## GMAT Problem Solving: There Can Be Only One

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?

11.5

12

12.5

13

13.5

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?

## GMAT Geometry: Central Angles

In poetry, a rose is a rose is a rose. On GMAT problems with central angle “slices” in circles, a fraction is a fraction is a fraction.

This may seem like common sense. Cut a pizza into six slices. If you cut it evenly, each slice now has one-sixth the cheese, one sixth the crust, and an angle of one sixth the way around a circle—that is, 60 degrees.  However, though this may seem obvious, it’s actually a very useful technique for resolving certain geometry problems.

Consider the following Data Sufficiency question:

In a radius 6 circle, two points A and B are connected to the center, point O. What is angle AOB?

1)    The length of the minor arc defined by sector O is 1.5π

2)    The area of the sector defined by angle AOB is 4.5π

It looks all GMAT-like and formal, but if you actually think about it…, it’s just the pizza I described above. We’re taking a slice, and want to know what’s the angle of that slice. The “cheese” of the pizza is the area of a radius 6 circle,  which is the radius squared times pi, or 36π. The “crust” is it’s circumference, which in this case would be the diameter times pi, 12π.

Statement 1) tells us that the length of the arc/crust is 1.5π, which is 1.5/12 = 1/8 of the circumference. And an eighth is an eighth is an eighth. Our slice, which goes one eighth of the way around the outside of the circle, and it goes an eighth of the way around the inside of our circle as well—its central angle is 1/8 * 360 = 45 degrees. Sufficient!

And statement 2) says that the area of our slice/sector is 4.5π, or 4.5/36 = 1/8 the total area of the whole pie. Once again, an eighth is an eighth is an eighth, so by the exact same reasoning we get 45 degrees. Sufficient!

This is a pretty basic rule, but it’s widely applicable to many circle problems. It can be especially useful with central triangles, which have central angles by definition. Take a look at today’s question of the day to test your skills, and when you get it right, treat yourself to a slice of pepperoni as a reward!

Question:

What is the area of the circle above with center O?
(1) The area of D AOC is 18.
(2) The length of arc ABC is 3π.

Step 1: Analyze the Question Stem

This is a Value question. For any Circle question, we
only need one defining parameter of a circle (area,
circumference, diameter, or radius) in order to calculate
any of the other parameters. Also, all radii of the same
circle will have the same length. So AO = CO. That makes
the triangle an isosceles right triangle (or 45-45-90) for
which we know the ratio of the sides. Not only would the
circle’s circumference, diameter, or radius be sufficient,
but information that gave us any side length of triangle
AOC would also be sufficient, as it would give us the length

Step 2: Evaluate the Statements

Statement (1): We are given the area, and we already know
that the base and the height are equal. So if we call the
radius of the circle r, then the area of the triangle is equal to
1/2 (Base)(Height) = 1/2 (r)(r) = 1/2 r ^2 = 18
Remember that we are not asked to calculate the actual
value of r. Because we have set up an equation with
one variable, and we know in this case that r can only
be positive since it is part of a geometry figure, we have
enough information to determine the area of the circle (the
solution would have been r ^2 = 36, r = 6).
Therefore, Statement (1) is sufficient. We can eliminate
choices (B), (C), and (E).

Statement (2): Because O is the center of the circle and
angle AOC measures 90 degrees, we know that the length
of arc ABC is one-fourth of the circumference. Because this
statement allows us to solve for the circumference, it is
sufficient.

(D) is correct.

## GMAT Quant: Not Revolutionary – Just Radical

One of the most common mistakes that I see students make when practicing for the GMAT is the misapplication of the rules that govern square roots.  When approaching a question that involves radicals, it is vital that you know not only the rules that you must follow, but also the operations that are commonly believed to be rules, but are not.  On test day, the wrong answer choices will almost always be derived from the latter.

If you need to manipulate a square root, you must remember two key rules.  First, that √(ab) = √a x √b and, second, that the √(a/b) = √a/√b.  For example, if you need to simplify √20, you can rewrite it as √(4×5).  When choosing which factors to use, always look for perfect squares.  Since 4×5 includes the perfect square 4, it is better than 2×10, which does not include a perfect square.  Next, you can break this down to √4√5, which in turn becomes 2√5.  You can follow the same steps in division problems.

