GMAT algebra note: Do you need to make x larger or smaller?


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Advanced Guessing Tactics To repeat, the tactics below are less universally useful than the strategies we just covered.


GMAT word problem note: However, when all else fails, “break the glass” and try one or more of these tactics. They’re almost always better than guessing completely randomly.


GMAT word problem note: If you’re way behind on time and you have to sacrifice a couple of problems, though, guess immediately and move on. Of course, if you can solve the problem directly, do so.


GMAT word problem note: But also study the tactic, so you’re ready to use it on a harder problem of the same type. Look for Answer Pairs Certainty:.


GMAT word problem note: The GMAT likes to put in a final obstacle. Say you do everything correct, except you solve Add up to 1 on a probability or fraction question Add up to 100% on questions involving percents Add up to 0 (be opposites of each other) Multiply to 1 (be reciprocals of each other).


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. For the wrong unknown or forget to subtract from 1.


GMAT statistics note: Under the pressure of the exam, people make this sort of penultimate error all the time (penultimate means next to last). Also, the way in which the answers are paired may provide clues about the correct solution method and/or traps in the problem.


GMAT statistics note: Try-It #2-9 At a certain high school, the junior class is twice the size of the senior class. If of the seniors and of the juniors study Japanese, what fraction of the students in both classes do not study Japanese?


GMAT statistics note: Note that two pairs of answers each add up to 1: , and .


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. The fact that these pairs sum to 1 also provides a clue to double-check the wording of the question: do vs.


GMAT fractions, decimals, and percents note: Do not study Japanese (the sum of the fractions of the students that do study Japanese and those who do not will equal 1). To solve, you could use a double-set matrix that shows juniors versus seniors and Japanese studiers versus non-Japanese studiers.


GMAT fractions, decimals, and percents note: Pick a smart number that is a multiple of the denominators in the problem:. If the junior class has 12 people, then the senior class thus has 6.


GMAT fractions, decimals, and percents note: If of the seniors study Japanese, then seniors study Japanese. If of the juniors study Japanese, then juniors study Japanese.


GMAT fractions, decimals, and percents note: There are 12 + 6 = 18 students total, and 2 + 3 = 5 of them study Japanese. Thus, of the students do study Japanese, so the fraction of the students that do not study Japanese is .


GMAT fractions, decimals, and percents note: High You may be able to eliminate answers above or below some easily calculated threshold value.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This strategy can sometimes be used in combination with an answer pairs strategy, as pairs of answers are o en composed of a high and a low value.


GMAT word problem note: Therefore, somewhere between and of the students overall, or less than half, must study Japanese. This implies that the fraction of students who do not study Japanese must be more than half.


GMAT word problem note: In 2013, 60% of the eSoroban devices sold were purchased by women, of whom purchased the orange device. If an equal number of orange and green eSoroban devices were sold in 2013, what fraction of men who purchased an eSoroban in 2013 purchased the green device?


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. More than half of the people purchasing this device are women, and more than half of them buy the green version.


GMAT statistics note: If an equal number of green and orange devices are sold, then more than half of the men must buy the orange version. Furthermore, women represent 60% of purchases, while men represent just 40%.


GMAT statistics note: Pretend for a moment that there were equal numbers of men and women. If that were the case, then of women would buy orange and would buy green.


GMAT statistics note: On the flip side, of men would buy green and would buy orange. In fact, there are more women than men, so the proportion of men buying orange devices has to be higher than .


GMAT statistics note: Fewer than must buy the green version. The actual split is , so the correct value is not that much lower than .


GMAT statistics note: But here’s the actual solution, using a double-set matrix and a smart number of 100 total people:.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Men Women Total Green 50 − 35 = 15 60 − 25 = 35 50 Total 40 60 100 Because 15 men purchased a green device out of 40 men total, the proportion is .


GMAT word problem note: Look at Positive/Negative Certainty:. (Again, this is something to do when you have run out of direct approaches or you are short on time!) Try-It #2-11 If x □ y is defined to equal for all x and y, then (−1 □ 2) □ 3 is equivalent to which of the following?


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The negative sign in the term −1 will be extinguished, because the term before the □ symbol is squared when this function is calculated.


GMAT word problem note: Therefore, the correct answer will be positive. Answer Properties Certainty:.


GMAT word problem note: Moderate You may not have time, while doing a Problem Solving problem, to figure out the correct answer. However, it might take less time to figure out something about the correct answer.


GMAT word problem note: Here are some other things you may be able to determine: 5. Moderate The estimation technique can be extended to some Geometry problems.


GMAT word problem note: You can approximate the length of a line segment, size of an angle, or area of an object by drawing it as accurately as possible on your scrap paper. Is the correct answer divisible by a certain value?


GMAT word problem note: What is the units digit of the correct answer? Does the correct answer include square roots?


GMAT word problem note: Does the correct answer include decimals?


GMAT geometry note: When you take the GMAT, the laminated scratch booklet is printed with a light grid. This grid can help you draw very accurate scale pictures, and o en the picture alone is enough to answer the question.


GMAT geometry note: A circle is then inscribed inside triangle ADE. What is the ratio of the area of the circle to the area of the square?


GMAT geometry note: The circle is about the height of the square and about the width. Therefore, the area of the circle should be approximately of the area of the square.


GMAT geometry note: The most efficient way to solve this problem fully is to assign a radius of 1 to the circle. If the center of the circle is O, then OF = 1.


GMAT geometry note: Because ADE is an equilateral triangle, angle ADE = 60°. OD bisects angle ADE, so angle ODE = 30°.


GMAT geometry note: Therefore, triangle OFD is a 30: 60: 90 triangle, and . DE must be twice the length of DF, meaning that , and the area of square BCDE = So the ratio of the area of the circle to the area of the square is Notice how much easier the draw to scale tactic was!


GMAT geometry note: But it works o en enough that you want to think about using it.


GMAT strategy note: Translate the wording into variables, relationships, and equations before calculating. Problem Set Solve problems 1–12.


GMAT strategy note: In each case, identify whether you could use advanced strategies (Choose Smart Numbers, Work Backwards, Test Cases, or Avoid Needless Computation) and guessing tactics (Look for Answer Pairs, Apply Cutoffs, Look at Positive-Negative, or Draw to Scale) in any beneficial way. Of course, there are textbook ways to solve these problems directly.


GMAT strategy note: Instead of trying the textbook method first, focus on applying the strategies and tactics described in this chapter. A popular yoga studio that is always filled to capacity moves to a new location that is able to serve 45% more students.


GMAT strategy note: Unfortunately, 20% of the current students will no longer attend classes at the new location. If classes at the new location are also filled to capacity, what fraction of the students at the new location will be new students?


GMAT quant note: If x < 10, y < 8, and y < x, what must be true? If , which of the following could be the value of x?


GMAT quant note: If , which of the following could be the value of x? If 154 is of x, approximately what is the value of 2x?


GMAT quant note: In a certain clothing store, the most expensive pair of socks sells for $1 less than twice the price of the cheapest pair of socks. A customer notices that for exactly $18, she can buy three fewer pairs of the most expensive socks than the cheapest socks.


GMAT quant note: What could be the number of pairs of the cheapest socks she could have purchased? If , then m must equal which of the following?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If 3x + 3x + 3x = 1, what is x?


GMAT number properties note: If a and b are integers and a is a factor of b, what must be true? The distinct prime factors of a2 are also factors of b.II.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Without solving problems 13–16, which answers could you confidently eliminate and why?


GMAT number properties note: The integer k is positive but less than 400. If 21k is a multiple of 180, how many unique prime factors does k have?


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Is equivalent to which of the following?


GMAT fractions, decimals, and percents note: In the 7-inch square to the right, another square is inscribed. What fraction of the larger square is shaded?


GMAT fractions, decimals, and percents note: If Mason is now twice as old as Gunther was 10 years ago, and G is Gunther’s current age in years, which of the following represents the sum of Mason’s and Gunther’s ages 4 years from now?


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Choose a smart number of 100 for this Percents problem.


GMAT fractions, decimals, and percents note: If the new studio is filled to capacity, at 145 students, but only 80 old students continue to attend, then the studio will have 65 new students. Old capacity: 100 students New capacity: 145 students Old students who will stay with the studio: 80 2.


GMAT fractions, decimals, and percents note: Make sure to choose only values that are allowed by the problem: x < 10, y < 8, and y < x. If x = 9 and y = 7, then xy = 63, which is less than 80.


GMAT fractions, decimals, and percents note: If, on the other hand, x = −10 and y = −20, then xy = 200, which is greater than 80. Statement I does not have to be true.


GMAT fractions, decimals, and percents note: Positive values for both x and y will make , but if x = 9 and y = −1, then , which is not greater than 1.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If x = 3 and y = 2, then x2 + y2 is 13, which is greater than 1.


GMAT algebra note: If x = −2 and y = −3, then x2 + y2 is still 13, which is still greater than 1. Don’t forget about fractions!


GMAT algebra note: If x = 0.3 and y = 0.2, then x2 + y2 = 0.09 + 0.04, which is not greater than 1. Fractions between 0 and 1 get smaller when squared.


GMAT algebra note: None of the statements must be true. You’d actually need to use the quadratic formula, and the equation would turn out to be really nasty.


GMAT algebra note: Much easier: 122 = 144, and that goes very nicely with 288.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. It would be much easier if 154 were 150.


GMAT fractions, decimals, and percents note: First of all, put it over 100:. This is almost , so use that fraction instead.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Translate the equation with the estimated values:.


