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Math Review for Standardized Tests 2nd Edition Targeted math review for many tests, including: *SAT ACT đ ASVAB • GMAT® GRE® • CBEST® • PRAXIS I® • GED® And More! *SAT is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product Jerry Bobrow, Ph.D CliffsNotes ® Math Review for Standardized Tests 2ND EDITION CliffsNotes ® Math Review for Standardized Tests 2ND EDITION by Jerry Bobrow, Ph.D revised by Ed Kohn, M.S Contributing Authors Peter Z Orton, Ph.D Ray Shiflett, Ph.D Michael Clapp, Ph.D Consultants Dave Arnold, M.A Dale Johnson, M.A Pam Mason, M.A Editorial Composition Acquisitions Editor: Greg Tubach Proofreader: Project Editor: Elizabeth Kuball Wiley Publishing, Inc Composition Services Technical Editors: Mary Jane Sterling, Abraham Mantell CliffsNotes® Math Review for Standardized Tests, 2nd Edition Published by: Wiley Publishing, Inc 111 River Street Hoboken, NJ 07030-5774 www.wiley.com Note: If you purchased this book without a cover, you should be aware that this book is stolen property It was reported as “unsold and destroyed” to the publisher, and neither the author nor the publisher has received any payment for this “stripped book.” Copyright © 2010 Jerry Bobrow Published by Wiley, Hoboken, NJ Published simultaneously in Canada Library of Congress Cataloging-in-Publication data is available from the publisher upon request ISBN: 978-0-470-50077-4 Printed in the United States of America 10 No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 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products and services or to obtain technical support, please contact our Customer Care Department within the U.S at (877) 762-2974, outside the U.S at (317) 572-3993, or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books For more information about Wiley products, please visit our web site at www.wiley.com About the Author Jerry Bobrow, Ph.D., was a national authority in the field of test preparation As founder of Bobrow Test Preparation Services, he administered testpreparation programs at over 25 California institutions for over 30 years Dr.Bobrow authored over 30 national best-selling test preparation books, and his books and programs have assisted over million test takers Each year, the faculty at Bobrow Test Preparation Services lectures to thousands of students on preparing for graduate, college, and teacher credentialing exams Table of Contents Introduction Why You Need This Guide What This Guide Contains Range of Difficulty and Scope A General Guideline How to Use This Guide PART I: BASIC SKILLS REVIEW Arithmetic and Data Analysis Diagnostic Test Questions Arithmetic Data Analysis Answers Arithmetic Data Analysis Arithmetic Review 10 Preliminaries 10 Groups of Numbers 10 Ways to Show Multiplication 11 Common Math Symbols 11 Properties of Basic Mathematical Operations 11 Some Properties (Axioms) of Addition 11 Some Properties (Axioms) of Multiplication 12 A Property of Two Operations 13 Place Value 13 Expanded Notation 14 Grouping Symbols: Parentheses, Brackets, Braces 15 Parentheses ( ) 15 Brackets [ ] and Braces { } 15 Order of Operations 16 Rounding Off 18 Signed Numbers: Positive Numbers and Negative Numbers 19 Number Lines 19 Addition of Signed Numbers 20 Subtraction of Signed Numbers 21 vii CliffsNotes Math Review for Standardized Tests, 2nd Edition Minus Preceding Parenthesis 23 Multiplying and Dividing Signed Numbers 24 Multiplying and Dividing Using Zero 24 Divisibility Rules 25 Examples: Divisibility Rules 25 Common Fractions 27 Numerator and Denominator 27 Negative Fractions 27 Proper Fractions and Improper Fractions 27 Mixed Numbers 28 Equivalent Fractions 29 Reducing Fractions 29 Enlarging Denominators 30 Factors 31 Common Factors 32 Greatest Common Factor 32 Multiples 33 Common Multiples 34 Least Common Multiple 34 Adding and Subtracting Fractions 35 Adding Fractions 35 Adding Positive and Negative Fractions 37 Subtracting Fractions 38 Subtracting Positive and Negative Fractions 39 Adding and Subtracting Mixed Numbers 40 Adding Mixed Numbers 40 Subtracting Mixed Numbers 42 Multiplying Fractions and Mixed Numbers 43 Multiplying Fractions 43 Multiplying Mixed Numbers 45 Dividing Fractions and Mixed Numbers 46 Dividing Fractions 46 Dividing Complex Fractions 46 Dividing Mixed Numbers 47 Simplifying Fractions and Complex Fractions 48 Decimals 50 Changing Decimals to Fractions 50 Adding and Subtracting Decimals 51 Multiplying Decimals 52 Dividing Decimals 53 Changing Fractions to Decimals 53 viii Data Sufficiency Is ᭝ABC an isosceles triangle? (1) (2) is perpendicular to ∠ACB is bisected by C A B D What is the measure of ∠ACB