However, you must also remember that √(a+b) ≠ √a + √b and that √(a-b) ≠ √a – √b.  To understand why you cannot break apart a radical across an addition or subtraction sign, consider the expression √(4+9).  We could add the four and nine together, giving us √13, but since 13 is not a perfect square and is also prime, we are unable to break the expression down any further.  If we mistakenly believed we could break apart the radical around the addition sign, we would end up with √4 + √9 = 2 + 3 = 5.  Since the square root of 13 does not equal 5, we know that this operation is not allowed.  If we tried a similar exercise with subtraction, we would find a similar outcome.

Keep in mind when you can and cannot break apart a radical as you give the problem below a try:

Problem:

Solution:

Analyze

What do we see in this Q-stem? A complex set of operations under a radical: multiplication, division, and addition.

We’re just simplifying the original expression.

Approach Strategically

Take this piece-by-piece. First, separate the numerator and denominator under their own radicals:

Now, the denominator is easily simplified. It’s a perfect square:

Next we can start simplifying the numerator. We can factor the 9s out of the expression to get another perfect square:

The square root of 9 is 3, so we can put that outside the radical:

It’s now clear that (C) is incorrect. To get one step closer, factor 20 into 4 x 5. Since 4 is a perfect square, we can get a little more out from under the radical:

Are you all clear on this one?  Use the comments to let me know where you’re getting stuck, and I can help.

## GMAT Data Sufficiency: Say Yes, Saying No

The trickiest question type in the quantitative section of the GMAT for most students is yes/no data sufficiency questions.  When approaching these problems, it is imperative that you keep in mind the purpose of data sufficiency.

Let’s start with a review of data sufficiency.  On these questions, your goal is not to find the answer.  Rather, it is to determine if you have enough information to find the answer, regardless of what the answer is.  On value questions, this is fairly straightforward.  If you are asked for the value of x and you know it is 4, that’s sufficient, but if it could be 4 or 6, that’s not sufficient.  In the former case we could narrow down the possibilities to one answer.  In the latter, we could not.

Now let’s see how this same concept applies to yes/no questions.  If a question asks us if x is positive and we know the answer is “yes,” that’s sufficient, because we only have one possible answer.  Likewise, if we know that the answer is “no”, that’s also sufficient, because we still have only one possible answer.  Since this is data sufficiency, it does not matter if the answer is “yes” or “no”, it only matters that we have one possible answer, just as was the case with a value question.  If you must answer the question “sometimes yes, sometimes no”, that will be insufficient, because you have two possible solutions.

Thus, when you encounter a yes/no data sufficiency question on test day, see if you can answer it with an “always yes” or “always no.”  If so, remind yourself that, since you are able to give a definitive answer to the question, the statement is sufficient.

Problem:

Is the sum of x , y , and z equal to 3?

(1) xyz = 1

(2) x , y , and z are each greater than zero.

Solution:

The stem does not tell you anything, so move on to the statements.

Statement 1: Don’t assume that x, y, and z are positive integers! Pick numbers:

If x = y = z = 1, then x × y × z = 1 and x + y + z = 3.

If x = 1 and y = z = – 1, then x × y × z = 1 but x + y + z = – 1.

Statement 1 gives a “sometimes yes, sometimes no” answer.  Statement 1 is insufficient.

Statement 2: x, y, and z can be any positive numbers.

This yields a “sometimes yes, sometimes no” answer.  Statement 2 is insufficient.

Statements 1 & 2: Pick numbers.

If x = y = z = 1, then x × y × z = 1 and x + y + z = 3.

If x = 1, , and z = 2, then x × y × z = 1, but .

Combined, we still get an insufficient answer, “sometimes yes, sometimes no.”

Did you have trouble with the above question? Post your questions in the comments area so we can help you work through it.

## When Time is not on your Side during the GMAT

It is essential to remember that the GMAT is about more than just doing the math correctly.  The GMAT is really a test of your critical thinking abilities – that is, your ability to not just do the work, but to figure out exactly what that work is.

To that end, the GMAT will often present you with problems that would take too long to solve if you do all of the math that is possible.  I have had countless students approach me to tell me that if they were not timed, they could solve all of the math questions.  However, they just cannot find a way to complete the problems in time.  Additionally, all the extra math provides opportunities for careless errors.  I always tell these students the same thing – do only the math you absolutely need to in order to reach the correct answer.

This is especially true on data sufficiency questions.  Data sufficiency questions are all about your ability to answer the question, rather than determine exactly what that answer is.  Thus, your focus should be on deciding if the information given could be used to solve, rather than actually solving.  On many data sufficiency problems, though certainly not all of them, you will not need to do any math at all to arrive at the answer.

Below, you will find one such question.  Give it a shot and see if you can reach the correct result without doing any math at all.

Problem:

Team X won 40 basketball games.  What percent of its basketball games did team X win?

