PLUG IN NUMBERS FOR “DEFINED OPERATION” QUESTIONS At least one of your 25–26 Problem Solving questions will probably be an example of what’s called a “defined operation.” These questions look weird and therefore might strike you as difficult. But they’re really not. In fact, the math turns out to be ridiculously easy. What’s being tested is your ability to understand what the problem requires and then to perform the simple arithmetical calculations—carefully! 12. Let ac b d be defined for all numbers a, b, c, and d by ac b d 5 ac 2 bd. If x 5 52 4 1 , what is the value of x 2 10 1 ? (A) 1 (B) 2 (C) 18 (D) 38 (E) 178 The correct answer is (B). In defining the diamond-shaped figure as “ac 2 bd,” the test makers are saying that whenever you see four numbers in a diamond like this, you should plug them into the mathematical expression shown in the order given. The question itself then requires you to perform this simple task twice. First, let’s figure out the value of x.Ifx is the diamond labeled as x, then a 5 5, b 5 4, c 5 2, and d 5 1. Now, we plug those numbers into the equation given, and do the simple math: x 5~5 3 2!2~4 3 1! x 510 2 4 x 5 6 Now, we tackle the second step. Having figured out the value of x, we can plug it into our second diamond, where a 5 6, b 5 10, c 5 2, and d 5 1. Again, plug in the numbers and do the math: (6 3 2) 2 (10 3 1) 5 12 2 10 5 2 As you can see, the math is very easy; the trick is understanding what the test makers are doing, which is “defining” a new math operation and then carefully plugging in the numbers and working out the solution. With a little practice, you’ll never get a “defined operation” question wrong. KEYS TO SUCCESSFUL GMAT PROBLEM SOLVING Here are some basic tips you should follow for any type of Problem Solving question. Apply these “keys” to the Practice Tests in Part VI, and then review them again just before exam day. Chapter 7: Problem Solving 173 www.petersons.com Narrow Down Answer Choices Up Front by Sizing Up the Question If the question asks for a number value, you can probably narrow down the answer choices by estimating the value and type of number you’re looking for. Use your common sense and real-world experience to formulate a “ballpark” estimate for word problems. QUESTION 1 You can narrow down answer choices by looking at the problem from a “commonsense” viewpoint. The five answer choices in this question provide some useful clues. Notice that they range in value from 4.8 to 13.0. That’s a wide spectrum, isn’t it? But what general value should you be looking for in a correct answer to this question? Without crunching any numbers, it’s clear that most of the juice will still remain in the bottle, even after lunch. So you’re looking for a value much closer to 13 than to 4. So you can safely eliminate (A) and (B). Common Sense Can Sometimes Reveal the Right Answer In many questions, you can eliminate all but the correct answer without resorting to precise calculations. QUESTION 1 Look at the question from a broader perspective. If you subtract 10% from a number, then 20% from the result, that adds up to a bit less than a 30% decrease from the original number. Thirty percent of 16 ounces is 4.8 ounces. So the solution must be a number that is a bit greater than 11.2 (16 2 4.8). Choice (D), 11.5, is the only choice that fits the bill! QUESTION 3 In Question 3, notice that we made c a much greater number than either p or q. Only a fraction with c in the numerator and a large number in the denominator (or vice versa) is likely to yield a quotient you’re looking for. With this in mind, choice (B) jumps off the paper at you as the likely choice! Scan the Answer Choices for Clues to Solving the Problem Scan the answer choices to see what all or most of them have in common—such as radical signs, exponents, factorable expressions, or fractions. Then try to formulate a solution that looks like the answer choices. QUESTION 3 Notice that each answer choice includes all three letters (p, q, and c). So the solution you’re aiming for must also include all three letters. Also, notice that every answer but choice (E) is a fraction. So anticipate building a fraction to solve the problem. 174 PART IV: GMAT Quantitative Section www.petersons.com Don’t Be Fooled by Too-Obvious Answer Choices The test makers will intentionally tempt or “bait” you with wrong-answer choices that result from making common errors in calculation and in setting up and solving equations. Don’t assume that your response is correct just because your solution appears among the five answer choices! Rely instead on your sense for whether you understood what the question called for and performed the calculations and other steps carefully and accurately. QUESTION 1 In this question, each of the four incorrect choices is sucker bait: (A) 4.8 You performed the wrong calculation: 30% of 16 ounces 5 4.8 ounces (B) 5.5 This is the number of ounces Susan drank. (The question asks for the amount remaining.) (C) 11.2 You performed the wrong calculation: 30% of 16 ounces 5 4.8 ounces 16 2 4.8 5 11.2 (D) 11.5 This is the correct answer. (E) 13.0 You confused percentages with raw numbers, erroneously converting 30% (10% 1 20%) into 3.0: 