Please visit tai lieu hoc www types math questions gmat

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Standard Math 19

Substitution 23

Coordinate Geometry 62

Elimination Strategies 71

Ratio & Proportion 96

Exponents & Roots 101

Word Problems 127

Sequences & Series 135

Data Sufficiency 287

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The Three Step Method 383

Extra Reading 397

Extra Arguments 487

Sentence Correction 501

Faulty Verb Tense 519

The Use of Adverbs 521

ORIENTATION

• WHAT DOES THE GMAT MEASURE?

• GUESSING

• ORDER OF DIFFICULTY

Shortened Study Plan

• QUESTIONS AND ANSWERS

Orientation	11

OK, the GMAT is an aptitude test. The question is—does it measure aptitude for business school? The GMAT’s ability to predict performance in school is as poor as the SAT's. This is to be expected since the tests are written by the same company (ETS) and the problems are quite similar (though the formats are different). However, the GMAT also includes two types of questions—Arguments and Data Sufficiency—that the SAT does not. Many students struggle with these questions because they are unlike any material they have studied in school. However, the argument and data sufficiency questions are not inherently hard, and with sufficient study you can raise your performance on these questions significantly.

No test can measure all aspects of intelligence. Thus any admission test, no matter how well written, is inherently inadequate. Nevertheless, some form of admission testing is necessary. It would be unfair to base acceptance to business school solely on grades; they can be misleading. For instance, would it be fair to admit a student with an A average earned in easy classes over a student with a B average earned in difficult classes? A school’s reputation is too broad a measure to use as admission criteria: many students seek out easy classes and generous instructors, in hopes of inflating their GPA. Furthermore, a system that would monitor the academic standards of every class would be cost prohibitive and stifling. So until a better system is proposed, the admission test is here to stay.

Section	Type	Questions	Time
1	Analysis of Issue Essay		30 minutes
2	Analysis of Argument Essay		30 minutes
3	Math	37	75 minutes
4	Verbal	41	75 minutes

The GMAT is a standardized test. Each time it is offered, the test has, as close as possible, the same level of difficulty as every previous test. Maintaining this consistency is very difficult—hence the experimental questions (questions that are not scored). The effectiveness of each question must be assessed before it can be used on the GMAT. A problem that one person finds easy another person may find hard, and vice versa. The experimental questions measure the relative difficulty of potential questions; if responses to a question do not perform to strict specifications, the question is rejected.

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12

Knowing that the experimental questions can be disproportionately difficult, if you do poorly on a particular question you can take some solace in the hope that it may have been experimental. In other words, do not allow a few difficult questions to discourage your performance on the rest of the test.

The CAT and the Paper & Pencil Test

To do well in your undergraduate classes, you had to attempt to solve every, or nearly every, problem on a test. Not so with the GMAT. For the vast majority of people, the key to performing well on the GMAT is not the number of questions they solve, within reason, but the percentage they solve correctly.

Scoring the GMAT

Orientation	13

On the CAT, you cannot skip questions; each question must be answered before moving on to the next question. However, if you can eliminate even one of the answer-choices, guessing can be advantageous. We’ll talk more about this later. Unfortunately, you cannot return to previously answered questions.

On the test, your first question will be of medium difficulty. If you answer it correctly, the next question will be a little harder. If you again answer it correctly, the next question will be harder still, and so on. If your GMAT skills are strong and you are not making any mistakes, you should reach the medium-hard or hard problems by about the fifth problem. Although this is not very precise, it can be quite helpful. Once you have passed the fifth question, you should be alert to subtleties in any seemingly simple problems.

The “2 out of 5” Rule

It is significantly harder to create a good but incorrect answer-choice than it is to produce the correct answer. For this reason usually only two attractive answer-choices are offered. One correct; the other either intentionally misleading or only partially correct. The other three answer-choices are usually fluff. This makes educated guessing on the GMAT immensely effective. If you can dismiss the three fluff choices, your probability of answering the question successfully will increase from 20% to 50%.

(B) Ford which conducted a similar survey with similar results continued to build large cars and also lost more of their market to the Japanese.