(1)  Team X played the same number of basketball games as Team Y.

(2)  Team Y won 45 games, representing 25 percent of the basketball games it played.

Solution:

This problem illustrates how important it is to approach data sufficiency questions strategically.  By following a set method, we can find our answer without doing much, if any, math.

First, we want to determine what information we would need to find out in order to answer our question.  By identifying this, we can check our statements for sufficiency more quickly.  Since we are asked what percent of games Team X won and are told that Team X won 40 games, we know that we need to find out how many games Team X played in order to answer the question.  Now when we check our statements, all we need to do is determine whether we know how many games Team X played – any additional math is unnecessary and is not a good use of our time.

Statement 1 tells us that Teams X and Y played the same number of games, but not the actual number of games.  Thus, statement 1 is not sufficient.

Statement 2 tells us that Team Y won 45 games, which represents 25% of the games Team Y played.  From this, we could calculate the number of games Team Y played, but we do not know anything about Team X.  Therefore, statement 2 is not sufficient.

Together, though, we know how many games Team Y played from statement 2 and that Team X played the same number of games from statement 1.  Since we now could calculate the number of games Team X played, the statements are sufficient together.

Notice that we did not have to do any math to reach the solution.  On GMAT math problems in general, and especially on data sufficiency questions, you want to do as little math as possible.  Math takes time and opens up opportunities for errors, so we only want to do math that is essential to arriving at the answer.

I’ve got a new crop of eager test-preppers just starting out on their road to GMAT success.  Often, my classes will contain one or two folks with at least some prior GMAT experience, but this group is all green and it’s gonna be fun.  However, before the fun can really get rolling I have to spend time drawing blood.

Many of the battles I must wage (and win) in the GMAT classroom are focused on preconceptions.  Whether a student has GMAT history or is coming in blind, all will inevitably have several strongly held beliefs about the test and everything that comes along with it, including their ability to crack it.  Sometimes, these beliefs will be their most significant road block on the way to hitting their target.

After last night’s class, a student stayed behind for some clarification on Data Sufficiency questions.  This student—let’s call her Jane—believes that she does not know/remember how to do any math whatsoever.  I have worked with many students that hold this belief, and I am always right about how ludicrous it is.  Jane may well not remember much; she may not have had to use any of this kind of stuff since she was a sophomore in high school.  The last time Jane was messing around with quadratic equations might not have ended so well.  But, regardless of past experience, regardless of how rusty and dusty this content knowledge is, Jane can absolutely learn (or re-learn) what she needs to in order to become successful in her bid for b-school acceptance.

Jane and I covered the Data Sufficiency concepts that were tripping her up and she is now ready to move forward on those little buggers.  But, I couldn’t let the conversation end there.  Unless she gets over her preconception about her perceived lack of GMAT potential and ability, moving forward will be much harder than it could be.  In addition, she simply won’t get as far.

So, we talked.  We talked about her education, work, past, future, etc.  We talked about what her application package is going to look like and where she wants to submit it.  We talked about ways she can get up to speed on some of the content so she can really push on the core competencies, methods, and strategies she needs to learn in order to excel on the GMAT.

By the end of the conversation, she felt better—a little more confident, a little more capable.  I am not going to let this go, however.  Jane is not where she needs to be.  But, she did end by saying something that was absolute music to my ears:

“I know that my biggest problem is my attitude.  I know that.  I’m gonna work on that.”

## Kaplan GMAT Sample Problem: Data Sufficiency Averages

GMAT Data Sufficiency questions can take simple concepts like averages and have test-takers pausing or falling into traps because of the way they are worded, and the fact that you have to keep in mind what your goal is with Data Sufficiency—to find out whether or not you have sufficient information to answer the question!

Problem:

Each of the 8 numbers s, t, u, v, w, x, y and z is positive.  Is the average (arithmetic mean) of s, t, u, v, w, x, y and z greater than 46?

(1) The average (arithmetic mean) of s, t, u, v and w is greater than 74.

(2) The average of x, y and z is greater than 120.

Solution:

Before evaluating the statements, you should reword the question.  We are asked if the average of a list of numbers is greater than 46.  Since average is equal to the sum of the terms divided by the number of terms, we can write this question as: is sum/8 > 46?  This can be simplified to: is sum > 46(8) or is sum > 368?

Statement 1 tells us that the average of the first five numbers in our set is greater than 74.  At first, this may seem insufficient, as it tells us nothing about x, y and z.  However, if five of our numbers have an average greater than 74, it means that those numbers must sum to a result greater than 74 x 5, which equals 370.  If the sum of five numbers is greater than 370 and the other three numbers must all be positive, the overall sum must still be greater than 370.  If the sum is greater than 370, then it is also greater than 368.  Therefore, based on statement 1, we can answer the question as ‘always yes,’ which is sufficient.

