly Specific Question Stem

For most of our lives, we ask and answer relatively generic questions: “How’s it going?” “What are you up to this weekend?” “What time do the Cubs play tonight?”

And think about it, what if those questions were more specific: “Are you in a melancholy mood today?” “Are you and Josh going to dinner at Don Antonio’s tonight and ordering table-side guacamole?” “Do the Cubs play at 7:05 tonight on WGN?” If someone is asking those questions instead, you’re probably a bit suspicious. Why so specific? What’s your angle?

The same is true on the GMAT. Most of the question stems you see are relatively generic: “What is the value of x?” “Which of the following would most weaken the author’s argument?” So when the question stem get a little too specific, you should become a bit suspicious. What’s the test going for there? Why so specific?

The overly-specific Critical Reasoning question stem is a great example. Consider the problem:

Raisins are made by drying grapes in the sun. Although some of the sugar in the grapes is caramelized in the process, nothing is added.
Moreover, the only thing removed from the grapes is the water that evaporates during the drying, and water contains no calories or nutrients.
The fact that raisins contain more iron per food calorie than grapes do is thus puzzling.

Which one of the following, if true, most helps to explain why raisins contain more iron per calorie than do grapes?

(A) Since grapes are bigger than raisins, it takes several bunches of grapes to provide the same amount of iron as a handful of raisins does.
(B) Caramelized sugar cannot be digested, so its calories do not count toward the food calorie content of raisins.
(C) The body can absorb iron and other nutrients more quickly from grapes than from raisins because of the relatively high water content of grapes.
(D) Raisins, but not grapes, are available year-round, so many people get a greater share of their yearly iron intake from raisins than from grapes.
(E) Raisins are often eaten in combination with other iron-containing foods, while grapes are usually eaten by themselves.

Look at that question stem: a quick scan naturally shows you that you need to explain/resolve a paradox, but the question goes into even more detail for you. It reaffirms the exact nature of the paradox – it’s not about “iron,” but instead that that raisins contain more iron per calorie than grapes do. By adding that extra description into the question stem, the testmaker is practically yelling at you, “Make sure you consider calories…don’t just focus on iron!” And therefore, you should be prepared for the correct answer B, the only one that addresses calories, and deftly avoid answers A, C, D, and E, which all focus only on iron (and do so tangentially to the paradox).

Strategically speaking, if a Critical Reasoning question stem gets overly specific, you should pay particular attention to the specificity there…it’s most likely directing you to the operative portion of the argument.

Overly specific questions are most helpful in Data Sufficiency questions (and that same logic will help on Problem Solving too, as you’ll see). The testmaker knows that you’ve trained your entire algebraic life to solve for individual variables. So how can a question author use that lifetime of repetition against you? By asking you to solve for a specific combination that doesn’t require you to find the individual values. Consider this example, which appears courtesy the Official Guide for GMAT Quantitative Review:

If x^2 + y^2 = 29, what is the value of (x – y)^2?

(1) xy = 10
(2) x = 5

Two major clues should stand out to you that you need to Leverage Assets on this problem. For one, using both statements together (answer choice C) is dead easy. If xy = 10 and x = 5 then y = 2 and you can solve for any combination of x and y that anyone could ever ask for. But secondly and more subtly, the question stem should jump out as a classic way-too-specific, Leverage Assets question stem. They asked for a really, really specific value: (x – y)^2.

Now, immediately upon seeing that specificity you should be thinking, “That’s too specific…there’s probably a way to solve for that exact value without getting x and y individually.” That thought process alone tells you where to spend your time – you want to really leverage Statement 1 to try to make it work alone.

And if you’re still unconvinced, consider what the specificity does: the “squared” portion removes the question of negative vs. positive from the debate, removing one of the most common reasons that a seemingly-sufficient statement just won’t work. And, furthermore, the common quadratic (x – y)^2 shares an awful lot in common with the x^2 and y^2 elsewhere in the question stem. If you expand the parentheses, you have “What is x^2 – 2xy + y^2?” meaning that you’re already 2/3 of the way there (so to speak), since they’ve spotted you the sum x^2 + y^2.

The important strategy here is that the overly-specific question stem should scream “LEVERAGE ASSETS” and “You don’t need to solve for x and y…there’s probably a way to solve directly for that exact combination.” Since you know that you’re solving for the expanded x^2 – 2xy + y^2, and you already know that x^2 + y^2 = 29, you’re really solving for 29 – 2xy. Since you know from Statement 1 that xy = 20, then 29 – 2xy will be 29 – 2(10), which is 9.

Statement 1 alone is sufficient, even though you don’t know what x and y are individually. And one of the major signals that you should recognize to help you get there is the presence of an overly specific question stem.

So remember, in a world of generic questions, the oddly specific question should arouse a bit of suspicion: the interrogator is up to something! On the GMAT, you can use that to your advantage – an overly specific Critical Reasoning question usually tells you exactly which keywords are the most important, and an overly specific Data Sufficiency question stem begs for you to leverage assets and find a way to get the most out of each statement.

By Brian Galvin.

GMAT Tip of the Week: The Least Helpful Waze To Study

If you drive in a large city, chances are you’re at least familiar with Waze, a navigation app that leverages user data to suggest time-saving routes that avoid traffic and construction and that shave off seconds and minutes with shortcuts on lesser-used streets.

And chances are that you’ve also, at some point or another, been inconvenienced by Waze, whether by a devout user cutting blindly across several lanes to make a suggested turn, by the app requiring you to cut through smaller streets and alleys to save a minute, or by Waze users turning your once-quiet side street into the Talladega Superspeedway.

To its credit, Waze is correcting one of its most common user  that it often leads users into harrowing and time-consuming left turns. But another major concern still looms, and it’s one that could damage both your fender and your chances on the GMAT:

Beware the shortcuts and “crutches” that save you a few seconds, but in doing so completely remove all reasoning and awareness.

With Waze, we’ve all seen it happen: someone so beholden to, “I must turn left on 9th Street because the app told me to!” will often barrel through two lanes of traffic – with no turn signal – to make that turn…not realizing that the trip would have taken the exact same amount of time, with much less risk to the driver and everyone else on the road, had he waited a block or two to safely merge left and turn on 10th or 11th. By focusing so intently on the app’s “don’t worry about paying attention…we’ll tell you when to turn” features, the driver was unaware of other cars and of earlier opportunities to safely make the merge in the desired direction.

The GMAT offers similar pitfalls when examinees rely too heavily on “turn your brain off” tricks and techniques. As you learn and practice them, strategies like the “plumber butt” for rates and averages may seem quick, easy, and “turn your brain off” painless. But the last thing you want to do on a higher-order thinking test like the GMAT is completely turn your brain off. For example, a “turn your brain off” rate problem might say:

John drives at an average rate of 45 miles per hour. How many miles will he drive in 2.5 hours?

And using a Waze-style crutch, you could remember that to get distance you multiply time by rate so you’d get 112.5 miles. That may be a few seconds faster than performing the algebra by thinking “Rate = Distance over Time”; 45 = D/2.5; 45(2.5) = D; D = 112.5.

