Now that you’ve seen how Problem Solving questions are constructed, let’s look at how to handle them. Kaplan has developed a Method for Problem Solving that you can use to attack each and every question of this type.
Begin your analysis of the problem by getting an overview. If it’s a word problem, what’s the basic situation? Or is this an algebra problem? An overlapping sets problem? A permutations problem? Getting a general idea of what’s going on will help you to organize your thinking and know which rules or formulas you’ll have to use.
If there’s anything that can be quickly simplified, do so—but don’t start solving yet. For example, if a question stem gives you the classic quadratic x2 – y2 = 64, it would be fine to rewrite it immediately in your scratchwork as (x + y) (x − y) = 64. But you wouldn’t want to start solving for values just yet; doing so would likely cause you to miss an important aspect of the problem or to overlook an efficient alternative solution.
Also, make sure to glance at the answer choices. Can they help you choose an approach? If they are variable expressions, you can use picking numbers (a strategy we’ll cover later in this chapter). If they are widely spread, you can estimate. If they are numerical values, you might be able to plug them back into the question, a strategy we call backsolving. Looking at the choices may trigger an important strategic insight for you.
Before you choose your approach, make sure you know what you’re solving for. The most common wrong answer trap in Problem Solving is the right answer—but to the wrong question. And perhaps you won’t have to do as much work as you might think—you may be able to solve for what you need without calculating the value of every single variable involved in the problem.
The operative word here is strategically. Resist the temptation to hammer away at the problem, hoping for something to work. Use your analysis from steps 1 and 2 to find the most straightforward approach that allows you to make sense of the problem.
There is rarely a single “right approach.” Choose the easiest for you given the current problem. Broadly speaking, there are three basic approaches.
Approach 1: A Kaplan Strategy. Frequently there will be a more efficient strategic approach to the math for those who analyze the problem carefully. Consider picking numbers or backsolving, which you will learn about later in this chapter. These approaches can simplify some tough problems and should always be on your mind as possible alternatives.
Approach 2: Straightforward Math. Sometimes simply doing the math is the most efficient approach. But remember, only do math that feels straightforward. To be sure, the hardest problems will make you sweat during the analysis, but you should never find yourself performing extremely complicated calculations.
Remember, the only thing that matters is that you select the correct answer. There is no human GMAT grader out there who’s going to give you extra points for working out a math problem the hard way.
Approach 3: Guess Strategically. If you notice that estimation or simple logic will get you the answer, use those guessing techniques. They will often be faster than doing the math. If you have spent 60 to 90 seconds analyzing the problem and still haven’t found a straightforward path to a solution, make a guess. You only have an average of 2 minutes per question, so you need to keep moving.
And if you have fallen behind pace, you need to guess strategically to get back on track. Hunt actively for good guessing situations and guess quickly on them (within 20–30 seconds). Don’t wait until you fall so far behind that you are forced into random guesses or forced to guess on questions that you could otherwise have easily solved. Never try to make up time by rushing!
Because the GMAT is adaptive, you aren’t able to return to questions you’ve already seen to check your work, so you need to build that step into your process on each question. The most efficient way to do this is to reread the question stem as you select your answer. If you notice a wrinkle in the problem that you missed earlier, you should redo the problem (if you have time) or change your answer. If you got everything right the first time, move on to the next problem with confidence.
Now let’s use the Kaplan Method and the strategic principles of Problem Solving to answer a sample question:
The phrase “total population” near the end of the question stem should stand out as key. One of the classic tricks used by the GMAT test makers is to give you a part-to-part ratio and ask for an answer expressed as a part-to-whole ratio.
You need to use the ratio of those born in-state to those born out-of-state to determine the ratio of those born in-state to the total population.
Having four times as many people born in-state as out-of-state means a ratio of 4:1. You are looking for the ratio of those born in the state to the total. You are told that 4 is the part of the ratio for those who were born in the state. You can find the total by adding the two parts, in-state (4) and out-of-state (1), getting 5. The part-to-whole ratio is then 4:5, choice (E).
A good natural-language paraphrase of the information is “Four of every five residents were born in-state.” That matches choice (E). You need to avoid traps that restate or distort the part-to-part information. Be wary of answer choice (A), which is just a distortion of the original ratio.
Now, let’s apply the Kaplan Method for Problem Solving to the question you saw at the beginning of this chapter:
George has a bunch of unpaired socks and is pulling them out at random. This doesn’t sound like the best way to get dressed, but that’s the situation the question gives you.
What’s the smallest number of socks George has to remove to be sure of getting a matching pair? In other words, how many socks does George need to pick before he is guaranteed to have 2 matching socks? The question doesn’t specify which color, so any matching color will do.
There’s no equation to set up here. Despite the fact that the question starts off sounding like a probability question, you’re not asked to calculate probabilities directly.
So think logically. Obviously, George can’t have a matching pair of socks if he only removes 1 sock. You need at least 2 for a pair. So what if George removes 2 socks? He could get a pair, of course; there’s nothing stopping him from randomly drawing, for example, 2 blue socks. But he also could get an unmatched pair—1 blue and 1 black, perhaps—so removing 2 socks doesn’t guarantee a matching pair.
What if he takes 3 socks? Again, he could match a color—1 blue and 2 whites, for example. But could he still have no match with 3 socks? Yes, as it’s possible (and happily you are not asked to calculate exactly how possible) for him to get 1 of each color—1 white, 1 blue, and 1 black.
But after that, he’s guaranteed to match 1 of the colors. After all, there are only 3 colors. So even if he didn’t have a match after taking out 3 (1 white, 1 blue, and 1 black), he’s guaranteed to have a pair when he selects a 4th sock. You can’t know whether that pair will be white, blue, or black, but the question didn’t ask you to get a specific color . . . any one will do. The correct answer is (B).
Reread the question stem, making sure that you didn’t miss anything about the problem or give the right answer to the wrong question—a common GMAT trap. For example, (E) is what you would have selected if you solved for the total number of socks minus 1 pair.