Statement 2 we must approach in a similar manner.  Now we know that the final three numbers in our set must have an average greater than 120.  This means they must have a sum greater than 360.  The other five numbers in the set can be equal to 1 at the smallest, therefore the total sum must be greater than 365.  As 365 is smaller than 368, the average may or may not be greater than 46.  ‘Sometimes yes, sometimes no’ is insufficient.  So our final Data Sufficiency answer choice is (A) or choice (1); the first statement is sufficient to answer the question, the 2nd is not.

## Paraphrasing on GMAT Data Sufficiency Questions

Math on the GMAT can be hard enough when it is presented without trickery. However, the test-maker knows that even simple math can be made challenging by including complex verbal descriptions or by disguising one mathematical concept as another. Kaplan considers paraphrasing one of the core competencies necessary to crack the GMAT, and this is never more true then when you are approaching complex Data Sufficiency question stems.

At the most basic level, you should ‘paraphrase’ any algebraic word problems in DS question stems as mathematical equations. Consider the following DS stem:

Prior to a reorganization, every manager in a company supervised 17 workers. Under the new company structure, several new managers were hired, and each manager now supervises 14 workers. If the number of workers did not change, how many managers were there after the reorganization?

Inexperienced and overeager test-takers may jump straight to the statements from here, but it’s worth the time to analyze this. We have three variables—the number of managers Before the reorganization (we’ll call this B) and the number of managers After (A), and the unchanging number of Workers (W). We also have to ratios, which we can write out as follows:

17A = W

14B = W

And our question:

A = ?

Well, wait a second. We know this rule—three variables, two equations. The n-variables n-equations rule tells us that all we need is one more equation, any equation, for sufficiency! So, while other test-takers may be struggling to plug in actual values or to get a number solution, we’ll be able to tell at a glance that

1) There are 714 workers at the company

and

2) 9 new managers were hired by the company

are both sufficient! They both give us a third equation to complete our system of three variables, and so the answer must be (D)

Paraphrasing can be more than just simplifying, however. Number properties questions, in particular, give skilled test-takers a chance to recognize a mathematical concept that’s hidden from plain view. Consider this problem:

Is 6 + n/8 an integer?

Looks like a fraction problem, right? That’s what the test-makers want you to think! But this first impression doesn’t withstand scrutiny. The key deduction is that an integer plus an integer will always be an integer, and an integer plus a non-integer will always be a non-integer. Or, to put it another way, that 6 is irrelevant! We get the same answer with 100 + n/8 or, more usefully, 0 + n/8. So really, all we care about is this:

Is n/8 an integer?

But wait. This isn’t a fraction problem at all. In fact, when asking if a variable divided by a number is an integer, we’re really dealing with divisibility rules. So, we can paraphrase the question as follows:

Is n divisible by 8?

Rephrasing the question makes the statements much easier to analyze. If our statements look like this:

1) n is even

2) n divided by 8 has a remainder of 2

Then we can see right away how the paraphrasing helps us! These rules have nothing to do with the fraction problem that this question was disguised as, and everything to do with divisibility. We can evaluate them much more effectively having paraphrased the stem. Statement 1) doesn’t help, since n could be 2 or 8, “No” or “Yes.” (Remember, on a Yes/No question, whenever the answer is “Maybe” the statement is insufficient!) However, statement 2) will never be divisible by 8. Never, or “Always No,” is sufficient to answer a Yes/No DS question, so the answer is (B), 2) alone is sufficient.

Many students will try to “save time” by skipping straight to the statements, but that’s counterproductive. Take the time to ask yourself if there is a better way to express the information presented to you in the question stem. Doing so can pay the time back twofold if it makes the statements easier to analyze.

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## Kaplan GMAT Sample Question: Data Sufficiency Multiples

Try this GMAT Data Sufficiency question dealing with multiples and divisibility.  Remember on Data Sufficiency questions to take each statement individually first, and then look at them together if needed.

Sample Question:

Is the integer y a multiple of 4?

1) 3y2 is a multiple of 18.

2) y = p/q, where p is a multiple of 12 and q is a multiple of 3.

