• Does either statement establish a solvable system of equations (for example, two equations in two variables)? • Does a statement seem to merely repeat (paraphrase) all or some of the information in the question? (If so, you can’t answer the question with that statement alone.) Asking yourself questions such as these may in some cases enable you to determine the correct answer choice without doing any more work. Otherwise, proceed to step 3. Step Three: Consider Statement (1) Alone If the information provided in statement (1) suffices to answer the question, eliminate choices (B), (C), and (E) as viable answer choices. On the other hand if statement (1) is insufficient alone, eliminate choices (A) and (D) as viable answer choices. Step Four: Consider Statement (2) Alone If the information provided in statement (2) answers the question, eliminate choices (A), (C), and (E) as viable answer choices. On the other hand if statement (2) is insufficient alone, eliminate choices (B) and (D) as viable answer choices. Step Five: If Neither Statement Alone Answers the Question, Consider Both Together Now if you can answer the question, the correct answer choice is (C). If you still don’t have enough information, the correct answer choice is (E). Sample Questions 1. If a jewelry merchant bought a particular ring for $10,000 and sold the ring to Judith, how much did Judith pay for the ring? (1) The merchant’s profit from the sale was 50%. (2) The amount that the merchant paid for the ring was two-thirds the amount that Judith paid for the ring. (A) Statement (1) ALONE is sufficient, but statement (2) alone is NOT sufficient to answer the question asked. (B) Statement (2) ALONE is sufficient, but statement (1) alone is NOT sufficient to answer the question asked. (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. (D) Each statement ALONE is sufficient to answer the question asked. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed. TIP If you’re pressed for time, take your best guess and move on after step 3. Your odds of selecting the correct answer choice are fairly good at this point. NOTE On the actual GMAT CAT screen, you’ll select your choice by clicking on one of five blank ovals (instead of lettered answer choices). Chapter 8: Data Sufficiency and Analysis 183 www.petersons.com 2. The symbol N represents the third digit in the 5-digit number 62,N 79. What number does N represent? (1) 62,N79 is a multiple of 3. (2) The sum of the digits of 62,N79 is divisible by 4. (A) Statement (1) ALONE is sufficient, but statement (2) alone is NOT sufficient to answer the question asked. (B) Statement (2) ALONE is sufficient, but statement (1) alone is NOT sufficient to answer the question asked. (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. (D) Each statement ALONE is sufficient to answer the question asked. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed. 3. If xy Þ 0, is x . y? (1) |x|.|y| (2) x 5 2y (A) Statement (1) ALONE is sufficient, but statement (2) alone is NOT sufficient to answer the question asked. (B) Statement (2) ALONE is sufficient, but statement (1) alone is NOT sufficient to answer the question asked. (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. (D) Each statement ALONE is sufficient to answer the question asked. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed. 4. In the figure above, is AB equal in length to AC? (1) x 1 y 5 z (2) y 5 180 2 z (A) Statement (1) ALONE is sufficient, but statement (2) alone is NOT sufficient to answer the question asked. (B) Statement (2) ALONE is sufficient, but statement (1) alone is NOT sufficient to answer the question asked. (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. (D) Each statement ALONE is sufficient to answer the question asked. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed. 184 PART IV: GMAT Quantitative Section NOTE Most Data Sufficiency questions will not include diagrams (geometry figures, graphs, and charts). www.petersons.com Apply the 5-Step Plan Let’s review the four sample questions one at a time using the 5-step plan you just learned. By now, you’re probably familiar with the five answer choices, so we won’t bother including them with the questions from now on. QUESTION 1 Question 1 is a relatively easy question. Approximately 85% of test takers respond correctly to questions like it. Here’s the question again: 1. If a jewelry merchant bought a particular ring for $10,000 and sold the ring to Judith, how much did Judith pay for the ring? (1) The merchant’s profit from the sale was 50%. (2) The amount that the merchant paid for the ring was two-thirds of the amount that Judith paid for the ring. Step 1: The focus of this question is the concept of percent increase—in the context of a word problem involving profit. This type of question is usually fairly easy, so you can expect to determine the correct response within a minute—without resorting to an educated guess. It should be worth investing your time on this one. Step 2: Notice that the two statements (1 and 2) provide the same information—only in different ways! This is a huge clue that the correct answer choice is either (D) or (E). You’ll still have to consider one of the two statements alone, but that should suffice. Step 3: Consider the premise, along with statement (1) alone. (Disregard statement (2) for now.) Given that the merchant paid $10,000 for the ring, if the merchant earned a 50% profit from the sale to Judith, determining Judith’s ring price is a simple matter of adding 50% of $10,000 to $10,000: $10,000 1 .5($10,000) 5 Judith’s ring price At this point, it’s clear that you can determine Judith’s ring price by simple multiplication and addition. Don’t waste time actually computing Judith’s ring price. You know that statement (1) alone suffices to answer the question and that’s all you need to know! Eliminate choices (B), (C), and (E) from consideration. The correct choice must be either (A) or (D). Step 4: If you’re not convinced that both statements say essentially the same thing, go ahead and consider the premise along with statement (2) alone. (Disregard statement (1) for now.) If the merchant’s cost was 2 3 the amount Judith paid, then Judith paid 3 2 of the merchant’s cost. Determining Judith’s ring price is a simple matter of multiplying $10,000 by 3 2 : $10,000 3 3 2 5 Judith’s ring price At this point, it’s clear that you can determine Judith’s ring price by simple multiplication. As in step 3, don’t waste time actually computing that price. You know that statement (2) alone suffices to answer the question, and that’s all you need to know. Chapter 8: Data Sufficiency and Analysis 185 www.petersons.com Step 5: This step is unnecessary here. There’s no need to consider both statements together. You know that either statement (1) or (2) alone suffices to answer the question, so you can eliminate choice (A). The correct answer is (D). QUESTION 2 Question 2 is average in difficulty level. Approximately 65% of test takers respond correctly to questions like it. Here’s the question again: 2. The symbol N represents the third digit in the 5-digit number 62,N 79. What number does N represent? (1) 62,N79 is a multiple of 3. (2) The sum of the digits of 62,N79 is divisible by 4. Step 1: This question is testing factors and divisibility. The peculiar use of a “placeholder” is a typical GMAT technique for testing your understanding of integers and digits. Questions such as these are usually straightforward once you know the basic rules, as well as a few shortcuts for divisibility. Step 2: Both statements appear to add different information to the question. So there’s no obvious shortcut here. (Go on to step 3.) Step 3: Consider statement (1) alone. If the sum of the digits of a number is divisible by 3, the number is also divisible by 3. Excluding the digit represented by N, the sum of the digits in the number 62,N79 is 24. Accordingly, if the number is a multiple of (divisible by) 3, the missing digit must be 0, 3, 6, or 9. Since there’s more than one possible value for N , statement (1) alone is insufficient to answer the question. Eliminate answer choices (A) and (D). Step 4: Consider statement (2) alone. The number that N represents can be 0, 4, or 8. Thus, statement (2) alone is insufficient to answer the question. Eliminate answer choice (B). Step 5: Consider statements (1) and (2) together. The two statements together establish that the missing digit is 0, because 0 is the only common number in the two lists of possible values for N. Thus, statements (1) and (2) together are sufficient to answer the question. The correct answer is (C). QUESTION 3 Question 3 is moderately difficult. Approximately 45% of test takers respond correctly to questions like it. Here’s the question again: 3. If xy Þ 0, is x . y? (1) |x| .|y| (2) x 5 2y Step 1: This is a typical absolute value question. Whenever you see inequalities and variables but no numbers, that’s a clue that you’ll need to consider different types of numbers—such as negative numbers, positive numbers, fractions, and perhaps the numbers 0 and 1—to determine the correct answer choice. Getting to the answer might entail performing some simple calculations, and perhaps a bit of trial and error (plugging in possible values). 