GMAT algebra note: Finally, make sure to answer the right question that was asked! Call the number of cheap pairs c and the number of expensive pairs e.


GMAT algebra note: Call the cost of a cheap pair $c and the cost of an expensive pair $e.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Note that you don’t actually have to figure out the value of the right-hand side of the equation, as long as you know that it will not equal 9.


GMAT algebra note: Here’s how the algebra would have to be set up.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. From sentence 2: 18 = cx and 18 = ey = (c − 3)y From here, you would solve the equations from sentence 2 for x and y, respectively, and plug them into the equation from sentence 1:.


GMAT algebra note: With the correct manipulation (over multiple, complicated steps!), that equation would eventually become c2 − 21 + 108 = 0 and you could solve for the two solutions, c = 9 and c = 12. In this circumstance, start with the middle number of the remaining choices.


GMAT algebra note: (Avoid Needless Computation) 10.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Testing choices is fast, too.


GMAT number properties note: Stop when you find the correct answer. Which number should you try next, −1 or 1?


GMAT number properties note: Since 0 leads to a number that’s too big, try −1 next. A factor can be smaller than the main number, but a factor can also equal the main number: 6 is a factor of 6.


GMAT number properties note: Statement I 2 is a factor of 6 6 is not a factor of 2 1, 2, 3, and 6 are all factors of 6.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The distinct prime factors of a2 are also factors of b.


GMAT number properties note: There is just one distinct prime factor of a2: 2. In this case, a is a factor of b, yes.


GMAT number properties note: There are two distinct prime factors of a2: 2 and 3. In this case, a is still a factor of b.


GMAT number properties note: The distinct prime factors of a2 are the same as the distinct prime factors of a, since a2 is made up of a multiplied by itself. So, first, this statement is really saying that the distinct prime factors of a are also factors of b.


GMAT number properties note: If a is a factor of b, then by definition all of a’s prime factors also have to be factors of b. No matter what numbers you try for statement II, a will be a factor of b.


GMAT number properties note: If b is 6, then a could be 6, in which case . Alternatively, if b is 6, then a could be 2, in which case So far, this statement looks good.


GMAT number properties note: Don’t forget about negative numbers! The problem doesn’t specify positive integers.


GMAT number properties note: In this case, a could still be 2, so . Statement III does not have to be true.


GMAT number properties note: The GMAT doesn’t o en test the factors of negative numbers, but the definiti.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. An easier technique is to pick a smart number: one value of k that satisfies the constraints.


GMAT number properties note: Any value of k that fits the constraints must have the same number of different prime factors as any other legal value of k. Otherwise, the problem could not exist as written.


GMAT number properties note: There would be more than one correct answer. The problem states that 21k is a multiple of 180, so must be an integer.


GMAT number properties note: In other words, k must be divisible by 60. The easiest number to choose is k = 60.


GMAT number properties note: The prime factorization of 60 is 2 × 2 × 3 × 5, so 60 has the unique prime factors 2, 3, and 5. Thus, k has three unique prime factors.


GMAT number properties note: If you ignore the 103 in the denominator, the division is . You have slightly overstated the denominator, thus slightly understated the result.


GMAT geometry note: Ignoring the dimensions 3 and 4 for a moment, think about the types of squares that might be inscribed in the larger square:. The shaded area can be at most of the larger square, which occurs when the smallest possible square is inscribed in the larger square.


GMAT geometry note: This gives you a great cutoff. The larger the inscribed square, the smaller the shaded area and the more the inscribed square must be rotated from the vertical orientation of the minimum inscribed square.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. There is a way to arrive at the exact answer: compute the relevant areas and take the ratio.


GMAT fractions, decimals, and percents note: The point of this exercise, though, is to practice guessing tactics. Gunther must be at least 10 years old for him to have had a non-negative age “10 years ago” and for Mason to have a non-negative age now.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Follow a Consistent Process Principle #2:.


GMAT quant note: Never Rephrase Yes/No as Value Principle #3:. Work from Facts to Question Principle #4:.


GMAT quant note: Be a Contrarian Principle #5:.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This gives you a significant advantage.


GMAT word problem note: Once you know whether a piece of information lets you answer the given question, you can stop calculating. You do not have to waste time finishing that calculation.


GMAT word problem note: However, this type of problem presents its own challenges. That is, unlike Problem Solving (PS), DS may contain math problems that cannot be solved!


GMAT word problem note: This aspect of DS is unsettling. Practice with other material and then return here when you feel fully comfortable with how DS works.) On DS problems, the issue being tested is “answer-ability” itself—can the given question be answered, and if so, with what information?


GMAT word problem note: So the GMAT disguises “answer-ability” as best it can. The given facts and the question.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Itself are generally presented in ways that make this determination difficult.


GMAT word problem note: For instance, information that seems to be sufficient may actually be insufficient if it permits an alternative scenario that leads to a different answer. Likewise, information that seems to be insufficient may actually be sufficient if all the possible scenarios lead to the same answer for the question.


GMAT word problem note: Advanced DS problems require you to step up your game. You will have to get really good at Testing Cases, which is even more important for advanced DS than for advanced PS problems.


GMAT word problem note: You will also have to get really good at simplifying the given facts and the given question. The GMAT increases the trickiness of the phrasing of the question and/or statements even more than the complexity of the underlying concepts.


GMAT word problem note: So how do you approach advanced DS problems? Unfortunately, there is no “one-size-fits-all” approach:.


GMAT word problem note: All that said, there are a few guiding principles you can follow. The best approach may involve precise application of theory, or it may involve a “quick and dirty” approach.


GMAT word problem note: The statements may be easy or difficult to interpret. The question may require no rephrasing or elaborate rephrasing.


GMAT word problem note: In fact, the crux of the problem may rest entirely on a careful rephrasing of the question.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Follow a Consistent Process This is the most important principle.


GMAT word problem note: A consistent process will prevent the most common errors. It will focus your efforts at any stage of the problem.


GMAT word problem note: Perhaps most importantly, it will reduce your stress level, because you will have confidence in the approach that you’ve practiced. Follow these rules of the road:.


GMAT word problem note: Do your work on paper, not in your head. Writing down each thought as it occurs helps you keep track of the work you’ve done.


GMAT word problem note: Your mind is also freed up to think ahead. Many DS questions are explicitly designed to confuse you if you do all the work in your head.


GMAT word problem note: Label everything and separate everything physically on your paper. If you mix up the elements of the problem, you will o en mess up the problem itself.


GMAT word problem note: Keep these four elements straight:. Facts given in the question stem.


GMAT word problem note: You can leave them unlabeled or you can put them with each statement. Label the question with a question mark.


GMAT word problem note: Amazingly, many people fail to take this simple step. Without a question mark, you might think the question is a fact—and you will.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. It may be worth rewriting the facts from the question stem alongside each statement.


GMAT word problem note: Although this may seem redundant, the time is well spent if it prevents you from forgetting to use those facts. Keep the question mark as you rephrase.


GMAT word problem note: It’s also helpful to keep the helping verb in a Yes/No question:. Rephrase the question and statements whenever possible.


GMAT word problem note: Question stems and statements are o en more complex than they need to be. If you can simplify the information up front, you will save yourself time and effort later in the problem.


GMAT word problem note: In particular, try to rephrase the question before you dive into the statements. Evaluate the easier statement first.


GMAT word problem note: If the second statement looks much easier to you than the first, then start with the second statement. Physically separate the work that you do on the individual statements.


GMAT word problem note: Doing so can help reduce the risk of statement carryover— unintentionally letting one statement influence you as you evaluate the other.


GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. Scrapwork for Rephrasing: ~~~~ Constraints: ~~~~ AD BCE FACT from question stem: ~~~~ FACT from (1): ~~~~ ...


GMAT fractions, decimals, and percents note: And any work you do to combine these facts FACT from question stem: ~~~~ FACT from (2): ~~~~ ... And any work you do to combine these facts QUESTION:.


GMAT fractions, decimals, and percents note: Also observe that this schema explicitly parses out the facts given in the question stem and evaluates those facts alongside each statement.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #3-1 If x and y are integers and 4xy = x2y + 4y, what is the value of xy?


GMAT number properties note: Because the question stem contains an equation, simplify it before considering the statements:. Therefore, either x = 2 or y = 0, or both.


GMAT number properties note: One of the following scenarios must be true: x y xy = ? 2 Not 0 2y Not 2 0 0 y − x = 0(1) x3 < 0(2).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. It turns out that you only need to know the value of y.


GMAT algebra note: Otherwise, x must equal 2, in which case the value of xy is still determined by the value of y. Rather than trying to combine this algebraically with the equation in the question stem, try a couple of the possible scenarios that fit the statement y − x = 2.


GMAT algebra note: Construct scenarios using the earlier table as inspiration:. Since there are two possible answers, this statement is not sufficient.


GMAT algebra note: If x does not equal 2, then y must equal 0, according to the fact from the question stem. Notice how valuable it was to evaluate the fact in the question stem first and to use it to rephrase the question.


GMAT algebra note: Then, when you reach the statements, your work is made much easier. Here is approximately how your paper could look:.


GMAT algebra note: X = 2, y = 0 or both If y = 0, value of xy = 0. If x = 2, value of xy depends on y.


GMAT algebra note: If x = 2, then y = 2 + x = 4, so xy = (2)(4) = 8. If y = 0, then x = y − 2 = −2, so xy = (−2)(0) = 0.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Constraints: x and y integers Case 1:.