But where a shortcut crutch saves you time on easier problems, it can leave you helpless on longer problems that are designed to make you think. Consider this Data Sufficiency example:

A factory has three types of machines – A, B, and C – each of which works at its own constant rate. How many widgets could one machine A, one Machine B, and one Machine C produce in one 8-hour day?

(1) 7 Machine As and 11 Machine Bs can produce 250 widgets per hour

(2) 8 Machine As and 22 Machine Cs can produce 600 widgets per hour

Here, simply trying to plug the information into a simple diagram will lead you directly to choice E. You simply cannot separate the rate of A from the rate of B, or the rate of B from the rate of C. It will not fit into the classic “rate pie / plumber’s butt” diagram that many test-takers use as their “I hate rates so I’ll just do this trick instead” crutch.

However, those who have their critical thinking mind turned on will notice two things: that choice E is kind of obvious (the algebra doesn’t get you very close to solving for any one machine’s rate) so it’s worth pressing the issue for the “reward” answer of C, and that if you simply arrange the algebra there are similarities between the number of B and of C:

7(Rate A) + 11(Rate B) = 250
8(Rate A) + 22(Rate C) = 600

Since 11 is half of 22, one way to play with this is to double the first equation so that you at least have the same number of Bs as Cs (and remember…those are the only two machines that you don’t have “together” in either statement, so relating one to the other may help). If you do, then you have:

14(A) + 22(B) = 500
8(A) + 22(C) = 600

Then if you sum the questions (Where does the third 22 come from? Oh, 14 + 8, the coefficients for A.), you have:

22A + 22B + 22C = 1100

So, A + B + C = 50, and now you know the rate for one of each machine. The two statements together are sufficient, but the road to get there comes from awareness and algebra, not from reliance on a trick designed to make easy problems even easier.

The lesson? Much like Waze, which can lead to lack-of-awareness accidents and to shortcuts that dramatically up the degree of difficulty for a minimal time savings, you should take caution when deciding to memorize and rely upon a knee-jerk trick in your GMAT preparation.

Many are willing (or just unaware that this is the decision) to sacrifice mindfulness and awareness to save 10 seconds here or there, but then fall for trap answers because they weren’t paying attention or become lost when problems are more involved because they weren’t prepared.

So, be choosy in the tricks and shortcuts you decide to adopt! If a shortcut saves you a or two of calculations, it’s worth the time it takes to learn and master it (but probably never worth completely avoiding the “long way” or knowing the general concept). But if its time savings are minimal and its grand reward is that, “Hey, you don’t have to understand math to do this!” you should be wary of how well it will serve your aspirations of scores above around 600.

Don’t let these slick shortcut waze of avoiding math drive you straight into an accident. Unless the time savings are game-changing, you shouldn’t make a trade that gains you a few seconds of efficiency on select, easier problems in exchange for your awareness and understanding.

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

By Brian Galvin.

Quarter Wit, Quarter Wisdom: Squares and Square Roots on the GMAT

In today’s post, we will try to clear up your doubts regarding positive and negative solutions in the case of squares and square roots. We will explain the reasons behind each case, which will help you recall the fundamentals when you need to use them. While preparing for the GMAT, you have probably come across a discussion that says x^2 = 4 has two roots, 2 and -2, while √4 has only one value, 2.

Now, let’s try to understand why this is so:

1) x^2 = 4
Basic algebra tells us that quadratics have two roots. Here, x can be either 2 or -2; each, when squared, will give you 4.

x^2 – 4 = 0 and (x + 2)*(x – 2) = 0 when x equals -2 or 2.

2) √x is positive, only
Now this is odd, right? √4 must be 2. Why is that? Shouldn’t it be 2 or -2. After all, when we square both 2 and -2, we get 4 (as discussed above). So, √4 should be 2 or -2.

Here is the concept: √x denotes only the principal square root. x has two square roots – the positive square root (or principal square root) written as √x and the negative square root written as -√x. Therefore, when you take the square root of 4, you get two roots: √4 and -√4, which  is 2 and -2 respectively.

On a GMAT question, when you see √x, this is specifically referring to the positive square root of the number. So √4 is 2, only.

3) (√x)^2 = x
This is fairly straightforward – since x has a square root, it must be non-negative. When you square it, just the square root sign vanishes and you are left with x.

4) √(x^2) = |x|
Now this isn’t intuitive either. √(x^2) should simply be x – why do we have absolute value of x, then? Again, this has to do with the principal square root concept. First you will square x, and then when you write √, it is by default just the principal square root. The negative square root will be written as -√(x^2). So, irrespective of whether x was positive or negative initially, √(x^2) will definitely be positive x. Therefore, we will need to take the absolute value of x.

Here’s a quick recap with some examples:

• √9 = 3
• x^2 = 16 means x is either 4 or -4
• √(5^2) = 5
• √(-5^2) = 5
• (√16)^2 = 16
• √100 = 10

To see this concept in action, let’s take a look at a very simple official problem:

If x is not 0, then √(x^2)/x =

(A) -1
(B) 0
(C) 1
(D) x
(E) |x|/x

We know that √(x^2) is not simply x, but rather |x|. So, √(x^2)/x = |x|/x.

Depending on whether x is positive or negative, |x|/x will be 1 or -1 – we can’t say which one. Hence, there is no further simplification that we can do, and our answer must be E.

Now that you are all warmed up, let’s examine a higher-level question:

Is √[(x – 3)^2] = (3 – x)?

Statement 1: x is not 3
Statement 2: -x * |x| > 0

We know that √(x^2) = |x|, so √[(x – 3)^2] = |x – 3|.

This means that our question is basically:

Is |x – 3| = 3 – x?

Note that 3 – x can also be written as -(x – 3).

Is |x – 3| = -(x – 3)?

Recall the definition of absolute values: |a| = a if a is greater than or equal to 0, and -a if a < 0.

So, “Is |x – 3| = -(x – 3)?” depends on whether (x – 3) is positive or negative. If (x – 3) is negative (or 0), then |x – 3| is equal to -(x – 3).

So our question now boils down to:

Is (x – 3) negative (or 0)?

Statement 1: x is not 3

This means we know that (x – 3) is not 0, but we still don’t know whether it is negative or positive. This statement is not sufficient.

Statement 2: -x * |x| > 0

|x| is always non-negative, so for the product to be positive, “-x” must also be positive. This means x must be negative. If x is negative, x – 3 must be negative, too.

If (x – 3) is negative, |x – 3| is equal to -(x – 3). Hence, this statement alone is sufficient, and our answer is B.

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: Mother Knows Best

This weekend is Mother’s Day here in the United States, and also, as the first full weekend in May, a weekend that will kick off a sense of study urgency for those intent on the September Round 1 MBA admissions deadlines. (If your mother were here she’d tell you why: if you want two full months to study for the GMAT and two full months to work on your applications, you have to start studying now!)