(C) The surveyed owners who preferred big cars also preferred big homes.

14

The other choices can be quickly dismissed. Choice (B) simply states that Ford made the same mistake that GM did. Choice (C) is irrelevant. Finally, choice (D), rather than explaining the discrepancy, would give even more reason for GM to continue making large cars.

Computer Screen Options

The Help button will present a short tutorial showing how to use the program.

You select an answer-choice by clicking the small oval next to it.

• Arrive at the test center 30 minutes before your test appointment. If you arrive late, you might not be admitted and your fee will be forfeited.

You will be provided with scratch paper. Do not bring your own, and do not remove scratch•
paper from the testing room.

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Orientation	15

This book contains the equivalent of a six-week, 50-hour course. Ideally you have bought the book at least four weeks before your scheduled test date. However, if the test is only a week or two away, there is still a truncated study plan that will be useful.

Arguments
Study: Logic I
Logic II (Diagramming)
Classification

Sentence Correction
Study: All

How important is the GMAT and how is it used?

It is crucial! Although business schools may consider other factors, the vast majority of admission decisions are based on only two criteria: your GMAT score and your GPA.

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16	GMAT Prep Course

Or online: www.ets.org

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• SUBSTITUTION

• MATH NOTES

• ELIMINATION STRATEGIES

• INEQUALITIES

• EXPONENTS & ROOTS

• FACTORING

• SEQUENCES & SERIES

• COUNTING

Standard Math	21

Types of Math Questions

The Math section consists of 37 multiple-choice questions. The questions come in two formats: the standard multiple-choice question which we will study in this section and the Data Sufficiency question which we will study in the next section. The math section is designed to test your ability to solve problems, not to test your mathematical knowledge.

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	(A)	n3
	(B)
	(B)
	(C)
	(D)
	(E)	n

We are told that n is an odd integer. So choose an odd integer for n, say, 1 and substitute it into


Next,

		1 = 1, which is not even—eliminate (E). The answer is (D).

When using the substitution method, be sure to check every answer-choice because the number you choose may work for more than one answer-choice. If this does occur, then choose another number and plug it in, and so on, until you have eliminated all but the answer. This may sound like a lot of computing, but the calculations can usually be done in a few seconds.

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Example 3:	If	x y is a fraction greater than 1, then which of the following must be less than 1?
(A)
(A)
(B)
(B)			3y
(C)				x y
(D)
(D)
(E)
						x y> 1. So choose x = 3 and y = 2. Now,		3y x= 3⋅2
than 1, so eliminate (A). Next,					x 3y=
							x= 2 3< 1. So it too may be our answer. Next, y = 2 > 1;

eliminate (E). Hence, we must decide between answer-choices (B) and (D). Let x = 6 and y = 2.

Then

1.	If n is an odd integer, which of the fol-		If y is an even integer and x is an odd

			sions could be an even integer?
			(A)	3x +y 2

	(B)
	(B)
	(C)	2n
	(C)	2n			2
	(D)				2
	(D)		(C)
	(E)		(C)
	(E)		(D)
2.	If x and y are perfect squares, then		(D)
2.	If x and y are perfect squares, then		(E)	x2 + y2


(A)		must be less than k?
(B)	xy
(C)
(D)
(D)		(B)	1
(E)			1


		(D)	k
			k2

5.		11.	Substitution	25
5.		11.	If x is an integer, then which of the	25

following is the product of the next two integers greater than 2(x + 1)?

(A)

(B)

h – k

(C)

(E)

k – h + 1

If the integer x is divisible by 3 but not

(B)

(B)

x

(D)

x2

must be true?

3

13.

III.

x3 is odd.

(C)

III only

(E)

(A)

(C)

(E)

I, II, and III

to the equation x

−

= − ?

smaller than p(q + 2)?

(B)

(D)

(E)

10.

If x and y are prime numbers, then

equal x – y ?

(E)

4x −3

26

problem, we can use the actual answer-choices. This is called Plugging In. It is a very effective technique but not as common as Substitution.

Example 1: The digits of a three-digit number add up to 18. If the ten’s digit is twice the hun- dred’s digit and the hundred’s digit is 1/3 the unit’s digit, what is the number?

