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Introductory Technical Mathematics

John Peterson, Robert D. Smith

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With an emphasis on real-world math applications, the Sixth Edition of INTRODUCTORY TECHNICAL MATHEMATICS provides readers with current and practical technical math applications for today's sophisticated trade and technical work environments. Straightforward and easy to understand, this hands-on book helps readers build a solid understanding of math concepts through step-by-step examples and problems drawn from various occupations. Updated to include the most current information in the field, the sixth edition includes expanded coverage of topics such as estimation usage, spreadsheets, and energy-efficient electrical applications.

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Cengage Learning

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                Introductory  Technical Mathematics 6th Edition  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Introductory  Technical Mathematics 6 th Edition John C. Peterson Robert D. Smith  Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous;  editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest.  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Introductory Technical Mathematics, 6th Edition John C. Peterson and Robert D. Smith Vice President, Careers & Computing: Dave Garza Director of Learning Solutions: Sandy Clark Associate Acquisitions Editor: Katie Hall  © 2013 Delmar, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher.  Managing Editor: Larry Main Senior Product Manager: Mary Clyne Editorial Assistant: Kaitlin Murphy Vice President, Marketing: Jennifer Ann Baker Marketing Director: Deborah Yarnell Associate Marketing Manager: Erica Glisson  For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at cengage.com/permissions Further permissions questions can be emailed to permissionrequest@cengage.com  Senior Director, Education Production: Wendy A. Troeger  Library of Congress Control Number: 2012941079  Production Manager: Mark Bernard  ISBN-13: 978-1-111-54200-9  Project Manager: Sara Dovre Wudali  ISBN-10: 1-111-54200-7  Senior Content Project Manager: Kara A. DiCaterino Senior Art Director: David Arsenault Media Editor: Deborah Bordeaux Cover Image: © iStockphoto/Christoph Völlmy  Delmar 5 Maxwell Drive Clifton Park, NY 12065-2919 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: international.cengage.com/region Cengage Learning products are represented in Canada by Nelson ­Education, Ltd. To learn more about Delmar, visit www.cengage.com/delmar Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Notice to the Reader Publisher does not warrant or guarantee any of the products described herein or perform any independent analysis in connection with any of the product information contained herein. Publisher does not assume, and expressly disclaims, any obligation to obtain and include information other than that provided to it by the manufacturer. The reader is expressly warned to consider and adopt all safety precautions that might be indicated by the activities described herein and to avoid all potential hazards. By following the instructions ­contained herein, the reader willingly assumes all risks in connection with such instructions. The publisher makes no representations or warranties of any kind, including but not limited to, the warranties of fitness for particular purpose or merchantability, nor are any such representations implied with respect to the material set forth herein, and the publisher takes no responsibility with respect to such material. The publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or part, from the readers' use of, or reliance upon, this material.  Printed in the United States of America 1 2 3 4 5 6 7 16 15 14 13 12  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Contents  Preface.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv  Section I Fundamentals of General Mathematics 1 UNIT 1  Whole Numbers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1–1 Place Value.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1–2 Expanding Whole Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1–3 Estimating (Approximating).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1–4 Addition of Whole Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1–5 Subtraction of Whole Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1–6 Problem Solving—Word Problem Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1–7 Adding and Subtracting Whole Numbers in Practical Applications. . . . . . . . . . . . . . . . . . . . 10 1–8 Multiplication of Whole Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1–9 Division of Whole Numbers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1–10 Multiplying and Dividing Whole Numbers in Practical Applications. . . . . . . . . . . . . . . . . . . . 19 1–11 Combined Operations of Whole Numbers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1–12 Combined Operations of Whole Numbers in Practical Applications. . . . . . . . . . . . . . . . . . . 24 UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1–13 Computing with a Calculator: Whole Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1–14 Computing with a Spreadsheet: Whole Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  UNIT 2  Common Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2–1 2–2 2–3 2–4 2–5 2–6 2–7 2–8 2–9 2–10 2–11 2–12 2–13 2–14  Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Fractional Parts.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A Fraction as an Indicated Division .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Equivalent Fractions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Expressing Fractions in Lowest Terms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Expressing Mixed Numbers as Improper Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Expressing Improper Fractions as Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Division of Whole Numbers; Quotients as Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Use of Common Fractions in Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Addition of Common Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Subtraction of Common Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Adding and Subtracting Common Fractions in Practical Applications.. . . . . . . . . . . . . . . . 56 Multiplication of Common Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Multiplying Common Fractions in Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64  v Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  vi  Contents 2–15 Division of Common Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2–16 Dividing Common Fractions in Practical Applications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2–17 Combined Operations with Common Fractions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2–18 Combined Operations of Common Fractions in Practical Applications.. . . . . . . . . . . . . . . 75 UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2–19 Computing with a Calculator: Fractions and Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . 82 2–20 Computing with a Spreadsheet: Fractions and Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . 88  UNIT 3  Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3–1 Meaning of Fractional Parts.