Copyright © 2017 by McGraw-Hill Education, Inc All rights reserved Printed in the United States of America Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher ISBN: 978-1-259-85950-2 MHID: 1-259-85950-9 The material in this eBook also appears in the print version of this title: ISBN: 978-1-259-85951-9, MHID: 1-259-85951-7 eBook conversion by codeMantra Version 1.0 All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark Where such designations appear in this book, they have been printed with initial caps McGraw-Hill Education eBooks are available at special quantity 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terminated if you fail to comply with these terms THE WORK IS PROVIDED “AS IS.” McGRAW-HILL EDUCATION AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw-Hill Education and its licensors not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free Neither McGraw-Hill Education nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom McGraw-Hill Education has no responsibility for the content of any information accessed through the work Under no circumstances shall McGraw-Hill Education and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise CONTENTS About the Author Acknowledgment Preface Part I: Introduction CHAPTER 1: The GRE Quantitative Reasoning Sections CHAPTER 2: The Mathematics You Need to Review CHAPTER 3: Calculators on the GRE Quantitative Reasoning Sections 3.1 Overview 3.2 Calculator for the Computer Version of the GRE 3.3 Calculator for the Paper Version of the GRE 3.4 Some General Guidelines for Calculator Usage 3.5 Online Calculator Examples 3.6 Handheld Calculator Examples Part II: Types of GRE Math Questions CHAPTER 4: GRE Quantitative Comparison Questions 4.1 Quantitative Comparison Item Format 4.2 Examples 4.3 Solution Strategies 4.4 Exercises 4.5 Solutions CHAPTER 5: GRE Multiple-Choice Questions 5.1 Multiple-Choice Item Format 5.2 Examples 5.3 Solution Strategies 5.4 Exercises 5.5 Solutions CHAPTER 6: Other GRE Math Question Formats 6.1 Numeric Entry Item Format 6.2 Examples 6.3 Solution Strategies 6.4 Exercises 6.5 Solutions 6.6 Multiple-Response Item Format 6.7 Examples 6.8 Solution Strategies 6.9 Exercises 6.10 Solutions CHAPTER 7: GRE Data Interpretation Questions 7.1 Data Interpretation Item Format 7.2 Examples 7.3 Solution Strategies 7.4 Exercises 7.5 Solutions Part III: GRE Mathematics Review CHAPTER 8: Number Properties 8.1 The Number Line 8.2 The Real Numbers 8.3 Rounding Numbers 8.4 Expanded Notation 8.5 Practice Problems 8.6 Solutions 8.7 Odd and Even Numbers 8.8 Number Properties Test 8.9 Solutions 8.10 Solved GRE Problems 8.11 Solutions 8.12 GRE Practice Problems 8.13 Primes, Multiples, and Divisors 8.14 GCD and LCM Revisited 8.15 Practice Problems 8.16 Solutions 8.17 Number Properties Test 8.18 Solutions 8.19 Solved GRE Problems 8.20 Solutions 8.21 GRE Practice Problems CHAPTER 9: Arithmetic Computation 9.1 Symbols 9.2 Order of Operations 9.3 Properties of Operations 9.4 Practice Problems 9.5 Solutions 9.6 Fractions 9.7 Practice Problems 9.8 Solutions 9.9 Operations with Fractions 9.10 Practice Problems 9.11 Solutions 9.12 Decimals 9.13 Practice Problems 9.14 Solutions 9.15 Arithmetic Computation Test 9.16 Solutions 9.17 Solved GRE Problems 9.18 Solutions 9.19 GRE Practice Problems 9.20 Word Problems 9.21 Practice Problems 9.22 Solutions 9.23 Ratio and Proportion 9.24 Practice Problems 9.25 Solutions 9.26 Motion and Work Problems 9.27 Practice Problems 9.28 Solutions 9.29 Percentage 9.30 Practice Problems 9.31 Solutions 9.32 Percentage Word Problems 9.33 Practice Problems 9.34 Solutions 9.35 Types Of Averages 9.36 Practice Problems 9.37 Solutions 9.38 Powers and Roots 9.39 Standard Deviation 9.40 Practice Problems 9.41 Solutions 9.42 Simple Probability 9.43 Practice Problems 9.44 Solutions 9.45 Arithmetic Computation Test 9.46 Solutions 9.47 Solved GRE Problems 9.48 Solutions 9.49 GRE Practice