Now that you’re familiar with how Data Sufficiency questions are constructed, let’s look at two important strategic approaches to these questions. The first you’re already familiar with from the Problem Solving chapter: we’ll start by looking at how to effectively utilize the Picking Numbers strategy. After that, we’ll discuss in greater detail how to most effectively combine statements.
You’ve already seen the power of the Kaplan strategy of picking numbers for Problem Solving questions. You can also use this strategy for many Data Sufficiency questions that contain variables, unknown quantities, or percents of an unknown whole. When using this strategy, you always pick at least two different sets of numbers, trying to prove that the statements are insufficient by producing two different results. It’s usually easier to prove insufficiency than sufficiency. But as you practice this strategy, the Core Competency of Pattern Recognition will also come into play. You will become adept at recognizing the types of numbers that can produce different results: positives vs. negatives, fractions vs. integers, odds vs. evens, and so on. Also, don’t hesitate to use the numbers 0 and 1, as they have unique properties that make them great candidates for the picking numbers strategy.
Now let’s use the Kaplan Method on a Data Sufficiency question that involves picking numbers:
Here’s a Value question, so you need one, and only one, value for the expression c − d. There’s nothing much to simplify here, but keep in mind that you are given a value for another expression, a + b. To obtain sufficiency, you’ll need values for both c and d or a way to relate the equation a + b = 20 to the expression c − d.
As always, think strategically. Since the GMAT doesn’t present the statements in any particular order, it’s sometimes wise to start by evaluating Statement (2) if it looks easier to evaluate than Statement (1). Here, Statement (2) gives you a value for d but not for c. There’s also no way to relate the equation a + b = 20 to the expression c − d. Eliminate (B) and (D). Notice that you’ve eliminated 50 percent of the wrong answer choices very quickly.
Now let’s tackle Statement (1), remembering to use the information it provides in conjunction with the question stem. You’re given a + b = 20, so let’s use picking numbers here. Pick a = 10 and b = 10.
So, the expression c − d can equal 3. But you’re not finished yet. You have to pick a different set of numbers to see whether you can produce a different answer.
What permissible numbers might be likely to produce a different answer? Since Statement (1) involves subtraction, try negative numbers. Try a = 25 and b = −5.
After picking two sets of numbers that have different properties and receiving the same result, you can say with reasonable confidence that Statement (1) is sufficient. Eliminate (C) and (E). The correct answer is (A).
If—and only if—each statement on its own is insufficient, you must then consider the statements together. The best way to do this is to think of the statements as one long sentence. At this stage, you can essentially approach the question as you would a Problem Solving question, using all the information you’re given to answer the question. The main difference is that, since this is still Data Sufficiency, you will stop solving as soon as you know that you can solve. The time saved by avoiding unnecessary calculations is better spent on questions later in the section.
As you’ve learned earlier in this chapter, information is always consistent between the statements. The statements never contradict each other. For example, you’ll never see a question in which Statement (1) says that x must be negative and Statement (2) says that x is positive. You might, however, learn from Statement (1) that x is greater than −5 and learn from Statement (2) that it’s positive. Learning to recognize how one statement does or does not limit the information in the other is the key in deciding between choices (C) and (E).
Answers follow the exercises.
The following exercise contains a single question stem—“What is the value of x?”— and many sets of sample statements, which have already been simplified and evaluated for you. Imagine that you had analyzed the statements and gotten the possible values for x listed below: Which statements, either separately or combined, are sufficient to answer the question? Choose the appropriate Data Sufficiency answer choice—(A), (B), (C), (D), or (E)—for each pair of statements.
What is the value of x?
Now let’s use the Kaplan Method on a Data Sufficiency question that involves picking numbers:
an integer?
First, this is a Yes/No question. Any statement that gives you a “sometimes yes; sometimes no” answer is insufficient. Now, how can you paraphrase the question in the stem? You can say either, “Is 2x a multiple of y?” or “Does 2x divide evenly by y?”
