December 18, 2012
The holidays are upon us, and with them come a flurry of seasonal activities: shopping trips, parties, and visits with family and friends. If you’re planning on taking your GMAT in January, you’re probably struggling with the challenge of fitting your studies into your holiday schedule. Here are a few tips to help you make the most of this busy time.
First, acknowledge your limitations. Because of your holiday obligations, you’ll probably need to scale back your GMAT study time. The holidays provide you with a great opportunity to recharge mentally and emotionally, so there’s nothing wrong with cutting back a little on your studies to give yourself some more personal time. You’ll be able to create a study schedule–and stick to it–if you’re realistic with yourself about how much time you’ll actually have for studying over the holidays.
Second, since you’ll have less time to study, plan out carefully what you’re going to study and when. A specific agenda for each study session will help guarantee that you use your productively. If you’re used to longer study sessions (two or three hours), but you’ll be studying for shorter periods during the holidays, limit your agenda to one key topic at a time. For instance, you might spend an hour reviewing just proportions, rather than two hours reviewing proportions, averages, and ratios. This is also a good time to give yourself short quizzes (5 to 10 questions at time); in an hour, you can complete a short quiz in 15 to 30 minutes, giving you an equal amount of time to use for review of the quiz. As always, though, be sure to balance your studying between Quant and Verbal. A good plan is alternating between the two every day, or depending on your individual strengths and weaknesses, spending two days on one, followed by one day on the other.
Finally, recognize that there are benefits to a mini-vacation from your studies. Taking time away from intense studying gives you time to digest the material. There’s a lot to learn for the GMAT, and all that material takes time to “settle in” to your brain. Slowing down for a little while allows you to master concepts and helps prevent burn out. So take advantage of the pleasures of the holidays: enjoy your time spent not studying, maximize the time you do spend studying, and rest assured that your brain will benefit from the holiday as well.
October 1, 2012
A few months ago I had a student in one of my GMAT classes tell me her study plan. She was very diligent and committed to the study process, and the plan was a very well thought out and detailed. Furthermore, she was executing the plan brilliantly. The problem was that her score was going nowhere. She wasn’t gaining any ground from her masterful execution. What was the problem?
After digging a bit deeper, one thing stood out. She was using all the tools: practice tests, online quizzes, workshops, workbooks etc. None of this seemed odd. In fact, it was all commendable. However, there was a fatal flaw in the way she was using these resources. She wanted to makes sure that she had the endurance to answer these questions on test day. Therefore, when she sat down to do quantitative problems, she would create a set of 37, do them all, and then review the answers. She would do 41 questions for the GMAT verbal section. This seems like a great idea, right? It’s very realistic. Wrong!!!
This is the same challenge that Toyota solved with lean processes and the Lean Startup movement is busy solving in the entrepreneurial world. Working in large batches seems reasonable and efficient. However, when our goal is learning and validation, it is counterproductive in a big way. Now, we could spend a lot of time diving deep into either lean manufacturing or lean startup methods, and that would be a lot of fun. However, let’s stay on point and look at how it works with GMAT studying.
To complete 37 questions will take you about 75 minutes. During this time you are busy answering the questions. This practice is good, but you aren’t adding new knowledge to the mix. You are just moving along the experience curve and getting faster at what you know. But what if what you know is wrong? In that case, you will continue to make the same mistakes all the way through, without the benefit of learning from early mistakes.
Now imagine that you take them in batches of 5 questions and then review the answers. In this case, if you are lacking some crucial piece of knowledge, you will learn that in the first batch. Even if you got a question right, you may learn a better way to approach it. You will then be able to apply that knowledge in subsequent sets and move on to higher level challenges. By working in small batches you will do this over and over again. In this way you can compound your rate of learning and move to higher and higher scores.
As a final note on this, I thought I’d share a recent success story. I had a student who was scoring around 650 on his practice tests right up to the week before his test. His goal was mid 700’s. He was using a large batch approach as well. After making the switch to small batch study, he spent a week compounding his learning. On test day he scored a 750! Try studying in small batches….
September 17, 2012
This task probably won’t be given to you directly in the question stem—more likely, this would be an intermediate step after translating a word problem or plugging in numbers for variables. But it’s certain you’ll see something like this at some point on some GMAT problem.
In real life, you might plug these straight into a calculator. Doing so would give us this:
Ugly, huh? A five-digit number divided by a three-digit number. But the result is a nice even 30. There must be a better way to get there if the division is so neat! The shortcut is to divide. Any time you have numbers over numbers, you should always cancel, cancel, cancel. Dividing first keeps your numbers small and your arithmetic simple. Check out what happens if we cancel first in this problem:
Easy as pie! 7 goes evenly into 21, 9 goes evenly into 45, and 11 goes evenly into 22. Reducing fractions and ratios to their simplest form before multiplying will save you mountains of work on test day.
