June 6, 2012
One of our wonderful student guest bloggers, Candice Batts, recently wrote about her on-going fight to conquer GMAT word problems. Translating word problems into quantifiable, workable information is her GMAT Achilles’ heel. Candice called me out at the tail end of that post and my response to her is something everyone wrestling with the same GMAT Problem Solving demon should hear.
First of all, the strategies Candice lists in the last paragraph of her post are spot on. Creating and taking quizzes in her online account (who doesn’t love the Quiz Banks tool?!?) and then reviewing BOTH the right AND wrong answers is critical to increasing your problem solving (PS) question skill level. Also, making flashcards for the “Big Two” strategies (Picking Numbers and Backsolving) is a good idea to help train up on what type of PS question set-ups are good candidates for these strategic approaches. Further, it sounds like Candice is exhibiting an even more effective prep behavior: looking for opportunities to employ these strategies. After all, the more you look for them, the more you’ll see them.
Keep this in mind for GMAT word problems, as well…
Your approach to GMAT word problems is limited to just five options:
1. Do the math.
2. Pick some numbers to use in place of variables.
3. Use the numbers in the answer choices to solve.
4. Critically think our way around the math.
5. Guess strategically.
That’s it. You will have to use each one of these on test day, but the infrequency of which you’ll use #1 compared to the frequency of which you’ll use #4 might surprise you. Approaches #2 and #3 will get you through about half of the problems you’ll see on Test Day. Also, never be afraid of #5. No one knows how to do every single question on the GMAT. At some point during the test—at several points, actually—you are going to have to guess. The trick to guessing is to not do it blindly. It is very likely that after a modest amount of critical thinking you’ll be able to whittle down the five answer choices to four, three, or even two. And 50% is much better odds than 20%.
In terms of preventing multiple re-reads of a word problem, I suggest unpacking the information from the question onto your scratch paper as you go. Test takers will often read a problem in its entirety, have a look at the answer choices, and do some thinking about it before ever doing any work on it. Always put your pen to work. Don’t make the GMAT any harder than it already is by trying to do everything in your head. Plus, working through the information on paper may very well lead to insights you would have missed had you only been using gray matter.
Often times, students are left stunned by a problem, like a deer in headlights. As I mention in that linked post, I recommend Step 1 drills (Step 1, by the way, is problem analysis). Step 1 drills will increase your command, deepen your insight, and decrease your overall time spent on a word problem. Here’s how you do it:
Compile a bunch of challenging questions covering all sorts of content and concepts, then focus only on taking control of the question. As you read and record info, ask the problem some questions: What is happening? What information are you giving me? What information are you missing? What do you want me to figure out? What do your answer choices look like? What type of math are you testing (e.g., algebra, arithmetic, geometry)? What math concept will I need to use to solve you (e.g., linear equations, percents, volume)?
Talk out loud. Jot down notes. And then, move on to the next problem as quickly as you can. Go back and solve these problems later. Here, I don’t care about you turning the crank; I just want you to get the machine loaded. The focus of this exercise is to get you moving, get you working, and avoid standing still. Learn how to take the first step on a GMAT word problem no matter what it says quickly and confidently. After all, you can’t find the gold in the mud without getting dirty, right?
What is your Achilles’ Heel? Post in the comments below, and we’ll talk about how you can overcome it on test day.
May 20, 2012
Greetings, I would like to announce that I have officially figured out that my biggest challenge on the GMAT is translating word problems into workable equations that allow me to get to the right answer quickly.
When I first started studying for the GMAT, the words from these long and confusing paragraphs would float in one ear and out the other. I would desperately try to hold on to each word, analyzing each one looking for a clue that would help me. Then I would read one question over and over and over like the answer was going to suddenly appear and say ‘I was hear all along!!’ That’s the habit that Kaplan instructors train you NOT to do first. With this strategy, no wonder the words were just swimming around in my head with no meaning!
As I said, my Kaplan instructor showed me how to strategically read questions once and sort through the information provided to isolate exactly what is was needed to answer the question. In particular, he showed me how to identifying keywords in order to extract data and rephrase the facts in my own words. Additionally, he showed me how to pinpoint patterns that appear throughout each GMAT question type. This helped me a great deal on the verbal section, but I realized I was still re-reading quantitative questions. I felt stuck. From my diagnostics and CATs, I could see it was really affecting my score.
