GRE Math Review 1 ArithmeticWord文档格式.docx

1、 website. Other downloadable practice and test familiarization materials in large print and accessible electronic formats are also available. Tactile figure supplements for the 4 chapters of the Math Review, along with additional accessible practice and test familiarization materials in other format

2、s, are available from ETS Disability Services Monday to Friday 8:30 a m to 5 p m New York time, at 1609-771780, or 1866-3782 （toll free for test takers in the United States, US Territories, and Canada）, or via email at stassdets.org.The mathematical content covered in this edition of the Math Review

3、 is the same as the content covered in the standard edition of the Math Review. However, there are differences in the presentation of some of the material. These differences are the result of adaptations made for presentation of the material in accessible formats. There are also slight differences b

4、etween the various accessible formats, also as a result of specific adaptations made for each format.Information for screen reader users:This document has been created to be accessible to individuals who use screen readers. You may wish to consult the manual or help system for your screen reader to

5、learn how best to take advantage of the features implemented in this document. Please consult the separate document, GRE Screen Reader Instructions.doc, for important details.FiguresThe Math Review includes figures. In accessible electronic format （Word） editions, figures appear on screen. Following

6、 each figure on screen is text describing that figure. Readers using visual presentations of the figures may choose to skip parts of the text describing the figure that begin with “Begin skippable part of description of ” and end with “End skippable part of figure description”.Mathematical Equations

7、 and ExpressionsThe Math Review includes mathematical equations and expressions. In accessible electronic format （Word） editions some of the mathematical equations and expressions are presented as graphics. In cases where a mathematical equation or expression is presented as a graphic, a verbal pres

8、entation is also given and the verbal presentation comes directly after the graphic presentation. The verbal presentation is in green font to assist readers in telling the two presentation modes apart. Readers using audio alone can safely ignore the graphical presentations, and readers using visual

9、presentations may ignore the verbal presentations.Table of ContentsOverview of the Math Review 5Overview of this Chapter 51.1 Integers 61.2 Fractions 111.3 Exponents and Roots 161.4 Decimals 201.5 Real Numbers 241.6 Ratio 301.7 Percent 31Arithmetic Exercises 39Answers to Arithmetic Exercises 44Overv

10、iew of the Math ReviewThe Math Review consists of 4 chapters: Arithmetic, Algebra, Geometry, and Data Analysis.Each of the 4 chapters in the Math Review will familiarize you with the mathematical skills and concepts that are important to understand in order to solve problems and reason quantitativel

11、y on the Quantitative Reasoning measure of the GRE revised General Test.The material in the Math Review includes many definitions, properties, and examples, as well as a set of exercises with answers at the end of each chapter. Note, however, that this review is not intended to be all inclusive. The

12、re may be some concepts on the test that are not explicitly presented in this review. If any topics in this review seem especially unfamiliar or are covered too briefly, we encourage you to consult appropriate mathematics texts for a more detailed treatment.Overview of this ChapterThis is the Arithm

13、etic Chapter of the Math Review.The review of arithmetic begins with integers, fractions, and decimals and progresses to real numbers. The basic arithmetic operations of addition, subtraction, multiplication, and division are discussed, along with exponents and roots. The chapter ends with the conce

14、pts of ratio and percent.1.1 Integers The integers are the numbers 1, 2, 3, and so on, together with their negatives, negative 1, negative 2, negative 3, dot dot dot, and 0.Thus, the set of integers is dot dot dot, negative 3, negative 2, negative 1, 0, 1, 2, 3, dot dot dot.The positive integers are

15、 greater than 0, the negative integers are less than 0, and 0 is neither positive nor negative. When integers are added, subtracted, or multiplied, the result is always an integer; division of integers is addressed below. The many elementary number facts for these operations, such as+=15, 78 minus 8