September 15, 2012
The Wrentham Village Premium Outlets are a great place to stop for cheap brand-name clothes, and they’re a popular tourist destination for visitors to Massachusetts. Like all tourist/retail locations, they need to get people in the door. They’ve tried lot of things, but their latest gimmick has interesting implications for GMAT students. They’ve started stacking discounts.
Nearly every store in the mall has signs that say something like, “65% off, PLUS take an additional 20% off!” Moreover, a coupon book gives additional discounts—the particular store with that sign also offered 15% off purchases over a certain value.
To the unenlightened, this seems too good to be true. After all, 65% + 20% + 15% = 100%. Are we seriously to believe that the outlet store is giving away things for free?
Well, that might be a trap answer on the GMAT—and it’s a trap answer for the unwary consumer as well. But because we have been practicing GMAT quant, we know better. Even though the signs say “additional” and “plus,” we’re not really adding. 65% off means that the baseline price 35% of the retail value, and a further 20% off means we pay 80% of that discounted value. When translating from English to Math, the word “of” means “times.” So, when we take a percentage “of” a percent, we multiply; the results of the previous example are as follows:
(1 – 0.65)(1 – 0.2)(1 – 0.15) = (0.35)(0.8)(0.85) = 0.238
We end up with a 76.2% discount all told; that’s a pretty good deal, but hardly the 100% sale that some might have mistakenly expected!
When stacking percentage increases or decreases on the GMAT, you need to multiply—or, you can pick 100 and plug it into the equation. But however you solve, you cannot just add the numbers together; and you can quickly rule out any answer choice that is just a sum of the percents in the stem.
September 4, 2012
On the GMAT, there is only one correct answer to each question (How many caught the Highlander reference in the title? Be honest!).
I know, big surprise, right?
But that simple, obvious statement leads us to a powerful deduction. Some Problem Solving questions on the Quantitative section will have terms, variables, or unknowns that are unsolvable—they could take multiple values on the basis of the information in the stem. And we’re not talking Data Sufficiency here. “Not sufficient” isn’t a choice (Occasionally, “Cannot be determined” is a choice on problem solving questions. This answer is usually a trap, but you can use Data Sufficiency solving techniques to see if multiple answers are possible). So if the answer choices are numbers or proportions, and some term in the question stem is unsolvable, that undetermined x-factor can’t affect the outcome. Some ratio or mathematical step in the solution has to result in that variable “canceling out,” because otherwise the problem would have multiple correct solutions and therefore could not appear on the GMAT!
This is one of the ways that the Kaplan strategy of Picking Numbers works. Once you’ve identified an unknown that cancels out, you can plug in any value for that unknown, and be confident that your result is the right one. For instance, consider the following problem:
A runner runs downhill from point A to point B at 15 kilometers per hour, then runs uphill along the same path from point B to point A at 10 kilometers per hour. Assuming the time spent turning was negligible, what was the runner’s average speed during the round trip?
Here, the distance the runner travels is unknown and unknowable. But we’re asked for average speed. That’s a ratio; if we double the distance, we’ll double the time, and get the same speed. The answer to this question won’t change if he’s running one meter or one million!
Of course, neither one nor one million makes this problem particular easy to solve. Since every distance will give us the same average speed, we should pick the distance that takes the least effort when we plug it into the question. What’s the right distance to choose to make the arithmetic work? And, how can we use it to find the average speed?
August 20, 2012
Take a look at the picture with this blog. It’s an iconic optical illusion. Stare at it—what do you see? The picture is called the Great Wave off Kanagawa, painted by Katsushika Hokusai, a Japanese artist famed for his brilliant compositions. This drawing is of a wave, of course, but do you see the other wave, the reverse wave in the sky?
This image utilizes negative space. You take the whole frame, the great big rectangle, you block out that actual image—and what remains is, in its own right, an interesting picture.
To find the area of the shaded region, we need to subtract the area of the smaller inner circle from the large outer circle—the difference is the area of the ring.
But the concept extends beyond simple pictures and geometry questions. Probability problems sometimes operate on a similar principle, subtracting an easy-to-find probability of failure from 1, the total of all probabilities, the “whole frame,” as it were. Once you’ve subtracted all the failures, then whatever remains, the “negative space,” must be the chances of success!