GMAT number properties note: If y = 0, x = −2, so xy = 0. If x is negative, y must equal 0.


GMAT number properties note: You can lay out your paper in many other ways. For instance, you might go with this version:.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. In this second layout, facts go on the le, while the question and any rephrasing go on the right.


GMAT quant note: Then the process is always to see whether you can bridge the gap, going from le to right. The important thing is that you develop a consistent layout that you always use.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Never Rephrase Yes/No as Value All DS questions can be divided into two types:.


GMAT quant note: Value questions (such as “What is a?”) and Yes/No questions (such as “Is x an integer?”). Value questions and Yes/No questions are fundamentally different: they require different levels of information to answer the question.


GMAT quant note: Therefore, never rephrase a Yes/No question as a Value question. Value questions usually require more information than Yes/No questions.


GMAT quant note: Try-It #3-2 Is the integer n odd? You don’t need to know which value n might be, just whether n is odd.


GMAT quant note: Therefore, do not rephrase this question to “What is integer n?” Doing so unnecessarily increases the amount of information you need to answer the question. Of course, if you happen to know what n is, then great, you can answer any Yes/No question about n.


GMAT quant note: But you generally don’t need to know the value of n to answer Yes/No questions about n, and the GMAT loves to exploit that truth at your expense. N2 − 2n is not a multiple of 4.(1) n is a multiple of 3.(2).


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. (1) SUFFICIENT: n2 − 2n = n(n − 2).


GMAT quant note: If n is even, both terms in this product will be even and the product will be divisible by 4. Since n2 − 2n is not a multiple of 4, integer n cannot be even—it must be odd.


GMAT quant note: Multiples of 3 can be either odd or even. Rephrasing a Yes/No question into a Value question makes the question unnecessarily picky.


GMAT quant note: Yes/No questions can o en be sufficiently answered despite having multiple possible values for the answer. Note that the converse of this principle is not always true.


GMAT quant note: Occasionally, it’s okay to rephrase a Value question as a Yes/No question—specifically, when it turns out that there are only two possible values. Try-It #3-3 If x is a positive integer, what is the remainder of Some quick analysis will show that x2 − 1 can be factored into (x + 1)(x − 1).


GMAT quant note: If x is odd, then both of these terms will be even and the product will be divisible by 4, yielding a remainder of 0 when divided by 4. If x is even, then x2 will be divisible by 4, so the remainder of x2 − 1 will be 3.


GMAT quant note: There are only two possible values of the remainder: 0 and 3.


GMAT quant note: Separate total outcomes from desired outcomes and count carefully. “Is x even?” Although this Value question seemed at first to have several different potential outcomes, only two are possible, so you are able to change the question to a Yes/No format by suitable rephrasing.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Work from Facts to Question Especially for simple Yes/No questions, people o en assume the answer to the question is Yes before looking at the statements.


GMAT word problem note: They only test cases for which the answer is Yes, rather than testing all of the possible cases allowed by the statements. This line of thinking is backwards—and tempting, because of the order in which things are presented.


GMAT word problem note: Instead ask, “If I start with the applicable facts and consider all possibilities, do I get a definitive answer to the question?” Always work from the given facts to the question—never the reverse! This is why you have to keep the facts separated from the question and why you should always clearly mark the question on your paper.


GMAT word problem note: Try-It #3-4 If x ≠ 0, is x = 1?


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If you work from the question to the facts, you would assume a Yes for the question, then plug this information into the statements.


GMAT algebra note: For instance, you would plug x = 1 into each statement. That’s especially easy to do in this case because this particular question is much simpler to think about than the statements (which are nasty little equations).


GMAT algebra note: Don’t start from the question! No matter what, when you are judging sufficiency, always proceed from the facts to the question.


GMAT algebra note: It doesn’t matter how easy or hard the question is at that point. A er you’ve rephrased, put the question on hold and work from the statements and any other given facts to the question:.


GMAT algebra note: In a Yes/No question, when evaluating the statements, always try to determine whether the question can be answered the same way under any possibility that is consistent with.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. A Yes answer means Always Yes, for all allowed scenarios.


GMAT statistics note: Likewise, a No answer means Always No.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Be a Contrarian To avoid statement carryover and to gain insight into the nature of a problem, deliberately try to violate one statement as you evaluate the other statement.


GMAT word problem note: This will make it much harder for you to make a faulty assumption that leads to an incorrect answer. Think outside the first statement’s box.


GMAT word problem note: Try-It #3-5 If x ≠ 0, is xy > 0? The question is actually asking whether x and y have the same sign.


GMAT word problem note: This indicates nothing about the sign of y. Besides, it is natural to assume that a given variable will have a positive value, because positive numbers are much more intuitive than negative numbers.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. In this case, it is necessary to flip the sign of the inequality when you cross-multiply.


GMAT sequences and patterns note: That is, if x < 0, then means that 1 > xy, and the answer to the question is MAYBE. When evaluating individual statements, deliberately trying to violate the other statement can help you see the full pattern or trick in the problem.


GMAT sequences and patterns note: You will be less likely to fall victim to statement carryover.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Assume Nothing This principle is a corollary of the previous principle: avoid assuming constraints that aren’t actually given in the problem—particularly assumptions that seem natural to make.


GMAT number properties note: Try-It #3-6 Is z an even integer? The wording of this question has a tendency to bias people toward integers.


GMAT number properties note: A er all, the “opposite” of even is odd, and odd numbers are integers, too. However, the question does not state that z must be an integer in the first place, so do not assume that it is.


GMAT number properties note: The fact that is an even integer implies that z = 2 × (an even integer), so z must be an even integer. The fact that 3z is an even integer implies that z = , so z might not be an integer at all.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Equal One way to avoid assuming is to invoke Principle #3: work from facts to question.


GMAT number properties note: No even integers have been skipped over, nor have you allowed the question to suggest z values. That is how assumptions sneak in.


GMAT number properties note: Next, divide the numbers in your list by 3: , etc. Note that you could have stopped computing a er the second value, since you already achieved a Yes and a No.


GMAT number properties note: Another common assumption is that a variable must be positive. Do not assume that any unknown is positive unless it is stated as such in the information given (or if the unknown counts physical things or measures some other positive-only quantity).


GMAT number properties note: 3 × even = even, so z could be even. 3 × odd = odd, so z is definitely not odd.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Your process may be different.


GMAT quant note: There are 19 batters on a baseball team. Every batter bats either right-handed only, le -handed only, or both right-handed and le - handed.


GMAT quant note: How many of the 19 batters bat le -handed? Seven of the batters bat right-handed but do not bat le - handed.


GMAT quant note: Four of the batters bat both right-handed and le -handed.(2) 2. If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?


GMAT quant note: The remainder when a is divided by 40 is 0.(1) The remainder when 40 is divided by a is 40.(2).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Apply Principle #2 (Never Rephrase Yes/No as Value).


GMAT number properties note: Describe why these Yes/No questions cannot be rephrased as Value questions. If a, b, c, d, and e are positive integers such that is an integer?


GMAT number properties note: If a, b, and c are each integers greater than 1, is the product abc divisible by 6? The product ab is even.(1) The product bc is divisible by 3.(2) 5.


GMAT number properties note: If n is a positive integer, is n − 1 divisible by 3? N2 + n is not divisible by 6.(1) 3n = k + 3, where k is a positive multiple of 3(2) 6.


GMAT number properties note: Revisit problems #4 and #5 above, this time deliberately violating Principle #3 (Work from Facts to Question). Determine the incorrect answer you might have selected if you had reversed the process and worked from the question to the facts.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Then, re-solve the problem by applying Principle #4 (be a contrarian).


GMAT algebra note: Verify that applying Principle #4 leads to the correct answer, whereas not following the principle could lead to an incorrect answer. For problems 8 and 9, apply Principle #5 (assume nothing) by identifying the explicit constraints given in the problem.


GMAT algebra note: What values are still permissible? Next, solve using these constraints.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Your notes may be different.) 1.


GMAT number properties note: Rephrasing: 19 batters total → integers only! Some R only Some L only Some R & L 0 “neither” How many le -handed batters?


GMAT number properties note: L Not L Total R Not R 0 Total x 19 What is the integer x?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. L Not L Total R 7 Not R 0 Total x 7 19 Question:.


GMAT number properties note: Rephrasing: 81 divided by a → remainder of 1. A goes evenly into 80, and a ≠ 1.


GMAT number properties note: A is one of the factors of 80 other than 1. A = 2, 4, 5, 8, 10, 16, 20, 40, or 80.


GMAT number properties note: Which of the numbers listed above is the value of a?


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Which of the listed numbers is a?


GMAT quant note: (INSUFFICIENT) a must be 80. Yes (SUFFICIENT) Yes (SUFFICIENT).


GMAT word problem note: Watch divisibility, factors, remainders, parity, and prime structure. Each of the below is an accurate Yes/No rephrase:.


GMAT word problem note: Alternatively, you could ask “Is there an even integer and a multiple of 3 among a, b, and c?” (1) INSUFFICIENT: ab is divisible by 2, but it’s unclear whether it is divisible by 3 (or whether c is divisible by 3). (2) INSUFFICIENT: bc is divisible by 3, but it’s unclear whether it is divisible by 2 (or whether a is divisible by 2).


GMAT word problem note: Therefore abc is divisible by both 2 and 3. Is Is abc = 6, 12, 18, 24, 30, 36, 42, 48, etc.?