Solution:

Assessing Statement 1:

If 3y2 is a multiple of 18, we know that y2 must be a multiple of 6, because 18 divided by 3 equals 6.  If y2 is a multiple of 6 and y must be an integer, y2 could be 36, in which case y = 6, or it could be 144, in which case y = 12.  If y = 6. then y is not a multiple of 4 and the answer to our question is ‘no’; if y = 12, then y is a multiple of 4 and the answer to our question is ‘yes.’  As we end up with ‘sometimes yes, sometimes no’, statement 1 is insufficient.

Assessing Statement 2:

Here we are told that y = p/q, p is a multiple of 12 and q is a multiple of 3.  Therefore, p could equal 36 and q could equal 6.  In this case y = 36/6 = 6.  If y = 6, then the answer to our question is ‘no.’ However, if p still equals 36, but we make q equal 3, we find that y = 36/3 = 12.  If y = 12, then our answer is ‘yes,’ since 12 is a multiple of 4.  Again, we have ‘sometimes yes, sometimes no,’ so statement 2 is insufficient.

Assessing both statements together:

When we consider the statements together, we see that in both cases y can equal 6 and y can equal 12.  Therefore, we still get an answer of ‘sometimes yes, sometimes no’ to the question, ‘is y a multiple of 4?’  Thus, the statements are insufficient when taken together, or answer choice “E” or “5”.

## GMAT Data Sufficiency: The Lazy Person’s Favorite Question Type?

By Guest Author Kurt Keefner

About 100 years ago there lived a man famous for pioneering a field he called “motion study.”  His name was Frank B. Gilbreth.  He specialized in making factories more efficient.  When Gilbreth first walked into a factory he was helping, what he did was to ask to meet the laziest worker there, because he figured that that person had already figured out how to be efficient.

We can all take a cue from Mr. Gilbreth when it comes to the GMAT.  A general principle is:  Do no more work than is necessary to get the answer. Nowhere does this principle apply more than on that unique question type known as Data Sufficiency.

Data Sufficiency is like a Zen puzzle:  the answer to the question is not the answer to the problem.  The answer to the problem is what combination of data statements would be sufficient to answer the question.  You may or may not need to answer the question in order to answer the problem.  If you answer the question when you don’t need to, you are failing to apply our principle from above—only do the minimum amount of work required.

Let’s try an example from the Kaplan course to make this point more concrete:

Hallie has only nickels, dimes and quarters in her pocket.  If she has at least 1 of each kind of coin and has a total of \$2.75 in change, how many nickels does she have?

(1)  She has a total of 21 coins, with twice as many dimes as nickels.

(2)  She has \$1.50 in quarters.

Now this question is certainly answerable.  You might have to play with the information, do a little substitution, a little combination, but eventually you’d figure out that she had 5 nickels. Now what do you do with that number?  There is no answer choice for 5.  There is no way to enter your discovery at all.  Strictly speaking the information is useless.

Furthermore, if you focused on answering the question about Hallie’s nickels instead of the data sufficiency question that is being asked, chances are that you would use all the information given in both statements instead of considering them separately.  (The problem is easier to solve if you do.)  In that case, you will perhaps conclude that the statements taken together are sufficient to answer the question, when in fact, statement 1 by itself is sufficient.

Lastly, this whole process of solving for the number of nickels takes a lot of time– time you probably wish you could use on other problems.

So how would a “lazy person” approach this question?  The lazy person would never just dive in and start sifting information, but would take a bird’s-eye view.  He or she would look for the concept. Data Sufficiency is concept math.

Our lazy student would look at the question stem and see that there are three unknowns (nickels, dimes and quarters), that they total up to a definite amount and that he is being asked for the value of one of them separately.  That would strongly suggest a system of equations.

Next our lazy student would recall the rule for systems of equations, which is that you must have as many distinct equations as you have variables in order to solve for the individual variables.  How many equations does the stem give us?  One, for the total dollars.  So we need two more.

Statement 1 gives us two more.  That there are 21 coins would give us one equation, and that there are twice as many dimes as nickels would give us the other.  (Note, an equation to be useful does not have to have all the variables in it, so the lack of quarters in the second equation doesn’t matter.)  Together with the stem, that makes three equations, which allow us to solve the system (which we needn’t bother to do).

Statement 2 only gives us one more equation, so it is not sufficient, and the answer is Statement 1 alone is sufficient.

Notice that our lazy student did not even bother to translate the words into formal equations.  It’s enough to know that you COULD do so.

So if you find yourself tempted to do extensive mathematical calculations on a GMAT Data Sufficiency problem—remember the “lazy student”, and that the secret on Data Sufficiency problems is that the answer to the actual mathematical question doesn’t matter in most instances.  Keep your eye on the concept being tested rather than the details of the execution.