Use the Kaplan strategy of picking numbers. Pick permissible, manageable numbers. If you pick x = 5 and y = 5, that will give you a yes to the original question, since 2x, or 10, divided by y, or 5, will yield an integer. But if you pick a different set of numbers, say x = 2 and y = 42, that will yield a fraction. Statement (1) is insufficient; eliminate (A) and (D).
Pick numbers for Statement (2), making sure they’re permissible. For example, 36 and 6 have the same distinct prime factors: 2 and 3. Picking the numbers x = 36 and y = 6 gives you an integer; choosing x = 6 and y = 36, on the other hand, gives you a fraction. Therefore, Statement (2) is insufficient; eliminate (B).
Now, combining the statements, you will notice something interesting. Statement (2) is more restrictive than Statement (1). So as you combine the statements, ask yourself, “Will any numbers that satisfy Statement (2) also satisfy Statement (1)?” Yes, they will. For instance, x = 36 and y = 6, which you picked for Statement (2), also work for Statement (1): these numbers will again yield a yes answer. And x = 6 and y = 36 also work for Statement (1), yielding a no answer. Since combining the statements didn’t add any new information and the information presented was insufficient, the answer must be (E). Even combined, you get a “sometimes yes, sometimes no” answer to the question in the stem.
When you run into a very complicated question, like the following, don’t forget to use a sound Data Sufficiency guessing strategy. That means perhaps skipping a statement that looks too daunting and trying to eliminate some answer choices by looking at the easier statement. Try your hand at the difficult question below. Don’t try to solve it; instead, see if you can narrow down the possibilities quickly.
Example:
This is a Value question. To obtain sufficiency, you need an exact maximum temperature.
Statement (1) is long and complicated. Skip it and go straight to Statement (2); it’s much easier.
Statement (2) tells you that the value you’re looking for is 5 degrees more than the temperature on some other day. Without knowing the temperature on that other day, you don’t have the information you need. This statement is insufficient. You can eliminate (B) and (D) and give yourself a one-in-three chance to get the right answer without even evaluating Statement (1).
Perhaps you feel comfortable evaluating Statement (1). Perhaps you don’t. But let’s pretend for a moment that you aren’t sure how to evaluate Statement (1). Keep in mind that on the GMAT, complicated or hard-to-evaluate statements are more likely to be sufficient than insufficient. For this reason, you should avoid (E) and lean toward (A), unless you have a logical reason to suspect that Statement (1) alone is insufficient. Of course, this doesn’t guarantee you a correct answer, but if you’re falling behind on time, it will help you move through the Quantitative section most efficiently. Remember, no one particular question will make or break your GMAT score, but spending too much time on one question and having to rush through several others just to make up for the lost time will hurt your score. As it turns out, (C) is the right answer to this particular problem.
Fortunately, you’re unlikely to run into anything as complicated as Statement (1) on Test Day. The point of this exercise is to show you that even if you do encounter something this difficult, you’re still in control. If you work the odds and look for the strategic approach, you will increase your likelihood of picking up the points even for questions you’re not sure how to solve.
For the record, let’s analyze Statement (1). If the average maximum temperature from May 8 to May 14 was 72 degrees, then the sum of the maximum temperatures of those days is 7 × 72 = 504 degrees. If the average maximum temperature from May 9 to May 13 was 72 + 2, or 74 degrees, then the sum of the maximum temperatures of those days was 5 × 74 = 370 degrees.
The difference between those two sums is simply the sum of the maximum temperature on May 8, which you can call x, and the maximum temperature on May 14, which you can call y (since these two days were left out of the second time period). So x + y = 504 − 370 = 134. Statement (1) by itself is insufficient. But Statement (2) tells you that y − x = 5. You have the two distinct linear equations, x + y = 134 and y − x = 5. These equations can be solved for a single value for y, so the statements taken together are sufficient: choice (C).
This example demonstrates how guessing can be a good alternative approach for certain questions. By looking at only one statement, you can narrow down the possibilities to two or three choices. This can be a great help, particularly on difficult or time-consuming problems for which you think you might have to guess. But you must be sure you know the rules for eliminating answer choices absolutely cold by Test Day.