The first task in this section is to learn the five basic steps for handling any GMAT Problem Solving question. You’ll apply these steps to three sample questions.

Sample Questions

Question 1 is a word problem involving changes in percent. (Word problems account for about half of the Quantitative questions.)

1. If Susan drinks 10% of the juice from a 16-ounce bottle immediately before lunch and 20% of the remaining amount with lunch, approximately how many ounces of juice are left to drink after lunch?
   
   A. 4.8%
   B. 5.5%
   C. 11.2%
   D. 11.5%
   E. 13.0%
   

This next Problem Solving question involves the concept of arithmetic mean (simple average).

2. The average of 6 numbers is 19. When one of those numbers is removed, the average of the remaining 5 numbers is 21. What number was taken away?
   
   A. 2
   B. 8
   C. 9
   D. 11
   E. 20
   

Here’s a somewhat more difficult Problem Solving question. This one involves the concept of proportion.

3. If p pencils cost 2q dollars, how many pencils can you buy for c cents? [Note: 1 dollar = 100 cents.]
   
   A. pc / 2q
   B. pc / 200q
   C. 50pc / q
   D. 2pc / c
   E. 200pcq
   

The 5-Step Plan

Here’s the 5-step approach that will help you to handle any Problem Solving question. Just a few pages ahead, we’ll apply this approach our three sample Problem Solving questions.

Step 1: Size up the question. Read the question and then pause for a moment to ask yourself:

• What specific subject area is being covered?
• What rules and formulas are likely to come into play?
• How complex is this question? (How many steps are involved in solving it? Does it require setting up equations, or does it require merely a few quick calculations?)
• Do I have a clue, off the top of my head, how I would begin solving this problem?

Determine how much time you’re willing to spend on the problem, if any. Recognizing a “toughie” when you see it may save you valuable time; if you don’t have a clue, take a guess and move on.

Step 2: Size up the answer choices. Before you attempt to solve the problem at hand, examine the answer choices. They can provide helpful clues about how to proceed in solving the problem and about what sort of solution you should be aiming for. Pay particular attention to the following:

• Form: Are the answer choices expressed as percentages, fractions, or decimals? Ounces or pounds? Minutes or hours? If the answer choices are expressed as equations, are all variables together on one side of the equation? As you work through the problem, rewrite numbers and expressions to the same form as the answer choices.



• Value: Are the answer choices extremely small valued numbers? Numbers between 1 and 10? Greater numbers? Negative or positive numbers? Do the answer choices vary widely in value, or their values clustered closely around an average? If all answer choices are tightly clustered in value, you can probably disregard decimal points and extraneous zeros in performing calculations. At the same time, however, you should be more careful about rounding off your figures where answer choices do not vary widely. Wide variation in value suggests that you can easily eliminate answer choices that don’t correspond to the general value of number suggested by the question.



• Other distinctive properties and characteristics: Are the answer choices integers? Do they all include a variable? Do one or more include radicals (roots)? Exponents? Is there a particular term, expression, or number that they have in common?

Step 3: Look for a shortcut to the answer. Before plunging headlong into a problem, ask yourself if there’s a quick, intuitive way to get to the correct answer. If the solution is a numerical value, perhaps only one answer choice is in the right ballpark. Also, some questions can be solved intuitively, without resort to equations and calculations. (You’ll see how when we apply this step to our sample questions.)

Step 4: Set up the problem and solve it. If your intuition fails you, grab your pencil, roll up your sleeves, and do whatever computations, algebra, or other procedures are needed to solve the problem at hand. Simple problems may require just a few quick calculations, while complex algebra and geometry questions may require setting up and solving a series of equations.

Step 5: Verify your response before moving on. After solving the problem, if your solution does not appear among the answer choices, go back and check your work. (You obviously made at least one mistake.) If your solution does appear among the choices, don’t celebrate quite yet. Although there’s a good chance your answer is correct, it’s possible your answer is wrong, and that the test-maker anticipated your error by including a “sucker bait” answer choice—just for you and other test-takers who made the same mistake. (We’ll look at some “sucker-bait” answer choices a few pages ahead.) So check the question to verify that your response corresponds to what the question calls for—in terms of value, expression, units of measure, and so forth. If it does, and you’re confident that your work was careful and accurate, don’t spend any more time checking your work. Confirm your response and move on to the next question.