GMAT word problem note: Is abc divisible by 2 and by 3? An accurate Yes/No rephrase is the following.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. This narrows down the values of interest to a certain type of number, which follows a pattern: 1, 4, 7, 10, etc.


GMAT sequences and patterns note: If n2 + n = n (n + 1) is not divisible by 6, you can rule out certain values for n. N n + 1 n (n + 1) not divisible by 6 The pattern from the rephrasing is apparent here: n can only be 1, 4, 7, 10, etc., all integers that are one greater than a multiple of 3.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The integers n − 1, n, and n + 1 must be consecutive.


GMAT number properties note: If n(n + 1) is not divisible by 3, then n − 1 must be divisible by 3, since in any set of three consecutive integers, one of the integers must be divisible by 3. If 3n = 3 × pos integer + 3, then n = pos integer + 1.


GMAT number properties note: This does not resolve whether n is definitely one greater than a multiple of 3. Revisiting #4, working incorrectly from the Question to the Facts:.


GMAT number properties note: Manipulating the question to abc = 6 × integer (and losing track of the question mark), someone might be tempted to check whether it is possible for abc to be a multiple of 6, instead of whether abc is definitely a multiple of 6:. Be sure to revisit how to do this problem correctly!


GMAT number properties note: Revisiting #5, working incorrectly from the Question to the Facts:. Someone working incorrectly might try multiples of 3 for n − 1 to see whether they “work” with the statements: (1) abc = 6 × integer, so ab is even.


GMAT number properties note: ✓ (2) abc = 6 × integer, so bc is divisible by 3.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. (1) SUFFICIENT: n − 1 n n + 1 n2 + n = n(n+ 1) not divis by 6 9 10 11 110 ✓ (2) SEEMS SUFFICIENT (incorrectly):.


GMAT quant note: Be sure to revisit how to do this problem correctly, so you are certain how to do so for the future!


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Non-Contrarian Approach Contrarian Approach (1) INSUFFICIENT: m > 0 or m = pos, so “Is pos3 > pos2?” (1) INSUFFICIENT: m > 0 or m = pos, so “Is pos3 > pos2?” (2) SEEMS SUFFICIENT: m2 > m implies that m is not a fraction or 1, therefore m3 > m2.


GMAT fractions, decimals, and percents note: Or, if you assume that m > 0, carrying over from (1), you might do the following: m2 > m m > 1 (dividing by m) Therefore, m3 > m2. (2) INSUFFICIENT: m2 > m implies that m is not a fraction or 1, so if m > 1 then the answer is Yes.


GMAT fractions, decimals, and percents note: BUT, contradicting (1), what if m is negative? That is possible according to (2), since neg2 > neg.


GMAT fractions, decimals, and percents note: No, if m is a proper fraction or 1.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Non-Contrarian Approach Contrarian Approach (1) AND (2) SUFFICIENT:.


GMAT fractions, decimals, and percents note: Combined, the statements eliminate negative, zero, positive proper fractions, and 1 for the value of m. Note that both solutions were hampered by inadequate rephrasing.


GMAT fractions, decimals, and percents note: Ideally, you would first rephrase as follows:. With proper rephrasing, other errors are less likely.


GMAT fractions, decimals, and percents note: But even with inadequate rephrasing, taking a contrarian approach can save you from a wrong answer. (It’s okay to divide by m2, which must be positive: a square is never negative and m is also not 0, according to the question stem.) 8.


GMAT fractions, decimals, and percents note: The explicit constraint is yz ≠ 0, which indicates that y ≠ 0 and z ≠ 0. Both y and z could be any nonzero value, including positive integers, negative integers, positive fractions, negative fractions, etc.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. For (1), a faulty assumption could be made by those who plug in values for z.


GMAT fractions, decimals, and percents note: That would yield a definite No answer to the question, as all the y values are at least as great as 1. The (unverbalized) assumption is that z is an integer, but that’s not necessarily so.


GMAT fractions, decimals, and percents note: Correct solution: (1) INSUFFICIENT: y must be positive, but is it a fraction or an integer? If z = 2, then y = 4, and the answer is No.


GMAT quant note: Since you’re assuming y < 0, the rephrasing is y < −1. YES NO (1) AND (2) SUFFICIENT:.


GMAT quant note: If z = , then y = , and the answer is Yes.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Avoid limiting yourself to cases in which x is greater than y.


GMAT fractions, decimals, and percents note: Since “Is x > y?” is a question and not a statement, x might be greater than or less than y. Actively seek cases that give a No answer.


GMAT fractions, decimals, and percents note: 3 4 |–1| < |3| (valid) No 4 3 |1| < |3| (valid) Yes Since x could be greater than y or less than y, given the information in the statement, this statement is insufficient. The second case tested above also fits this statement and yields an answer of Yes.


GMAT fractions, decimals, and percents note: Find another case in which |x| > |y|, but x is not greater than y. 4 3 |4| > |3| (valid) Yes –4 3 |–4| > |3| (valid) No (1) AND (2) INSUFFICIENT:.


GMAT fractions, decimals, and percents note: A case that fits both statements and gives an answer of Yes has already been found. Try to find another case for which both statements are true, but which gives an answer of No.


GMAT fractions, decimals, and percents note: As you test different values for x and y, you may find certain cases for which one or both statements are not true. Cross these off on your paper, and do not take them into consideration when choosing your answer.


GMAT quant note: X y (1): |x – y| < |x| (2): |x| > |y| Is x > y?x y (1): |x – y| < |x| (2): |x| > |y| Is x > y? 4 3 |1| < |3| (valid) |4| > |3| (valid) Yes –4 3 |–7| < |–4|: (not valid) n/a n/a The answer to the question can be either Yes (if x = 4 and y = 3) or No (if x = –4 and y = –3), so the statements together are insufficient.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Advanced Strategies Advanced Guessing Tactics Summary Common Wrong Answers.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Strategies are broad: they apply to a wide variety of problems, they provide a complete approach, and they can be used safely in most circumstances.


GMAT word problem note: Moreover, a particular tactic may only be useful in special situations or for parts of a problem. Compute to Completion For some problems, you won’t necessarily be able to tell whether the answer can be calculated until you follow through on the calculations all.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Extract the Equation For many Word Problems, you need to represent the problem with algebraic equations to avoid embedded tricks that can be difficult to spot otherwise.


GMAT word problem note: Know the Code The most challenging part of some DS problems is figuring out what the question and statements are actually saying. If the problem is written in “GMAT code,” translate it into simple language before you do anything else.


GMAT word problem note: Use the Constraints Many DS problems provide explicit constraints on the variables. In other problems, these constraints will be implicit (e.g., a variable that refers to a number of people, houses, or airplanes must be both positive and an integer).


GMAT word problem note: In either case, these constraints frequently determine the correct answer, so you must identify and use them. Beware of Inequalities Whenever a DS problem involves inequality symbols, be especially careful—the GMAT loves to trick people with inequalities.


GMAT word problem note: Test Cases One of the best ways to show that a statement is insufficient is to test different scenarios, or cases, in which that statement is true.


GMAT strategy note: Watch divisibility, factors, remainders, parity, and prime structure. These tactics are listed later in the chapter.


GMAT strategy note: As with Problem Solving (PS) tactics, some of these D S tactics work wonders when used correctly. Others only slightly improve your guessing odds.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. For some problems, however, you cannot determine whether a single answer can be obtained until you’ve calculated the problem all the way through.


GMAT algebra note: This is particularly common in the following situations:. Try-It #4-1 What is the value of ab?


GMAT algebra note: Multiple equations are involved—particularly if they are non-linear. A complicated inequality expression is present.


GMAT algebra note: Variables hidden within a Geometry problem are related. A = b + 1 (1) a2 = b + 1 (2).


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Evaluating both statements together is trickier, however:.


GMAT quant note: Based on this work, either a = 0 or a = 1. It would be tempting at this stage to decide that since a can have two different values, statements (1) and (2) together are insufficient.


GMAT quant note: Look at the values b can hold in these two scenarios: a b ab 1 a − 1 = 0 0 0 a − 1 = −1 0 While it is true that a can take on different values, ab is equal to zero in either case. When a = 1, b = 0, and ab = 0.


GMAT quant note: When a = 0, b = −1, and ab = 0.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Therefore, (1) and (2) combined are SUFFICIENT to answer this specific question:.


GMAT word problem note: In a multiple-scenario problem, be sure to compute for the specific question asked (in this case, ab) in order to determine whether the end result for each scenario is actually different. EXTRACT THE EQUATION For Word Problems, setting up an algebraic representation of the question is essential.


GMAT word problem note: It is very easy to get intellectually lazy and miss an embedded trick in the problem. These tricks are usually much easier to spot if you are looking at the underlying algebra behind the problem.


GMAT word problem note: Try-It #4-2 A store sells two types of bird feeders:. Alphas feed 1 bird at a time, whereas Bravos feed 2 birds at a time.


GMAT word problem note: The total number of birds that can be fed at one time by bird feeders sold last month is 50. What is the total revenue generated by birdfeeders sold last month?


GMAT word problem note: Last month, the price of each Alpha was $15 and the price of each Bravo was $30. 40 Alphas were sold last month.(2).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. From the words in the question stem, Extract the Equation.


GMAT algebra note: The problem indicates that Alphas can feed 1 bird at a time and Bravos 2 birds at a time. The problem also indicates that last month 50 birds could be fed at a time, so you have this:.


GMAT algebra note: To calculate the revenue, it seems you will need the prices of the bird feeders and the number of bird feeders, A and B. Again extract the equation from the wording of the question:.