Problems CHAPTER 10: Algebra 10.1 Algebraic Expressions 10.2 Exponents Revisited 10.3 Roots Revisited 10.4 General Laws of Exponents 10.5 Practice Problems 10.6 Solutions 10.7 Tables of Powers and Roots 10.8 Radical Expressions 10.9 Practice Problems 10.10 Solutions 10.11 Operations with Radicals 10.12 Practice Problems 10.13 Solutions 10.14 Algebra Test 10.15 Solutions 10.16 Solved GRE Problems 10.17 Solutions 10.18 GRE Practice Problems 10.19 Translating Verbal Expressions into Algebraic Expressions 10.20 Evaluating Algebraic Expressions 10.21 Evaluating Formulas 10.22 Practice Problems 10.23 Solutions 10.24 Addition and Subtraction of Algebraic Expressions 10.25 Multiplication of Algebraic Expressions 10.26 Division of Algebraic Expressions 10.27 Practice Problems 10.28 Solutions 10.29 Algebraic Fractions 10.30 Factoring Algebraic Expressions 10.31 Practice Problems 10.32 Solutions 10.33 Operations with Algebraic Fractions 10.34 Practice Problems 10.35 Solutions 10.36 Algebra Test 10.37 Solutions 10.38 Solved GRE Problems 10.39 Solutions 10.40 GRE Practice Problems 10.41 Linear Equations 10.42 Literal Equations 10.43 Equations with Fractions 10.44 Equations That Are Proportions 10.45 Equations with Radicals 10.46 Practice Problems 10.47 Solutions 10.48 Systems of Linear Equations 10.49 Practice Problems 10.50 Solutions 10.51 Linear Inequalities 10.52 Practice Problems 10.53 Solutions 10.54 Quadratic Equations and Inequalities 10.55 Practice Problems 10.56 Solutions 10.57 Functions 10.58 Practice Problems 10.59 Solutions 10.60 Algebraic Word Problems 10.61 Practice Problems 10.62 Solutions 10.63 Algebra Test 10.64 Solutions 10.65 Solved GRE Problems 10.66 Solutions 10.67 GRE Practice Problems CHAPTER 11: Geometry 11.1 Symbols 11.2 Points, Lines, and Angles 11.3 Practice Problems 11.4 Solutions 11.5 Polygons 11.6 Practice Problems 11.7 Solutions 11.8 Triangles 11.9 Practice Problems 11.10 Solutions 11.11 Quadrilaterals 11.12 Practice Problems 11.13 Solutions 11.14 Perimeter and Area 11.15 Practice Problems 11.16 Solutions 11.17 Circles 11.18 Practice Problems 11.19 Solutions 11.20 Solid Geometry 11.21 Practice Problems 11.22 Solutions 11.23 Coordinate Geometry 11.24 Practice Problems 11.25 Solutions 11.26 Geometry Test 11.27 Solutions 11.28 Solved GRE Problems 11.29 Solutions 11.30 GRE Practice Problems Part IV: GRE Math Practice Sections GRE Math Practice Section GRE Math Practice Section GRE Math Practice Section GRE MATH PRACTICE SECTION 20 Questions Time—35 Minutes The numbers on this test are real numbers You may assume that positions of points, lines, and angles are in the order shown You can NOT assume measures of line segments or angles unless the figure is drawn to scale Figures are in a plane unless stated otherwise You may assume that lines that appear to be straight lines are straight lines Lines are NOT parallel or perpendicular unless you are told they are or you can deduce it from information given DIRECTIONS: Quantitative comparison questions provide two quantities, Quantity A and Quantity B You are to compare the two quantities Then choose A if Quantity A is greater B if Quantity B is greater C if the two quantities are equal D if the relationship can not be determined from the given information NOTE: There is no answer E for these questions, so marking E is always wrong Common Information: In a question, information concerning one or both of the quantities to be compared is centered above the two columns A symbol that appears in both columns represents the same thing for both Quantity A and in Quantity B , what is the value of x ? If 5.5 2.2 4.8 0.075 What is the selling price of a lamp if a store bought it for $118 and marked it up 30%? $35.40 $82.60 $148 $153.40 $200.60 What is the value of 10 If the diameter (d ) of a circle is an integer and cm ≤ d ≤ 10 cm, which could be the area of the circle? 