In honor of mothers everywhere and in preparation for your GMAT, let’s consider one of the things that makes mothers so great. Even today as an adult, you’ll likely find that if you live a flight or lengthy drive/train from home, when you leave your hometown, your mother loads you up with snacks for the plane, bottled water for the drive, hand sanitizer for the airport, etc. Why is that? When it comes to their children – no matter how old or independent – mothers are prepared for every possible situation.

What if you get hungry on the plane, or you’re delayed at your connecting airport and your credit card registers fraud because of the strange location and you’re unable to purchase a meal?! She doesn’t want you getting sick after touching the railing on an escalator, so she found a Purell bottle that’s well less than the liquid limit at security (and also packed a clear plastic bag for you and your toiletries). Moms do not want their children caught in a unique and harmful (or inconvenient) situation, so they plan for all possible occurrences.

And that’s how you should approach Data Sufficiency questions on the GMAT.

When a novice test-taker sees the problem:

What is the value of x?

(1) x^2 = 25

(2) 8 < 2x < 12

He may quickly say “oh it’s 5” to both of them. 5 is the square root of 25, and the second equation simplifies to 4 < x < 6, and what number is between 4 and 6? It’s 5.

But your mother would give you caution, particularly because her mission is to avoid *negative* outcomes for you. She’d be prepared for a negative value of x (-5 satisfies Statement 1) and for nonintegers (x could be 4.00001 or 5.9999 given Statement 2). Knowing those contingencies, she’d wisely recognize that you need both statements to guarantee one exact answer (5) for x.

Just like she’d tie notes to your mittens or pin them on your shirt when you were a kid so that you wouldn’t forget (and like now she’ll text you reminders for your grandmother’s birthday or to RSVP to your cousin’s wedding), your mom would suggest that you keep these unique occurrences written down at the top of your noteboard on test day: Negative, Zero, Noninteger, Infinity, Biggest/Smallest Value. That way, you’ll always check for those unique situations before you submit your answer, and you’ll have a much better shot at a challenge-level problem like this:

The product of consecutive integers a, b, c, and d is 5,040. What is the value of d?

(1) d is prime

(2) d < c < b < a

So where does mom come in?

Searching for consecutive integers, you’ll likely factor 5,040 to 7, 8, 9, and 10 (the 10 is obvious because 5,040 ends in a 0, and then when you see that the rest is 504 and know that’s divisible by 9, and you’re just about done). And so with Statement 1, you’ll see that the only prime number in the bunch is 7, meaning that d = 7 and Statement 1 is sufficient. And Statement 2 seems to support that exact same conclusion – as the smallest of the 4 integers, d is, again, 7.

Right?

Enter mom’s notes: did you consider zero? (irrelevant) Did you consider nonintegers? (they specified integers, so irrelevant) Did you consider negative numbers?

That’s the key. The four consecutive integers could be -10, -9, -8, and -7 meaning that d could also be -10. That wasn’t an option for Statement 1 (only positives are prime) and so since you did the “hard work” of factoring 5,040 and then finally got to where Statement 2 was helpful, there’s a high likelihood that you were ready to be finished and saw 7 as the only option for Statement 2.

This is why mom’s reminders are so helpful: on harder problems, the “special circumstances” numbers that mom wants to make sure you’re always prepared for tend to be afterthoughts, having taken a backseat to the larger challenges of math. But mother knows best – you may not be stranded in a foreign airport without a snack and your car might not stall in the desert when you don’t have water, but in the rare event that such a situation occurs she wants you to be prepared. Keep mom’s list handy at the top of your noteboard (alas, the Pearson/Vue center won’t allow you to pin it to your shirt) and you, like mom, will be prepared for all situations.

By Brian Galvin.

How to Avoid Trap Answers On GMAT Data Sufficiency Questions

When I’m not teaching GMAT classes or writing posts for our fine blog, I am, unfortunately, writing fiction. Anyone who has taken a stab at writing fiction knows that it’s hard, and because it’s hard, it is awfully tempting to steer away from pain and follow the path of least resistance.

This tendency can manifest itself in any number of ways. Sometimes it means producing a cliché rather than straining for a more precise and original way to render a scene. More often, it means procrastinating – cleaning my desk or refreshing espn.com for the 700th time – rather than doing any writing at all. The point is that my brain is often groping for an easy way out. This is how we’re all wired; it’s a dangerous instinct, both in writing and on the GMAT.

This problem is most acute on Data Sufficiency questions. Most test-takers like to go on auto-pilot when they can, relying on simple rules and heuristics rather than proving things to themselves – if I have the slope of a line and one point on that line, I know every point on that line; if I have two linear equations and two variables I can solve for both variables, etc.

This is not in and of itself a problem, but if you find your brain shifting into path-of-least-resistance mode and thinking that you’ve identified an answer to a question within a few seconds, be very suspicious about your mode of reasoning. This is not to say that you should simply assume that you’re wrong, but rather to encourage you to try to prove that you’re right.

Here’s a classic example of a GMAT Data Sufficiency question that appears to be easier than it is:

Joanna bought only \$.15 stamps and \$.29 stamps. How many \$.15 stamps did she buy?

1) She bought \$4.40 worth of stamps
2) She bought an equal number of \$.15 stamps and \$.29 stamps

Here’s how the path-of-least-resistance part of my brain wants to evaluate this question. Okay, for Statement 1, there could obviously be lots of scenarios. If I call “F” the number of 15 cent stamps and “T” the number of 29 cent stamps, all I know is that .15F + .29T = 4.40. So that statement is not sufficient. Statement 2 is just telling me that F = T. Clearly no good – any number could work. And together, I have two unique linear equations and two unknowns, so I have sufficiency and the answer is C.

This line of thinking only takes a few seconds, and just as I need to fight the urge to take a break from writing to watch YouTube clips of Last Week Tonight with John Oliver because it’s part of my novel “research,” I need to fight the urge to assume that such a simple line of reasoning will definitely lead me to the correct answer to this question.

So let’s rethink this. I know for sure that the answer cannot be E – if I can solve for the unknowns when I’m testing the statements together, I clearly have sufficiency there. And I know for sure that the answer cannot be that Statement 2 alone is sufficient. If F = T, there are an infinite number of values that will work.

So, let’s go back to Statement 1. I know that I cannot purchase a fraction of a stamp, so both F and T must be integer values. That’s interesting. I also know that the total amount spent on stamps is \$4.40, or 440 cents, which has a units digit of 0. When I’m buying 15-cent stamps, I can spend 15 cents if I buy 1 stamp, 30 cents if I buy two, etc.

Notice that however many I buy, the units digit must either be 5 or 0. This means that the units digit for the amount I spend on 29 cent stamps must also be 5 or 0, otherwise, there’d be no way to get the 0 units digit I get in 440. The only way to get a units digit of 5 or 0 when I’m multiplying by 29 is if the other number ends in 5 or 0 . In other words, the number of 29-cent stamps I buy will have to be a multiple of 5 so that the amount I spend on 29-cent stamps will end in 5 or 0.