GMAT algebra note: It turns out that you don’t need to know A and B individually, since the question stem equation indicated that A + 2B = 50. Therefore, total revenue equals $15(A + 2B) = $15(50) = $750.


GMAT algebra note: It’s true that there are many possible values of A and B that satisfy the condition that A + 2B = 50. However, mathematically every possible combination that satisfies this equation would lead to the same revenue of $750.


GMAT algebra note: The number of each type of bird feeder sold is irrelevant. In this sense, Extract the Equation can be similar to the Compute to Completion strategy because once the equation has been extracted, you may find that the multiple possibilities for the variables might converge to a single answer to the specific question that’s been asked.


GMAT algebra note: Total number of birds fed = A + 2B = 50, where A and B represent the number of birdfeeders of each type that have been sol.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If there were 40 Alphas sold, there were 5 Bravos sold.


GMAT number properties note: But you still don’t know the prices, so you can’t compute revenue. Be sure to translate all Word Problems into math so they can be properly evaluated.


GMAT number properties note: These issues are sensitive to the exact numbers given. KNOW THE CODE Many DS questions and statements are intentionally written in a complicated way.


GMAT number properties note: Take a look at the two statements below:. These statements look very different, but from the perspective of someone who’s solving a DS problem, they say exactly the same thing.


GMAT number properties note: In other words, the first statement is just saying that a and b are both odd—but it’s saying it in code to keep you from noticing. There are two clues that the statement is really about odd and even numbers.


GMAT number properties note: First, it specifies that a and b are integers. Second, if 2 is not a factor of a certain integer, that’s just another way of saying that the integer is odd.


GMAT number properties note: You can write that out as follows: a and b are integers such that 2 is not a factor of 7a2b3.1. A and b are both odd integers.2.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The next thing to notice is that multiplying a number by 7 has no effect on whether it’s odd or even.


GMAT number properties note: Multiplying an even number by 7 gives you an even number, and multiplying an odd number by 7 gives you an odd number. So the 7 can be removed entirely:.


GMAT number properties note: The exponents also don’t matter. Squaring or cubing an odd integer gives you an odd integer, and squaring or cubing an even integer gives you an even integer.


GMAT number properties note: So saying that a2b3 is odd is no different from saying that ab is odd. Finally, if the product of two integers is odd, both of those integers have to be odd.


GMAT number properties note: Multiplying an even integer by another integer always results in an even value. Here’s what the statement really says:.


GMAT number properties note: The two statements were really the same all along. If you see something in a DS problem that looks like it could be GMAT code, try to crack the code before you go further.


GMAT number properties note: You may find that a complicated statement is disguising something much easier to work with. USE THE CONSTRAINTS a and b are integers, and 7a2b3 is odd.


GMAT number properties note: A and b are integers, and a2b3 is odd.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. When this information is given, it is usually essential to the problem.


GMAT algebra note: Try-It #4-3 If 8x > 3x + 4x, what is the value of the integer x? On top of that, there is also another constraint given: x must be an integer.


GMAT algebra note: This limits the scope of the potential values of x even further. Make note of this type of constraint in your work on paper.


GMAT algebra note: Write “x = int” or something similar. You could also incorporate this information by rephrasing the question to include the constraint: “If the integer x is positive, what is the value of x?” (1) SUFFICIENT:.


GMAT algebra note: Solve this inequality for x: 8x can only be greater than 7x when x is positive.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Since you know from the question stem that x > 0, you can conclude that 0 < x < 2.


GMAT word problem note: The only integer between 0 and 2 is 1. Manipulate this compound inequality as follows:.


GMAT word problem note: Note that it’s fine to manipulate all the parts of the compound inequality at the same time as long as you perform each manipulation to all three parts of the inequality. Since the question stem indicates that x > 0, it must be the case that 0 < x ≤ 1.


GMAT word problem note: The only integer that fits this criteria is 1. If you had overlooked the fact that x is an integer, you would have determined that there are many values between 0 and 1 or between 0 and 2.


GMAT word problem note: Watch divisibility, factors, remainders, parity, and prime structure. Make a note of any additional information given to you in the question stem (e.g., “x is positive” or “x is an integer”).


GMAT word problem note: You o en will have to use this information properly to get the correct answer. Integer constraints in particular are very potent: they o en limit the possible solutions for a problem to a small set.


GMAT word problem note: Sometimes this set is so small that it contains only one item. Constraints will not always be explicitly given.


GMAT word problem note: The ones the GMAT doesn’t explicitly give you can be called hidden constraints. Hidden constraints are most prevalent in Word Problems and Geometry questions.


GMAT word problem note: The number of countable items must be a non-negative integer. Note that zero is only a possibility if it is possible for the items not to exist at all—if the problem clearly assumes that the items exist, then the number of items must be positive.


GMAT word problem note: Number of people Number of yachts Number of books Many non-countable quantities must be non-negative numbers, though not necessarily integers. Again, zero is only an option if the underlying object might not exist.


GMAT word problem note: If the problem clearly assumes the existence and typical definition of an object, then these quantities must be positive.


GMAT geometry note: If these constraints are important in a P S problem, then failing to take a constraint into account may make you unable to solve the problem. That will alert you to the existence of the constraints, since every P S problem must be solvable.


GMAT geometry note: In contrast, you will get no such signal on a DS problem. A er all, solvability is the very issue that D S tests!


GMAT geometry note: BEWARE OF INEQUALITIES Whenever a D S question involves inequality symbols, be especially careful. The GMAT can employ a variety of different inequality-specific tricks.


GMAT geometry note: The side of a triangle must have a positive length. The only exception is negative coordinates in a coordinate plane problem.) The weight of a shipment of products must be positive in any unit.


GMAT geometry note: The height of a person must be positive in any unit. Many other non-countable quantities are theoretically allowed to take on negative values.


GMAT geometry note: The profit of a company (However, if a company made a profit, then that profit is positive!) The growth rate of a population The change in the value of essentially a.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. One inequality can imply another seemingly unrelated inequality, depending on the situation.


GMAT word problem note: If x is greater than 5, then it must be positive, thus x > 0. However, the opposite is not the case.


GMAT word problem note: If you knew that x < 5, then you would not be able to determine whether x > 0. A er all, x could be positive but less than 5, or x could be negative.


GMAT word problem note: Inequalities can combine with integer constraints to produce a single value. Some Word Problems can create a hidden constraint involving inequalities.


GMAT word problem note: These inequalities may come into play in determining the correct answer. The next oldest student in the class...


GMAT word problem note: The youngest student in the class...” This can be translated to the following inequality: youngest < middle < oldest. Inequalities involving a variable in a denominator o en involve two possibilities: a positive and a negative one.


GMAT word problem note: However, this may not be correct. It depends on whether y is a positive or negative number.


GMAT word problem note: If y > 0, then it is correct to infer t.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #4-4 If is a prime number, what is the value of x?


GMAT number properties note: If is a prime number, then possible values are 2, 3, 5, and so on. Therefore, x must be a perfect square of a prime.


GMAT number properties note: Possible values include 4, 9, 25, and so on. Manipulate the inequality to isolate x:.


GMAT number properties note: Many questions involving inequalities are actually disguised positive/negative questions. What is interesting is that the product is positive, meaning both x and y are positive or both x and y are negative.


GMAT number properties note: Thus, x and y have the same sign. Here, the inequality symbol is used to disguise the fact that x and y have the same sign.


GMAT number properties note: −16 < −3x + 5<22(1) x2 is a two-digit number(2).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Since x is the square of a prime, it can’t be negative or zero.


GMAT number properties note: The smallest possible square of a prime is 4 and the next smallest possible square of a prime is 9. This inequality allows just one possible value: x = 4.


GMAT number properties note: In addition, from the question stem, is a prime number. Determine the possible value(s) for x:.


GMAT number properties note: Because there are two possible values for x, this statement is not sufficient. Try-It #4-5 If mn ≠ 0, is m > n?


GMAT number properties note: If x = 4, then is a prime number, 2, and x2 is a two-digit number, 16. If x = 9, then is a prime number, 3, and x2 is a two-digit number, 81.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The constraint in the question stem indicates that neither m nor n equals zero.


GMAT algebra note: You can solve algebraically/theoretically or you can test cases. If you solve algebraically, be careful: you have to account for multiplying the inequality by a negative:.


GMAT algebra note: Alternatively, test cases: (2) INSUFFICIENT:. This statement indicates nothing about the signs of the two variables.


GMAT algebra note: Either one could be positive or negative. If m and n are both positive, then m > n.


GMAT algebra note: If m and n are both negative, the sign flips twice, so m > n again. If only one is negative, then the sign flips once and m < n.


GMAT algebra note: In this case, m must be the negative number, since any positive is greater than any negative. If m and n are both positive, then m > n and m2 > n2.


GMAT algebra note: Both statements allow this scenario. If m and n are both negative, then m > n but m2 is not greater than n2.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Because there are scenarios in which m > n and m < n, both statements together are still insufficient to answer the question.


GMAT algebra note: One way to show that this is true is by Testing Cases. Write down several scenarios, or cases, in which the statement is true.


GMAT algebra note: For instance, you might think of different possible values for the variables in the statement. For each of these cases, determine the answer to the question.


GMAT algebra note: If m is negative and n is positive, then m < n. It could also be true that m2 > n2, as long as m’s magnitude is larger than n’s.


GMAT algebra note: If you’re not sure, test cases (see below). If m = 3 and n = 2, then m > n and m2 > n2.