in the circle with center O? (1) (2) CA = CB The arc AB is the circumference of the circle C O B A What is the area of ᭝ABC? (1) (2) AC = BD = 10 C A B D 479 Part II: Strategies and Practice What is the measure of ∠x? (1) (2) ∠y is twice ∠x ∠x = ∠z x y z What is the measure of ∠DBC? (1) (2) ∠ADB = 100° AD = DB D A 40º C B The small circle with center B is tangent to the large circle with center D What is the area of the small circle? (1) (2) The large circle has area 9π and CD = BD = A 480 D B C E Data Sufficiency 10 What is the area of the shaded portion? (1) (2) is an arc of a circle with center A ABCD is a square and AD = D C A B 11 The volume of a box is 216 cubic inches What are the dimensions of the box? (1) (2) Length, width, and height are equal The length and width are each 6" 12 What is the length of the hypotenuse of a 45°-45°-90° triangle? (1) (2) The area is The perimeter equals 13 What is the volume of the right circular cylinder? (1) (2) The height is The radius is an integer that is less than the height 14 The circle centered at O has radius of the circle? (1) (2) passing through O What is the AB = BD = B A O D 481 Part II: Strategies and Practice 15 Does AB = CD? (1) (2) C is the midpoint of BC = CD A B C D E Answer Key for Data Sufficiency Practice The boldface page number in parentheses following each answer will direct you to the complete explanation for that specific problem Read this explanation first The other page numbers in parentheses refer to the suggested basic review section Arithmetic B (p 483, 10) C (p 484, 326) E (p 483, 329) 10 C (p 484, 94) D (p 483, 83) 11 C (p 484, 43) D (p 483, 58) 12 C (p 484, 10) E (p 483) 13 E (p 485, 83) D (p 484, 35) 14 D (p 485, 320) C (p 484) 15 A (p 485) B (p 484, 58) Algebra A (p 485, 136) A (p 486, 182) C (p 485, 339) 10 D (p 486, 206) C (p 485, 122) 11 D (p 486, 136) D (p 485, 64, 122) 12 E (p 487) C (p 486, 136) 13 B (p 487, 122) B (p 486, 200) 14 C (p 487, 32) D (p 486, 120, 125) 15 B (p 487, 125) E (p 486) 482 Data Sufficiency Geometry B (p 488, 262, 270) A (p 489, 284) B (p 488, 246) 10 C (p 489, 277, 284) E (p 488, 289) 11 D (p 489, 289, 290) C (p 488, 252) 12 D (p 489, 265) B (p 488, 283, 287) 13 E (p 489, 290) E (p 488, 277) 14 C (p 489, 262, 281) A (p 488, 241) 15 E (p 490, 245) D (p 488, 258) Data Sufficiency Answers and Explanations Arithmetic B can be shown as the product of two integers in the following manner: (1)(6), (–1)(–6), (2)(3), or (–2)(–3) Statement (1) says that both integers are positive, but that still leaves two choices Statement (2) says that one of the integers is Therefore, the other must be Statement (2) suffices E Without your knowing the rate at which the airplane flew, statement (1) does not suffice Statement (2) provides more information but does not say how long the total travel time was or the speed after eight hours The statements together not provide sufficient data to answer the question, because you don’t know the speed for the final hour D Knowing that the two numbers average 5, you can use either statement alone to determine both numbers For example, using statement (1), which says that is one of the numbers, you can determine that the other number must be 3, because the average of the two numbers is The same process works for statement (2) alone D Statement (1) is sufficient to determine an answer Just multiply $15,000 by 0.06 Statement (2) is also sufficient to determine an answer Just divide $15,900 by 1.06 E John’s being shorter than his brother has nothing to with John’s jump; so statement (1) alone does not solve the problem 483 Part II: Strategies and Practice Because no information is given about how John’s jump compares to his brother’s jump, statement (2) is not useful either D Statement (1) implies that the fractions are both Statement (2) implies that the first fraction is ; therefore, the second is Each statement is separately sufficient C Statement (1) gives the ratio of men to women, but because no numbers are given, it is insufficient to determine the number of men Statement (2) alone is insufficient to determine the number of men If you take them together, however, you know that the number of men is twice the number of women, Therefore, there are 18 men at the party B Statement (1) tells you nothing of value for this problem Statement (2) suffices Since 75% of the total is 12 ounces, then 0.75T = 12, or C Statement (1) tells you that Bob has read half the book by midnight But because you don’t know when he started or how long he has been reading, you can’t calculate when he’ll finish Statement (2) alone is also insufficient but together with statement (1), it allows you to calculate