186 PART IV: GMAT Quantitative Section www.petersons.com Step 2: Both statements appear to add different information to the question. So there’s no obvious shortcut here. But a good reasoned guess at this point would be that the correct answer choice is (E). Why? Because the question doesn’t restrict the value of either x or y (except that neither can equal 0). So if you’re pressed for time, guess choice (E) and move on to the next question. Otherwise, go on to step 3. Step 3: You must consider both positive and negative values for x and y. Given |x| . | y|,an x-value of either 4 or 24 and a y-value of 2, for example, satisfies the inequality but results in two different answers to the question. Thus, statement (1) alone is insufficient to answer the question. Eliminate answer choices (A) and (D). Step 4: Similarly, given x 5 2y, if you use negative values for both x and y (for example, x 524 and y 522), the answer to the question is no; but if you use positive values (for example, x 5 4 and y 5 2), the answer to the question is yes. Thus, statement (2) alone is insufficient. Eliminate answer choice (B). Step 5: Statements (1) and (2) together are still insufficient. For example, if x 524 and y 522, both statements (1) and (2) are satisfied, x , y, and the answer to the question is no. However, if x 5 4 and y 5 2, statements (1) and (2) are both satisfied, but x . y, and the answer to the question is yes. Eliminate answer choice (C). The correct answer is (E). QUESTION 4 Question 4 is a relatively difficult question. Approximately 30% of test takers respond correctly to questions like it. Here’s the question again: 4. In the figure above, is AB equal in length to AC? (1) x 1 y 5 z (2) y 5 180 2 z Step 1: This question is a geometry problem involving the isosceles triangle. (You’ll see anywhere from five to eight geometry questions on your GMAT.) This question involves three distinct rules of geometry. Two of these rules (A and C below) apply specifically to triangles: Rule A: If two angles of a triangle are congruent, then the two sides opposite those angles are congruent. Rule B: If angles formed from the same vertex form a straight line, their degree measures total 180 (and they are known as “supplementary” angles). Rule C: In any triangle, the sum of the degree measures of the three interior angles is 180. Chapter 8: Data Sufficiency and Analysis 187 www.petersons.com If you’re unfamiliar with any of the three rules in step 1, you won’t get very far with this question! So if you’re pressed for time and if you’re particularly weak in this area of geometry, consider taking a guess and moving on. Step 2: Intuition alone probably won’t get you very far on this question. If you’re really on your toes, you’ll notice that statement (1) merely restates Rule C (see step 1) in a different form. Also because statement (2) includes a number, this statement is probably more likely than statement (1) to suffice in answering the question. (This amounts to little more than a guess, however.) So let’s move on to step 3. Beware: Don’t shortcut the analysis by simply measuring the lengths with your eye. Data Sufficiency figures are not necessarily drawn to scale, so analyze these problems using your knowledge of mathematics, not your eye. Step 3: Consider statement (1) alone. Given Rule A (see step 1) to answer the question, you need to know whether angle y is congruent to the triangle’s unidentified angle—the interior angle at point C. Let’s call this angle a.Ifa 5 y, then the answer to the question is yes. Otherwise, the answer is no. In either case, we need to know whether a 5 y in order to answer the question. Together, angles a and z form a straight line—the line passing through points A and C: a 1 z 5 180 a 5 180 2 z The sum of x, y, and a is 180 (Rule C). You can substitute (180 2 z)fora in this equation, and manipulate the result so that it is identical to the equation in statement (1): x 1 y 1 a 5 180 Rule C (sum of angle measures is 180°) x 1 y 1~180 2 z!5180 substituting (180 2 z)fora x 1 y 2 z 5 0 subtract 180 from each side x 1 y 5 z add z to each side Statement (1) essentially restates a rule that is true for any triangle, so it is insufficient alone to answer the question. Eliminate answer choices (A) and (D) as viable choices. Notice that you could have “shortcut” this entire analysis had you already been aware of the rule that the measure of an exterior angle of a triangle is always equal to the sum of the measures of the two remote interior angles. Step 4: Now consider statement (2), disregarding statement (1) for now. The expression (180 2 z) equals our third unidentified angle, which we called a in step 3. Given that (180 2 z) also equals y, the two angles a and y are congruent (equal in degree measure). The two sides opposite a and y must also be congruent (see Rule A). Thus, statement (2) alone suffices to answer the question. Step 5: Because statement (1) alone is insufficient to answer the question, while statement (2) alone is sufficient, the correct answer is (B). There’s no need to consider the two statements together. Based on statement (2), the answer to the question itself is yes, but you don’t need to go this far. Had neither (1) nor (2) alone been sufficient to answer the question, you would have then considered both statements together to determine whether the correct answer choice was (C) or (E). 