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3–2 Reading Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3–3 Simplified Method of Reading Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3–4 Writing Decimal Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3–5 Rounding Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3–6 Expressing Common Fractions as Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3–7 Expressing Decimal Fractions as Common Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3–8 Expressing Decimal Fractions in Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3–9 Adding Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3–10 Subtracting Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3–11 Adding and Subtracting Decimal Fractions in Practical Applications.. . . . . . . . . . . . . . . 103 3–12 Multiplying Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3–13 Multiplying Decimal Fractions in Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3–14 Dividing Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3–15 Dividing Decimal Fractions in Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3–16 Powers and Roots of Decimal Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3–17 Decimal Fraction Powers and Roots in Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . 120 3–18 Table of Decimal Equivalents .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3–19 Combined Operations of Decimal Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3–20 Combined Operations of Decimal Fractions in Practical Applications.. . . . . . . . . . . . . . 128 UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3–21 Computing with a Calculator: Decimals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3–22 Computing with a Spreadsheet: Decimal Fractions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145  UNIT 4  Ratio and Proportion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4–1 Description of Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–2 Order of Terms of Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–3 Description of Proportions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–4 Direct Proportions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–5 Inverse Proportions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  152 153 155 158 159 162  UNIT 5  Percents.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5–1 5–2 5–3 5–4 5–5 5–6 5–7 5–8  Definition of Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressing Decimal Fractions as Percents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressing Common Fractions and Mixed Numbers as Percents. . . . . . . . . . . . . . . . . . . Expressing Percents as Decimal Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressing Percents as Common Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Simple Percent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finding Percentage in Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finding Percent (Rate) in Practical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  166 167 167 168 169 169 172 174  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  vii  Contents  5–9 Finding the Base in Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5–10 More Complex Percentage Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180  UNIT 6  Signed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6–1 Meaning of Signed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–2 The Number Line.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–3 Operations Using Signed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–4 Absolute Value.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–5 Addition of Signed Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–6 Subtraction of Signed Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–7 Multiplication of Signed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–8 Division of Signed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–9 Powers of Signed Numbers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–10 Roots of Signed Numbers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–11 Combined Operations of Signed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–12 Scientific Notation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6–13 Engineering Notation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  183 185 186 186 187 191 192 193 195 198 201 204 210 213  SECTION II Measurement 219 UNIT 7  PRECISION, ACCURACY, AND TOLERANCE.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7–1 Exact and Approximate (Measurement) Numbers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–2 Degree of Precision of Measuring Instruments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–3 Common Linear Measuring Instruments.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–4 Degree of Precision of a Measurement Number.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–5 Degrees of Precision in Adding and Subtracting Measurement Numbers.. . . . . . . . . . 7–6 Significant Digits.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–7 Accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–8 Accuracy in Multiplying and Dividing Measurement Numbers. . . . . . . . . . . . . . . . . . . . . . . . 7–9 Absolute and Relative Error.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–10 Tolerance (Linear).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–11 Unilateral and Bilateral Tolerance with Clearance and Interference Fits. . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  220 221 222 223 224 224 225 226 227 228 229 232  UNIT 8  Customary Measurement Units.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8–1 8–2 8–3 8–4 8–5 8–6 8–7 8–8 8–9 8–10 8–11 8–12  Customary Linear Units.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressing Equivalent Units of Measure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arithmetic Operations with Compound Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Linear Measure Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Units of Surface Measure (Area). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Area Measure Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Units of Volume (Cubic Measure).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Volume Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Units of Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Capacity Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Units of Weight (Mass). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary Weight Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  237 238 241 244 248 249 250 252 252 253 254 255  viii  Contents 8–13 Compound Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 8–14 Compound Units Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259  UNIT 9  Metric Measurement Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 9–1 Metric Units of Linear Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–2 Expressing Equivalent Units within the Metric System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–3 Arithmetic Operations with Metric Lengths .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–4 Metric Linear Measure Practical Applications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–5 Metric Units of Surface Measure (Area).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–6 Arithmetic Operations with Metric Area Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–7 Metric Area Measure Practical Applications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–8 Metric Units of Volume (Cubic Measure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–9 Arithmetic Operations with Metric Volume Units .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–10 Metric Volume Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–11 Metric Units of Capacity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–12 Metric Capacity Practical Applications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–13 Metric Units of Weight (Mass).