Apply the 5-Step Plan

It’s time to go back to the three sample questions you looked at a few pages back. Let’s walk through them—one at a time–using the 5-step game plan you just learned.

Question 1

Question 1 is a relatively easy question. Approximately 80% of test-takers respond correctly to questions like it. Here’s the question again:

1. If Susan drinks 10% of the juice from a 16-ounce bottle immediately before lunch and 20% of the remaining amount with lunch, approximately how many ounces of juice are left to drink after lunch?
   
   A. 4.8%
   B. 5.5%
   C. 11.2%
   D. 11.5%
   E. 13.0%
   

Step 1: This problem involves the concept of percent—more specifically, percentage decrease. The question is asking you to perform two computations—in sequence. (The result of the first computation is used to perform the second one.) Percent questions tend to be relatively simple. All that is involved here is a two-step computation.

Step 2: The five answer choices in this question provide two useful clues:

1. Notice that they range in value from 4.8 to 13.0. That’s a broad spectrum, isn’t it? But what general value should we 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. Eliminate (A) and (B).



2. 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.

Step 3: You already eliminated (A) and (B) in step 1. But if you’re on your toes, you can eliminate all but the correct answer without resort to precise calculations. 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 – 4.8). Answer choice (D), 11.5, is the only choice that fits the bill!

Step 4: If your intuition fails you, go ahead and crunch the numbers. First, determine 10% of 16, then subtract that number from 16:

16 × .1 = 1.6

16 - 1.6 = 14.4

Susan now has 14.4 ounces of juice. Now perform the second step. Determine 20% of 14.4, then subtract that number from 14.4:

14.4 × 2 = 2.88

Round off 2.88 to the nearest tenth: 2.9

14.4 - 2.9 = 11.5

Step 5: The decimal number 11.5 is indeed among the answer choices. Before moving on, however, ask yourself whether your solution makes sense—in this case, whether the value of our number (11.5) “fits” what the question asks for. If you performed step 2, you should already realize that 11.5 is in the right ballpark. If you’re confident that your calculations were careful and accurate, confirm your response (D), and move on to the next question.

Question 2

Question 2 is average in difficulty. Approximately 60% of test-takers respond correctly to questions like it. Here’s the question again:

2. The average of 6 numbers is 19. When one of those numbers is removed, the average of the remaining 5 numbers is 21. What number was taken away?
   
   A. 2
   B. 8
   C. 9
   D. 11
   E. 20
   

Step 1: This problem involves the concept of arithmetic mean (simple average). To handle this question, you need to be familiar with the formula for calculating the average of a series of numbers. But notice that the question does not ask for the average, but rather for one of the numbers in the series. This curveball makes the question a bit tougher than most arithmetic mean problems.

Step 2: Take a quick look at the answer choices for clues. Notice that the middle three are clustered closely together in value. So take a closer look at the two aberrations: (A) and (E). Choice (A) would be the correct answer to the question: “What is the difference between 19 and 21?” But this question is asking something entirely different, so you can probably rule out (A) as a sucker bait answer choice. Choice (E) might also be a sucker bait choice, since 20 is simply 19 + 21 divided by 2. If this solution strikes you as too simple, you’ve got good instincts! The correct answer is probably either (B), (C), or (D). If you’re pressed for time, guess one of these, and move on to the next question. Otherwise, go to step 3.

Step 3: If you’re on your “intuitive toes,” you might recognize a shortcut to the answer here. You can solve this problem quickly by simply comparing the two sums. Before the sixth number is taken away, the sum of the numbers is 114 (6 × 19). After removing the sixth number, the sum of the remaining numbers is 105 (5 × 21). The difference between the two sums is 9, which must be the value of the number removed.