188 PART IV: GMAT Quantitative Section www.petersons.com DATA SUFFICIENCY STRATEGIES In this section, you’ll learn strategies for handling Data Sufficiency problems and see examples of each strategy. Plug In “Easy” Numbers, but Don’t Forget Negatives, Fractions, Zero, and One If a Data Sufficiency question involves variables, you can easily confuse yourself by thinking about the problem purely abstractly. You should also experiment with different numerical values. Take pencil to paper and scratch out some scenarios. This technique will help you see what’s behind the problem at hand. Just be sure to try all the different “types” of numbers that the problem allows (greater numbers, lesser numbers, positive and negative numbers, non-integers, as well as 0 and 1). If the answer to the question depends on what kind of values you plug in, then the correct answer choice must be (E). 5. If a, b, c, and d are all positive integers, is a b greater than c d ? (1) a . c (2) b . d The correct answer is (E). Neither statement alone allows you to compare the values of the two fractions. To see this, try plugging in some simple numbers. For example, let a 5 4 and c 5 2, in accordance with statement (1). Since you can choose any values for b and d, the possible values of either fraction are infinite in number, and so you can see that statement (1) alone is insufficient to answer the question. By the same reasoning, statement (2) alone is also insufficient. Now, consider the two statements together. Again, let a 5 6 and c 5 2, in accordance with statement (1). Now, start plugging in some values for b and d that meet the condition in statement (2), which is that b . d. Can you answer yes to the question? Easily; for example, b 5 2 and d 5 1. Can you answer no to the question? Easily; here’s just one possible way: b 5 4 and d 5 1. Stop here! You’ve found two different answers to the question and so you know the correct answer must be choice (E). Look for Two Statements That Say Essentially the Same Thing In most Data Sufficiency questions, one numbered statement will provide information that is different from the other statement. But this isn’t always the case. One skill that the test makers will test is your ability to recognize two statements that provide essentially the same information—just in a slightly different form. If they’re the same, you know the correct answer choice must be either (D) or (E). Chapter 8: Data Sufficiency and Analysis 189 www.petersons.com 6. A BC x° Is the triangle in the figure above equilateral? (1) Minor arc AB has a degree measure of 120°. (2) x 5 60 The correct answer is (E). The two statements tell us the same thing—that angle C measures 60°, as it would if the triangle were equilateral. But without further information, we can’t tell that angles A and B are also 60° angles. It’s possible, for instance, that angle A measures 59° while angle B measures 61°—in which case the triangle is not equilateral. Since we can’t know for sure, the answer must be choice (E). Take note: The triangle inscribed in the circle certainly looks equilateral (which means that all three sides are equal in length). But the apparent dimensions of the triangle in the diagram are irrelevant to answering the question. Only the facts given in the question and in the two numbered statements are important. Focus on Quantitative Concepts, Not on Number Crunching Data Sufficiency focuses more on mathematical concepts than on working toward a quantitative solution (which is what Problem Solving questions are primarily about). So be sure to size up the problem at hand. Ask yourself: “What rule, principle, or formula is the question covering?” Once you’ve figured this out, you should be able to handle the problem relatively easily. 7. What is the value of p 2 2 q 2 ? (1) p 1 q 524 (2) p 2 q 5 4 The correct answer is (C). Many test takers would jump headlong into trying various values for p and q in trial-and-error fashion. That’s not the way to approach this problem or, for that matter, any Data Sufficiency problem. Before you evaluate either statement alone, look at the expressions given in the problem. Did you notice that p 2 2 q 2 is the difference of two squares, and that the expressions given in the two statements provide its two binomial factors? In other words: p 2 2 q 2 5 (p 1 q)(p 2 q). This is the concept that the question is designed to cover. Once you see this, handling the problem is a snap. Although neither statement alone suffices to answer the question (because you’re dealing with a quadratic rather than a linear equation), statements (1) and (2) together provide the two binomials, allowing you to answer the question. (To calculate the answer, you would simply multiply: 24 3 4 5216.) 190 PART IV: GMAT Quantitative Section www.petersons.com Don’t Do More Work Than Necessary Keep in mind that the Data Sufficiency format does not require you to answer the question. So once you’ve convinced yourself which statement(s), if any (1, 2, both, neither), suffices to answer the question, stop right there! You’d only be wasting your precious time by figuring out the answer itself. 