It’s not that I can’t extract the right information, but once I have the information I don’t know what to do with it. Should I make an equation? Am I dealing with ratios here or am I just adding & subtracting? Maybe I should draw a graph? For some reason my brain is having a hard time remembering and utilizing the strategies I learned during my course. I am taking quizzes and reviewing the wrong and right answers, and it has been helping. I also made notecards to memorize the strategies and use them as I review the quizzes and tests. What I notice is that I am getting better at translating word problems into workable equations, but I am still re-reading questions. This affects my time per question. Currently, I plan to concentrate a bulk of my preparation on word problems trying to skim down my overall time per question.
What other strategies should I employ? Is there a better approach? Hey Lucas and Brett, any advice?
Do any of you struggle with this problem? What’s your Achilles’ heel on the GMAT?
For about a year, I always used the same method to solve the following GMAT problem:
How many liters of water must be evaporated from 50 liters of a 3 percent sugar solution to get a 5 percent sugar solution?
“This is simple percentages,” I would say. “Just start by taking 3% of 50 liters, which is 3 over 100 times 50, which comes out to 1.5 liters sugar…”
But one day, teaching this same quantitative problem, a student’s hand shot straight up. “Yes, James?” I said. (That wasn’t his real name, by the way, but it will do.)
“Eli, who cares about the sugar?”
I paused. “Well, the sugar will help us figure out the solution.”
“But you don’t need it!” James explained. “I’ve been a chemical engineer for years, so I do this problem all the time. The sugar is a constant. The amount of sugar doesn’t change, and that amount is always equal to the concentration times the volume. So just do CV = CV; 50 times 3 is equal to the final volume times 5!”
I paused, impressed, and amazed—and have taught his timesaving shortcut ever since.
However, there is a bigger lesson here than simple mixture problems. I had approached that problem uncritically. I “knew” how to find the right answer, so I never gave it a second thought. I spent far more time prepping the combination and probability problems given their complexities and hidden challenges. As a result of my complacency, I made extra work for myself.
In other words, the takeaway is this: don’t focus exclusively on your mistakes. Review your correct answers as carefully as your errors. There might be a time-saving shortcut that you missed the first time through. Or better yet, you might have used such a technique that you can notice, reconstruct, and repeat! It’s very tempting to spend all of your time studying those red X’s on your quizzes and CATs. But avoid that trap; looking at your right answers as well as your errors will help you stay positive, and more importantly, will give you many opportunities to raise your score.
May 12, 2012
Everyone studying for the GMAT wants to identify the skills that will lead directly to the greatest point increases. While this can be difficult to do, given the adaptive nature of the exam, some skills definitely do come into play more often than others.
One of the most important skills to master for the GMAT is prime factorization. Finding prime factors can be useful on many different types of questions. On test day, if you are stuck on a question and unsure of how to solve, remember the big number rule. The big number rule is simply this: if you see a big number, one that is so large it is unreasonable to work with, find its prime factors. Once you have those factors, you should be able to simplify.
Every positive integer that is not prime, with the exception of 1, can be broken down into a series of prime numbers multiplied together. Additionally, each series of primes is unique and will only result in a single integer.
For example, let’s say you want to find the prime factors of 20. Start by identifying any two numbers that multiply to equal 20. We will choose 4 x 5. 5 is prime, so we cannot break it down further, but 4 is not, so we repeat the process. If you try to think of two numbers that multiply to equal 4, you will only come up with 2 x 2, which are both prime. Thus, the prime factorization of 20 is 2 x 2 x 5.
Now let’s say that instead of choosing 4 x 5, you choose 2 x 10. 2 is prime, so we leave it alone, but we need to break down 10. 10 equals 2 x 5, thus we again find the prime factorization of 20 is 2 x 2 x 5. Notice that no matter what numbers we choose, as long as the math is correct, we reach the prime factors.
Try the question below and see if you can figure out how prime factorization can help you solve.
Is q a multiple of 48?
(1) q is a multiple of 6.
(2) q is a multiple of 8
The stem asks whether q is a multiple of 48. It’s a yes/no question so you should see if you can answer the question with both a yes and a no. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, so we need four 2’s and one 3 to know that q is a multiple of 48.
Statement 1: This tells you that q is a multiple of 6. The prime factorization of 6 is 2 x 3, so we know q has one 2 and one 3, but we do not know if it has the additional three 2’s we need, so this statement is not sufficient.
Statement 2: If q is a multiple of 8, we know that it must include the prime factors 2 x 2 x 2, which gives us three 2’s. Since we do not know if it will have the additional 2 and one 3, it is insufficient.