Step 4: Lacking a burst of intuition (step 3), you can solve this problem in a conventional (and slower) manner. The formula for arithmetic mean (simple average) can be expressed this way:

AM = (sum of terms in the set) / (number of terms in the set)

In the question, you started with six terms. Let a through f equal those six terms:

19 = (a + b + c + d + e + f) / 6

114 = (a + b + c + d + e + f)

f = 114 – (a + b + c + d + e)

Letting f = the number removed, here’s the arithmetic-mean formula, applied to the remaining five numbers:

21 = (a + b + c + d + e) / 5

105 = a + b + c + d + e

Substitute 105 for (a + b + c + d + e) in the first equation:

f = 114 – 105

f = 9

The correct answer is C.

Step 5: If you have time, check to make sure you got the formula right, and check your calculations. Also make sure you didn’t inadvertently switch the numbers 19 and 21 in your equations. (It’s remarkably easy to commit this careless error under time pressure!) If you’re satisfied that your analysis is accurate, confirm your answer and move on to the next question.

Question 3

Question 3 is moderately difficult. Approximately 50% of test-takers respond correctly to questions like it. Here’s the question again:

3. If p pencils cost 2q dollars, how many pencils can you buy for c cents? [Note: 1 dollar = 100 cents.]
   
   A. pc / 2q
   B. pc / 200q
   C. 50pc / q
   D. 2pc / c
   E. 200pcq
   

Step 1: The first step is to recognize that this question involves a literal expression. Although it probably won’t be too time-consuming, it may be a bit confusing. You should also recognize that the key to this question is the concept of proportion. It might be appropriate to set up an equation to solve for c. Along the way, expect to convert dollars into cents.

Step 2: The five answer choices provide a couple of useful clues:

• Notice that each answer choice includes all three letters (p, q, and c). So the solution you’re shooting for must also include all three letters.



• Notice that every answer choice but (E) is a fraction. So anticipate building a fraction to solve the problem algebraically.

Step 3: Is there any way to answer this question besides setting up an algebraic equation? You bet! In fact, there are two ways. One is to use easy numbers for the three variables—for example, p = 2, q = 1, and c = 100. These simple numbers make the question easy to work with: “If 2 pencils cost 2 dollars, how many pencils can you buy for 100 cents?” Obviously, the answer to this question is 1. Therefore, plug in the numbers into each answer choice to see which choice provides an expression that equals 1.

Only choice (B) fits the bill. Another way to shortcut the algebra is to apply some intuition to this question. If you strip away the pencils, p’s, q’s and c’s, in a very general sense the question is asking:

“If you can by an item for a dollar, how many can you buy for one cent?”

Since one cent (a penny) is 1/100 of a dollar, you can buy 1/100 of one item for a cent. So you’re probably looking for a fractional answer with a large number in the denominator—something on the order of 100 (as opposed to a number such as 2, 3, or 6). Answer choice (B) is the only choice that appears to be in the correct ballpark. (B) is indeed the correct answer.

Step 4: You can also answer the question in a conventional manner using algebra. (This is easier said than done.) Here’s how to approach it:

1. Express 2q dollars as 200q cents (1 dollar = 100 cents).



2. Let x equal the number of pencils you can buy for c cents.



3. Think about the problem “verbally,” then set up an equation and solve for x:

   “p pencils is to 200q cents as x pencils is to c cents”

   The ratio of p to 200q is the same as the ratio of x to c (in other words, the two ratios are proportionate).

   p / 200q = x / c
   
   pc / 200q = x

Step 5: Our solution, pc / 200q, is indeed among the answer choices. If you arrived at this solution using the conventional algebraic approach (step 4), you can verify your solution by substituting simple numbers for the three variables (as we did in step 3). Or, if you arrived at your solution by plugging in numbers, you can check your work by plugging in a different set of numbers or by thinking about the problem conceptually (as in step 3). Once you’re confident you’ve chosen the correct expression among the five choices, confirm your choice, and then move on to the next question. The correct answer is indeed B.

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