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–14 Metric Weight Practical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–15 Compound Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–16 Compound Units Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–17 Metric Prefixes Applied to Very Large and Very Small Numbers. . . . . . . . . . . . . . . . . . . . . . 9–18 Conversion Between Metric and Customary Systems.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  262 264 266 266 267 269 270 270 272 272 273 274 275 276 276 278 279 282 286  UNIT 10  Steel Rules and Vernier Calipers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 10–1 Types of Steel Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–2 Reading Fractional Measurements.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–3 Measurements that do not Fall on Rule Graduations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–4 Reading Decimal-Inch Measurements.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–5 Reading Metric Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–6 Vernier Calipers: Types and Description.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–7 Reading Measurements on a Customary Vernier Caliper.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–8 Reading Measurements on a Metric Vernier Caliper.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–9 Reading Digital Calipers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  289 289 291 292 293 294 296 298 300 303  UNIT 11  Micrometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 11–1 11–2 11–3 11–4 11–5 11–6 11–7 11–8 11–9 UNIT  Description of a Customary Outside Micrometer.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading a Customary Micrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Customary Vernier Micrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading a Customary Vernier Micrometer .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of a Metric Micrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading a Metric Micrometer.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Metric Vernier Micrometer .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading a Metric Vernier Micrometer.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading Digital Micrometers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  305 306 307 308 310 310 311 312 314 316  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  ix  Contents  Section III Fundamentals of Algebra 319 UNIT 12  Introduction to Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 12–1 Symbolism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–2 Algebraic Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12–3 Evaluation of Algebraic Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  320 321 323 328  UNIT 13  Basic Algebraic Operations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 13–1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–2 Addition.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–3 Subtraction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–4 Multiplication.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–6 Powers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–7 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–8 Removal of Parentheses.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–9 Combined Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13–10 Basic Structure of the Binary Numeration System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  331 332 334 336 340 344 347 350 351 352 356  UNIT 14  Simple Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 14–1 Expression of Equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–2 Writing Equations from Word Statements.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–3 Checking the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–4 Principles of Equality.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–5 Solution of Equations by the Subtraction Principle of Equality. . . . . . . . . . . . . . . . . . . . . . . 14–6 Solution of Equations by the Addition Principle of Equality. . . . . . . . . . . . . . . . . . . . . . . . . . . 14–7 Solution of Equations by the Division Principle of Equality.. . . . . . . . . . . . . . . . . . . . . . . . . . . 14–8 Solution of Equations by the Multiplication Principle of Equality. . . . . . . . . . . . . . . . . . . . . 14–9 Solution of Equations by the Root Principle of Equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14–10 Solution of Equations by the Power Principle of Equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  361 362 364 366 366 369 372 374 377 379 380  UNIT 15  Complex Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 15–1 Equations Consisting of Combined Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–2 Solving for the Unknown in Formulas.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–3 Substituting Values Directly in Given Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15–4 Rearranging Formulas .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  383 386 387 390 394  UNIT 16  The Cartesian Coordinate System and Graphs of Linear Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 16–1 16–2 16–3 16–4 16–5  Description of the Cartesian (Rectangular) Coordinate System .. . . . . . . . . . . . . . . . . . . . . Graphing a Linear Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slope of a Linear Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slope Intercept Equation of a Straight Line.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Point-Slope Equation of a Straight Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  396 397 400 401 401  x  Contents 16–6 Determining an Equation, Given Two Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 16–7 Describing a Straight Line.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406  UNIT 17  Systems of Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Graphical Method of Solving Systems of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Substitution Method of Solving Systems of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Addition or Subtraction Method of Solving Systems of Equations.. . . . . . . . . . . . . . . . . . Types of Systems of Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determinants.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cramer's Rule.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Writing and Solving Systems of Equations from Word Statements, Number Problems, and Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–1 17–2 17–3 17–4 17–5 17–6 17–7  409 410 411 415 416 417 418 424  UNIT 18  Quadratic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 General or Standard Form of Quadratic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incomplete Quadratic Equations (ax2 5 c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complete Quadratic Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical Applications of Complete Quadratic Equations. Equations Given... . . . . . . Word Problems Involving Complete Quadratic Equations. Equations Not Given.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18–1 18–2 18–3 18–4 18–5  427 428 432 436 442 446  Section IV Fundamentals of Plane Geometry 449 UNIT 19  INTRODUCTION TO PLANE GEOMETRY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 19–1 Plane Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–2 Axioms and Postulates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19–3 Points and Lines.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  450 451 454 455  UNIT 20  Angular Measure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Units of Angular Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units of Angular Measure in Degrees, Minutes, and Seconds. . . . . . . . . . . . . . . . . . . . . . . . Expressing Degrees, Minutes, and Seconds as Decimal Degrees. . . . . . . . . . . . . . . . . . . Expressing Decimal Degrees as Degrees, Minutes, and Seconds. . . . . . . . . . . . . . . . . . . Arithmetic Operations on Angular Measure in Degrees, Minutes, and Seconds.