that it takes Bob four hours to read half the book; therefore, he will finish in four more hours The answer is (C) Note that your knowledge of Bob’s reading speed is useless, because you don’t know the page length of the book 10 C Statement (1) gives the brother’s height in feet and inches Statement (2) gives Alice’s height in centimeters Neither statement alone will suffice Converting centimeters to feet and inches and subtracting the height in statement (1) from that in statement (2) shows the answer is (C) 11 C To answer the question, you need to know how much gas was used Statement (1) says how much of the tank was used but not how much gas Statement (2) says how much gas is in a full tank Neither alone is enough, but taking the statements together, you find that Bruce used gallons You can then divide 300 miles by gallons of gas 12 C From statement (1), you know that the smaller integer must be 2, 3, 5, or 7, because those are the only one-digit prime numbers This statement alone is insufficient From statement (2), you know that times the smaller integer will give the second integer, greater than 30 Statement (2) alone is insufficient Together, however, the statements provide the information that only could be the smaller integer, because × > 30 484 Data Sufficiency 13 E Because you know only that Hal’s salary is over $800 per month, no precise average weekly salary may be determined from statement (1) Statement (2) is also insufficient, because you don’t know how many days each week Hal works 14 D By using the simple interest formula (I = prt), either statement alone will suffice to determine rate of interest (15% in this case) 15 A Statement (1) alone allows you to extrapolate back 20 minutes at a time to determine when volume was cubic meter (Each previous 20 minutes the mixture is halved.) Statement (2) is insufficient because you aren’t given any information regarding what the present time is Algebra A The problem states that 3a – 2b = Statement (1) gives a = 2b, resulting in two equations in two unknowns, which leads to a unique solution Statement (2) does not lead to a unique solution, however, because many pairs of integers a and b will satisfy 3a – 2b = C Let x = John’s brother’s age, and 2x = John’s age Statement (1) tells you that John’s brother is two years older than their sister Hence, x – = the sister’s age But you have no value for x Thus, statement (1) alone is insufficient Statement (2) tells you the sister’s age, years old, but you have no exact relationship between the sister and either of her brothers So statement (2) alone is insufficient Taking the statements together you have x = John’s brother 2x = John x – = = sister Thus, you can solve for x (5 years) and find John’s age, 10 C Statement (1) says that x2 = + 2x; so x2 – 2x – = 0, which factors to give (x – 3)(x + 1) = This has two answers, x = or x = –1, which in data sufficiency problems does not constitute a solution Statement (2) says that the answer is negative, which alone does not suffice However, taken together, the data are sufficient to determine a unique solution, x = –1 D Because anam = a15 and you know that anam = an + m, you have n + m = 15 Statement (1) says that n is even and m is 3, which results in n + = 15, or n = 12, a solution Statement (2) says n = 4m This together with n + m = 15 gives two equations in two unknowns, which lead to a solution also 485 Part II: Strategies and Practice C Statement (1) alone is not sufficient because it is an equation with two unknowns Statement (2) allows you to solve for the unknown, h, which you may then plug into statement (1) to determine the value of p Both statements taken together will answer the question B The fact that implies that ab = 16 Statement (1) says ab = 16 and b is even This does not allow you to find a, because several choices are possible : (2)(8), (4)(4), and so on If statement (2) is used, you have a2 = 16, and there is only one positive integer solution, D If you let x represent the original school record, then John’s first jump was x – feet and his second jump was x + feet Statement (1) says that the average of the two jumps was 18 feet This means that 18 feet was the old record Statement (1) will suffice Statement (2) says that (x – 1) + (x + 1) = 36 Solving this for x gives 18 feet So this, too, gives a solution E The question implies that x + y must be 45 or larger Statement (1) implies that (x + y) – 45 = So x + y = 48, which does not allow you to find x Statement (2) implies that x < y, but neither this nor the previous piece of information allows you to determine x A Statement (1) alone is sufficient to determine the coordinates of point