## GMAT Coordinate Geometry

The key to many GMAT coordinate geometry questions is to remember that coordinate geometry is just another way of expressing the possible solutions to a two variable equation.  Each point on the line in a coordinate plane corresponds to a solution for the equation of that line.

The base equation for a line is y = mx + b, where b is the y intercept, or the point at which the line crosses the y-axis, and m is the slope, or the steepness of the line.  More specifically, the slope of a line is the change in the y coordinates divided by the change in the x coordinates between any two points on the line.

While understanding the basic format for an equation of a line can be very useful on the GMAT quantitative section, you will encounter GMAT problems in which it is faster and easier to think of the problem in algebraic terms.  In such cases you should think of the equation as an algorithm that will produce the y value given any x value.  This is the reason that the x values are sometimes referred to as inputs and the y values as outputs.

For example, if your answer choices are solution sets and you are asked to determine which option is on the line given in the y = mx + b form, rather than graphing the line and trying to determine which point falls on it, which is especially difficult as you will not have graph paper, you can plug each x value into the equation and determine which one produces the appropriate y value.

On test day, the key is to remember that coordinate geometry is just a way of expressing algebraic concepts visually.  Thus, we can often treat these problems as algebra rather than as geometry.  To see this in action, try the problem below.

Question:

In the xy-coordinate system, if (m, n) and (m 1 2, n 1 k) are two points on the line

with the equation x 5 2y 1 5, then k 5

(A) 1/2

(B) 1

(C) 2

(D) 5/2

(E) 4

Solution:

Step 1: Analyze the Question

For any question involving the equation of a line, a good

place to start is the slope-intercept form of the line,

y = mx 1 b. Remember that if you have two points on a

line, you can derive the entire equation, and if you have an

equation of the line, you can calculate any points on that

line.

We are solving for k, which is the amount by which the

y-coordinate increases when the x-coordinate increases

by 2.

Step 3: Approach Strategically

The slope of a line is the ratio between the change in y and

the change in x. In other words, every time the x-coordinate

increases by 1, the y-coordinate increases by the amount

of the slope.

The equation of the line in the question stem is defined as

x = 2y + 5. We must isolate y to have slope-intercept form:

So the slope of this line is 1/2 . This means that for every

change of +1 in the x direction, there is a change of + 1/2

in the y direction. Then we know that, because there is an

increase in 2 units in the x direction when moving from

m to m + 2, there must be a change of 1 unit in the y

direction when moving from n to n + k. So k = 1.

Since there are variables that eventually cancel (m and n

are not part of the answers), we can Pick Numbers. Let’s

say that you choose the y-coordinate of the point (m, n) to

be 0 to allow for easier calculations. Using the equation

we’re given to relate x- and y-coordinates, we can calculate

the x-coordinate:

So (m, n) is the point (5, 0).

Now we’ll plug our values of m and n into the next point:

(m + 2, n + k). That yields (7, k). All we have to do is plug

an x-coordinate of 7 into the equation to solve for k, the

y-coordinate:

## GMAT Test Day: What to Expect

“I knew it. We should have practiced with these helmets on. I hear Darth everywhere…”

Most students, after careful study, know what to expect on test day in terms of GMAT content.  However, it also important to know what to expect when you arrive at the Pearson Center.  Just as you have learned and practiced GMAT strategies, you should have a plan for handling your breaks and using your scratch sheets wisely.

When you first arrive at the Pearson Center, you will use your ID to check in and register a digital scan of the vein patterns in your palm.  Afterwards, you will place all of you personal items in a locker.  These include ID’s, watches, phones, wallets, keys, and even tissues.  You will not be able to bring anything with you into the testing room.  Furthermore, you will not be able to access these items during breaks in the test.

Once you are ready to get started, you will scan your palm at the door to the testing room, and you will be assigned a computer on which to take your exam.  Any time you reenter the test room you will need to provide a palm scan to prove you are still the same person.

Once seated, you will begin the GMAT, but keep in mind that other test takers will not be starting at the exact same time as you.  Some will be in the middle of their exams when you begin, and some may start after you.  Furthermore, some test takers will be taking tests other than the GMAT.  This means that everyone’s breaks will be at different times.  While no one will talk in the testing room, be ready for people to move around while you are taking your exam.  The proctor will offer you noise-canceling headphones when you arrive.  These can help to minimize these distractions, but you may want to take one of your GMAT practice tests with headphones to get used to the sound of your breathing.  It can be a bit distracting…kind of sounds like Darth Vader.