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–6 Simple Semicircular Protractor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–7 Complements and Supplements of Scale Readings.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20–1 20–2 20–3 20–4 20–5  456 457 458 459 461 468 472 472  UNIT 21  Angular Geometric Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 21–1 Naming Angles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–2 Types of Angles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–3 Angles Formed by a Transversal.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21–4 Theorems and Corollaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  475 475 476 478 484  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xi  Contents  UNIT 22  Triangles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 22–1 Types of Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22–2 Angles of a Triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22–3 Isosceles and Equilateral Triangles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22–4 Isosceles Triangle Practical Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22–5 Equilateral Triangle Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22–6 The Pythagorean Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22–7 Pythagorean Theorem Practical Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  488 490 494 494 495 496 496 499  UNIT 23  CONGRUENT AND SIMILAR FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 23–1 Congruent Figures .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23–2 Similar Figures.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23–3 Practical Applications of Similar Triangles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  503 505 508 514  UNIT 24  POLYGONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 24–1 Types of Polygons.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24–2 Types of Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24–3 Polygon Interior and Exterior Angles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24–4 Practical Applications of Polygon Interior and Exterior Angles .. . . . . . . . . . . . . . . . . . . . . . 24–5 Practical Applications of Trapezoid Median.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  517 519 520 521 526 528  UNIT 25  CIRCLES.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 25–1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–2 Circumference Formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–3 Arc Length Formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–4 Radian Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–5 Circle Postulates .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–6 Chords, Arcs, and Central Angles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–7 Practical Applications of Circle Chord Bisector .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–8 Circle Tangents and Chord Segments.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–9 Practical Application of Circle Tangent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–10 Practical Applications of Tangents from a Common Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–11 Angles Formed Inside and on a Circle.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–12 Practical Applications of Inscribed Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–13 Practical Applications of Tangent and Chord. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–14 Angles Outside a Circle.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–15 Internally and Externally Tangent Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–16 Practical Applications of Internally Tangent Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25–17 Practical Applications of Externally Tangent Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  531 533 534 536 539 540 541 544 545 545 548 549 550 552 554 555 556 560  xii  Contents  SECTION V Geometric Figures: Areas and Volumes 567 UNIT 26  Areas of Common Polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 26–1 Areas of Rectangles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26–2 Areas of Parallelograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26–3 Areas of Trapezoids.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26–4 Areas of Triangles Given the Base and Height.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26–5 Areas of Triangles Given Three Sides.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unit Exercise and Problem Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  568 572 576 579 581 585  UNIT 27  Areas of Circles, Sectors, Segments, and Ellipses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 27–1 Areas of Circles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27–2 Ratio of Two Circles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27–3 Areas of Sectors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27–4 Areas of Segments.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27–5 Areas of Ellipses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  590 591 594 596 598 600  UNIT 28  Prisms and Cylinders: Volumes, Surface Areas, and Weights. . . . . . . . . . . . . . 604 28–1 Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28–2 Volumes of Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28–3 Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28–4 Volumes of Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28–5 Computing Heights and Bases of Prisms and Cylinders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28–6 Lateral Areas and Surface Areas of Right Prisms and Cylinders. . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  604 604 608 608 610 612 615  UNIT 29  Pyramids and Cones: Volumes, Surface Areas, and Weights. . . . . . . . . . . . . . . 617 29–1 29–2 29–3 29–4 29–5 29–6 29–7 29–8 Unit  Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cones.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volumes of Regular Pyramids and Right Circular Cones.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computing Heights and Bases of Regular Pyramids and Right Circular Cones.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lateral Areas and Surface Areas of Regular Pyramids and Right Circular Cones.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frustums of Pyramids and Cones.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volumes of Frustums of Regular Pyramids and Right Circular Cones. . . . . . . . . . . . . . . Lateral Areas and Surface Areas of Frustums of Regular Pyramids and Right Circular Cones .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise and Problem Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  617 618 618 620 621 624 625 627 631  UNIT 30  Spheres and Composite Figures: Volumes, Surface Areas, and Weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 30–1 Spheres.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30–2 Surface Area of a Sphere.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30–3 Volume of a Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30–4 Volumes and Surface Areas of Composite Solid Figures.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  633 634 634 636 641  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xiii  Contents  Section VI Basic Statistics 643 UNIT 31  GRAPHS: BAR, CIRCLE, AND LINE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 31–1 Types and Structure of Graphs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–2 Reading Bar Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–3 Drawing Bar Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–4 Drawing Bar Graphs with a Spreadsheet .