Statements 1 & 2: If q is a multiple of 6 and 8, it must have one 2 and one 3 in its prime factors, so it will divide by 6, and three 3’s in its prime factors, so it will divide by 8. This means that q must have three 2’s and one 3. Note that one of those 2’s is being used to divide by 6 and by 8, since we need to be able to divide by both, but not at the same time. However, to reach 48 the prime factors would need to include one additional 2. Since we do not know if q would include a fourth 2 or not, the statements are still not sufficient. Therefore, the answer is (E).
May 2, 2012
Not surprisingly, most GMAT test takers have a background in the business world. As such, many readers have worked on a committee formed from a larger group of employees. Every time a committee is formed in this fashion, you are, in fact, doing a GMAT problem. More specifically, you are attempting one of the most dreaded question types on the GMAT quantitative section – combinations.
While these questions can be tough, by thinking about the real life experience of forming a committee, you can more easily understand exactly what a combinations question is asking you to do. Let’s say that your business has ten employees and needs to create a committee of four people. If you want to determine how many different possible committees you could create, you would use the combinations formula, n!/[k!(n - k)!], where n is the number of people with which you start (in this case 10) and k is the number of people you want in your group (in this case 4).
Now, let’s take a step back and see how we know that this is the appropriate formula to use. A combinations question will ask you how many different groups you can create. For example, if our people are A, B, C, D, E, F, G, H, I, and J, one group we could make is A, B, C, and D and another group we could create is B, C, H, and I. However, in combinations problems, order will not matter; that is, changing the order of the entities in the group will not make it a different outcome. In our example, A, B, C, and D is the same group as D, C, B, and A. Thus, we are grouping, but not arranging.
On GMAT test day, when this occurs, you know that you are dealing with a combinations question and should apply the formula outlined above.
Corporation Z has 2 locations, one in New York and one in Los Angeles. The New York location has 6 executives and the Los Angeles location has 4 executives. If a meeting of 4 executives is to take place, and exactly 2 executives from New York must attend, how many different groups of 4 executives are possible?
The question asks for the number of combinations of 4 executives, 2 from New York and 2 from Los Angeles, which can be formed starting with a larger group in each city.
Use the combination formula to find the number of pairs of executives from each city.
New York has 6 executives.
Los Angeles has 4 executives.
Then multiply the two results.
15 × 6 = 90 – answer choice (A)
April 21, 2012
One of the big GMAT skills that is often overlooked by students is translation. Any time you decide approach a word problem using algebra, you will need to translate the English in the question stem into an algebraic equation. While this seems as if it would usually be fairly straightforward, the GMAT will often find ways to make it more difficult. A translation error will often lead to a trap answer, so it is essential that you learn how to translate difficult statements before test day.
To understand why translation can be more difficult than it seems, think about translating a foreign language. If you only need to translate one word, you can usually just find the equivalent word in English. Similarly, if a GMAT problem uses the phrase “more than” you know that it must translate to addition.
However, when you try to translate an entire sentence from a foreign language to English, the process becomes much more complicated. You do not need to only determine the matching English words, but also need to consider the structure and meaning of the sentence as whole, since the foreign language might construct sentences differently than English. Along the same lines, when you translate a word problem to math, you must consider the relationships between the different parts of the sentence and check to see if any words are used in unusual ways. You might get lucky and have a problem that uses the word “equals”, but you are just as likely to see the words “costs”, “weighs”, and “sells” used. Identifying when these other words translate to “equals” will be key to writing a correct equation.c
Below, you will find a word problem that can be solved algebraically. Be careful on your translation and then check out the solution to see how you did.
Jacob is now 12 years younger than Michael. If 9 years from now Michael will be twice as old as Jacob, how old will Jacob be in 4 years?
Step 1: Analyze the Question
We must first translate the question into algebraic equations, then apply the techniques of combination or substitution to solve for Jacob’s age.
Step 2: State the Task
Translate the question stem, then solve for Jacob’s age using substitution.
Step 3: Approach Strategically
Equation for Michael’s and Jacob’s current ages: M = J + 12.
Equation for Michael’s and Jacob’s ages 9 years from now: M + 9 = 2(J + 9)
We now have two distinct equations with two unknowns that we can solve. We can substitute the first equation into the second for M, yielding the equation:
( J + 12) + 9 = 2 (J + 9)
J + 12 + 9 = 2J + 18
J + 21 = 2J + 18
Thus, J = 3. So if Jacob is 3 years old now, then in 4 years, he will be 7 years old. Don’t forget to add these 4 years at the end.
Step 4: Confirm Your Answer
We can plug in J = 3 into the original question stem to confirm our answer. Be careful with your calculations, since (A) is a trap for those who don’t add 4 years to his current age.