Your breaks will be 8 minutes long and are optional.  This is a tricky move on the part of the test maker.  We tend to have a sense of what 10 minutes feels like, but 8 minutes is a different story.  Make sure you locate the bathrooms before you start your exam, so that you can find them quickly once you are on your break.  The test will start without you if you are not back in time.  Remember you have some time-consuming security hoops to jump through to get back to the computer, so make sure you don’t take too long of a break.

Finally, you will be given four bound, double-sided wet erase sheets and a wet erase pen.  If you run out of room you can receive a new set of sheets; to do so you must raise your hand, and the proctor will bring a new set to you and take away your used set.  Since this process takes time, you want to minimize how often you trade.  The best move is to always trade during the breaks so that it doesn’t take up any of your test time.   Additionally, you may want to switch sets once in the middle of the quantitative section.  You should try not to trade out your scratch sheets during any of the other sections.

The vast majority of your prep time should be on GMAT content, but you don’t want to run into unnecessary test day stress because you are not ready for all of the rules at the Pearson Center.

## GMAT Quantitative: Two Types of Mixture Problems

Mixture problems show up frequently on the quantitative section of the GMAT and fall into two basic categories.  As each type of mixture question will be approached in fairly different ways, it is important that you know the difference between them.

First, there are mixture problems that ask you to alter the proportions of a single mixture.  These questions could, for example, tell you that you have a 200 liter mixture that is 90% water and 10% bleach and ask how much water you would need to add to make it 5% bleach.  The key in this type of question is the part of the mixture that is constant – in this case the bleach.  While we are adding water, the amount of bleach stays the same.  First, determine how much bleach we have.  10% of 200 is 20 liters.  Next, we know we want those 20 liters to equal 5% of our total.  Since 20 is 5% of 400, our new total should be 400 liters.  To go from 200 liters to 400 liters, you would need to add 200 liters of water, which would be the answer (For yet another way to solve this type of GMAT quantitative problem, check out this post).

The other type of mixture problem will ask you to combine two mixtures.  For example, you could be told that mixture A is 20% bleach and 80% water, while mixture B is 5% bleach and 95% water.  You could then be asked in what ratio these mixtures should be combined to achieve a mixture that is 10% bleach.  You should solve problems such as this algebraically.

Both sides of your equation will represent the amount of bleach in the combined mixture.  On one side you will represent the amount of bleach in terms of the individual mixtures.  This will give you .2A + .05B.  On the other side of the equation you will represent the amount of bleach overall, which is .1(A + B).  Note that in these expressions A represents the total amount of mixture A and B represents the total amount of mixture B.  Because these expressions both represent the total amount of bleach, we can set them equal to each other.  This gives us .2A + .05B = .1(A + B).  The ratio of A to B can be solved as follows:

.2A + .05B = .1(A + B)

.2A + .05B = .1A + .1B

.1A = .05B

A/B = .05/.1

A/B = 1/2

Now try the problem below to see how you do on your own.

Question:

Two brands of detergent are to be combined. Detergent X contains 20 percent bleach

and 80 percent soap, while Detergent Y contains 45 percent bleach and 55 percent

soap. If the combined mixture is to be 35 percent bleach, what percent of the final

mixture should be Detergent X?

(A) 10%

(B) 32_ 1_ 2 %

(C) 35%

(D) 40%

(E) 60%

Solution:

Step 1: Analyze the Question

This is a complex question, but there is a straightforward

solution. We are creating a new mixture from two others,

X and Y. X is 20% bleach, and Y is 45% bleach. The new

mixture is to be 35% bleach.

In other words, some amount of a 20% bleach mixture plus

some amount of a 45% bleach mixture will balance each

other out to a 35% bleach mixture.

Because this involves finding a particular balance between

Detergents X and Y, you can use the balance approach to

solve. We could use Algebra or Backsolving, but balance is

the most efficient. This will let us calculate the proportion

of Detergent X in the final mixture.

Step 3: Approach Strategically

The question does not state how many parts of Detergent

X are used, so call this x. And the question does not state

how many parts of Y are used, so call this y.

So 0.10y = 0.15x.  To solve for a proportional amount, view

this as a ratio. Divide both sides by y and by 0.15 to get

the ratio:

0.10y = 0.15x

0.10 / 0.15 = x / y

10 / 15 = x / y

2 / 3 = x / y

So x:y is 2:3. Add the total to the ratio to determine how x

relates to the total: x:y:total = 2:3:5.

Thus x:total = 2:5. That’s 2 /5 , or 40%.