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–5 Circle Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–6 Drawing Circle Graphs with a Spreadsheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–7 Line Graphs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–8 Reading Line Graphs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–9 Reading Combined-Data Line Graphs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–10 Drawing Line Graphs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–11 Drawing Broken-Line Graphs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–12 Drawing Broken-Line Graphs with a Spreadsheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–13 Drawing Straight-Line Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31–14 Drawing Curved-Line Graphs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  644 645 650 652 657 661 663 664 666 670 670 672 674 675 679  UNIT 32  STATISTICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 32–1 Probability.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–2 Independent Events.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–3 Mean Measurement.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–4 Other Average Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–5 Quartiles and Percentiles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–6 Grouped Data.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–7 Variance and Standard Deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–8 Statistical Process Control: X-Bar Charts.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32–9 Statistical Process Control: R Charts.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  683 685 687 690 691 694 699 705 709 713  Section VII Fundamentals of Trigonometry 715 UNIT 33  Introduction to Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 Ratio of Right Triangle Sides.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identifying Right Triangle Sides by Name. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric Functions: Ratio Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Customary and Metric Units of Angular Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determining Functions of Given Angles and Determining Angles of Given Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33–1 33–2 33–3 33–4 33–5  716 717 718 720 720 726  UNIT 34  Trigonometric Functions with Right Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 Variation of Functions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions of Complementary Angles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determining an Unknown Angle When Two Sides of a Right Triangle Are Known. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34–4 Determining an Unknown Side When an Acute Angle and One Side of a Right Triangle Are Known.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34–1 34–2 34–3  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  728 730 732 735 738  xiv  Contents  UNIT 35  Practical Applications with Right Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 35–1 Solving Problems Stated in Word Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35–2 Solving Problems Given in Picture Form That Require Auxiliary Lines.. . . . . . . . . . . . . . 35–3 Solving Complex Problems That Require Auxiliary Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  740 745 754 764  UNIT 36  Functions of Any Angle, Oblique Triangles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 Cartesian (Rectangular) Coordinate System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determining Functions of Angles in Any Quadrant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternating Current Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determining Functions of Angles Greater Than 3608. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instantaneous Voltage Related to Time Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving Oblique Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Law of Sines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving Problems Given Two Angles and a Side, Using the Law of Sines. . . . . . . . . . . Solving Problems Given Two Sides and an Angle Opposite One of the Given Sides, Using the Law of Sines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36–10 Law of Cosines (Given Two Sides and the Included Angle).. . . . . . . . . . . . . . . . . . . . . . . . . . 36–11 Solving Problems Given Two Sides and the Included Angle, Using the Law of Cosines.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36–12 Law of Cosines (Given Three Sides). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36–13 Solving Problems Given Three Sides, Using the Law of Cosines . . . . . . . . . . . . . . . . . . . . 36–14 Practical Applications of Oblique Triangles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36–1 36–2 36–3 36–4 36–5 36–6 36–7 36–8 36–9  768 769 772 775 777 778 779 779 781 785 785 788 789 793 801  UNIT 37  Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 Scalar and Vector Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description and Naming of Vectors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector Ordered Pair Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector Length and Angle Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adding Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphic Addition of Vectors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Addition of Vectors Using Trigonometry.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General (Component Vector) Procedure for Adding Vectors Using Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UNIT EXERCISE AND PROBLEM REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37–1 37–2 37–3 37–4 37–5 37–6 37–7 37–8  806 806 807 808 808 810 814 820 825  Appendix A  United States Customary and Metric Units of Measure.. . . . . . . . . . . . . . . . . . . . . . . . . . 829  Appendix B  Formulas for Areas (A) of Plane Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831  Appendix C  Formulas for Volumes and Areas of Solid Figures.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832  Appendix D	  Answers to Odd-Numbered Exercises.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833  INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  P r e fac e  Introductory Technical Mathematics, Sixth edition, is written to provide practical vocational and technical applications of mathematical concepts. Presentation of concepts is followed by applied examples and problems that have been drawn from diverse occupational fields. Both content and method have been used by the authors in teaching related technical mathematics on both the secondary and postsecondary levels. Each unit is developed as a learning experience based on preceding units. The applied examples and problems progress from simple to those whose solutions are relatively complex. Many problems require the student to work with illustrations such as are found in trade and technical manuals, handbooks, and drawings. The book was written from material developed for classroom use, and it is designed for classroom purposes. However, the text is also very appropriate for self-instruction use. Great care has been taken in presenting explanations clearly and in giving easy-tofollow procedural steps in solving examples. One or more examples are given for each mathematical concept presented. Throughout the book, practical application examples from various occupations are shown to illustrate the actual on-the-job uses of the mathematical concept. Students often ask, "Why do we have to learn this material and of what practical value is it?" This question was constantly kept in mind in writing the book, and every effort was made to continuously provide an answer.  