16 2 3.0 5 13.0 QUESTION 2 This question contains two sucker answer choices: (A) 2 This would be the correct answer to the question: “What is the difference between 19 and 21?” But this question is asking something entirely different. (E) 20 20 is simply 19 1 21 divided by 2. If this solution strikes you as too simple, you’ve got good instincts. Don’t Do More Work Than Needed to Get to the Answer If the question asks for an approximation, that’s a huge clue that precise calculations aren’t necessary. QUESTION 1 Notice that each answer choice is carried to exactly one decimal place, and that the question asks for an approximate value. These two features are clues that you can probably round off your calculations to the nearest tenth as you go. Chapter 7: Problem Solving 175 www.petersons.com Look for Shortcuts to Conventional Ways of Solving Problems The adage “There’s more than one way to skin a cat” applies to many GMAT Problem Solving questions. QUESTION 2 You can solve this problem quickly by simply comparing the two sums. Before the sixth number is removed, the sum of the numbers is 114 (6 3 19). After removing the sixth number, the sum of the remaining numbers is 105 (5 3 21). The difference between the two sums is 9, which must be the value of the number. Know When to Plug In Numbers for Variables If the answer choices contain variables (like x and y), the question might be a good candidate for the “plug-in” strategy. Pick simple numbers (so the math is easy) and substitute them for the variables. You’ll definitely need your pencil for this strategy. QUESTION 3 This question was a perfect candidate for the plug-in strategy. Instead of trying to figure out how to set up and solve an algebraic equation, in step 3 we used easy numbers for the three variables, then plugged those numbers into each answer choice to see which choice worked. Know When to Work Backward from Numerical Answer Choices If a Problem Solving question asks for a number value and if you draw a blank about how to set up and solve the problem, don’t panic. You might be able work backward by testing each answer choice. This might take a bit of time, but if you test the answer choices in random order, the statistical odds are that you’ll only need to test three choices to find the correct one. QUESTION 2 You already learned that comparing the two sums is the quickest shortcut to the answer. But if this strategy didn’t occur to you, working backward from the answer choices would be the next quickest method. After the sixth number is removed, the sum of the five remaining numbers is 21 3 5 5 105. So to test an answer choice, add this sum to the number provided in the choice, dividing the new sum by 6. If the result is 19, you’ve found the correct choice. Here’s how to do the math for choice (C), which is the correct answer: 105 1 9 6 5 114 6 5 19 Problem Solving questions always list numerical answer choices in ascending order of value. So if you use the strategy of working backward, start with the median value: choice (C). If (C) turns out too great, you know the correct answer must be either (A) or (B). Conversely, if (C) turns out too small, then either (D) or (E) must be correct. Of course, you might also be able to eliminate an answer choice right away by sizing up the questions (a previous strategy). Doing so would make your job even easier! 176 PART IV: GMAT Quantitative Section www.petersons.com Always Check Your Work Always check your work. Here are three suggestions for doing so: Do a reality check. Ask yourself whether your solution makes sense based upon what the question asks. (This check is especially appropriate for word problems.) For questions where you solve algebraic equations, plug your solution into the equation(s) to make sure it works. Confirm your calculations (except for the simplest no-brainers) with your calcula- tor. It’s amazingly easy to accidentally push the wrong button. Checking your calculations is especially crucial for questions asking for an approximation. Why? If your solution doesn’t precisely match one of the five answer choices, you might conclude that you should just pick the choice that’s closest to your solution—a big mistake if you miscalculated! QUESTION 1 A reality check on this question will tell you that answer choice (C), 11.5, seems about right, but that most of the other choices don’t. Read the Question One Last Time Before Moving On Among GMAT test takers, simple carelessness in reading a Problem Solving question is by far the most likely cause of an incorrect answer. So even if your solution is among the choices and you’re confident your calculations are accurate, don’t move on quite yet. Read the question again. Make sure you answered the precise question asked. For example, does the question ask for: • Arithmetic mean or median? • A circumference or an area? • A sum or a difference? • A perimeter or a length of one side only? • An aggregate rate or a single rate? • Total time or average time? Also check to make sure you: • Used the same numbers provided in the question • Didn’t inadvertently switch any numbers or other expressions • Didn’t use raw numbers where percentages were provided or vice-versa Chapter 7: Problem Solving 177 www.petersons.com QUESTION 1 The question asked for the amount of juice remaining, not the amount Susan drank. Also, a careless test taker might subtract 10 ounces instead of 10%. QUESTION 2 A careless test taker might inadvertently switch the numbers 19 and 21. QUESTION 3 The question asks for an answer in cents, not dollars. 