8. What is the average weight of the 5 members of a football team? (1) The average weight of the 3 heaviest team members is 340 pounds. (2) The 2 lightest team members weigh 275 and 290 pounds. The correct answer is (C). To calculate the average of a group of numbers, you must have two pieces of information: The total of the numbers and the number of numbers. In this case, the only missing piece of information is the total weight of the team members. (You already know the number of numbers involved: 5.) Neither statement (1) nor (2) alone gives you the players’ total weight. But if you combine statements (1) and (2), you can determine it. You’d multiply 340 by 3 (to get the combined weight of the heaviest team members) and add 275 and 290 (the weights of the two lightest members). But there’s no need to actually perform these steps. All that matters is that you can tell that it would be theoretically possible to make these calculations and so determine the average. This is enough to get the correct answer, choice (C). Don’t Assume Any Information Not Stated in the Problem One of the skills the test makers are measuring is your ability to distinguish facts provided in a Data Sufficiency problem from unsupported assumptions made out of carelessness or inattention. There’s a natural tendency to “invent” facts that aren’t really there so that you can answer the question. No test taker is immune to this tendency. You might be surprised how many “smart” GMAT test takers slip up in this way, robbing themselves of precious Quantitative score points. 9. What percentage of the female students in a certain history class is majoring in economics? (1) 50 percent of the students in the class are male and 50 percent are female. (2) 50 percent of all students in the class are majoring in economics. The correct answer is (E). Many test takers would carelessly assume that the percent of students majoring in economics is the same for the class’s male students as for its female students. If this were the case, then you could easily answer the question. (The answer would be 50.) But the problem provides no information to support this assumption! Thus, the correct answer must be choice (E). Chapter 8: Data Sufficiency and Analysis 191 www.petersons.com Beware of Statements That Are Irrelevant or Provide Facts Already Supplied in the Question Ask yourself what kind of information each statement provides. A statement is more likely to be sufficient to answer the question if: • It provides specific numerical values not given in the premise. • It adds something new to the premise. • It provides information that strikes you as relevant to the question. On the other hand a statement is more likely to be insufficient to answer the question if: • It does not provide any specific numerical values that the premise leaves unknown. • It seems redundant—simply paraphrasing the premise (or some part of it). • The information strikes you as irrelevant to the premise or question. 10. A certain granola recipe calls for a simple mixture of raisins costing $3.50 per pound with oats. At a cost of $2.00 per pound for the granola mixture, how many pounds of oats must be added to 10 pounds of raisins? (1) The granola mixture is packaged in one-pound bags. (2) Oats cost $1.00 per pound. The correct answer is (B). The question itself provides two of the three facts you need to answer it: the cost per pound of raisins and the cost per pound of the mixture. Statement (1) alone provides no useful information for answering the question. So without even looking at statement (2), you’ve eliminated answer choices (A), (C), and (D)! Statement (2) provides the third needed fact: the cost per pound of oats. Although you don’t need to do the math, here’s how you would answer the question with the additional information provided by statement (2). Think of the quantities as costs per pound and multiply the cost by the weight. The total mixture will consist of 10 pounds of raisins at $3.50 per pound, or ($3.50)(10), plus “x” pound of oats at $1.00 per pound, or ($1.00)(x). The mixture costs $2.00 per pound, and it will be (10 1 x) pounds: ~3.50!~10!1~1.00!~x!5~2.00!~10 1 x! 35 1 x 5 20 1 2x 15 5 x 15 pounds of oats are needed. Don’t Assume That Diagrams Are Accurate Although a diagram will conform to the information in the question, it won’t necessarily conform to either statement (1) or (2). So don’t use a Data Sufficiency figure to estimate or measure values, shapes, lengths, or other sizes. For example, don’t rely on a figure’s appearance to determine: • Whether one line segment is longer than another 192 PART IV: GMAT Quantitative Section www.petersons.com . you to answer the question. (To calculate the answer, you would simply multiply: 24 3 4 5216 .) 190 PART IV: GMAT Quantitative Section www.petersons.com Don’t Do More Work Than Necessary Keep. some simple calculations, and perhaps a bit of trial and error (plugging in possible values). 186 PART IV: GMAT Quantitative Section www.petersons.com Step 2: Both statements appear to add different. both statements together to determine whether the correct answer choice was (C) or (E). 188 PART IV: GMAT Quantitative Section www.petersons.com DATA SUFFICIENCY STRATEGIES In this section,