36π cm2 25π cm2 14π cm2 6π cm2 3π cm2 Questions 11–15 refer to the following graph 11 What is the ratio of local taxes to donations for Springfield Schools in 2006? to 32 to to 16 to 16 to 12 What percentage of the funds for Springfield Schools came from state, federal, and local government agencies? 40% 53% 77% 85% 90% DIRECTIONS: Questions 13 and 14 are quantitative comparison questions Choose A if Quantity A is greater B if Quantity B is greater C if the two quantities are equal D if the relationship can not be determined from the given information 13 Local taxes plus federal funds 50% 14 For this question, enter your answer in the box 15 According to the graph, if Springfield Schools had a budget of $7,200,000 in 2006, how many dollars were raised from fund raising? 16 Which is the supplement of a 52° angle? 38° 52° 78° 68° 128° 17 If the length of a rectangle is 16 cm and its width is 10 cm, what is the perimeter of the rectangle? 26 cm 42 cm 52 cm 80 cm 160 cm 18 If a bag contains 18 yellow tiles and 22 black tiles, what is the probability that a tile selected at random from the bag will be yellow? 19 After a 5% raise, Kim has a salary of $40,530 What was Kim’s salary before the raise? $2,026.50 $38,503.50 $38,600.00 $42,556.50 $42,663.16 20 Which is equal to 0.0001? Choose all that apply (0.1)3 (0.1)4 × 10−4 GRE MATH PRACTICE SECTION Answer Key 10 11 12 13 14 B C A B C C C D A B D D B A 15 16 17 18 19 20 E C B C A,C,E SOLUTIONS B Quantity B is greater 121 : 1, 11, 121, so divisors 24 : 1, 2, 3, 4, 6, 12, 24, so divisors B > A The answer is B C The two quantities are equal The answer is C A Quantity A is greater x − 9x = 36 x − 9x − 36 = (x − 12)(x + 3) = x = 12 or x = −3 P + Q = 12 + (−3) = PQ = 12(−3) = −36 P + Q > PQ The answer is A B Quantity B is greater r>0 , Cross-multiply 7r < 7r + 56 so The answer is B C The two quantities are equal Since two sides are given as equal, the angles opposite these sides are equal Thus, all three of the angles are equal because two of the angles were given to be a ° Since all angles of the triangle are equal, all sides of the triangle are equal Thus z = x and (x + z ) = x The answer is C C The quantities are equal The surface area of the sphere is 4πr The area of the circle is πr 4πr : πr = 4:1 The answer is C C 4.8 , 5x = 24, x = 24 ÷ 5, x = 4.8 The answer is C D $153.40 Original price + mark-up = Selling price $118 + 0.3($118) = $118 + $35.40 = $153.40 A 10 B 25π cm2 d is an integer and cm ≤ d ≤ 10 cm so ≤ r ≤ cm Thus, r = cm, cm, or cm, and A = 9π cm2 ,16π cm2 , or 25π cm2 Therefore of the areas listed, only 25π cm2 is a possible area Note: Since r is an integer in all the answers, you can ignore values of r when d is odd 11 D 16 to Donations 6% Local taxes 32% 32%to6% = 16 to 12 D 85% State 45% Federal 8% Local taxes 32% 45% + 8% + 32% = 85% 13 B Quantity B is greater Local taxes + federal funds = 32% + 8% = 40% 14 A Quantity A is greater Donations = 6% Other = 5% 15 Because 4% of funds are raised by fund raising, fund raising = 0.04(7,200,000) = 288,000 16 E 128° Supplementary angles have a sum of 180° The supplement of a 52° angle = 180° − 52° = 128° 17 C 52 cm P = 2l + 2w = 2(16 cm) + 2(10 cm) = 32 cm + 20 cm = 52 cm 18 B 19 C $38,600 S = previous salary; S + 0.05S = $40, 530; 1.05S = $40, 530; S = $38, 600 20 A, C, and E Choices A, C, and E are all correct ... Solutions 11.28 Solved GRE Problems 11.29 Solutions 11.30 GRE Practice Problems Part IV: GRE Math Practice Sections GRE Math Practice Section GRE Math Practice Section GRE Math Practice Section... resulting therefrom McGraw- Hill Education has no responsibility for the content of any information accessed through the work Under no circumstances shall McGraw- Hill Education and/or its licensors... Examination, generally called the GRE The GRE is developed and administered by Educational Testing Service (ETS) There are nine exams that can be referred to as GRE tests; the GRE General Test and eight