P Because the coordinates are given for endpoint Q (–3,–6), using the midpoint, you can derive coordinates for the other endpoint, P (The midpoint coordinates are the averages of the endpoint coordinates Thus, coordinates of point P are (7,14) Statement (2) tells you nothing of value 10 D Either statement (1) or (2) alone is sufficient to answer the question conclusively Statement (1) tells you that x = y or x = –y In either case, (or ) will be an integer, as it will equal either or –1 Statement (2) tells you that x = Therefore, also an integer 11 D The average of a, b, and c is 486 , which is Data Sufficiency Statement (1) and the information in the question give you three equations: a + c = 20 a – c = 12 a + b = 18 If you add the first two equations, you get 2a = 32, so a = 16 This allows you to find b and c So statement (1) is sufficient Statement (2) says b = So this together with a + c = 20 allows you to find a + b + c = 12 + 20 = 32 Thus, statement (2) will suffice 12 E If you let x represent Larry’s height three years ago and y represent Bill’s height three years ago, statement (1) says x = y – This does not allow you to find x or Larry’s present height Statement (2) gives no relationship between the heights Putting both statements together allows you to write the second condition as (x + 3) = Larry’s height now (y + z) = Bill’s height now Since Larry was inches shorter, he is now only inches shorter than Bill, so (x + 3) = (y + z) – x=y–4 Thus, no new information is gained and the problem cannot be solved 13 B Statement (1) alone is insufficient Substituting and as values of y will result in a “no” then a “yes” answer for the original question Statement (2), however, is alone sufficient Any value of y greater than will yield a consistent answer (“yes”) to the question 14 C Statement (1) allows you to know that x is a multiple of both and Thus, x may be 35, 70, 105, and so on Since statement (2) alone tells you that x may be 28, 35, or 42 (all multiples of 7), it, too, is insufficient Both statements taken together, however, will suffice to determine that x is 35 15 B Statement (1) is insufficient, because it contains two unknowns Statement (2) is sufficient alone Although it contains two unknowns, notice that xy – 10 is simply two less than xy – Therefore, if xy – 10 equals 22, xy – equals 24 487 Part II: Strategies and Practice Geometry B The key here is that the diagonals of a parallelogram bisect each other Statement (1) does not provide enough information to compute However, the fact that ᭝AEB is a right triangle the length of or with hypotenuse 10 does not determine the lengths of Statement (2) allows you to determine at once B Although statement (1) allows you to conclude that both ∠1 and ∠2 are right angles, this does not suffice to conclude that l1 and l2 are parallel You can conclude from statement (2), however, that l1 and l2 are parallel because ∠1 and ∠2 are supplementary angles E In order to conclude that the triangles are similar, you would need to know that and are parallel Neither statement (1) nor statement (2) gives enough information to allow this conclusion C An isosceles triangle has two sides of equal length To conclude this, the information provided in statement (1) does not suffice because no conclusions about the sides AC and BC can be drawn Similarly, statement (2) alone is insufficient The information combined, however, leads to the conclusion that ᭝ACD and ᭝BCD are congruent (angle-side-angle) Hence AC = BC B ∠ACB is an inscribed angle Hence, its measure is equal to onehalf of ∠AOB Statement (1) gives no information about the number of degrees in ∠ACB Statement (2), however, tells you that ∠AOB is of 360° This, together with the initial observation, is sufficient to answer the question E The area of a triangle is equal to one-half the product of a base and an altitude Neither statement (1) nor statement (2) allows you to determine the altitude of the triangle A The key here is that ∠x and ∠y are supplementary angles Thus, statement (1) implies that ∠x is 60° and ∠y is 120° Statement (2) is always true (because vertical angles are equal) D ∠DBC is an exterior angle of ᭝ABD and, hence, it is equal to the sum of the angles BAD and ADB Since ∠BAD is given as 40° and statement (1) tells you that ∠ADB is 100°, you can determine ∠DBC Statement (2) tells you that ᭝ABD has two equal sides Therefore, it is isosceles Thus, ∠BAD = ∠ABD, and because ∠ABD and ∠DBC are supplementary, you can determine ∠DBC based on statement (2) alone 488 Data Sufficiency A You would need either the radius or the diameter of the small circle to find its area Statement (1) tells you that the radius