## GMAT Prep: When to Take the Test

For those of you who recently received your undergraduate degree, you may already know that you want to go to business school and get your MBA some day but are not sure exactly when.  If this is the case, you may be unsure of the best plan for taking the GMAT.

GMAT scores are good for five years.  If you expect to go to business school further than five years in the future, you can’t take the GMAT yet.  However, if you plan to start within this time frame, you will do yourself a favor by taking the GMAT sooner rather than later.

First, you are still used to studying for school.  While this may not seem like a big deal after 16 years of education, just a few years in the workforce and away from academia can make it difficult to jump back into studying.  Additionally, as you move up in your career and your responsibilities increase, you will have more and more difficulty finding the time you need to prepare properly for the GMAT .

Additionally, the GMAT quantitative section focuses on topics that many students have not studied since high school.  The further away you get from those math classes, the more difficulty you will have reviewing and relearning those topics.  This is true of the sentence correction portion of the test as well, but to a lesser extent, as you will continue to use grammar on a daily basis in a way you do not use math.

Finally, the business school application process can be extremely stressful.  You will be struggling to find the time to research schools, write compelling essays, manage people writing recommendations, and complete applications.  Having the GMAT in the rear view mirror can significantly lighten your load and free up valuable time to engage in meaningful interactions with admissions officers and students at the schools you are researching.  This can not only increase the likelihood that you are accepted but also help ensure that the school you choose is a good cultural fit.

Thus, if you are unsure if you should take the GMAT now, your best bet is to just go ahead and sign up for it.  Give yourself a few months to study, and you will be able to improve in the most efficient manner possible.

## GMAT Critical Reasoning: The Denial Test

In your GMAT preparation you have probably learned to tackle critical reasoning assumption questions by identifying the conclusion of the argument, followed by the evidence and then looking for the missing link between these, which will be the central assumption.  However, you have also probably encountered GMAT problems in which you either cannot figure out what the assumption is before you go to the answer choices or the assumption you found is not listed as an option.  When this happens you want to be ready with a backup strategy.

The standard backup strategy for assumption questions – and do keep in mind this should not be used as a primary strategy, since it is more time consuming than the usual approach – is the denial test.

The denial test is based on the idea that the assumption is something that must be true in order to link the evidence to the conclusion.  Another way to think about this is that if the assumption were not true, the evidence would no longer lead to the conclusion; that is, the argument would fall apart.

Therefore, as long as you have identified both the conclusion and evidence you can apply the denial test by negating each answer choice.  Once you negate the option, see if the argument can still be true, even though the answer choice is false.  If the argument cannot be true once the choice has been negated, you have found your assumption

For example, in the argument “poisons are harmful, therefore chemical X is harmful,” the conclusion is “chemical X is harmful” and the evidence is “poisons are harmful.”  If an answer choice for the assumption said “chemical X is a poison,” we would negate this by making it “chemical X is not a poison.”  If we know that chemical X is not a poison, then knowing that poisons are harmful tells us nothing about chemical X and the argument falls apart.  Thus, we have found our assumption.

By using this strategy on GMAT test day when you get stuck on an assumption question you will be able to find the right answer without either guessing or using a method that is not working for you on that problem. Give it a try on the question below.

Question:

Politician: It is important for members of the State Assembly to remember that Governor Norman’s proposed new state thruway was part of her platform during her landslide re – election campaign last year. This means that if the thruway plan is defeated, its opponents will have much to answer for in next November’s State Assembly elections.

The politician’s argument relies upon which of the following assumptions?

 o A. Many of those who voted for Governor Norman oppose the thruway proposal. o B. The thruway proposal is likely to be defeated by the State Assembly. o C. Many of those who voted for Governor Norman supported the thruway proposal. o D. Everyone who voted for Governor Norman last year will vote in the State Assembly elections. o E. Those members of the State Assembly who oppose the thruway proposal do not have valid reasons for opposing it.

The question stem asks us to identify an assumption. Read the stimulus and find the evidence and conclusion. How do they differ? The assumption holds the evidence and conclusion together despite their apparent differences.

When the Governor won by a landslide, her platform included a thruway proposal. Based on this evidence, the politician concludes that if the thruway plan is shot down in the State Assembly, those responsible for its defeat will be in big trouble come election time.

The author assumes that the Governor won because her platform included a thruway proposal. But for all we know, the Governor may have won despite, not because of, the proposal. If the November threat to thruway opponents is real, it must be true that many of those who contributed to the landslide also support the project.

Choice (C) is a perfect replica of the paraphrase above. If, in fact, many who voted for Norman support the thruway, then the politician’s conclusion is surely reasonable — opponents of the thruway may be in hot water with the voters, at least over this issue.