GMAT algebra note: Both statements allow this scenario.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. If the question has different answers depending on which case you’re testing, the statement is insufficient.


GMAT word problem note: If the question always has the same answer in every case you try, the statement is likely sufficient. However, the theory required to answer a question may be cumbersome to figure out in two minutes.


GMAT word problem note: If implemented correctly, Testing Cases can be fast, easy, and accurate, so it should be part of your toolbox. Particularly, testing cases is o en the simplest way to show that a statement is not sufficient.


GMAT word problem note: It’s easy to overlook a salient scenario. The key is to have a systematic approach to testing cases.


GMAT word problem note: This relies on three approaches: the standard number set for testing, discrete number listing, and case testing with concepts. Sometimes testing enough cases will also allow you to notice a pattern to help you deduce whether the statement is sufficient.


GMAT word problem note: Standard Number Set for Testing The GMAT o en tests.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Odd Even Proper Fraction Improper Fraction Negative −1 −2 − Zero 0 Positive 1 2 This set includes integers, non-integers, positive and negative numbers, and numbers greater than and less than 1.


GMAT number properties note: Thus, a comprehensive set of test numbers (to memorize and apply) would be as follows:. Remember this list as “every integer and half-integer between −2 and 2.” Not all of these numbers will be relevant or possible on every problem.


GMAT number properties note: The question itself may suggest certain values to test, but always keep in mind the potential need to test a value of each relevant type—and if a problem really might entail testing nine different cases, consider whether that problem is really worth your time.


GMAT number properties note: Using the Standard Set ensures that you don’t get tripped up by forgetting to try a particular type of value that will give you a different answer, thereby allowing you to prove definitively that the statement is not sufficient.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. This problem presents inequalities, and the question asks whether a is negative.


GMAT number properties note: Therefore, test different values of a to see which values fit the statements. The question stem doesn’t provide any constraints, so begin by testing some easier integers and then move to fractions if needed: (1) INSUFFICIENT:.


GMAT number properties note: Test the possible integers from the standard number set: a a3 − a2 − 2a < 0 a < 0? −2 (−2)3 − (−2)2 − 2(−2) = −8 (valid) Yes 1 (1)3 − (1)2 − 2(1) = −2 (valid) No a3 < a2 + 2a(1) a2 > a3(2) −1 (−1)3 − (−1)2 − 2(−1) = 0 (invalid) 0 (0)3 − (0)2 − 2(0) = 0 (invalid).


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Stop when you find two valid cases that return different answers to the question.


GMAT fractions, decimals, and percents note: Try integers first: a a2 − a3 > 0 a < 0? −2 (−2)2 − (−2)3 = 12 (valid) Yes −1 (−1)2 − (−1)3 = 2 (valid) Yes Be careful:.


GMAT fractions, decimals, and percents note: −2 (−2)2 − (−2)3 = 12 (valid) Yes −1 (−1)2 − (−1)3 = 2 (valid) Yes 0 (0)2 − (0)3 = 0 (invalid) 1 (1)2 − (1)3 = 0 (invalid) 2 (2)2 − (2)3 = -4 (invalid) 0 (0)2 − (0)3 = 0 (invalid) 1 (1)2 − (1)3 = 0 (invalid) 2 (2)2 − (2)3 = −4 (invalid).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. When , the constraint is fulfilled, but a is positive.


GMAT algebra note: Combining statements (1) and (2) shows that whenever a = −2, − , or both conditions are fulfilled. The variable a could thus be positive or negative.


GMAT algebra note: Notice that you did not need to test every possible value for a. That means that testing − was unlikely to be necessary, since that value is not much different from −2.


GMAT algebra note: Furthermore, when you do find a contradictory answer, you can stop testing. For a Yes/No question, all you need to do is find one valid Yes and one valid No to prove insufficiency.


GMAT algebra note: This problem can also be solved algebraically, but it takes more conceptual work. First solve the inequality as though it were an equation, then map the solutions on a number line and test to see in which regions the inequality is true.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This number line demonstrates that either a < −1 or 0 < a < 2.


GMAT word problem note: Since a can’t be zero, and a2 must be positive, you can divide by a2. Overlapping the possible ranges, either a < −1 or 0 < a < 1.


GMAT word problem note: This is still not enough information to tell whether a is negative. In some cases, the standard number testing list may not quite suffice.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. However, x could be 1.1, in which case x2 = 1.21, which is less than 1.3.


GMAT algebra note: You could figure out that x could be greater than 1 upon a quick inspection of this problem. You might think to try a number slightly larger than 1.


GMAT algebra note: However, if the problem were more complicated, it might not be so obvious. In cases like these, use the boundary principle: test values that are close to boundaries given in the problem.


GMAT algebra note: In this case, the boundary value is 1, so add 0.9 and 1.1 to your list of numbers to test. You might even try 0.99 or 1.01.


GMAT algebra note: You should know that −1, 0, and 1 are natural boundaries, because numbers behave differently on either side of them (that’s why the standard list contains numbers in the ranges defined by −1, 0, and 1). Discrete Number Listing In the previous problems, the variables a and x were not constrained, so you had to test a series of different numbers to solve the problem.


GMAT algebra note: By contrast, many questions suggest.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. By contrast, some problems describe a smooth range of potential values for a variable or expression (e.g., 0 < x < 1 or x must be negative).


GMAT number properties note: In these cases, don’t list consecutive values to test because the set of possible values is not discrete. If the variable can take on any real number in a range, then rely on the standard number testing list, potentially with some modifications, as described in the previous section.


GMAT number properties note: A key to the discrete number listing process is to test consecutive values that fit the criteria—it would be too easy to leave out the one exception that proves insufficiency. Never skip numbers that fit the constraint.


GMAT number properties note: This is especially important if you are listing discrete numbers to equal an expression, not just a variable. By the way, remember to work from the facts to the question, not the other way around!


GMAT number properties note: Don’t assume that the question should be answered Yes and only test values that make it so. Try-It #4-8 Is x a multiple of 12?


GMAT number properties note: Any set that is “integer-like,” with well-defined, separated values is odd.(1) x is a multiple of 3.(2).


GMAT geometry note: Watch divisibility, factors, remainders, parity, and prime structure. Notice that you’re picking values for , not x.


GMAT geometry note: It would be far too much work to test different values for x to determine which make odd, and you could potentially miss some values that fit the statement. Do not plug in numbers for x here!


GMAT geometry note: Instead, list consecutive odd values for , a quick and easy process. Then, solve for x in each case.


GMAT geometry note: For this problem, your work on paper may look something like this: (1) AND (2) SUFFICIENT:. Combine these statements by selecting only the values for x that are in both lists.


GMAT geometry note: On your paper, circle the following values: x = 12 and x = 84. This seems to be SUFFICIENT—the values for x that fit both statements are multiples of 12.


GMAT geometry note: At this point, if you wanted to check another value, you could, or you could go with the trend, which is likely going to be correct a er testing this many cases. INSUFFICIENT: = odds = 1, 3, 5, 7, 9, etc.


GMAT geometry note: X − 3 = 1, 9, 25, 49, 81, etc. INSUFFICIENT: x = multiples of 3 = 3, 6, 9, 12, 15, etc.


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. Thus, x is an odd perfect square plus 3.


GMAT algebra note: One might conclude that x is even, which is a true but incomplete description! Listing numbers is an easy way to see that these numbers are all multiples of 4.


GMAT algebra note: Yes, it’s worth knowing how to do this algebra. The point is that a discrete number testing process is quick and simple, so it’s also worth knowing how to do.


GMAT algebra note: Case Testing with Concepts You don’t need to use actual numbers to do case testing. In fact, it’s sometimes more efficient to case test using concepts, rather than numbers.


GMAT algebra note: (1) INSUFFICIENT: where k is an integer. (2) INSUFFICIENT: x is a multiple of 3, so x must be divisible by 3.


GMAT algebra note: (1) AND (2) SUFFICIENT: x is divisible by 3 and by 4, so x is divisible by 12.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #4-9 If a, b, and c are integers, is abc > 0?


GMAT number properties note: You could test cases in the traditional way: find specific values for a, b, and/or c that fit one statement or the other, then multiply those three values together and check whether their product was greater than zero. However, doing that much actual multiplication would be overkill.


GMAT number properties note: Instead of case testing by checking specific numbers, try case testing with concepts. Instead of choosing appropriate values for a, b, and c, think about whether they could be positive or negative.


GMAT number properties note: For instance, one scenario that fits the statement is that a, b, and c are all positive. Another scenario is that a and b are positive, but c is negative.


GMAT number properties note: You can put this information in a chart, just as you would with ordinary case testing: a b c ab > 0 abc > 0? + + + (+)(+) = + (valid) Yes + + — (+)(+) = + (valid) No (2) INSUFFICIENT: ab > 0(1) bc > 0(2).


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. A b c bc > 0 abc > 0?a b c bc > 0 abc > 0?


GMAT word problem note: + + + (+)(+) = + (valid) Yes — + + (+)(+) = + (valid) No Neither statement is sufficient on its own, so try them together. (1) AND (2) INSUFFICIENT: a b c (1): ab > 0 (2): bc > 0 abc > 0?


GMAT word problem note: It was never necessary to work with specific numbers to solve this problem. This strategy doesn’t only work on positive/negative or odd/even problems.


GMAT word problem note: Before reading the explanation, try to figure out how to solve the following problem using concepts (not specific numbers) to test cases. Try-It #4-10 At a certain company, the bonus pool is divided among a group of employees consisting entirely of engineers and managers so that.


GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. Each engineer receives a bonus of e dollars and each manager receives a bonus of m dollars.


GMAT statistics note: Is m at least 20% greater than e? At least 20% of the employees in the group are managers, but the amount earned by each employee is unknown.


GMAT statistics note: There could be a lot of managers who each earn a large bonus or there could be a lot of managers who each earn a small bonus. Think about scenarios on the extreme end of the range, ones that could result in a different answer to the question.


GMAT statistics note: # of Engineers # of Managers e m At least 20% managers? Few Many Small Large Lots of managers, and each manager earns a large bonus: (valid) Yes Few Many Large Small Lots of managers, but each manager earns almost no bonus: (valid) No (2) INSUFFICIENT:.


GMAT statistics note: This statement tells you about the total amount awarded to the managers. However, this total amount could be received by only a single manager or split across a huge number of managers.


GMAT statistics note: At least 20% of the employees in the group are managers.(1) The total amount awarded to managers is at least 20% greater than the total amount awarded to engineers.


GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. # of Engineers # of Managers e m m total 20% + greater than e total?


GMAT quant note: Many One Very small Very large Only one manager, but that manager gets most of the bonus pool: (valid) Yes Few Many Equal Equal Everybody gets the same bonus, but there are significantly more managers, so most of the pool goes to managers: (valid) No (1) AND (2) INSUFFICIENT:. Even considering both statements, there are multiple possible situations.


GMAT quant note: For instance, the group could be composed mostly of managers. In this case, since e = m, the answer to the question is No.


GMAT quant note: Another possibility is that the group contains an equal number of managers and engineers. Finally, since m is much greater than e, the answer to the question is Yes.


GMAT quant note: # of Engineers # of Managers e m (1) 20% + managers? (2) m total 20% + greater than e total?


GMAT quant note: Few Many Equal Equal Yes: (valid) Yes: (valid) No Equal Equal Very small Very large Yes: (valid) Yes: (valid) Yes.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try one more problem, this time Testing Cases based on odd and even numbers.


GMAT number properties note: Try-It #4-11 If a, b, and c are integers, is abc divisible by 4? Evaluate the possible odd/even combinations of a, b, and c without testing specific numbers.


GMAT number properties note: (1) INSUFFICIENT: 2c must be even because c is an integer. This statement implies that a + b is even, which occurs when a and b have the same odd/even parity.


GMAT number properties note: (There is no constraint on c.) (2) INSUFFICIENT: 2b must be even because b is an integer. Thus, this statement implies that a + c is odd, which occurs when a and c have opposite odd/even parity.


GMAT number properties note: (There is no constraint on b.) a + b + 2c is even.(1) a + 2b + c is odd.(2).


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. From (1) you know that a and b have the same odd/even parity, while from (2) you know that a and c have opposite odd/even parity.


GMAT quant note: A b c (1) a + b + 2c = Even (2) a + 2b + c = Odd Is abc divisible by 4? Even Even Odd ✓ (valid) ✓ Yes Odd Odd Even ✓ (valid) ✓ (valid) Maybe Even with these constraints, you do not have a definitive answer to the question.


GMAT quant note: Notice that in evaluating statements (1) and (2) together, you would not need to test completely new cases. Know what case you’re considering.


GMAT strategy note: Translate the wording into variables, relationships, and equations before calculating. There will be fewer opportunities to apply these tactics than the strategies mentioned previously.


GMAT strategy note: However, when all else fails, these tactics may be your only friend. The tactics are listed according to their reliability: the earlier tactics nearly always work, while the later tactics provide only a modest improvement over random guessing.


GMAT strategy note: Spot Identical Statements Certainty:. Very High IF the two statements tell you exactly the same thing (a er rephrasing)...


GMAT strategy note: Depending on (1) 3y – 6 = 2x.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The question, either each statement will be sufficient or each will not— because they are identical, there cannot be any benefit from looking at the statements together.


GMAT algebra note: Spot Clear Sufficiency Certainty:. Very High IF the two statements are clearly sufficient together...


GMAT algebra note: Try-It #4-12 If and mn ≠ 0, what is the value of Z? Quite o en, some algebraic work will m = 5(2).


GMAT quant note: Reveal that one or both of the statements will be sufficient on their own. Spot One Statement Inside the Other Certainty:.


GMAT quant note: Very High IF one statement is “contained within” (i.e., is a subset of) the other... However, the reverse is not true.


GMAT quant note: If (2) is insufficient, (1) could still possibly be sufficient on its own. Either way, it is impossible for both of the statements to be required together to answer the question.


GMAT quant note: So you can x > 50!(1) x > 10!(2).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. This situation can occur with inequalities.


GMAT number properties note: Be careful with this tactic, though. It can be easy to think—incorrectly— that one statement is a subset of the other.


GMAT number properties note: Almost every prime number is odd, but not all. Even if just one value escapes, you cannot use this tactic.


GMAT number properties note: Spot One Statement Adding Nothing Certainty:. High IF one statement adds no information to the other...


GMAT number properties note: (Some question involving x, y, and z) x is an odd number.(1) x is a prime number.(2).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This tactic may seem identical to the previous one, but it is not.


GMAT algebra note: Therefore, neither statement is a subset of the other. If y = 4, then you’d get 4x + (−4)x = z.


GMAT algebra note: Those equations are the same! The sign of y doesn’t matter because y and −y are symmetric.


GMAT algebra note: Moderate IF it is very obvious that the combined statements would be sufficient, but you can’t eliminate the possibility that one statement alone is sufficient...


GMAT quant note: Focus on equations, expressions, variables, and step-by-step manipulation. It does this by giving you two statements that are obviously sufficient when you put them together.


GMAT quant note: The test writers hope that you’ll jump to putting the two statements together, without testing each statement individually first. Try-It #4-13 If and ab ≠ 0, what is the value of K?


GMAT quant note: You can prove that the two statements combined are sufficient without actually doing any math. Then, you could plug those values into the question to solve for K.


GMAT quant note: A er all, you didn’t do anything to the question stem. Your next thought should be that one of the statements alone may be sufficient.


GMAT quant note: In some problems, the individual statements may a = 3(1) b − 3(5 − a) = 0(2).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In these cases, the more complicated statement may be enough.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This is not a tactic to use at lower levels of the test.


GMAT algebra note: This is one of many reasons why the material in this book is only appropriate if you have achieved a certain level of proficiency on the math side of the GMAT. Use Basic Algebraic Reasoning Certainty:.


GMAT algebra note: Rely on basic algebraic reasoning. Quite o en, basic intuition about algebra can lead you to the correct answer.


GMAT algebra note: Try-It #4-14 A sales manager at an industrial company has an opportunity to switch to a new, higher-paying job in another state. If his current annual salary is $50,000 and his current state tax rate is 5%, how much income after state tax would he make at the new job?


GMAT algebra note: His new salary will be 10% higher than his old salary.(1) His annual state taxes will total $2,200 in the new state at his new job.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. You may reason that in order to answer this question, you need to know how much his new salary will be and how much his taxes will increase by.


GMAT algebra note: Of course, you need to be very careful about using this tactic. Many problems on the GMAT are designed to hoodwink your algebraic reasoning —usually to make you think that you need to know every value precisely to answer the question.


GMAT algebra note: The problem explicitly states that the new job is higher-paying, so you only need to check whether the change in state tax might lead to a lower a er-tax compensation. You do not need to find out how much higher his pre-tax salary would be to answer this question.


GMAT algebra note: Because the GMAT frequently uses traps involving basic algebraic reasoning, you should only resort to using it if you are truly stuck. Spot Cross-Multiplied Inequalities Certaint.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This might be the hidden trick.


GMAT algebra note: If it is negative, you would need to flip the sign: . On some GMAT problems, you can assume the sign of the variables because the variables represent countable quantities of physical x > 0(2).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In those cases, you can assume that the variables are positive, and you can cross-multiply inequalities involving those variables at will.


GMAT algebra note: Judge by Appearance Certainty:. Low IF a statement is tricky and you don’t know what to do with it...


GMAT algebra note: THEN take a guess by “judging by appearance.” Sometimes a statement will leave you completely bewildered. In that case, the best (and only!) tactic is to guess whether it will be sufficient judging by how it looks:.


GMAT algebra note: The general rule is this: if the information in a statement has a structure and complexity similar to the question, and has the right ingredients (variables, coefficients, etc.), it’s more likely to be sufficient than otherwise. This won’t crack every case by any means, but you’d be surprised at how much mileage you can get from this tactic.


GMAT algebra note: Try-It #4-16 Does it look like it might be sufficient, even if you can’t see how? Does it use the variables you are looking for?


GMAT algebra note: Could it likely be manipulated into a form similar to that of the question?


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. However, what if you find yourself at a temporary loss on the test?


GMAT algebra note: Your exponent engine may shut down for a problem or two. It’s a very complicated expression, but it seems to have the right ingredients.


GMAT algebra note: It contains a and b and uses a as an exponent.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. I’ll bet if I manipulate this equation right, it will answer the question.


GMAT algebra note: The GMAT never needs to know that you guessed.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Summary That was a lot of information!


GMAT algebra note: If you can’t tell for certain whether the answer can be calculated in theory, keep going on the calculations all the way. Represent Word Problems with algebraic equations to avoid embedded tricks that can be difficult to find otherwise.


GMAT algebra note: Translate complicated questions and statements into simpler language. Bring explicit and implicit constraints to the surface.