178 PART IV: GMAT Quantitative Section www.petersons.com SUMMING IT UP • For success in the GMAT Problem Solving questions, follow the 5-step approach in this chapter: size up the question, appraise the answer choices, check for shortcuts to finding the answer, set up the problem and solve it, and verify your response before moving to the next question. • Problem Solving questions are designed to “reward” you for recognizing easier or more intuitive ways of finding the correct answer, so be on the alert for possible shortcuts. • Don’t look for easy solutions to complex problems, however. Those that involve algebraic formulas generally aren’t solved by adding or subtracting a few numbers. • Always check your calculations. Careless mistakes are the leading cause of incorrect responses on the GMAT Quantitative section. • Problem Solving questions list numerical choices in ascending order of value. So if you have to work backward, start from the middle choice (C). If it turns out to be too great, you know the correct answer must be choice (A) or (B); if it turns out to be too small, you can focus on choice (D) or (E) as the correct answer. Chapter 7: Problem Solving 179 www.petersons.com Data Sufficiency and Analysis OVERVIEW • The 5-step plan for data sufficiency problems • Data sufficiency strategies • Keys to successful GMAT data sufficiency • The 5-step plan for data analysis problems • Keys to successful GMAT data analysis • Summing it up In this chapter, you’ll learn these basics: • A step-by-step approach to handling all Data Sufficiency and Analysis questions • Keys for successfully tackling Data Sufficiency and Analysis questions The Data Sufficiency format is unique to the GMAT; you won’t find it on any other standardized test. Each Data Sufficiency consists of a question followed by two statements—labeled (1) and (2). Your task is to analyze each of the two statements to determine whether it provides sufficient data to answer the question and, if neither suffices alone, whether both statements together suffice. These are your answer choices: (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 an- swer the question asked, and additional data specific to the problem are needed. chapter 8 181 You’ll also learn several more advanced techniques for achieving your highest possible score on the Quantitative section of the GMAT. These include: • Applying the basic techniques to more challenging Data Sufficiency questions • Learning additional ways to apply techniques to certain types of Data Sufficiency questions, with example questions for practice • Learning a step-by-step approach to handling any Data Analysis question • Learning how to tackle Data Analysis questions • Further exploring some of the strategies listed above by applying them to GMAT-style questions that are more challenging THE 5-STEP PLAN FOR DATA SUFFICIENCY PROBLEMS The first task in this chapter is to learn the five basic steps for handling any GMAT Data Sufficiency problem: Size up the question Size up the two statements and look for a shortcut Consider Statement (1) alone Consider Statement (2) alone If neither statement alone answers the question, consider both together Later in this chapter, we’ll apply this 5-step approach to four sample Data Sufficiency questions. Step One: Size Up the Question As with Problem Solving questions, assess what specific mathematical area is being tested (e.g., what mathematical rules and formulas come into play). By determining what you’re up against, you’re well on your way to dealing with the question. Data Sufficiency questions, just like Problem Solving questions, vary widely in difficulty level. Try to get a feel for your limitations in handling complex questions. Determine how much time you’re willing to spend on the question, if any. Step Two: Size Up the Two Statements and Look for a Shortcut Before you plunge into a full-blown analysis of statement (1), read both statements and ask yourself: • Do the statements provide essentially the same information? If so, the answer is probably either choice (D) (“Each statement ALONE is sufficient to answer the question asked”) or choice (E) (“Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed”). 182 PART IV: GMAT Quantitative Section www.petersons.com . question asks for an answer in cents, not dollars. 178 PART IV: GMAT Quantitative Section www.petersons.com SUMMING IT UP • For success in the GMAT Problem Solving questions, follow the 5-step. answer but choice (E) is a fraction. So anticipate building a fraction to solve the problem. 174 PART IV: GMAT Quantitative Section www.petersons.com Don’t Be Fooled by Too-Obvious Answer Choices The. the difference between 19 and 21?” But this question is asking something entirely different. (E) 20 20 is simply 19 1 21 divided by 2. If this solution strikes you as too simple, you’ve got good