April 19, 2012
Most students learn that absolute value is the positive version of a number. Thus, the absolute value of 7 is 7 and the absolute value of -7 is also 7. While these absolute values are correct, many GMAT problems will be more straightforward if you learn the true definition of absolute value, which is the distance a number is from zero on a number line. Thus, the absolute values of 7 and -7 are 7 because both numbers are 7 away from zero on a number line.
To understand how absolute value works, imagine you live in a house right in the middle of a block. The street has 5 houses to the left of your house and 5 houses to the right of your house. Whether you walk two houses to the left or two houses to the right you will be 2 houses away from your home. Now, instead of picturing a block, consider the same concept as a number line in which your house represents zero. If you walk 2 houses to the left, you are at -2 and if you walk 2 houses to the right, you are at 2. Normally, we use positive and negative numbers to distinguish these as different spots. However, if we are concerned with the distance from us rather than the location relative to us, we no longer need to distinguish between positive and negative numbers. Since that distance cannot be negative – it is either 2 to the right or 2 to the left – we must define the distance in positive terms. Thus, absolute value will always be positive.
Keeping the concept of absolute value in mind, try the problem below and then check out the solution to see how you did.
Which of the following could be the value of x, if |4x – 2| = 10?
Analyze: We see an equation with an absolute value sign in the Q-stem. Also, what important GMAT wording do we see? “could be the value”—that means that there are at least two possible solutions for x in the equation and the absolute value sign means the expression could equal 10 or –10.
Task: Pick the answer that makes the equation true. There may be another solution, too, but only one of the answer choices will work.
Approach Strategically: To solve a normal algebraic equation, we have to do the same thing to both sides in order to isolate the variable.
We’ll do the same here, but we’ll set up two equations because of the 2 possible answers that the expression can equal, like so: 4x – 2 =10 and 4x – 2 = –10.
Solving for 4x – 2 = 10 gives us x = 3.
That’s not one of our answer choices.
Solving for 4x – 2 = – 10 gives us x = -2 which is answer choice B.
You could also backsolve here but it’s important to know the rules of absolute value on Test Day.
April 4, 2012
The only thing you need to know about sequences is that a sequence is an ordered series of numbers. Really, that’s it. Many sequences are defined by specific mathematical formulae or properties, and most sequences of note continue on infinitely, but neither of those are necessary. All you need to have a sequence is an ordered list.
Of course, that raises the question of why they are tested at all. One might assume that the simplicity of sequences means that they should be trivial and a waste of the testmaker’s time. Conversely, it could seem that since simple definition covers an enormous range of possibilities, it would be impossible to expect students to understand them all. But in fact, sequences test a very specific skill: your ability to decode instructions.
If this sounds familiar, it might be because you read my earlier blog post on symbolism. Sequences and symbolism are closely linked by one principle: any sequence or symbol you must solve for will be presented in the stem with the question.
Unlike symbolism, though, sequences have a predetermined method of defining their properties: subscript. A sequence will be given a letter, like X or S, and the first term will be listed as X1 or S1. The second term would have a 2 in the subscript, and so on. But more importantly, the sequence will be defined in terms of the nth term.
Whenever you see Sn , you should read that as “the nth term,” or perhaps “some term” or “any term.” And since n denotes some a position on a list, you can probably deduce that the (n – 1)th position is just one spot earlier on the list! So when you see Sn-1 you can just say to yourself, “the term before the nth term.” So, for example:
- Each term Sn after S1 is equal to 2(Sn-1) + 3
In plain English becomes:
- Every term after the first is equal to three plus twice the previous term.
So if the first term were 2, then the second term would be 2(2) + 3 = 7 and the third term (S3 in our notation) would be 2(7) + 3 = 17.
Try one on your own—this is an advanced problem, so don’t worry if it’s a little tough, but start by translating the rules of the sequence into plain English!
In the infinite series S1, each term Sn after S2 is equal to the sum of the two terms Sn-1 and Sn-2. If S1 = 4, what is the value of S2 ?
(1) S3 = 7
(2) S4 = 10
Step 1: Analyze the Question Stem
This is a Value question. We must determine what information is needed to solve directly for the term S2. The question stem says that if Sn is the nth term of the sequence, that for n > 2, we have Sn = Sn-1 + Sn+2. That is, after the second term in the sequence, the next term is equal to the sum of the two previous terms. For example, the third term is equal the sum of the first and second terms.