1.	The ten’s digit of a two-digit number is	4.	The sum of the digits of a two-digit

2.					5.
	one of the following is the original					(A)	93
						(A)	93
						(B)
	(A)					(C)
	(B)		21			(D)	39
	(C)					(E)
	(D)							= 1, then x could be
	(E)		83
	If	N + N
		N + N				(A)
		N2				(A)
		N2				(B)
				1		(C)	3
						(D)
						(E)	8
3.				3	6.
	(C)
	(C)								4	−	2	x	2
	(D)		2						4	−	2	x	2
	(E)		3			(A)
						(B)
						(C)
	each stop and no additional passengers					(C)
	each stop and no additional passengers					(D)	3
						(D)	3
						(E)
						(E)

(A) 20 (B) 16 (C) 8 (D) 6 (E) 4

2.	Example: Given												5 6 vs.
													6 7 is larger than

	Example:						1 4 =					1 2 and
3.	Example:						1 4 =					1 2 and
	Caution: This is not true for fractions greater than 1. For example,																								3 2 <	9 4.

								1		2	=										1 2.
4.								2			=										1 2.
					)	2. In fact, a2x2 = ax											)
	Example: 3⋅22= 3⋅4 = 12. But 3⋅2																		)	2 = 62 = 36. This mistake is often seen in the following

														)52= −1	)25 = −25. But −5							)	2 = −5 (	) −5
	1a	/=	1	. In fact,									1a b= 1 ab and				1	=b a.
	b	/=	ab	. In fact,									1a b= 1 ab and				ab	=b a.
	Example:					12 3= 1 2⋅1 3= 1 6 . But										1
	Example:					12 3= 1 2⋅1 3= 1 6 . But										23

7. Memorize the following factoring formulas—they occur frequently on the GMAT.

A. a2 −b2 = a −b ( ) a + b )

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28	GMAT Prep Course

1.						)									(E) 6x2
1.	(A) 4x2					(B) 4x4			(C) 8x2				(D) 6x4		(E) 6x2
2.															(E)
2.	(A)		15	(B)			7	(C)		13		(D)
3.	(A)		16	(B)			9	(C)		15		(D)				11
3.	1+	1			=											11
	1+	1−1 2			=
	(A)					(B)	1	(C)		3	(D) 2
4.	(A)					(B)	2	(C)		2	(D) 2
									1 4 is equal to the ratio of						1 4 to x, then what is the value of x?
	(A)		1	(B)			1	(C)		5		(D)		4
5.	(A)		20	(B)			5	(C)		16		(D)
5.

I.		7	8	2> 1
I.		7		2> 1
		8	

7

II.		8	8
II.		8	
		7	

III.	8
III.	7

										(E) I, II, and III
If a#b is defined by the expression a#b = −b4, what is the value of x# −y										)?
	(B) −x4						)		(E) −y4
1	2=
1−.2 ( )	2=
(A)		(B)	1	(C)	1	(D)		25
		(B)	4	(C)	2	(D)		24

8.	If 0 < x < 1, which of the following expressions is greatest?						Math Notes
9.	(A)	1	(B) x	(C)	(D) x3
	(A)		(B) x	(C)	(D) x3

	I.	y + 1> x
	I.	y + 1> x					(E) II and III only
	II.
	III.
	(A) I only		(B) II only			(D) I and III only

10.	If rs = 4 and st = 10, then	r 10	=

(A)	1	(B) 1		(C)	5
	10	(B) 1			2

x	y =		xy	x y=
x	y =		xy	x y=

a	c

b

Example: What is the area of the triangle to the right?

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30	GMAT Prep Course
30	10.

		Interior angles on the same side of
angles are equal.	are equal.
a	c	b

13.	a		e = a + b and e > a and e > b
	e

	°

°
°

14.	There are 180° in a straight angle.			50˚		c
y˚		x˚
15.
b		a + b + c = 180˚
a
Example: What is the degree measure of angle c in the triangle
to the right?
(A) 30 (B) 35 (C) 40 (D) 41 (E) 45