Here’s the sample space of how much I could have spent on 29-cent stamps:

Five stamps: 5*29 = 145 cents
Ten stamps: 10*29 = 290 cents
Fifteen stamps: 15* 29 = 435 cents

Any more than fifteen 29-cent stamps and I ‘m over 440, so these are the only possible options when testing the first statement.

Let’s evaluate: say I buy five 29-cent stamps and spend 145 cents. That will leave me with 440 – 145 = 295 cents left for the 15-cent stamps to cover. But I can’t spend exactly 295 cents by purchasing 15-cent stamps, because 295 is not a multiple of 15.

Say I buy ten 29-cent stamps, spending 290 cents. That leaves 440 – 290 = 150. Ten 15-cent stamps will get me there, so this is a possibility.

Say I buy fifteen 29-cent stamps, spending 435 cents. That leaves 440 – 435 = 5. Clearly that’s not possible to cover with 15-cent stamps.

Only one option works: ten 29-cent stamps and ten 15-cent stamps. Because there’s only one possibility, Statement 1 alone is sufficient, and the answer here is actually A.

Takeaway: Don’t take the GMAT the way I write fiction. Following the path of least-resistance will often lead you right into the trap the question writer has set for unsuspecting test-takers. If something feels too easy on a Data Sufficiency, it probably is.

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

GMAT Tip of the Week: Death, Taxes, and the GMAT Items You Know For Certain

Here on April 15, it’s a good occasion to remember the Benjamin Franklin quote: “In this world nothing can be said to be certain, except death and taxes.” Franklin, of course, never took the GMAT (which didn’t become a thing until a little ways after his own death, which he accurately predicted above). But if he did, he’d have plenty to add to that quote.

On the GMAT, several things are certain. Here’s a list of items you will certainly see on the GMAT, as you attempt to raise your score and therefore your potential income, thereby raising your future tax bills in Franklin’s honor:

Integrated Reasoning
You will struggle with pacing on the Integrated Reasoning section. 12 prompts in 30 minutes (with multiple problems per prompt) is an extremely aggressive pace and very few people finish comfortably. Be willing to guess on a problem that you know could sap your time: not only will that help you finish the section and protect your score, it will also help save your stamina and energy for the all-important Quant Section to follow.

Word Problems
On the Quantitative Section, you will certainly see at least one Work/Rate problem, one Weighted Average problem, and one Min/Max problem. This is good news! Word problems reward repetition and preparation – if you’ve put in the work, there should be no surprises.

Level of Difficulty
If you’re scoring above average on either the Quant or Verbal sections, you will see at least one problem markedly below your ability level. Because each section contains several unscored, experimental problems, and those problems are delivered randomly, probability dictates that every 700+ scorer will see at least one problem designed for the 200-500 crowd (and probably more than that). Do not try to read in to your performance based on the difficulty level of any one problem! It’s easy to fear that such a problem was delivered to you because you’re struggling, but the much more logical explanation is that it was either random or difficult-but-sneakily-so, so stay confident and move on.

Data Sufficiency
You will see at least one Data Sufficiency problem that seems way too easy to be true. And it’s probably not true: make sure that you think critically any time the testmaker is directly baiting you into a particular answer.

Sentence Correction
You will have to pick an answer that you don’t like, that doesn’t catch the ear the way you’d write or say it. Make sure that you prioritize the major errors that you know you can routinely catch and correct, and not let the GMAT bait you into a decision you’re just not qualified to make.

You will see a passage that takes you a few re-reads to even get your mind to process it. Remember to be– get enough out of the passage to know where to look when they ask you a specific question, but don’t worry about becoming a subject-matter expert on the topic. GMAT passages are designed to be difficult to read (particularly toward the end of a long test), so know that your competitive advantage is that you’ll be more efficient than your competition.

Critical Reasoning
You will have the opportunity to make quick work of several Critical Reasoning problems if you notice the tiny that each argument provides, and if you’re able to notice the subtle-but-significant words that make conclusions extra specific (and therefore harder to prove).

Few things are certain in life, but as you approach the GMAT there are plenty of certainties that you can prepare for so that you eliminate surprises and proceed throughout your test day confidently. On this Tax Day, take inventory of the things you know to be certain about the GMAT so that your test day isn’t so taxing.

By Brian Galvin.

What Makes GMAT Quant Questions So Hard?

We know that the essentials of the GMAT Quant section are pretty simple: advanced topics such as derivatives, complex numbers, matrices and trigonometry are not included, while fundamentals we all learned from our high school math books are included. So it would be natural to think that the GMAT Quant section should not pose much of a problem for most test-takers (especially for engineering students, who have actually covered far more advanced math during their past studies).

Hence, it often comes as a shock when many test-takers, including engineering students, receive a dismal Quant score on the first practice test they take. Of course, with practice, they usually wise up to the treachery of the GMAT, but until then, the Quant section is responsible for many a nightmare!

Today, let’s see what kind of treachery we are talking about – problems like this make some people laugh out loud and others pull at their hair!

Is the product pqr divisible by 12?
Statement 1: p is a multiple of 3
Statement 2: q is a multiple of 4

This seems like an easy C (Statements 1 and 2 together are sufficient, but alone are not sufficient), doesn’t it? P is a multiple of 3 and q is a multiple of 4, so together, p*q would be a multiple of 3*4 = 12. If p * q is already a multiple of 12, then obviously it would seem that p*q*r would be a multiple of 12, too.

But here is the catch – where is it mentioned that r must be an integer? Just because p and q are integers (multiples of 3 and 4 respectively), it does not imply that r must also be an integer.

If r is an integer, then sure, p*q*r will be divisible by 12. Imagine, however, that p = 3, q = 4 and r = 1/12. Now the product p*q*r = 3*4*(1/12) = 1. 1 is not divisible by 12, so in this case, pqr is not divisible by 12. Hence, both statements together are not sufficient to answer the question, and our answer is in fact E!

This question is very basic, but it still tricks us because we want to assume that p, q and r are clean integer values.

Along these same lines, let’s try the another one:

If 10^a * 3^b * 5^c = 450^n, what is the value of c?
Statement 1: a is 1.
Statement 2:  b is 2.

The first thing most of us will do here is split 450 into its prime factors:

450 = 2 * 3^2 * 5^2

450^n = 2^n * 3^2n * 5^2n

And do the same thing with the left side of the equation:

10^a * 3^b * 5^c = 2^a * 3^b * 5^(a+c)

Bringing the given equation back, we get:

2^a * 3^b * 5^(a+c) = 2^n * 3^2n * 5^2n

Statement 1: a is 1.

Equating the power of 2 on both sides, we see that a = n = 1.

a + c = 2n (equating the power of 5 on both sides)

1 + c = 2

c = 1

Statement 2:  b is 2.

Equating the power of 3 on both sides, we see that b = 2n = 2, so n = 1.