Organization and Approach An understanding of mathematical concepts is emphasized in all topics. Much effort was made to avoid the mechanical plug-in approach often found in mathematics textbooks. A practical rather than an academic approach to mathematics is taken. Derivations and formal proofs are not presented; instead, understanding of concepts followed by the ­application of concepts in real situations is stressed. Student exercises and applied problems immediately follow the presentation of concept and examples. Exercises and occupationally related problems are included at the end of each unit. The book contains a sufficient number of exercises and problems to ­permit the instructor to selectively plan assignments. Illustrations, examples, exercises, and practical problems expressed in metric units of measure are a basic part of the content of the entire text. Emphasis is placed on the ability of the student to think and to work with equal ease with both the customary and the metric systems. An analytical approach to problem solving is emphasized in the geometry and trigonometry sections. The approach is that which is used in actual on-the-job trade and technical occupation applications. Integration of algebraic and geometric principles with  xv Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xvi	Preface trigonometry by careful sequencing and treatment of material also helps the student in solving occupationally related problems. The majority of instructors state that their students are required to perform ­basic arithmetic operations on whole numbers, fractions, and decimals prior to calculator or spreadsheet usage. Thereafter, the students use a calculator or spreadsheet almost exclusively in problem-­solving computations. The structuring of calculator and spreadsheet instructions and examples in this text ­reflect the instructors' preferences. The scientific calculator is introduced at the end of this ­Preface. Extensive calculator instruction and examples are given directly following each of the units on whole numbers, fractions and mixed numbers, and decimals. Further ­calculator instruction and examples are given throughout the text wherever calculator applications are appropriate to the material presented. Often there are differences in the methods of computation among various makes and models of calculators. Where there are two basic ways of performing calculations, both ways are shown.  Changes to the Sixth Edition An extensive survey of instructors using the fifth edition was conducted. Based on ­instructor comments and suggestions, significant changes were made. The result is an updated and improved sixth edition, which includes the following revisions: d Throughout the book content has been reviewed and revised to clarify and update wherever relevant. d Estimation usage has been expanded throughout the text to help students have a ­better realization of whether an answer is reasonable. d Instruction and examples are included for using digital micrometers and calipers. d The use of spreadsheets now includes basic instruction for calculations, solving equations, graphing, and statistics.  Help for Teaching and Learning The supplements package has been thoroughly revised and expanded for the sixth ­edition, offering instructors an array of products that allow them to tailor a course to their own student profile. These supplements are either part of the Instructor Companion Website or the Applied Math CourseMate®.  The Instructor Companion Website The Instructor Companion Website contains the four major items listed below: Solutions Manual containing fully worked solutions to all the exercises in the text. It is in pdf format so that it can be downloaded to your computer or printed. Computerized Test Bank in ExamView® software includes 25 questions per unit, giving the instructor hundreds of questions to choose from throughout the course. The test bank can be searched by section, style, or level of difficulty. PowerPointTM presentations of major concepts in the text. These can be downloaded and used in the classroom. Image Library containing most of the figures in the text. Instructors can select the ones they want and either print them for student use or project them.  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xvii  Preface  Applied Math CourseMate Each Delmar mathematics text includes access to Applied Math CourseMate®, Cengage Learning's on-line solution for building strong math skills. Students and instructors alike will benefit from the following CourseMate resources: d an interactive eBook, with highlighting, note-taking, and search capabilities, and d interactive learning tools including homework quizzes and tutorial PowerPoint® slides. Instructors will be able to use Applied Math CourseMate® to access the Instructor Resources and other classroom management tools. To access these resources, please visit www.cengagebrain.com. At the cengagebrain.com homepage, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page, where your resources can be found.  About the Authors John C. Peterson is a retired Professor of Mathematics at Chattanooga State ­Technical Community College, Chattanooga, Tennessee. Before he began teaching he worked on several assembly lines in industry. He has taught at the middle school, high school, ­two-year college, and university levels. Dr. Peterson is the author or coauthor of four other Delmar Cengage Learning books: Technical Mathematics, Technical Mathematics with ­Calculus, Mathematics with Machine Technology, and Math for the Automotive Trade. In addition, he has had over 80 papers published in various journals, has given over 200 presentations, and has served as a vice president of The American Mathematical Association of Two-Year Colleges. Robert D. Smith was Associate Professor Emeritus of Industrial Technology at Central Connecticut State University, New Britain, Connecticut. Mr. Smith has had experience in the manufacturing industry as tool designer, quality control engineer, and chief manufacturing engineer. He has also been active in teaching applied mathematics, physics, and industrial materials and processes on the secondary school level and in apprenticeship programs. He is the author of Delmar Cengage Learning's Mathematics for Machine Technology.  Acknowledgments The authors and publisher wish to thank the following individuals for their contribution to the review process: Andrew Bachman Pottstown School District Pottstown, PA  Stephanie Craig Newcastle School of Trades Pulaski, PA  Susan Berry Elizabethtown Community and Technical College Elizabethtown, KY  Dennis Early Wisconsin Indianhead Technical College New Richmond, WI  John Black Salina Area Technical School Salina, KS  Debbie Elder Triangle Tech Pittsburgh, PA  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xviii	Preface Steve Hlista Triangle Tech Pittsburgh, PA  Dr. Julia Probst Trenholm Technical College Montgomery, AL  Todd Hoff Wisconsin Indianhead Technical College New Richmond, WI  Tony Signoriello Newcastle School of Trades Pulaski, PA  Mary Karol McGee Metropolitan Community College Omaha, NE	  John Shirey Triangle Tech Pittsburgh, PA  Vicky Ohlson William Strauss Trenholm Technical College New Hampshire Community Technical Montgomery, AL	  College Berlin, NH Steve Ottmann Southeast Community College Lincoln, NE	 In addition, the following instructors reviewed the text and solutions for technical accuracy: Margaret Hairston Halifax Community College Weldon, NC Linda Willey Clifton Park, NY  Todd Hoff Wisconsin Indianhead Technical College New Richmond, WI  The authors and publisher also wish to extend their appreciation to the following companies for the use of credited information, graphics, and charts: L. S. Starrett Company Athol, MA 01331  Chicago Dial Indicator Des Plaines, IL 60016  Texas Instruments, Inc. P.O. Box 655474 Dallas, TX 75265  S-T