McGRAW-HILL’s CONQUERING SAT MATH SECOND EDITION Robert Postman Professor of Mathematics and Education Mercy College, New York Ryan Postman Mathematics Department Pascack Hills High School, New Jersey MCGRAW-HILL New York / Chicago / San Francisco / Lisbon / London / Madrid / Mexico City Milan / New Delhi / San Juan / Seoul / Singapore / Sydney / Toronto Copyright © 2007 The McGraw-Hill Companies, Inc. All rights reserved. Manufactured 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 database or retrieval system, without the prior written permission of the publisher. 0-07-151084-2 The material in this eBook also appears in the print version of this title: 0-07-149341-7. All trademarks are trademarks of their respective owners. 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DOI: 10.1036/0071493417 ACKNOWLEDGMENTS This book shows you how to get your highest possible score on the New SAT: Mathematics Test. Hundreds of high school students, just like you, field test- ed this book. Much of what is here reflects these students’ opinions about what was most helpful to them. The book has also been reviewed by many high school teachers and college faculty, and it incorporates many of their suggestions about what will be most helpful to you. This book has over 600 realistic SAT problems with explained answers. Many of those problems are in five SAT Math Practice Tests. The rest of the problems are in the chapters that review all of the mathematics concepts on the new SAT. These review chapters have hundreds more examples and prac- tice exercises to help you sharpen your mathematics skills. Special thanks go to Judy Brendel, Mathematics Supervisor, who sup- ported the development of this book. Particular thanks go to John Uhl, Janet Telesmanich, Charleen Martinelli, Kevin Killian, and Michelle Gaeta. These talented mathematics teachers reviewed the SAT Math Practice Tests. Thanks are due also to the hundreds of high school students who reviewed every part of this book. Thanks are also due to Texas Instruments for per- mission to use images of their calculators. Professional Want to learn more? We hope you enjoy this McGraw-Hill eBook! If you’d like more information about this book, its author, or related books and websites, please click here. For more information about this title, click here CONTENTS CHAPTER 1: INTRODUCTION / 1 CHAPTER 2: NUMBERS AND OPERATIONS ON NUMBERS / 8 CHAPTER 3: FACTORS AND MULTIPLES / 20 CHAPTER 4: RATIOS AND PROPORTIONS / 32 CHAPTER 5: PERCENTS / 43 CHAPTER 6: MEAN, MEDIAN, AND MODE / 54 CHAPTER 7: POWERS AND RADICALS / 65 CHAPTER 8: BASIC ALGEBRA / 76 CHAPTER 9: COORDINATE GEOMETRY / 91 CHAPTER 10: FUNCTIONS AND MATHEMATICAL MODELS / 103 CHAPTER 11: TRIANGLES / 122 CHAPTER 12: QUADRILATERALS AND OTHER POLYGONS / 148 CHAPTER 13: CIRCLES / 164 CHAPTER 14: INTERSECTING LINES / 179 CHAPTER 15: SOLIDS / 193 CHAPTER 16: COUNTING, PROBABILITY, AND LOGIC PROBLEMS / 210 CHAPTER 17: DATA INTERPRETATION / 224 iv CONTENTS CHAPTER 18: SAT WORD PROBLEMS / 245 CHAPTER 19: SAT MATH PRACTICE TEST 1 / 255 CHAPTER 20: SAT MATH PRACTICE TEST 2 / 285 CHAPTER 21: SAT MATH PRACTICE TEST 3 / 313 CHAPTER 22: SAT MATH PRACTICE TEST 4 / 343 CHAPTER 23: SAT MATH PRACTICE TEST 5 / 369 CHAPTER 1 INTRODUCTION Conquering the New SAT Math covers all the mathematics topics and all the problem types on the new version of the SAT. It is the SAT you will take. Every page in this book was field tested by high school students and reviewed by experienced high school math teachers. We listened very carefully to stu- dent and teacher recommendations to create a book that will work for you. You learn mathematics by doing mathematics. The review sections of this book contain hundreds of examples and practice problems. The review sec- tions also feature worked-out model SAT problems—more than 300 practice SAT problems with explained answers. There are another 270 SAT problems with explained answers in five SAT Mathematics Practice Tests. That’s more than 600 worked-out SAT practice problems in all. We review each of the 17 major areas on the SAT in 17 separate chapters. Each chapter begins with a clear review and practice with answers. Next come several worked-out SAT problems to show you how to apply the math- ematics concepts to the SAT format. Then comes a whole set of practice SAT problems in each chapter, with explained answers. The problems generally increase in difficulty. The five mathematics practice tests follow the review chapters. Each test reflects the mix of mathematics problems you will find on the real SAT. The tests are deliberately a little more difficult than the actual SAT. You will get an estimated mathematics score range for each test you complete. We show you strategies for answering SAT multiple choice and “grid-in” math questions. However, the most important strategy is to think mathe- matically, just like the people who make up the test do. These test writers al- ways have particular mathematics skills and concepts in mind as they