If n = 1, a = 1 by equating the powers of 2 on both sides.

a + c = 2n (equating the power of 5 on both sides)

1 + c = 2

c = 1

So it seems that both statements are separately sufficient. But hold on – again, the variables here don’t need to be cleanly fitting integers. The variables could pan out the way discussed in our first problem, or very differently.

Say, n = 1. When Statement 1 gives you that a = 1, you get 10^1 * 3^b * 5^c = 450^1.

3^b * 5^c = 45

Now note that value of c depends on the value of b, which needn’t be 2.

If b  = 3, then 3^3 * 5^c = 45.

5^c = 45/27

C will take a non-integer value here.

c = .3174

The question does not mention that all variables are integers, therefore there are infinite values that c can take depending on the values of b. Similarly, we can see that Statement 2 alone is also not sufficient. Using both statements together, you will get:

2^a * 3^b * 5^(a+c) = 450^n

2^1 * 3^2 * 5^(1 + c) = 450^n

5^(1 + c) = 450^n/18

By now, you’ve probably realized that depending on the value of n, c can take infinite different values. If n = 1, c = 1. If n = 2, c = 4.8. And so on… We don’t need to actually find these values – it is enough to know that different values of n will give different values of c.

With this in mind, we can see that both statements together are not sufficient, and therefore our answer must be E.

Hopefully, in future, this sneaky trick will not get you!

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!

2 Tips to Make GMAT Remainder Questions Easy

Several months ago, I wrote an article aboutBecause this concept shows up so often on the GMAT, I thought it would be useful to revisit the topic. At times, it will be helpful to know the kind of terminology we’re taught in grade school, while at other times, we’ll simply want to select simple numbers that satisfy the parameters of a Data Sufficiency statement.

So let’s explore each of these scenarios in a little more detail. A simple example can illustrate the terminology: if we divide 7 by 4, we’ll have 7/4 = 1 + ¾.

7, the term we’re dividing by something else, is called the dividend. 4, which is doing the dividing, is called the divisor. 1, the whole number component of the mixed fraction, is the quotient. And 3 is the remainder. This probably feels familiar.

In the abstract, the equation is: Dividend/Divisor = Quotient + Remainder/Divisor. If we multiply through by the Divisor, we get: Dividend = Quotient*Divisor + Remainder.

Simply knowing this terminology will be sufficient to answer the following official question:

When N is divided by T, the quotient is S and the remainder is V. Which of the following expressions is equal to N?

A) ST
B) S + V
C) ST + V
D) T(S+V)
E) T(S – V)

In this problem, N – which is getting divided by something else – is our dividend, T is the divisor, S is the quotient, and V is the remainder. Plugging the variables into our equation of Dividend = Quotient*Divisor + Remainder, we get N = ST + V… and we’re done! The answer is C.

(Note that if you forgot the equation, you could also pick simple numbers to solve this problem. Say N = 7 and T = 3. 7/3 = 2 + 1/3.  The Quotient is 2, and the remainder is 1, so V = 1. Now, if we plug in 3 for T, 2 for S, and 1 for V, we’ll want an N of 7. Answer choice C will give us an N of 7, 2*3 + 1 = 7, so this is correct.)

When we need to generate a list of potential values to test in a data sufficiency question, often a statement will give us information about the dividend in terms of the divisor and the remainder.

Take the following example: when x is divided by 5, the remainder is 4. Here, the dividend is x, the divisor is 5, and the remainder is 4. We don’t know the quotient, so we’ll just call it q. In equation form, it will look like this: x = 5q + 4. Now we can generate values for x by picking values for q, bearing in mind that the quotient must be a non-negative integer.

If q = 0, x = 4. If q = 1, x = 9. If q=2, x = 14. Notice the pattern in our x values: x = 4 or 9 or 14… In essence, the first allowable value of x is the remainder. Afterwards, we’re simply adding the divisor, 5, over and over. This is a handy shortcut to use in complicated data sufficiency problems, such as the following:

If x and y are integers, what is the remainder when x^2 + y^2 is divided by 5?

1) When x – y is divided by 5, the remainder is 1
2) When x + y is divided by 5, the remainder is 2

In this problem, Statement 1 gives us potential values for x – y. If we begin with the remainder (1) and continually add the divisor (5), we know that x – y = 1 or 6 or 11, etc. If x – y = 1, we can say that x = 1 and y = 0. In this case, x^2 + y^2 = 1 + 0 = 1, and the remainder when 1 is divided by 5 is 1. If x – y = 6, then we can say that x = 7 and y = 1. Now x^2 + y^2 = 49 + 1 = 50, and the remainder when 50 is divided by 5 is 0. Because the remainder changes from one scenario to another, Statement 1 is not sufficient alone.

Statement 2 gives us potential values for x + y. If we begin with the remainder (2) and continually add the divisor (5), we know that x + y = 2 or 7 or 12, etc. If x + y = 2, we can say that x = 1 and y = 1. In this case, x^2 + y^2 = 1 + 1 = 2, and the remainder when 2 is divided by 5 is 2. If x + y = 7, then we can say that x = 7 and y = 0. Now x^2 + y^2 = 49 + 0 = 49, and the remainder when 49 is divided by 5 is 4. Because the remainder changes from one scenario to another, Statement 2 is also not sufficient alone.

Now test them together – simply select one scenario from Statement 1 and one scenario from Statement 2 and see what happens. Say x – y = 1 and x + y = 7. Adding these equations, we get 2x = 8, or x = 4. If x = 4, y = 3. Now x^2 + y^2 = 16 + 9 = 25, and the remainder when 25 is divided by 5 is 0.

We need to see if this will ever change, so try another scenario. Say x – y = 6 and x + y = 12. Adding the equations, we get 2x = 18, or x = 9. If x =  9, y = 3, and x^2 + y^2 = 81 + 9 = 90. The remainder when 90 is divided by 5 is, again, 0. No matter what we select, this will be the case – we know definitively that the remainder is 0. Together the statements are sufficient, so the answer is C.

Takeaway: You’re virtually guaranteed to see remainder questions on the GMAT, so you want to make sure you have this concept mastered. First, make sure you feel comfortable with the following equation: Dividend = Divisor*Quotient + Remainder. Second, if you need to select values, you can simply start with the remainder and then add the divisor over and over again. If you internalize these two ideas, remainder questions will become considerably less daunting.

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

GMAT Tip of the Week: The Biggie Smalls Sufficiency Strategy

If it’s March, it must be Hip Hop Month in the Veritas Prep GMAT Tip of the Week space, where this week we’ll tackle the most notorious GMAT question type – Data Sufficiency – with some help from hip hop’s most notorious rapper – Biggie Smalls.

Biggie’s lyrics – and his name itself – provide a terrific template for you to use when picking numbers to test whether a statement is sufficient or not. So let’s begin with a classic lyric from “Big Poppa” – you may think Big is describing how he’s approach a young lady in a nightclub, but if you listen closely he’s actually talking directly to you as you attack Data Sufficiency:

“Ask you what your interests are, who you be with. Things to make you smile; what numbers to dial.”

