Master gmat 2010 part 18 ppsx

Problem SolvingOVERVIEW • The 5-step plan for problem solving • Some advanced techniques • Use commonsense “guesstimates” to narrow the field • When to plug in numbers for variables • Wh

SUMMING IT UP

• The GMAT places less emphasis on writing style and mechanics than on content and

organization—but these factors can influence the exam reader and affect your score if the

way you write interferes with the reader’s understanding of your ideas

• Keep your overall tone and voice relatively formal and try to vary sentence length

• Work on writing as clearly and concisely as possible

• If you feel as though you need to build your vocabulary to strengthen your essay writing,

consult the Word List in Appendix C of this book

• Watch your diction and use of idioms; make sure whatever you write is commonly

understood

• For stronger essays, use the tools of rhetoric, such as irony, punctuation, and effective key

words and phrases; connect your ideas with transitional words or phrases; and apply the

language of critical reasoning in your writing

Chapter 6: Writing Style and Mechanics 153

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P ART IV

GMAT QUANTITATIVE SECTION

.

CHAPTER 7 Problem Solving CHAPTER 8 Data Sufficiency and Analysis CHAPTER 9 Math Review: Number Forms,

Relationships, and Sets CHAPTER 10 Math Review: Number Theory

and Algebra CHAPTER 11 Math Review: Geometry

Problem Solving

OVERVIEW

• The 5-step plan for problem solving

• Some advanced techniques

• Use commonsense “guesstimates” to narrow the field

• When to plug in numbers for variables

• When—and when not—to work backward from numerical

answer choices

• Find the easiest route to the answer

• Search geometry figures for clues

• Sketch a geometry figure to solve a problem

• Plug in numbers for “defined operation” questions

• Keys to successful GMAT problem solving

• Summing it up

In this chapter, you’ll learn:

• A step-by-step approach to handling any Problem Solving question

• Keys for successfully tackling Problem Solving questions

To handle GMAT Problem Solving questions, you’ll need to be well versed in

the fundamental rules of arithmetic, algebra, and geometry Your knowledge

of these basics is, to a large extent, what’s being tested (That’s what the math

reviews in Chapters 9–11 are all about.)

But the test makers are just as interested, if not more interested, in gauging

your mental agility, flexibility, creativity, and efficiency in solving quantitative

problems More specifically, they design Problem Solving questions to help

determine the following:

• Can you manipulate numbers with a certain end result already in mind?

• Can you see the dynamic relationships between numbers as you apply operations to them?

chapter

157

• Can you visualize geometric shapes and relationships between shapes?

• Can you devise unconventional solutions to conventional quantitative problems?

• Can you solve problems efficiently, by recognizing the easiest, quickest, or most reliable route to a solution?

This chapter will help give you the skills you need to answer “yes” to these questions What follows might strike you as merely a series of tips, shortcuts, or secrets for GMAT Problem Solving However, the skills you’ll learn here are intrinsic to the test and, along with your knowledge of substantive rules of math, they’re precisely what Problem Solving questions are designed to measure

THE 5-STEP PLAN FOR PROBLEM SOLVING

The first task in this chapter is to learn the five basic steps for handling any GMAT Problem Solving question:

Size up the question Size up the answer choices Look for a shortcut

Set up the problem and solve it Verify your response before moving on We’ll apply this approach to three sample Problem Solving questions

Step One: 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 tough question when you see it may save you valuable time; if you don’t have a clue, take a guess and move on

Step Two: 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

NOTE

Remember: The

computerized

GMAT testing

system adjusts the

difficulty level of

your questions

according to

previous

responses If you

respond

incorrectly to

tough questions,

you’ll see fewer

of them later in

that section.

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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 are 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 numbers suggested by the question

OTHER DISTINCTIVE PROPERTIES AND CHARACTERISTICS

Are the answer choices integers? Do they all include a variable? Does one or more include

radicals (roots)? Exponents? Is there a particular term, expression, or number that they have

in common?

Step Three: Look for a Shortcut

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 resorting to

equations and calculations (You’ll see how when we apply this step to our sample questions.)

Step Four: Set Up the Problem and Solve It

If your intuition fails you, grab your pencil and do whatever computations, algebra, or other

procedures you need to do to solve the problem Simple problems may require just a few quick

calculations; complex algebra and geometry questions may require setting up and solving a

series of equations

Step Five: Verify Your Response Before Moving On

After solving the problem, if your solution does not appear among the answer choices, 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” answer choice (We’ll look at some of this type of answer choice in a little while.) So

check the question to verify that your response corresponds to what the question calls for in

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

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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

Question 2 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 Question 3 is a somewhat more difficult Problem Solving question involving the concept

of proportion.

3 If p pencils cost 2q dollars, how many pencils can you buy for c cents? [Tip: 1 dollar

5 100 cents]

(A) pc

2q

(B) pc

200q

(C) 50pc

q

(D) 2pq

c

(E) 200pcq

Notice that in question 3, instead of performing a numerical computation, your task is to

express a computational process in terms of letters Expressions such as these are known as literal expressions, and they can be perplexing On the GMAT, you’ll probably find two or three

of them among the 25–26 Problem Solving questions

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Apply the 5-Step Plan

Let’s review the three sample questions one at a time using the 5-step plan you just learned

QUESTION 1

Question 1 is a relatively easy question Approximately 80% of test takers respond correctly to

questions like this one Here it is 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:

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)

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 resorting 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 2 4.8) Answer choice (D), 11.5, is the only choice that works

Step 4: If your intuition fails you, work out the problem First, determine 10% of 16, then

subtract that number from 16:

16 3 0.1 5 1.6

16 2 1.6 5 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 3 0.2 5 2.88

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Round off 2.88 to the nearest tenth: 2.9 14.4 2 2.9 5 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 The

correct answer is (D).

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 choice, since 20 is simply

19 1 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 toes, you might recognize a shortcut 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 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 removed

Step 4: If you don’t see a shortcut, here’s how to solve this problem conventionally The

formula for arithmetic mean (simple average) can be expressed this way:

AM 5 sum of terms in the set

number of terms in the set

NOTE

Many Problem

Solving questions

are designed to

“reward” you for

recognizing

easier, more

intuitive ways to

find the correct

answer—so don’t

skip step 3 It’s

worth your time

to look for a

shortcut.

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