Choice (A) is the exact denial of correct Choice (C). The fact that many of Norman’s supporters oppose the thruway would substantially weaken the politician’s argument.

Choice (B) goes beyond the scope of the argument by assessing the thruway proposal’s chances. The argument is based on the hypothetical “If it is defeated . . .” So even if it is not likely to be defeated, the threat may still be real should the defeat actually occur. The word if ensures that the chance of defeat plays no role in the validity of the argument.

Choice (D) is also not necessary to the argument. Even if not everyone who voted for the Governor last year votes in the State Assembly elections, enough of them may vote to cause trouble for thruway opponents — if those voters support the project.

Choice (E) is irrelevant to the argument. No matter what reasons the members of the State Assembly have for opposing the thruway, the Governor’s voters may not forgive them for a thruway defeat. Nothing regarding the validity of the opposition is required here.

## GMAT Scores: The Essay is Still Important

Often times, the portion of the GMAT most neglected by students is the writing sample.  While this section of the test is certainly less important than your overall 200 to 800 score, you still want to make sure that you know how to handle it.

The essay is graded on a scale from 1 to 6 and most business schools are expecting you to achieve a score of 4 or higher.  While the difference between a 4, 5, or 6 is not all that influential on your admissions prospects, receiving a score lower than a 4 can have a negative impact on your application.

While the integrated reasoning section, which was recently added to the GMAT, replaced the issue essay, the argument essay remains a part of the test.  In fact, it will be the very first section you see on test day.

The key to the essay is answering the question that GMAT test maker is asking.  This can be trickier than you would think.  The writing sample is all about analyzing the argument made by the author, not providing your own viewpoint on the topic.  Therefore, it is essential that you do not agree or disagree with the author’s opinion.  Rather, you need to analyze the argument the author makes to reach his/her conclusion.

To do so, you will need to look for flaws in the author’s reasoning.  Specifically, you will want to identify any faulty assumptions that the author makes.  Additionally, you will want to offer potential strengtheners – facts that, if they were true, would make the argument more sound.

You may notice that these skills are similar to those employed in the critical reasoning portion of the verbal section.  This is not a coincidence.  Both parts of the test are all about breaking down the argument and not about the accuracy of the opinion presented.

In order to get an idea of the types of arguments that appear on the GMAT, you can visit the test makers website, mba.com, and view a complete list of possible essay topics.   It is a good idea to practice taking a few of these arguments apart and writing essays before test day.

If you want feedback on how to identify the flaws in an argument, post the argument and a bulleted list of the flaws you notice in the comments below.  We’ll help you fill in the gaps.

## 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 Average Speed Problems

Imagine you are driving from Chicago to Los Angeles, and you want to know what your average speed needs to be to reach Los Angeles in a certain number of hours.  You would probably start by determining the speed you will be able travel during certain parts of your journey.  Since most of the distance will be covered by highway, you might plan to travel most of the distance at 70 miles per hour.  However, you will also want to plan for some traffic when you are still in or near Chicago and when you get close to Los Angeles.  During these parts of your journey let’s say you can plan to travel at 30 miles per hour.

When calculating the average speed at which you will be traveling, you need to avoid the trap of just averaging these speeds together and planning on an average speed of 50 miles per hour.  Because the vast majority of your journey will take place at 70 miles per hour and only a relatively small portion will take place at 30 miles per hour, simply averaging the speeds is not sufficient.  You need to account for the difference in the amount of time you will be driving at each speed.  Once you do so, your average speed will be much closer to 70 miles per hour than 30 miles per hour.

The same principle will apply when you see average speed questions on the GMAT.  Average speed is defined as total distance divided by total time, rather than the average of the speeds.  Additionally, the average of the speeds will almost always be offered as an answer choice, so be sure to avoid it.

This can be especially tricky when a problem gives little information other than the two speeds.  On test day, you should think strategically and pick a number for the distance, calculate the times using this number, and then plug into the average speed formula described above.  Give it a try on the following problem.

Problem:

A canoeist paddled upstream at 10 meters per minute, turned around, and drifted downstream at 15 meters per minute.  If the distance traveled in each direction was the same, and the time spent turning the canoe around was negligible, what was the canoeist’s average speed over the course of the journey, in meters per minute?

(A)  11.5

(B)  12

(C)  12.5

(D)  13

(E)  13.5

Step 1: Analyze the Question

A canoeist goes one rate in one direction, turns around, and goes back at a different rate. Whenever you deal with one entity that has different rates at different times, set up a chart to track the data. Otherwise, you’ll find yourself in a six-variable, six-equation system that will take a long time to work through. Also, notice that although the distance and time are never mentioned, there are no variables in the answer choices. Whenever variables will cancel out, consider Picking Numbers.