“What numbers to dial” tends to be one of the biggest challenges that face GMAT examinees, so let’s examine the strategies that can take your score from “it was all a dream” to sipping champagne when you’re thirsty.

Biggie Smalls Strategy #1: Biggie Smalls
Consider this Data Sufficiency problem:

What is the value of integer z?

1) z is the remainder when positive integer y is divided by positive integer (y – 1)

2) y is not a prime number

Statistically, more than 50% of respondents in the Veritas Prep practice tests incorrectly choose answer choice A, that Statement 1 alone is sufficient but Statement 2 alone is not sufficient. Why? Because they’re not quite sure “what numbers to dial.” People know that they need to test numbers – Statement 1 is very abstract and difficult to visualize with variables – so they test a few numbers that come to mind:

If y = 5, y – 1 = 4, and the problem is then 5/4 which leads to 1, remainder 1.

If y = 10, y – 1 = 9, so the problem is then 10/9 which also leads to 1, remainder 1.

If they keep choosing random integers that happen to come to mind, they’ll see that pattern hold – the answer is ALMOST always 1 remainder 1, with exactly one exception. If y = 2, then y – 1 = 1, and 2 divided by 1 is 2 with no remainder. This is the only case where z does not equal 1, but that one exception shows that Statement 1 is not sufficient.

The question then becomes, “If there’s only one exception, how the heck does the GMAT expect me to stumble on that needle in a haystack?” And the answer comes directly from the Notorious BIG himself:

You need to test “Biggie Smalls,” meaning that you need to test the biggest number they’ll let you use (here it can be infinite, so just test a couple of really big numbers like 1,000 and 1,000,000) and you need to test the smallest number they’ll let you use. Here, that’s y = 2 and y – 1 = 1, since y – 1 must be a positive integer, and the smallest of those is 1.

The problem is that people tend to simply test numbers that come to mind (again, over half of all respondents think that Statement 1 is sufficient, which means that they very likely never considered the pairing of 2 and 1) and don’t push the limits. Data Sufficiency tends to play to the edge cases – if you get a statement like 5 < x < 12, you can’t just test 8, 9, and 10 – you’ll want to consider 5.00001 and 11.9999. When the GMAT gives you a range, use the entire range – and a good way to remind yourself of that is to just remember “Biggie Smalls.”

Biggie Smalls Strategy #2:  Juicy
In arguably his most famous song, “Juicy”, Biggie spits the line, “Damn right I like the life I live, because I went from negative to positive and it’s all…it’s all good (and if you don’t know, now you know).”

There, of course, Biggie is reminding you that you have to consider both negative and positive numbers in Data Sufficiency problems. Consider this example:

a, b, c, and d are consecutive integers such that the product abcd = 5,040. What is the value of d?

1) d is prime

2) a>b>c>d

This problem exemplifies why keeping Big’s words top of mind is so crucial – difficult problems will often “satisfy your intellect” with interesting math…and then beat you with negative/positive ideology. Here it takes some time to factor 5040 into the consecutive integers 7 x 8 x 9 x 10, but once you do, you can see that Statement 1 is sufficient: 7 is the only prime number.

But then when you carry that over to Statement 2, it’s very, very easy to see 7, 8, 9, and 10 as the only choices and again see that d = 7. But wait! If d doesn’t have to be prime – primes can only be positive – that allows for a possibility of negative numbers: -10, -9, -8, and -7. In that case, d could be either 7 or -10, so Statement 2 is actually not sufficient.

So heed Biggie’s logic: you’ll like the life you live much better if you go from negative to positive (or in most cases, vice versa since your mind usually thinks positive first), and if you don’t know (is that sufficient?) now, after checking for both positive and negative and for the biggest and smallest numbers they’ll let you pick, now you know.

By Brian Galvin.

Quarter Wit, Quarter Wisdom: Ratios in GMAT Data Sufficiency

We know that ratios are the building blocks for a lot of other concepts such as time/speed, work/rate and mixtures. As such, we spend a lot of time getting comfortable with understanding and manipulating ratios, so the GMAT questions that test ratios seem simple enough, but not always! Just like questions from all other test areas, questions on ratios can be tricky too, especially when they are formatted as Data Sufficiency questions.

Let’s look at two cases today: when a little bit of data is sufficient, and when a lot of data is insufficient.

When a little bit of data is sufficient!
Three brothers shared all the proceeds from the sale of their inherited property. If the eldest brother received exactly 5/8 of the total proceeds, how much money did the youngest brother (who received the smallest share) receive from the sale?

Statement 1: The youngest brother received exactly 1/5 the amount received by the middle brother.

Statement 2: The middle brother received exactly half of the two million dollars received by the eldest brother.

First impressions on reading this question? The question stem gives the fraction of money received by one brother. Statement 1 gives the fraction of money received by the youngest brother relative to the amount received by the middle brother. Statement 2 gives the fraction of money received by the middle brother relative to the eldest brother and an actual amount. It seems like the three of these together give us all the information we need. Let’s dig deeper now.

From the Question stem:

Eldest brother’s share = (5/8) of Total

Statement 1: Youngest Brother’s share = (1/5) * Middle brother’s share

We don’t have any actual number – all the information is in fraction/ratio form. Without an actual value, we cannot find the amount of money received by the youngest brother, therefore, Statement 1 alone is not sufficient.

Statement 2: Middle brother’s share = (1/2) * Eldest brother’s share, and the eldest brother’s share = 2 million dollars

Middle brother’s share = (1/2) * 2 million dollars = 1 million dollars

Now, we might be tempted to jump to Statement 1 where the relation between youngest brother’s share and middle brother’s share is given, but hold on: we don’t need that information. We know from the question stem that the eldest brother’s share is (5/8) of the total share.

So 2 million = (5/8) of the total share, therefore the total share = 3.2 million dollars.

We already know the share of the eldest and middle brothers, so we can subtract their shares out of the total and get the share of the youngest brother.

Youngest brother’s share = 3.2 million – 2 million – 1 million = 0.2 million dollars

Statement 2 alone is sufficient, therefore, the answer is B.

When a lot of data is insufficient!
A department manager distributed a number of books, calendars, and diaries among the staff in the department, with each staff member receiving x books, y calendars, and z diaries. How many staff members were in the department?

Statement 1: The numbers of books, calendars, and diaries that each staff member received were in the ratio 2:3:4, respectively.

Statement 2: The manager distributed a total of 18 books, 27 calendars, and 36 diaries.

First impressions on reading this question? The question stem tells us that each staff member received the same number of books, calendars, and diaries. Statement 1 gives us the ratio of books, calendars and diaries. Statement 2 gives us the actual numbers. It certainly seems that we should be able to obtain the answer. Let’s find out:

Looking at the question stem, Staff Member 1 recieved x books, y calendars, and z diaries, Staff Member 2 recieved x books, y calendars, and z diaries… and so on until Staff Member n (who also recieves x books, y calendars, and z diaries).

With this in mind, the total number of books = nx, the total number of calendars = ny, and the total number of diaries = nz.