of the large circle (AD or DE) is (because the area is π times the square of the radius) Because AE must, therefore, be and CE = CD + DE = + = 4, you know that AC = 2, and you can find the area of the small circle Statement (2) does not give you any information about the relationship between BD and the other parts of the circle 10 C Statement (1) tells you that if you knew the radius of the circle (or equivalently the length of the side of the square), you could find the area of the shaded portion by subtracting the area of onequarter of the circle from that of the square However, statement (1) does not alone suffice Statement (2) alone is of little use, because you know nothing of the nature of the geometric figures You need both statements to solve the problem 11 D Statement (1) tells you that your box is a cube From the known volume, the length of its edge can be found So statement (1) is sufficient Statement (2) tells you that (6)(6)(h) = 216 Thus, you can find the third dimension, height 12 D The right triangle is isosceles and, thus, its sides are in the Statement (1) tells you that if you multiply proportion of one-half the length of a base times the altitude, you get Because the legs in this case give you a base and an altitude, you can solve for the length of a leg and, hence, find the hypotenuse hypotenuse = Using statement (2), you know that Thus, s = and, again, hypotenuse = 13 E The volume of a right circular cylinder is πr2h, where r is the radius of the base and h is the height Hence, statement (1) does not give sufficient information to compute the volume Statement (2) does not specify the radius 14 C Because is a diameter, ᭝ABD is inscribed in a semicircle and is, therefore, a right triangle, because ∠ABD = 90° You can find the 489 Part II: Strategies and Practice radius OA if you can determine (the diameter) Statement (1) gives you one leg of the right triangle Statement (2) gives you the second leg The Pythagorean theorem allows you to find AD and, hence, the radius 15 E Although statement (1) shows you that AC = CE, even taken with statement (2), no relationship may be derived between AB and CD Note that with both pieces of data, a diagram may be drawn as follows A A 490 B C B C D D E E Data Sufficiency Final Suggestions Now that you have completed this guide you should Obtain the CliffsNotes preparation guide for your specific test This guide will key in on your specific math problem types, offer more practice problems, and prepare you for the nonmathematical questions on your exam Practice working problems under time pressure Now that you’ve reviewed the basic skills, the next hurdle is being comfortable using them under timed test pressure Use this guide as a reference should problems arise 491 The fast way to master math! CliffsNotes Practice Packs contain full-length practice tests and hundreds of practice problems with explanations to help you master your math subjects Bonus CD includes practice problems in a Q&A flashcard format! 978-0-470-53349-9 • $18.99 $ 978-0-470-49596-4 • $18.99 $ 978 470 49597 • $18.99 $18 99 978-0-470-49597-1 978-0-470-48869-0 • $ $18.99 Available wherever books are sold or visit us online at CliffsNotes.com® Your guide to a higher math score on standardized tests *SAT ACT đ ASVAB GMATđ GRE đ CBESTđ PRAXIS I đ GEDđ And More! Why CliffsNotes? Go with the name you know and trust • Get the information you need–fast! About the Contents: Introduction • How to use this book • Overview of the exams • Proven study strategies and test-taking tips • FAQs Part I: Basic Skills Review • Arithmetic and Data Analysis • Geometry • Algebra • Word Problems Part II: Strategies and Practice • Mathematical Ability • Quantitative Comparison • Data Sufficiency Each section includes a diagnostic test, explanations of rules, concepts with examples, practice problems with complete explanations, a review test, and a glossary! ® Jerry Bobrow, Ph.D., was a national authority in test prep His test-prep company, Bobrow Test Preparation Services, is facilitating this new edition $14.99 US/$17.99 CAN For more test-prep help, visit CliffsNotes.com® ... CliffsNotes ® Math Review for Standardized Tests 2ND EDITION CliffsNotes ® Math Review for Standardized Tests 2ND EDITION by Jerry Bobrow, Ph.D revised... a “fighting chance” on the math questions encountered on standardized tests CliffsNotes Math Review for Standardized Tests, 2nd Edition, is designed specifically to review, refresh, reintroduce,... on most standardized tests What This Guide Contains CliffsNotes Math Review for Standardized Tests, 2nd Edition, provides an excellent and extensive overview of the areas of concern for most