GMAT algebra note: These constraints will o en be necessary to determine the correct answer. Whenever a problem involves inequality symbols, be careful—there are many ways in which inequalities can be used to trick you.


GMAT algebra note: There may be many possible scenarios in which a particular statement is true. Methodically write down these cases and check to see whether they result in different answers to the question.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Spot Clear Sufficiency (High):.


GMAT algebra note: Spot One Statement Inside the Other (High):. Spot One Statement Adding Nothing (High):.


GMAT algebra note: Use Basic Algebraic Reasoning (Moderate):. Apply basic knowledge of algebra, such as considering the number of unknowns relative to the number of known equations, to guide your thought process.


GMAT algebra note: Spot Cross-Multiplied Inequalities (Low):. If you’re completely unsure what to do, then make your best guess as to whether it appears to be sufficient.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Let’s call the first type of error a Type 1 error: you thought that a statement was sufficient, but it wasn’t.


GMAT fractions, decimals, and percents note: In other words, you thought that it gave you an exact answer to the question, but it was actually possible to get at least two different answers to the question. Here’s why that might happen:.


GMAT fractions, decimals, and percents note: The second type of DS error is a Type 2 error. If you made this type of error, you thought that a statement wasn’t sufficient, but it actually was.


GMAT fractions, decimals, and percents note: It You thought that a statement or both statements together were sufficient, but they were actually insufficient. You thought that a statement or both statements together were insufficient, but they were actually sufficient.


GMAT fractions, decimals, and percents note: If you tested cases, you might have missed a critical case. For instance, maybe the question has a different answer when you plug in a fraction or a negative value.


GMAT fractions, decimals, and percents note: You might have used information from the other statement without meaning to. This can make it seem as though you have enough information to answer the question, when you actually don’t.


GMAT fractions, decimals, and percents note: Keep the two statements completely separate on your paper and test them individually before you put them together.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Looked like you didn’t have enough information to answer the question, but you actually had more information than you thought:.


GMAT word problem note: The common thread in many of these errors is that they stem from assumptions. The easiest way to miss a DS problem is to assume that a statement is or isn’t sufficient, rather than proving it.


GMAT word problem note: You won’t always have time on test day to meticulously prove whether a statement is sufficient. But neither should you decide whether a statement is sufficient just by glancing at it.


GMAT word problem note: At the very least, briefly think about how the statement bears on the specific question you’re being asked. And if you have time, prove your suspicions about each statement before moving on.


GMAT word problem note: You may have assumed that you needed a specific value to answer a Yes/No question. The question might have asked something like “is x even?” If you treat this question as though you need to know the exact value of x, you’ll make Type 2 errors.


GMAT word problem note: If you assume that you need both statements in order to answer the question, it’s easy to miss the fact that one statement is actually sufficient on its own as well. Not Testing Cases o en leads to this type of error.


GMAT word problem note: Many statements look insufficient, because they don’t appear to include a lot of information.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Problem Set For problems 1−3, list five values that satisfy each of the following constraints.


GMAT number properties note: X2 > 0 , where M and N are positive integers. For problems 4–6, solve the problem by Testing Cases.


GMAT number properties note: Note which method is appropriate for each problem: testing the standard number set, listing discrete numbers, or case testing with concepts rather than numbers.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. At a certain florist shop, roses can be purchased either individually or as a bouquet of 12 at a discount of p percent.


GMAT number properties note: What is the greatest number of roses that can be purchased with $45? The greatest number of roses that can be purchased with $30 is p = 20(2) 6.


GMAT number properties note: For problems 7–9, Test cases to show that one or both statements are insufficient. If a, b, and c are positive integers, is abc an even integer?


GMAT number properties note: A + b is even.(1) b + c is odd.(2) 8. Twenty-eight students were each assigned to one of five classes:.


GMAT number properties note: Anthropology, Biology, Geology, Musicology, and Sociology. Was at least one of the classes assigned fewer than 5 students?


GMAT strategy note: Watch divisibility, factors, remainders, parity, and prime structure. At least 6 students were assigned to each of Biology, Geology, and Sociology.


GMAT strategy note: Geology and Sociology were not assigned the same number of students. Set A consists of the four integers x, x2, x3, and x4, and set B consists of the four integers x, 2x, 3x, and 4x.


GMAT strategy note: What is the probability that a randomly selected integer from 1 to 100, inclusive, is a member of neither set A nor set B? For problems 10–15, solve the problem.


GMAT strategy note: Also, which guessing tactics would not work, and why? The following advanced strategies and guessing tactics were discussed in this chapter:.


GMAT strategy note: Advanced Strategies Guessing Tactics x > 5(1) x < 10(2).


GMAT algebra note: Percent problems usually relate part, whole, rate, and percent change. The average (arithmetic mean) of the original six prices for six coats at a clothing store was $85.


GMAT algebra note: A er two of the six coats were each discounted by 20%, the average price of the six coats was $76. Was the coat with the lowest original price one of the two coats that were discounted?


GMAT algebra note: One of the discounted coats was the one with the highest original price. Before the discount, none of the coats had a price greater than 11.


GMAT algebra note: Amanda and Todd purchase candy, popcorn, and pretzels at the stadium. If a package of candy costs half as much as a bag of popcorn, how much more money did Amanda and Todd spend on the candy than on the popcorn and pretzels combined?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The cost of a bag of popcorn is equal to the cost of a pretzel.(1) Amanda and Todd purchased 24 packages of candy, 6 pretzels, and 6 bags of popcorn.


GMAT number properties note: What is the value of |x + 4|? X2 + 8x + 12 = 0(1) x2 + 6x = 0(2) 13.


GMAT number properties note: If m and n are positive integers, is n a multiple of 24? N is a multiple of (m + 4).(2) 15.


GMAT number properties note: What is the value of x(1 – y)(1 + y)?


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X = 0.5, 2, 3, 6.7, 0, −1, etc.


GMAT number properties note: (Note that the list should include integers and non-integers, positive and negative values—although the items you choose do not need to match these exact values. Anything other than 0 will work!) 3.


GMAT number properties note: M = 10, 17, 24, 31, 38, etc., and N = 1, 2, 3, 4, 5, etc.:. M must have a remainder of 3 when divided by 7:.


GMAT number properties note: Note that successive values of M differ by 7 and the corresponding values of N are consecutive integers. One way to generate this list is to pick consecutive numbers for N and then calculate values for M:.


GMAT number properties note: M = 7 + 3, 14 + 3, 21 + 3, 28 + 3, 35 + 3, etc. M = 10, 17, 24, 31, 38, etc.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. A Yes (1) AND (2) SUFFICIENT:.


GMAT statistics note: Combining the two constraints, 0 < a < 1. Within the standard number set, only is within this range.


GMAT statistics note: In addition, test numbers close to the boundaries of the range. A Yes Yes Yes It seems that at the extreme edges, the values are greater than 2, so statements (1) and (2) appear to be SUFFICIENT.


GMAT statistics note: The question asks you to maximize the number of roses that can be purchased for $45. To do so, first purchase as many 12-rose bouquets as.


GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. Possible, then purchase as many individual roses as possible with the remaining money.


GMAT quant note: The greatest number of roses that can be purchased with $30 is 24. In other words, two bouquets can be purchased for $30, and any le over money is not enough to purchase one individual rose.


GMAT quant note: Test cases to determine how many roses could be purchased with $45. If a bouquet of 12 roses costs exactly $15, then exactly 24 roses can be purchased with $30, and exactly 36 roses can be purchased with Case 2:.


GMAT quant note: Try to find a case in which only 24 roses can be purchased with $30, but more than 36 roses can be purchased with $45. This will happen if two bouquets cost less than $30 and the le over money isn’t enough to purchase any more individual roses.


GMAT quant note: But the le over money from $45 when three bouquets are purchased is just enough to purchase an additional rose. For instance, suppose that a bouquet costs $14 and an individual rose costs $2.50 before the discount.


GMAT quant note: However, 37 roses could be purc.


GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. The price of an individual rose is unknown.


GMAT quant note: Therefore, any number of roses might be purchased for $45. First, use the case in which exactly 36 roses can be purchased for $45.


GMAT quant note: Then, look for a case in which more than 36 roses can be purchased for $45. If a bouquet of roses costs exactly $15, then exactly 36 roses can be purchased for $45.


GMAT quant note: (For reference, this implies each rose in a bouquet cost = $1.25, so the individual [not discounted] rose price is Case 2:. Determine whether more than 36 roses could be purchased for $45.


GMAT quant note: If roses cost only slightly less than they did in Case 1, it might still only be possible to purchase 24 roses for $30, but it might also be possible to purchase one extra individual rose with the le over money. Try a cost of $1.50 per rose.


GMAT quant note: If a rose costs $1.50 individually, then a bouquet costs 12($1.50)(0.80) = $14.40, Since two bouquets cost 2($14.40) = $28.80, there isn’t enough money le over to purchase another single rose. Now three bouquets cost 3($14.40) = $43.20, and one more rose can be purchased for $1.50, bringing the total purchase to $44.70.


GMAT quant note: In this case, 37 roses can be purchased with $45.


GMAT coordinate geometry note: Watch divisibility, factors, remainders, parity, and prime structure. Since either 36 or 37 roses could be purchased for $45, the two statements are insufficient together.


GMAT coordinate geometry note: Because there are no constraints on y, use the standard number set, displayed here in number line graphic form:. Notice that you could use Guessing Tactic 3: spot one statement inside the other.