Question: What is n?

Statement 1 tells us that x:y:z = 2:3:4. This means the values of x, y and z can be:

2, 3, and 4,

or 4, 6, and 8,

or 6, 9, and 12,

or any other values in the ratio 2:3:4.

They needn’t necessarily be 2, 3 and 4, they just need the required ratio of 2:3:4.

Obviously, n can be anything here, therefore, Statement 1 alone is not sufficient.

Statement 2 tell us that nx = 18, ny = 27, and nz = 36.

Now we know the actual values of nx, ny and nz, but we still don’t know the values of x, y, z and n.

They could be

2, 3, 4 and 9

or 6, 9, 12 and 3

Therefore, Statement 2 alone is also not sufficient.

Considering both statements together, note that Statement 2 tells us that nx:ny:nz = 18:27:36 = 2:3:4 (they had 9 as a common factor).

Since n is a common factor on left side, x:y:z = 2:3:4 (ratios are best expressed in the lowest form).

This is a case of what we call “we already knew that” – information given in Statement 1 is already a part of Statement 2, so it is not possible that Statement 2 alone is not sufficient but that together Statement 1 and 2 are. Hence, both statements together are not sufficient, and our answer must be E.

A question that arises often here is, “Why can’t we say that the number of staff members must be 9?”

This is because the ratio of 2:3:4 is same as the ratio of 6:9:12, which is same as 18:27:36 (when you multiply each number of a ratio by the same number, the ratio remains unchanged).

If 18 books, 27 calendars, and 36 diaries are distributed in the ratio 2:3:4, we could give them all to one person, or to 3 people (giving them each 6 books, 9 calendars and 12 diaries), or to 9 people (giving them each 2 books, 3 calendars and 4 diaries).

When we see 18, 27 and 36, what comes to mind is that the number of people could have been 9, which would mean that the department manager distributed 2 books, 3 calendars and 4 diaries to each person. But we know that 9 is divisible by 3, which should remind us that the number of people could also be 3, which would mean that the manager distributed 6 books, 9 calendars and 12 diaries to each person. As such, we still don’t know how many staff members there are, and our answer remians E.

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 MLK Study Challenge (Remove Your Biases)

As we celebrate Martin Luther King, Jr. this weekend, you may take some of your free time to study for the GMAT. And if you do, make sure to heed the lessons of Dr. King, particularly as you study Data Sufficiency.

If Dr. King were alive today, he would certainly be proud of the legislation he inspired to end much of the explicit bias – you can’t eat here, vote there, etc. – that was part of the American legal code until the 1960s. But he would undoubtedly be dismayed by the implicit bias that still runs rampant across society.

This implicit bias is harder to detect and even harder to “fix.” It’s the kind of bias that, for example, the movie Freaknomics shows; often when the name at the top of a resume connotes some sort of stereotype, it subconsciously colors the way that the reader of that resume processes the rest of the information on it.

While that kind of subconscious bias is a topic for a different blog to cover, it has an incredible degree of relevance to the way that you attack GMAT Data Sufficiency problems. If you’re serious about studying for the GMAT, you’ll probably have long enacted your own versions of the Voting Rights Act and Civil Rights Act well before you get to test day – that is to say, you’ll have figured out how to eliminate the kind of explicit bias that comes from reading a question like:

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

1) x > 0

2) y is a 3 digit number

Here, you’ll likely see very quickly that Statement 1 is not sufficient, and come back to Statement 2 with fresh eyes. You don’t know that x is positive, so you’ll quickly see that y could be 111 and x could be 2, or that y could be -111 and x could be -2, so Statement 2 is clearly also not sufficient. The explicit bias that came from seeing “x is positive” is relatively easy to avoid – you know not to carry over that explicit information from Statement 1 to Statement 2.

But you also need to be just as aware of implicit bias. Try this question, as it is more likely to appear on the actual GMAT:

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

1) x is a prime number

2) y is a 3 digit number

On this version of the problem, people become extremely susceptible to implicit bias. You no longer get to quickly rule out the obvious “x is positive.” Here, the first statement serves to pollute your mind – it is, on its own merit, sufficient (if y is odd and the product of x and y is even, the only prime number x could be is 2, the only even prime), but it also serves to get you thinking about positive numbers (only positive numbers can be prime) and integers (only integers are prime). But those aren’t explicitly stated; they’re just inferences that your mind quickly makes, and then has trouble getting rid of. So as you assess Statement 2, it’s harder for you to even think of the possibilities that:

x could be -2 and y could be -111: You’re not thinking about negatives!

x could be 2/3 and y could be 333: You’re not thinking about non-integers!

On this problem, over 50% of users say that Statement 2 is sufficient (and less than 25% correctly answer A, that Statement 1 alone is sufficient), because they fall victim to that implicit bias that comes from Statement 1 whispering – not shouting – “positive integers.”

Harder problems will generally prey on your more subtle bias, so you need to make sure you’re giving each statement a fresh set of available options. So this Martin Luther King, Jr. weekend, applaud the progress that you have made in removing explicit bias from your Data Sufficiency regimen – you now know not to include Statement 1 directly in your assessment of Statement 2 ALONE – but remember that implicit bias is just as dangerous to your score. Pay attention to the times that implicit bias draws you to a poor decision, and be steadfast in your mission to give each statement its deserved, unbiased attention.

By Brian Galvin.

How to Make Rate Questions Easy on the GMAT

I recently wrote about the reciprocal relationship between rate and time in “rate” questions. Occasionally, students will ask why it’s important to understand this particular rule, given that it’s possible to solve most questions without employing it.

There are two reasons: the first is that knowledge of this relationship can convert incredibly laborious arithmetic into a very straightforward calculation. And the second is that this same logic can be applied to other types of questions. The goal, when preparing for the GMAT, isn’t to internalize hundreds of strategies; it’s to absorb a handful that will prove helpful on a variety of questions.

The other night, I had a tutoring student present me with the following question:

It takes Carlos 9 minutes to drive from home to work at an average rate of 22 miles per hour.  How many minutes will it take Carlos to cycle from home to work along the same route at an average rate of 6 miles per hour?

(A) 26

(B) 33

(C) 36

(D) 44

(E) 48

This question doesn’t seem that hard, conceptually speaking, but here is how my student attempted to do it: first, he saw that the time to complete the trip was given in minutes and the rate of the trip was given in hours so he did a simple unit conversion, and determined that it took Carlos (9/60) hours to complete his trip.

He then computed the distance of the trip using the following equation: (9/60) hours * 22 miles/hour = (198/60) miles. He then set up a second equation: 6miles/hour * T = (198/60) miles. At this point, he gave up, not wanting to wrestle with the hairy arithmetic. I don’t blame him.

Watch how much easier it is if we remember our reciprocal relationship between rate and time. We have two scenarios here. In Scenario 1, the time is 9 minutes and the rate is 22 mph. In Scenario 2, the rate is 6 mph, and we want the time, which we’ll call ‘T.” The ratio of the rates of the two scenarios is 22/6. Well, if the times have a reciprocal relationship, we know the ratio of the times must be 6/22. So we know that 9/T = 6/22.