write an item. Knowing how to spot those concepts is usually the secret to getting a high score. You’ll learn that from this book. You Decide How Much Review You Need This book is designed to help you no matter how much SAT mathematics preparation you need. Which type of student are you? ✓I need mathematics review. ✓I need to practice SAT problems. ✓I need to take practice SAT Tests. Work through the review chapters and then take the practice tests. ✓I need to practice SAT problems. ✓I need to take practice SAT Tests. Skip the teaching and practice in the review chapters and go directly to the Model Problems in each chapter. Complete the practice problems in each chapter and take the practice tests. ✓I just need to take practice SAT Tests. Skip the review chapters and go to the practice tests. 1 Copyright © 2007 The McGraw-Hill Companies, Inc. Click here for terms of use. 2 CONQUERING SAT MATH MATHEMATICS TEST OVERVIEW There are two types of SAT mathematics problems, multiple choice and stu- dent response (grid-in). Here is an overview of the Math sections of the SAT Two 25-minute test sections One 20-minute test section Total 70 minutes Multiple-choice items 44 items Grid-ins 10 items Total 54 items Math Is Not my Favorite Subject OK, we understand that. Most of the other parts of the SAT reflect things you might do in a normal week—reading, understanding, and writing. The only time most people do SAT mathematics problems is on the mathematics SAT. But, that’s the way it is. If things ever change, we’ll be the first to tell you. Until then, we’ll show you the way to your highest possible score. MATHEMATICS TEST STRATEGIES There are different strategies for each problem type. Let’s take them in order. Multiple Choice Remember, a multiple choice item always shows you the correct answer. Each multiple choice item has five answer choices. SAT scoring deducts 1/4 point for each incorrect answer. That means that, on average, you’ll lose a small fraction of a point if you just guess an answer. However, if you are sure that one or more of the answers are wrong, on average, you will gain if you guess. It’s just simple mathematics. Eliminate and Guess If you know the answer, fill in that oval on the answer sheet. If you don’t know the answer, and you can definitely eliminate at least one answer, cross off the answers you know are incorrect right in the test booklet. Then guess from among the remaining answers. Use the Test Booklet The test booklet is yours. Do your work in the test booklet. Cross off incor- rect answers in the test booklet. Draw diagrams—whatever you like. Darken the Correct Oval The machine that scores multiple choice tests does not know if you are right or wrong. The machine just “knows” if you have filled in the correct oval, so it is possible to know the correct answer but be marked wrong. The fewer times your eyes go from the test sheet to the answer sheet, the less likely you are to make a recording error. We recommend this practice. Write all the answers next to the problem for a two-page spread of questions. Then transfer the answer to the Answer Sheet. CHAPTER 1 / INTRODUCTION 3 2. In which of the following choices would the symbol > create an incorrect statement? 7 3 A. 10 4 A B.X2 1 .66 .5 3 2 5 5 C. 7 9 7 3 D. 8 4 5 4 E. 6 9 Estimate Estimate the answer. An estimate may lead you to the correct answer or it may enable you to cross off some answers you know are incorrect. Work From the Answers You know that one of the answers must be correct. You may be able to find the correct answer by substituting answers in the problem. Grid-ins The answer format for grid-in items is shown below. Write your answer in the four spaces at the top of the grid. But the scoring machine just reads the ovals you fill in. You must fill in the correct oval to receive credit. / / . . . . 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 You may enter the digits and symbols shown in each column. The symbols represent the fraction line and the decimal point. These choices put some natural limits on the answers. You’ll read about those limits in this section. How to Grid-in an Answer Here are some examples of grid-in answers. Remember, you must fill in the ovals to receive credit. 5 Here’s how to enter the fraction in the grid. Notice that we entered the 13 numerator and the denominator separated by the slash, and then we filled in the corresponding ovals. 4 CONQUERING SAT MATH 5 / 1 3 / / . . . . 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 3 Say that your answer is . There are several ways to fill in the grid. 5 3 You can grid in . You can grid in the decimal equivalent. 0.6 5 3 / 5 0 . 6 / / / / / / . . . . . . . . 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 If your answer is 197, you can fill in the grid two different ways. 1 9 7 1 9 7 / / / / / / . . . . . . . . 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 2 The fraction is equivalent to the repeating decimal 0.666 . . . You can grid- 3 2 in , or .666 or .667. 3