Our task is to calculate her average speed for the whole journey.

Step 3: Approach Strategically

The formula is this:

Average Speed = Total Distance / Total Time

But we’re seemingly told nothing about time, and the only thing we know about distance is that it is the same in both directions. So what to do? As with almost every problem involving a multistage journey, set up this chart:

 Rate Time Distance part 1 of trip part 2 of trip entire trip

Now plug in the data we’re given:

 Rate Time Distance upstream 10 downstream 15 entire trip

Now we see clearly that we’ll be able to know the time if we know something about the distance. Since we know whatever variable we put in place will cancel out by the time we get to the answer choices, let’s just pick a number for distance—one that will fit neatly with a rate of 10 and a rate of 15. A distance of 30 should work well:

 Rate Time Distance upstream 10 30 downstream 15 30 entire trip

At this point, we can fill in the rest of the chart very straightforwardly. The entire distance is 60. The time taken upstream must be 3, and the time taken downstream must be 2. That makes the entire time 5.

 Rate Time Distance upstream 10 3 30 downstream 15 2 30 entire trip 5 60

The speed for the entire trip, then, is 60 / 5 = 12. Answer (B).

Reread the question stem, making sure that you didn’t

To all of you strategic thinkers out there, can you spot a way to quickly eliminate 3 of the answer choices without doing any calculations?

## GMAT Word Problems Unravelled

Sometimes, you’ll come across a GMAT problem that gives you a lot of information.  The relationship between the pieces of information may not even be clear at first.  In my experience as a GMAT teacher, I’ve found that these types of questions tend to confuse students, especially when students try to immediately write one big equation to solve.  Instead of attempting to reach a solution all at once, take a step back and follow a few standard steps.

First, do not panic.  The question may look like it has too much data to consider in the two minute time frame you are allotted per question.  You need to remind yourself that GMAT questions are written to be solved in about two minutes, so a strategy must exist to complete the problem in a timely manner.

Second, in order to find that strategy, start translating the question stem piece by piece.  Do not expect to find one big equation; many complex word problems require a system of equations to solve.  Even if you end up with four or five equations to start, you will be ok as long as those equations are relatively straightforward.

Third, solve any one variable equations.  Use those results to solve two variable equations.  Keep the variable you ultimately want to solve for in mind.  Doing so may allow you to bypass some of the equations you initially wrote down.

By following the steps outlined here, you will avoid getting the various pieces of information confused, and you will be less likely to put variables in the wrong place.  Additionally, you will not rush through the question, which leads to mistakes and ends up taking longer than the method described above.

Try the problem below.  As you do, be sure to follow the steps we discussed.

Problem:

A truck driver drove for 2 days.  On the second day, he drove 3 hours longer and at an average speed of 15 miles per hour faster than he drove on the first day.  If he drove a total of 1,020 miles and spent 21 hours driving during the 2 days, what was his average speed on the first day, in miles per hour?

(A)  25

(B)  30

(C)  35

(D)  40

(E)  45

Step 1: Analyze the Question

Another two-stage journey. So despite the intimidating presentation, we know that we will transfer the data from the question stem into this chart:

We’re solving for speed on the first day, which is the top left box of the chart.

Step 3: Approach Strategically

Before you get too worried about what your solution will be, plug the data into the chart to help organize your thinking. The first thing we read is “On the second day, he drove 3 hours longer . . . than he did on the first day.” We know the total time will be 21 hours, so we can’t just pick a number. Let’s use t for time on the first day. That makes time on the second day t + 3.

Similarly, “On the second day, he drove . . . at an average speed of 15 miles per hour faster than he drove on the first day,” allows us to say that if r is speed on the first day, then r +15 is speed on the second day. The rest of the data is simply numerical:

Since the total of the two days’ times will be the time for the entire trip, we can say:

t + (t + 3) = 21

2t + 3 = 21

2t = 18

t = 9

We can put that into the chart:

That allows us to find distance for each day by multiplying (Rate 3 Time = Distance).

Now we have an equation that will allow us to solve for r, which is what we’re looking for—speed on Day 1. Since the total of the two days’ distances will be the distance for the entire trip, we can say:

9r + 12(r + 15) = 1,020

9r + 12r + 180 = 1,020

21r = 840

r = 40

Reread the question stem, making sure that you didn’t miss anything about the problem.

## 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.”