Cross-multiply to get 6T = 9*22.

Divide both sides by 6 to get T = 9*22/6.

We can rewrite this as T = (9*22)/(3*2) = 3*11 = 33, so the answer is B.

The other point I want to stress here is that there isn’t anything magical about rate questions. In any equation that takes the form a*b = c, a and b will have a reciprocal relationship, provided that we hold c constant. Take “quantity * unit price = total cost”, for example. We can see intuitively that if we double the price, we’ll cut the quantity of items we can afford in half. Again, this relationship can be exploited to save time.

Take the following data sufficiency question:

Pat bought 5 lbs. of apples. How many pounds of pears could Pat have bought for the same amount of money?

(1) One pound of pears costs \$0.50 more than one pound of apples.

(2) One pound of pears costs 1 1/2 times as much as one pound of apples.

Statement 1 can be tested by picking numbers. Say apples cost \$1/pound. The total cost of 5 pounds of apples would be \$5.  If one pound of pears cost \$.50 more than one pound of apples, then one pound of pears would cost \$1.50. The number of pounds of pears that could be purchased for \$5 would be 5/1.5 = 10/3. So that’s one possibility.

Now say apples cost \$2/pound. The total cost of 5 pounds of apples would be \$10. If one pound of pears cost \$.50 more than one pound of apples, then one pound of pears would cost \$2.50. The number of pounds of pears that could be purchased for \$10 would be 10/2.5 = 4. Because we get different results, this Statement alone is not sufficient to answer the question.

Statement 2 tells us that one pound of pears costs 1 ½ times (or 3/2 times) as much as one pound of apples. Remember that reciprocal relationship! If the ratio of the price per pound for pears and the price per pound for apples is 3/2, then the ratio of their respective quantities must be 2/3. If we could buy five pounds of apples for a given cost, then we must be able to buy (2/3) * 5 = (10/3) pounds of pears for that same cost. Because we can find a single unique value, Statement 2 alone is sufficient to answer the question, and we know our answer must be B.

Takeaway: Remember that in “rate” questions, time and rate will have a reciprocal relationship, and that in “total cost” questions, quantity and unit price will have a reciprocal relationship. Now the time you save on these problem-types can be allocated to other questions, creating a virtuous cycle in which your time management, your accuracy, and your confidence all improve in turn.

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

Use Number Lines on the GMAT, Not Memory!

I’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.

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

GMAT Tip of the Week

Data Sufficiency – Where “No” Means “Affirmative”

(This is one of a series of GMAT tips that we offer on our blog.)

My hat is off to whomever created the Data Sufficiency question type, which holds within its format several delightfully crafty ways to elicit an incorrect answer. Perhaps none is more tricky and understated, however, than the method by which the test preys on our innate connection between the word “no” and its connotation of “negative”. (Author’s Note: This same connection was exploited recently-and-brilliantly on an episode of 30 Rock, in which Tracy Jordan exclaims that Jack’s medical test results were “positive” — meaning “good news” — because the actual results came back “negative”. But I digress…)

To illustrate, a Data Sufficiency question might ask:

Is x > 0?

A simple enough question, it would seem – is x positive – but then consider a potential first statement:

1) IxI = -x

This statement tells us that the absolute value of the number is equal to itself multiplied by negative one. Because of this, the number cannot be positive – any positive number multiplied by negative one becomes negative, but all absolute values are either positive or zero. A positive number simply cannot satisfy statement 1, and the answer to the overall question — Is x > 0? — is “no”.

Herein lies the rub — the answer to the question is “no,” which may lead you to believe that statement 1 is “negative” or “undesirable,” because of that connotation of the word “no” at which we just arrived. However, “no” is a definitive answer to the overall question. Given the information in statement one, we can prove one answer to that question, which means that “Statement (1) ALONE is sufficient”.

If you see, it would be easy, and somewhat intuitive, to “eliminate” statement 1 because it provided the answer “no”. But that’s not what Data Sufficiency questions ask — instead, they ask “do you have enough information to answer the question?”. Because of that, a definitive answer of “no” is, in fact, enough to answer the question, and so you must remind yourself that “no” means “sufficient”. To combat this common pitfall, I suggest writing the word “sufficient” at the top of your noteboard, and glancing at it each time you answer a Data Sufficiency question to remind yourself what the question is specifically asking.

Veritas Prep offers a full lesson on the Data Sufficiency question format, as well as hundreds of Data Sufficiency practice problems in its quantitative curriculum. For more information, please take a look at all of Veritas Prep’s GMAT preparation options.

GMAT Tip of the Week

Enough is Enough

(This is one of a series of GMAT tips that we offer on our blog.)

A format unique to the GMAT, Data Sufficiency questions comprise nearly half of the GMAT

GMAT Tip of the Week

(This is one of a series of GMAT tips that we offer on our blog.)

Think Like the GMAT Testmaker

The GMAT is a difficult exam; there’s simply no way around that fact. Knowing why it is difficult, however, will help you to better plan for it, and a fantastic way to gain that knowledge is to put yourself in the position of the writers of the exam.

As any GMAT examinee knows, Data Sufficiency questions can often be tricky or confusing simply because of the unique format of the question. It is rare, if it occurs at all, that you encounter situations in which you are not asked to solve a problem, but are asked if the tools provided would be sufficient to do so (however, as a manager, you’ll be asked this question quite a bit – what resources will you need to complete this project in a cost-effective manner?).

To gain experience with the tricks and traps that the format allows the writers of the exam to employ, try writing a few questions of your own with the intent of ‘tricking and trapping’ your friends. In order to do so, you will need to anticipate the mistakes that someone could easily make – and those are the mistakes that could trip you up, too. Does it seem as though someone could forget about the exception to the rule that you ‘hid’ in statement 1? Does statement 2 supply that one missing piece of information that was omitted from statement 1, baiting someone to select “B”, even though statement 2 is clearly insufficient on its own? Laying these traps for someone else is a great way to become more aware of them for yourself.

Note that you can craft questions in this format even without using mathematical concepts to do so. Here’s an example:

Question: Name this U.S. President.

(1) He has the same first name as his father.

(2) His presidency began after the American Civil War.

Answer: Many would be tempted to select (C), narrowing the choices down to John Quincy Adams and George W. Bush, then eliminating Adams with statement 2. However, doing so would assume that the father mentioned in statement 1 was also a President, which is not explicitly stated. Actually, the answer must be (E), as both George W. Bush and William McKinley, for example, had the same names as their fathers – McKinley’s father was not a President, but that wasn’t required by this question, even though some may have assumed it.

That assumption error is akin to a GMAT examinee assuming that a value must be an integer, or must be a positive number. By writing trivia questions designed to capitalize on potential errors, you can become more attuned to the mistakes that you may make on test day.

If you would like more GMAT help, take a look at the GMAT prep resources on our web site, and take a free practice GMAT exam.