We also know that S1 is 4. Because S3 = S2 + S1, S3 = S2 + 4. So, we have one linear equation with two variables, S3 and S2. To find the value of S2, we need more information that will lead to the single possible value for S2.
We could find the value of S2 if we are given the value of S3. We could also find the value of S2 if we were given another linear equation with the variables S3 and S2 that is different from the first equation. Let’s look at the statements.
Step 2: Evaluate the Statements
Statement(1) says that S3 = 7. Because S3 = S2 + 4, we have the equation 7 = S2 + 4. From this equation, we can find the single possible value of S2. Statement (1) is sufficient. We can eliminate choices (B), (C), and (E).
Statement (2) says that S4 =10. Because S4 = S3 + S2, we have the equation S3 + S2 = 10. This is another linear equation containing the terms S2 and S3. Because the two equations are distinct, we have enough information to determine the values for both S3 and S2. Statement (2) is sufficient and choice (D) is correct.
March 14, 2012
Despite the name, irrational numbers aren’t crazy. You might think they are, and you wouldn’t be alone. The followers of Pythagoras (yes, the right triangle guy!) believed that irrational numbers were heretical; supposedly, they threw the first person to prove irrationals existed into the ocean! But in fact, the word ‘irrational’ refers to the mathematical concept of the ‘ratio.’ Irrational numbers are numbers that cannot be expressed at the ratio of two integers, a/b.
There are an infinite number of irrational numbers, though far fewer are of mathematical note. If you’ve ever studied statistics or calculus, you’ve probably run across the natural number e, and ancient architects had a centuries-long love affair with φ, the golden ratio. But on the GMAT, there are only two types of irrational numbers that will be relevant to you: π and radicals.
Both of these irrationals will appear most often in geometry questions, and both of them have the same basic rule: you don’t need an exact numerical value. Because they can’t be reduced to a ratio (assuming, in the case of a radical, that the number underneath isn’t a perfect square), their values can’t be effectively determined with pen and paper. So generally, they will simply appear as-is in the answer choices. An answer choice to a circle problem will take the form of 3π, not 9.42.
As a result, we can treat these irrational terms just like we would a variable like an x or y. So on complex algebraic statements that feature irrational numbers, we can simplify by adding coefficients to irrationals and combining like terms. In other words, if we are given:
we can express the first term with an algebraic coefficient:
Then, we can combine like terms, getting:
Not only is this helpful and efficient, it’s necessary; GMAT answer choices will almost invariably be written in the simplest terms possible.
Similarly, Irrationals can (and sometimes must) be factored out as well. If the GMAT gives us a multi-irrational tangle like this one:
then anything we can do to make it more manageable will help us find the correct answer choice. In this case, we are adding two terms that are both multiplied by π. That means we can pull the π out. The result is a little simpler:
This could help us find the answer to a tough problem with circles and triangle. So get comfortable manipulating these indivisible terms, and you’re one step closer to acing the GMAT quantitative section.
A right circular cylinder has a height of 20 and a radius of 5. A rectangular solid with a height of 15, and a square base is placed in the cylinder such that each of the corners of the solid is tangent to the cylinder wall. Liquid is then poured into the cylinder such that it reaches the rim. What is the volume of the liquid?
(A) 500(π – 3)
(B) 500(π – 2.5)
(C) 500(π -2)
(D) 500(π – 1.5)
(E) 500(π- 1)
Step 1: Analyze the Question
For any Geometry question without a figure, our first step must always be to draw a quick sketch. This is particularly important for solids and complex word problems such as this question. The challenging element of this question is that a square solid is inscribed inside the right circular cylinder. Our sketch shows this:
Notice that we drew the radius of the circle to end at the corner where the circle and square meet. When working with multiple shapes, look for points or sides where the two shapes overlap, since you can use the common side or point to connect the equations of the shapes.
Step 2: State the Task
We must determine the volume of the liquid that has filled the cylinder. The volume of the liquid will be equal to the volume of the cylinder minus the volume of the rectangular solid. So we must calculate the volume of the cylinder and the volume of the rectangular solid.
Step 3: Approach Strategically
The volume of the cylinder is
To calculate the volume of the rectangular solid, we’ll need to determine the lengths of the sides of the square. We can draw a right triangle to help.
Recognizing a 45-45-90 triangle (always true of a bisected square), we know that the ratio of the side lengths of the triangle and can solve for the value of x because we know the hypotenuse is 10.
We can now calculate the volume of the liquid: Volume of Liquid = Volume of the Cylinder – Volume of the Rectangular Solid
Step 4: Confirm Your Answer
Reread the question, making sure that you did not misread any important information.