The GMAT is a test of critical thinking. It tests your ability to find the most effective path to the solution. Sometimes, you’ll pick numbers, sometimes you’ll do the math directly, sometimes you’ll guess strategically. And sometimes, you’ll define the negative space around the answer, and solve that way. Today’s problem of the day is best solved by that principle. You can’t figure out how many arrangements follow the rule in the question stem. But you can find out how many arrangements don’t follow the rule, and subtract it from the total. Good luck!
Six children, Arya, Betsy, Chen, Daniel, Emily, and Franco, are to be seated in
a single row of six chairs. If Betsy cannot sit next to Emily, how many different
arrangements of the six children are possible?
Step 1: Analyze the Question
We have to arrange six children in six chairs, but two of
the children can’t sit together. We’re asked to calculate the
number of different arrangements of children.
Step 2: State the Task
We’ll calculate the number of possible arrangements of the
children. Then, we’ll subtract the number of ways Betsy can
sit next to Emily.
Step 3: Approach Strategically
The possible number of arrangements of six elements is
6! = 6 x 5 x 4 x 3 x 2 x 1= 720.
Now we’ll have to calculate the number of ways that would
violate the question stem by putting Betsy next to Emily.
If we number our seats from left to right, there are 5 ways
they can sit together if Betsy is on the left and Emily is on
Seats 1 & 2
Seats 2 & 3
Seats 3 & 4
Seats 4 & 5
Seats 5 & 6
And there are 5 more ways if Emily is on the left and Betsy
is on the right, for a total of 10. Now, for any one of those
10 ways, the four remaining children can be seated in 4!
ways: 4! = 4 x 3 x 2 x 1 = 24. So we need to subtract
24 x 10 = 240 ways that have Betsy and Emily sitting
together from our original total of 720: 720 – 240 = 480.
Answer choice (B).
August 16, 2012
Tackling some of the tougher GMAT probability questions efficiently relies on both steady practice and your ability to make two key decisions well. First, you will need to quickly and accurately assess the total number of possible outcomes (the denominator of your probability equation). Second, within a multitude of possible approaches, you will need to determine the most efficient route to calculate the number of desired outcomes (the numerator of your probability equation).
With the clock ticking away on your GMAT CAT, figuring out the total number of possibilities can be time-consuming and fraught with room for error. For instance, if a question asks about the probability of getting at least 2 heads on 5 coin tosses, you could sit there all day writing out possibilities:
So forth and so on. I know I got dizzy with the possibilities just writing those three out. There is a better and more efficient way. For every coin that you toss there are 2 possibilities. You can think of the total possibilities like a permutation problem.
__2__ __2__ __2__ __2__ __2_
1st 2nd 3rd 4th 5th
Just like in a GMAT permutations question when we are trying to determine the total number of codes possible or 4-digit numbers, we would multiply these individual probabilities together. Therefore, there are 2x2x2x2x2 = 2^5 = 32 total possibilities.
Next, we need to look at the numerator (desired outcomes). We want to find the all of the possibilities that have at least 2 heads, which means that we could have 2 heads, 3 heads, 4 heads, or 5 heads. To do so, we would need to count all of the different ways that these possibilities could be arranged. Again, we find ourselves in a situation that will be time-consuming and fraught with error. Instead of going down this path, remember that the sum of the probabilities of a complete set of mutually exclusive possible outcomes is 1. Thus, as is often the case on “at least” probability questions, we can look for those options that are restricted. Then we only have to count the options that have 1 or 0 heads.
There are only 6 of those, instead of the 26 possibilities the other way.
Finally, we can either subtract 6/32 from 1 in order to remove all of the restricted possibilities from 1 or we can subtract 6 from 32 and use the result as the desired possibilities. Either way, the answer is 26/32, which you can reduce down to 13/16.
Let’s look at another to make sure we have this down.
A test has 5 multiple-choice questions. Each question has 4 answer options (A,B,C,D). What is the probability that a student will choose “B” for at least four questions if she leaves no questions blank? Pause a moment and try it for yourself first.
Step 1: Total number of possibilities
There are 5 questions and each has 4 possibilities, so our total possibilities would be 4x4x4x4x4 = 4^5 = 1024
Step 2: Approach Desired Strategically
Here there are far more possibilities for 0, 1, 2, or 3 “B’s,” so let’s get a total for 4 or 5 “B’s”.
All B’s — B,B,B,B,B
Four B’s — A,B,B,B,B – B,A,B,B,B – B,B,A,B,B – B,B,B,A,B — B,B,B,B,A
C,B,B,B,B – B,C,B,B,B – B,B,C,B,B – B,B,B,C,B – B,B,B,B,C
D,B,B,B,B – B,D,B,B,B – B,B,D,B,B – B,B,B,D,B – B,B,B,B,D
3 x 5 = 15 because we can repeat the same pattern for each letter other than B
We can also calculate the total possibilities of 4 B’s by calculating the possibilities for each “no-B” position.