Industries St. James, MN 56081  The publisher wishes to acknowledge the following contributors to the supplements package: Susan Berry PowerPoint presentations  Anthony Signoriello Computerized Test Bank  Introduction to the Scientific Calculator A scientific calculator is to be used in conjunction with the material presented in this textbook. Complex mathematical calculations can be made quickly, accurately, and easily with a scientific calculator. Although most functions are performed in the same way, there are some differences among different makes and models of scientific calculators. In this book, generally, where there are two basic ways of performing a function, both ways are shown. However, not all of the differences among the various makes and models of calculators can be shown. It is very important that you become familiar with the operation of your scientific calculator. An owner's manual or reference guide included with the purchase of a scientific calculator explains the essential features and keys of the specific calculator, as well as providing detailed information on the proper use. It is essential that the owner's manual be studied and referred to whenever there is a question regarding calculator usage. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xix  Preface  For use with this textbook, the most important feature of the scientific calculator is the Algebraic Operating System (AOS™). This system, which uses algebraic logic, permits you to enter numbers and combined operations into the calculator in the same order as the expressions are written. The calculator performs combined operations according to the rules of algebraic logic, which assigns priorities to the various mathematical operations. It is essential that you know if your calculator uses algebraic logic. Most scientific calculators, in addition to the basic arithmetic functions, have algebraic, statistical, conversion, and program or memory functions. Some of the keys with their functions are shown in the following table. Most scientific calculators have functions in addition to those shown in the table. Some Typical Key Symbols and Functions for a Scientific Calculator Keys  Functions  1/2 or ( 2)  (  Basic Arithmetic Change Sign  p  Pi  , )  Parentheses  EE or EXP  Scientific Notation  Eng  Engineering Notation  STO , RCL , EXC  Memory or Memories  x 2 , !x  Square and Square Root  y x or x y  Power  x x ! y , !  1 /x  a  b/c  Root  or x 21  Reciprocal  %  Percent  or A  b/c  DRG DMS or °  Fractions and Mixed Numbers Degrees, Radians, and Graduations  90  sin , cos , tan  Degrees, Minutes, and Seconds Trigonometric Functions  General Information About the Scientific Calculator Since there is some variation among different makes and models of scientific ­calculators, your calculator function keys may be different from the descriptions that follow. To ­repeat, it is very important that you refer to the owner's manual whenever there is a question regarding calculator usage. d Solutions to combined operations shown in this text are performed on a calculator with algebraic logic (AOS™).  Turning the Calculator On and Off d  The method of turning on a battery-powered calculators depends on the calculator make and model. When a calculator is turned on, 0 and/or other indicators are displayed. Basically, a calculator is turned on and off by one of the following ways.  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  © Cengage Learning 2013  1 , 2 , 3 , 4 , 5 , or EXE , or ENTER  xx	Preface  d  d  d  With calculators with an on/clear, ON / C , key, press ON / C to turn on. Press the OFF key to turn off. With calculators with an all clear power on/power off, AC , key, press AC to turn on. Generally, the AC key is also pressed to turn off. With calculators that have an on-off switch, move the switch either on or off. The switch is usually located on the left side of the calculator.  NOTE: In order to conserve power, most calculators have an automatic power off f­eature that automatically switches off the power after approximately five minutes of nonuse.  Clearing the Calculator Display and All Pending Operations d  d d  To clear or erase all entries of previous calculations, depending on the calculator, either of the following procedures is used. With calculators with an on/clear, ON / C , key, press ON / C twice. With calculators with the all clear, AC , key, press AC .  Erasing (Deleting) the Last Calculator Entry d  d d  d  A last entry error can be removed and corrected without erasing previously entered data and calculations. Depending on the calculator, either of the following procedures is used. With calculators with the on/clear, ON / C , key, press ON / C . With calculators with a delete, DEL , key, press DEL . If your calculator has a back arrow, b , key, use it to move the cursor to the part you want to delete. With calculators with a clear, CLEAR , key, press CLEAR .  Alternate–Function Keys d  Most scientific calculator keys can perform more than one function. Depending on the calculator, the 2nd and 3rd keys or the SHIFT key enable you to use alternate functions. The alternate functions are marked above the key and/or on the upper half of the key. Alternate functions are shown and explained in the book where their applications are appropriate to specific content. Decisions Regarding Calculator Use The exercises and problems presented throughout the text are well suited for solutions by calculator. However, it is felt decisions regarding calculator usage should be left to the discretion of the course classroom or shop instructor. The instructor best knows the unique learning environment and objectives to be achieved by the students in a course. Judgments should be made by the instructor as to the degree of emphasis to be placed on calculator applications, when and where a calculator is to be used, and the selection of specific problems for solution by calculator. Therefore, exercises and problems in this text are not specifically identified as calculator applications.  Calculator instruction and examples of the basic operations of addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals are presented at the ends of each of Units 1, 2, and 3. Further calculator instruction and examples of mathematics operations and functions are given throughout the text wherever calculator applications are appropriate to the material presented.  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Se c t io n  I  Fundamentals of General Mathematics  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  UNIT 1  OBJECTI V E S  Whole Numbers  After studying this unit you should be able to ■■  express the digit place values of whole numbers.  ■■  write whole numbers in expanded form.  ■■  estimate answers.  ■■  arrange, add, subtract, multiply, and divide whole numbers.  ■■  solve practical problems using addition, subtraction, multiplication, and division of whole numbers.  ■■  solve problems by combining addition, subtraction, multiplication, and division.  ■■  solve arithmetic expressions by applying the proper order of operations.  ■■  solve problems with formulas by applying the proper order of operations.  All occupations, from the least to the most highly skilled, require the use of mathematics. The basic operations of mathematics are addition, subtraction, multiplication, and division. These operations are based on the decimal system. Therefore, it is important that you understand the structure of the decimal system before doing the basic operations. The development of the decimal system can be traced back many centuries. In ancient times, small numbers were counted by comparing the number of objects with the number of fingers. To count larger numbers pebbles might be used. One pebble represented one counted object. Counting could be done more quickly when the pebbles were placed in groups, generally ten pebbles in each group. Our present number system, the decimal system, is based on this ancient practice of grouping by ten.  1–1  Place Value In the decimal system, 10 number symbols or digits are used. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 can be arranged to represent any number. The value expressed by each digit depends on its position in the written number. This value is called the place value. The chart shows the place value for each digit in the number 2,452,678,932. The digit on the far right is in the units (ones) place. The digit second from the right is in the tens place. The digit third from the right is in the hundreds place. The value of each place is ten times the value of the place directly to its right.  Billions 2  Hundred Ten Hundred Ten Millions Millions Millions Thousands Thousands Thousands Hundreds Tens Units 4  5  2  6  7  8  9  3  2  © Cengage Learning 2013  2 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Unit 1  3  Whole Numbers  Examples Write the place value of the underlined digit in each number. 1. 23,164 Hundreds Ans 2. 523 Units Ans 3. 143,892 Hundred Thousands Ans 4. 89,874,726 Millions Ans 5. 7623 Tens Ans  1–2  Expanding Whole Numbers The number 64 is a simplified and convenient way of writing 6 tens plus 4 ones. In its expanded form, 64 is shown as 1 6 3 10 2 1 1 4 3 1 2 . Example  Write the number 382 in expanded form in two different ways. 382 5 3 hundreds plus 8 tens plus 2 ones 382 5 3 hundreds 1 8 tens 1 2 ones Ans 382 5 1 3 3 100 2 1 1 8 3 10 2 1 1 2 3 1 2 Ans  Examples  Write each number in expanded form. 1 7 3 1000 2 1 1 0 3 100 2 1 1 2 3 10 2 1 1 8 3 1 2 Ans 1. 7028 2. 52 5 tens 1 2 ones Ans 3. 734 7 hundreds 1 3 tens 1 4 ones Ans 1 8 3 10,000 2 1 1 6 3 1000 2 1 1 2 3 100 2 1 1 7 3 10 2 1 1 9 3 1 2 Ans 4. 86,279 1 3 3 100 2 1 1 4 3 10 2 1 1 5 3 1 2 Ans 5. 345  Write the place value for the specified digit of each number given in the tables.  Digit  Number  1. 2. 3. 4.  7  6732  3  139  6  16,137  4  3924  5.  3  136,805  6.  2  427  7.  9  9,732,500  8.  5  4,578,190  Place Value Hundreds Ans  Digit  9. 10. 11. 12. 13. 14. 15. 16.  Write each whole number in expanded form. 17. 857 5 1 8 3 100 2 1 1 5 3 10 2 1 1 7 3 1 2 Ans 18. 32 20. 1372 22. 5047 19. 942 21. 10 23. 379  Number  1  10,070  0  15,018  9  98  7  782,944  5  153,400  9  98,600,057  2  378,072  4  43,728  Place Value  24. 23,813 25. 504  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  © Cengage Learning 2013  EXERCISE 1–2  4  Section I  26. 6376 27. 333 28. 59  1–3  29. 600 30. 685,412 31. 90,507  Fundamentals of General Mathematics  32. 7,500,000 33. 97,560 34. 70,001  35. 234,123 36. 17,643,000 37. 428,000,975  Estimating (Approximating) For many on-the-job applications, there are times when an exact mathematical answer is not required. Often a rough mental calculation is all that is needed. Making a rough calculation is called estimating or approximating. Estimating is widely used in practical applications. A painter estimates the number of gallons of paint needed to paint the exterior of a building; it would not be practical to compute the paint requirement to a fraction of a gallon. In ordering plywood for a job, a carpenter makes a rough calculation of the number of pieces required. An electrician approximates the number of feet of electrical cable needed for a wiring job; there is no need to calculate the exact length of cable. When computing an exact answer, it is also essential to estimate the answer before the actual arithmetic computations are made. Mistakes often can be avoided if approximate values of answers are checked against their computed values. For example, if digits are incorrectly aligned when doing an arithmetic operation, errors of magnitude are 1 or 10 times the value of what the answer should be are somemade. Answers that are 10 times carelessly made. First estimating the answer and checking it against the computed answer will tell you if an error of this type has been made. Examples of estimating answers are given in this unit. When solving exercises and problems in this unit, estimate answers and check the computed answers against the estimated answers. Continue to estimate answers for exercises and problems throughout the book. It is important also to estimate answers when using a calculator. You can press the wrong digit or the wrong operation sign; you can forget to enter a number. If you have approximated an answer and check it against the calculated answer, you will know if you have made a serious mistake. When estimating an answer, exact values are rounded. Rounded values are approximate values. Rounding numbers enables you to mentally perform arithmetic operations. When rounding whole numbers, determine the place value to which the number is to be rounded. Increase the digit at the place value by 1 if the digit that follows is 5 or more. Do not change the digit at the place value if the digit that follows is less than 5. Replace all the digits to the right of the digit at the place value with zeros. Examples 1. Round 612 to the nearest hundred. Since 1 is less than 5, 6 remains unchanged. 600 Ans 2. Round 873 to the nearest hundred. Since 7 is greater than 5, change 8 to 9. 900 Ans 3. Round 4216 to the nearest thousand. Since 2 is less than 5, 4 remains unchanged. 4000 Ans 4. Round 175,890 to the nearest ten thousand. Since 5 follows 7, change 7 to 8. 180,000 Ans  EXERCISE 1–3A  Round the following numbers as indicated. 1. 63 to the nearest ten 2. 540 to the nearest hundred 3. 766 to the nearest hundred  4. 2587 to the nearest thousand 5. 8480 to the nearest thousand 6. 32,403 to the nearest ten thousand  Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Unit 1  5  Whole Numbers  7. 46,820 to the nearest thousand 8. 53,738 to the nearest ten thousand  9. 466,973 to the nearest ten thousand 10. 949,500 to the nearest hundred thousand  Rounding to the Even Many technical trades use a process called rounding to the even. Rounding to the even can be used to help reduce bias when several numbers are added. When using rounding to the even, determine the place value to which the number is to be rounded. (This is the same as in the previous method.) The only change is when the digit that follows is a 5 followed by all zeros. Then increase the digits at the place value by 1 if it is an odd number (1, 3, 5, 7, or 9). Do not change it if it is an even number (0, 2, 4, 6, or 8). In both cases, replace the 5 with a 0. Examples 1. Round 4250 to the nearest hundred. Since 2 is an even number, it remains the same. 4200 Ans  2. Round 673,500 to the nearest thousand. Since 3 is an odd number, change the 3 to a 4. 674,000 Ans  Exercise 1–3B  Using rounding to the even to round the following numbers as indicated. 1. 2. 3. 4.  1–4  5. 6. 7. 8.  785 to the nearest ten 675 to the nearest ten 1350 to the nearest hundred 5450 to the nearest hundred  31,500 to the nearest thousand 24,520 to the nearest thousand 26,455 to the nearest hundred 26,455 to the nearest ten  Addition of Whole Numbers A contractor determines the cost of materials in a building. A salesperson charges a customer for the total cost of a number of purchases. An air-conditioning and refrigeration technician finds lengths of duct needed. These people are using addition. Practically every occupation requires daily use of addition.  Definitions and Properties of Addition The result of adding numbers (the answer) is called the sum. The plus sign (1) indicates addition. Numbers can be added in any order. The same sum is obtained regardless of the order in which the numbers are added. This is called the commutative property of ­addition. For example, 2 1 4 1 3 may be added in either of the following ways: 2141359  or  3141259  The numbers can also be grouped

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