16、7 = negative 9, 7 minus negative 18 = 25, and 7 times 8 = 56,should be familiar to you; they are not reviewed here. Here are three general facts regarding multiplication of integers.Fact 1: The product of two positive integers is a positive integer.Fact 2: The product of two negative integers is a p

17、ositive integer.Fact 3: The product of a positive integer and a negative integer is a negative integer.When integers are multiplied, each of the multiplied integers is called a factor or divisor of the resulting product. For example, 2 times 3 times 10 = 60,so 2, 3, and 10 are factors of 60. The int

18、egers 4, 15, 5, and 12 are also factors of 60, since 4 times 15 equals 60 and 5 times 12 = 60.The positive factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The negatives of these integers are also factors of 60, since, for example, negative 2 times negative 30 = 60.There are no other

19、factors of 60. We say that 60 is a multiple of each of its factors and that 60 is divisible by each of its divisors. Here are five more examples of factors and multiples.Example A: The positive factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.Example B: 25 is a multiple of only six integers: 1

20、, 5, 25, and their negatives.Example C: The list of positive multiples of 25 has no end: 0, 25, 50, 75, 100, 125, 150, etc.; likewise, every nonzero integer has infinitely many multiples.Example D: 1 is a factor of every integer; 1 is not a multiple of any integer except 1 and negative 1.Example E:

21、0 is a multiple of every integer; 0 is not a factor of any integer except 0.The least common multiple of two nonzero integers a and b is the least positive integer that is a multiple of both a and b. For example, the least common multiple of 30 and 75 is 150. This is because the positive multiples o

22、f 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, etc., and the positive multiples of 75 are 75, 150, 225, 300, 375, 450, etc. Thus, the common positive multiples of 30 and 75 are 150, 300, 450, etc., and the least of these is 150.The greatest common divisor （or greatest common factor） of two

23、nonzero integers a and b is the greatest positive integer that is a divisor of both a and b. For example, the greatest common divisor of 30 and 75 is 15. This is because the positive divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, and the positive divisors of 75 are 1, 3, 5, 15, 25, and 75. Thus,

24、the common positive divisors of 30 and 75 are 1, 3, 5, and 15, and the greatest of these is 15.When an integer a is divided by an integer b, where b is a divisor of a, the result is always a divisor of a. For example, when 60 is divided by 6 （one of its divisors）, the result is 10, which is another

25、divisor of 60. If b is not a divisor of a, then the result can be viewed in three different ways. The result can be viewed as a fraction or as a decimal, both of which are discussed later, or the result can be viewed as a quotient with a remainder, where both are integers. Each view is useful, depen

26、ding on the context. Fractions and decimals are useful when the result must be viewed as a single number, while quotients with remainders are useful for describing the result in terms of integers only.Regarding quotients with remainders, consider two positive integers a and b for which b is not a di

27、visor of a; for example, the integers 19 and 7. When 19 is divided by 7, the result is greater than 2, since 2 times 7 is less than 19, but less than 3, since 19 is less than 3 times 7. Because 19 is 5 more than 2 times 7, we say that the result of 19 divided by 7 is the quotient 2 with remainder 5,

28、 or simply 2 remainder 5. In general, when a positive integer a is divided by a positive integer b, you first find the greatest multiple of b that is less than or equal to a. That multiple of b can be expressed as the product qb, where q is the quotient. Then the remainder is equal to a minus that m

29、ultiple of b, or ra minus qb, where r is the remainder. The remainder is always greater than or equal to 0 and less than b.Here are three examples that illustrate a few different cases of division resulting in a quotient and remainder. 100 divided by 45 is 2 remainder 10, since the greatest multiple

30、 of 45 thats less than or equal to 100 is 2 times 45, or 90, which is 10 less than 100. 24 divided by 4 is 6 remainder 0, since the greatest multiple of 4 thats less than or equal to 24 is 24 itself, which is 0 less than 24. In general, the remainder is 0 if and only if a is divisible by b. 6 divided by 24 is 0 remainder 6