No-B,B, B, B, B = 3x1x1x1x1 = 3
B, No-B, B, B, B = 1x3x1x1x1 = 3
B, B, No-B, B, B = 1x1x3x1x1 = 3
B, B, B, No-B, B = 1x1x1x3x1 = 3
B, B, B, B, No-B = 1x1x1x1x3 =3
A total of 15 possibilities with 4 B’s in the mix
That gives a total of 16 different ways that a student can choose at least 4 B’s here.
16/1024 = 1/64 as our final probability.
Keep these two decisions in mind each time that you approach a tough probability question on the GMAT quantitive section. You don’t have to write out all of the possible outcomes in order to tackle these on test day!
The language of probability can take a while to learn, especially if you are unfamiliar or out of practice with it to start. Post your questions below, and we can help you get on track.
August 8, 2012
Translating word problems into algebra is a staple skill of GMAT test-takers, one that underlies countless problems in practice and on Test Day. But some challenging translations occur as part of probability and combinatorics problems. That’s because a pair of the most basic words in the English language, “And” and “Or,” suddenly become overburdened with mathematical significance.
“And” is the simpler of the two. When “And” represents independent choices—cases in which one option or arrangement has no impact on the other choice—just multiply the outcomes. For instance:
“The number of ways to purchase three board games and two video games” is an independent choice. The board games we pick have no impact on the video games we pick. So, to translate: [The number of ways to purchase three board games] × [the number of ways to select two video games]. Of course, we’d need the combination formula to find actual values—but we’d know what to do with those values once we got them.
“Or” is a little more complicated. It’s confusing even in conversation, after all—if I say that you can have cake or ice cream for dessert, can you have both if you want? When you CAN have both, you can treat the problem similarly to an overlapping sets problem. But in most cases on the GMAT, the “Or”s will be mutually exclusive—for instance, if you want to know the odds of drawing a heart or a diamond out of a deck of cards, there is no card that is both a heart or diamond.
A mutually exclusive OR can be translated as a “plus.” That’s all you have to do. So:
“The probability of drawing a heart or a diamond from a deck of cards,” which is the odds of one of two mutually exclusive events occurring, translates to: [The probability of drawing a heart] + [The probability of drawing a diamond].
Today’s problem of the day hinges on those same ideas. Read carefully—you’re solving for the odds of one of two outcomes (an OR), but each of those two outcomes is the specific result of two independent events (an AND). Be systematic in your translation, and I’m sure you’ll get the right result.
Post your answers below before you read the solution, and we can go over them…
Each person in Room A is a student, and 1/6 of the students in Room A are
seniors. Each person in Room B is a student, and 5/7 of the students in Room
B are seniors. If 1 student is chosen at random from Room A and 1 student is
chosen at random from Room B, what is the probability that exactly 1 of the
students chosen is a senior?
Step 1: Analyze the Question
This is a complex question, but it can be broken down into
simple steps. As with any probability question, we must first
consider all of the scenarios in which the desired outcome
can be true. In this question, there are two different ways
in which exactly one of two students chosen is a senior.
Either (i) a senior is chosen from Room A and a non-senior
is chosen from Room B or (ii) a non-senior is chosen from
Room A and a senior is chosen from Room B.
Step 2: State the Task
Determine the probabilities of the two scenarios above and
add them together.
Step 3: Approach Strategically
Let’s start with (i) and find the probability that a senior is
chosen from Room A and a nonsenior is chosen from Room B.
The probability that the student chosen from Room A is a
senior is 1/6 .
The probability that the student chosen from Room B is not
a senior is 1- 5/7=2/7
So the probability that the student chosen from Room A
is a senior and the student chosen from Room B is not a
senior is (1/6) x (2/7) = 2/42 .
Let’s not simplify this yet, because we can expect that the
probability we will find when working with (ii) will also
have a denominator of 42.
Now let’s work with (ii). Let’s find the probability that a
nonsenior is chosen from Room A and a senior is chosen
from Room B.
The probability that the student chosen from Room A is not
a senior is 1 – 1/6 = 5/6 .
The probability that the student chosen from Room B is a
senior is 5/7 .
So the probability that the student chosen from Room A
is a not a senior and the student chosen from Room B is a
senior is (5/6) x (5/7) = 25/42 .
Now we sum the total desired outcomes. The probability
that exactly one of the students chosen is a senior
is (2/42) + (25/42) = 27/42 = 9/14 .
(C) is correct.