Mathematics for the Health Sciences PDF

MATHEMATICS FOR THE HEALTH SCIENCES A COMPREHENSIVE APPROACH Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. MATHEMATICS FOR THE HEALTH SCIENCES A COMPREHENSIVE APPROACH Joel R. Helms Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Mathematics for the Health Sciences: © 2010 Delmar, Cengage Learning A Comprehensive Approach ALL RIGHTS RESERVED. No part of this work covered by the copyright Joel R. 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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 1 2 3 4 5 XX 12 11 10 09 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. To my wife, Erin, and my children, Kelsey, Brianne, Deirdre, Corey, Michael, and Sean. To my Mom and Dad. v Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. CONTENT S CHAPTER 1: MATHEMATICAL ESSENTIALS 1 1.1 – Operations with Integers 1 1.2 – Fractions 9 1.3 – Order of Operations 21 1.4 – Decimals 23 1.5 – Percents 32 1.6 – Roman Numerals 37 Chapter Summary & Chapter Test CHAPTER 2: ALGEBRA 43 2.1 – Solving Linear Equations 43 2.2 – Mixture Problems 50 2.3 – Solving Rational Equations 55 2.4 – Formula Manipulation 62 2.5 – Ratios and Proportions 68 2.6 – Solving Percent Problems 75 2.7 – Properties of Exponents 79 2.8 – Scientific Notation 85 2.9 – Significant Digits 89 2.10 – Using the Scientific Calculator 92 Chapter Summary & Chapter Test CHAPTER 3: MEASUREMENT SYSTEMS AND CONVERSION PROCEDURES 99 3.1 – Basic Dimensional Analysis 101 3.2 – Conversions within the Metric System 104 3.3 – Conversions between Metric and Nonmetric 110 vii Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. viii Contents 3.4 – Apothecary and Household Systems 115 3.5 – Temperature Conversions 119 Chapter Summary & Chapter Test CHAPTER 4: DILUTIONS, SOLUTIONS, AND CONCENTRATIONS 125 4.1 – Dilutions 125 4.2 – Concentrations 132 4.3 – Concentrations and Volumes of Two Solutions 135 4.4 – Percent Solutions 139 Chapter Summary & Chapter Test CHAPTER 5: DRUG DOSAGES AND INTRAVENOUS CALCULATIONS 149 5.1 – Reading and Interpreting Drug Orders and Drug Labels 149 5.2 – Dosage Calculations: Formulas, Proportions, and Dimensional Analysis 156 5.3 – Parenteral Dosage Calculations 163 5.4 – Reconstitution of Solutions 169 5.5 – Intravenous Flow Rates 174 5.6 – Titration of Intravenous Medications 180 5.7 – Dosages Based on Weight 185 5.8 – Dosages Based on Body Surface Area 190 Chapter Summary & Chapter Test CHAPTER 6: LINEAR EQUATIONS, GRAPHING, AND VARIATION 201 6.1 – The Coordinate Plane 201 6.2 – Slope and Rate of Change 205 6.3 – Graphing Linear Equations Using the Slope 217 6.4 – Graphing Linear Equations Using Tables 222 6.5 – Interpreting Linear and Nonlinear Graphs 226 6.6 – Direct and Inverse Variation 235 Chapter Summary & Chapter Test Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Contents ix CHAPTER 7: EXPONENTIAL AND LOGARITHMIC FUNCTIONS 247 7.1 – Functions and Inequalities 247 7.2 – Exponential Functions 257 7.3 – Applications of Exponential Functions: Growth and Decay 265 7.4 – Logarithms 270 7.5 – Applications of Logarithms 278 Chapter Summary & Chapter Test CHAPTER 8: GEOMETRY 287 8.1 – Angles and Lines 287 8.2 – Geometric Figures 292 8.3 – Understanding Area and Volume 303 8.4 – Surface Area 307 8.5 – Density 312 Chapter Summary & Chapter Test CHAPTER 9: CHARTS, TABLES, AND GRAPHS 317 9.1 – Collecting Data 317 9.2 – Organizing Data Using Frequency Distribution Tables 318 9.3 – Reading and Interpreting Tables and Charts 322 9.4 – Constructing Charts and Graphs from Tables 330 Chapter Summary & Chapter Test CHAPTER 10: INTRODUCTORY STATISTICS 343 10.1 – Measures of Central Tendency 343 10.2 – Standard Deviation 348 10.3 – Normal Distribution 355 10.4 – z-Score 358 10.5 – Percentiles 360 Chapter Summary & Chapter Test Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. x Contents CHAPTER 11: TRIGONOMETRY 365 11.1 – Square Roots and the Pythagorean Theorem 365 11.2 – Special and Similar Triangles 369 11.3 – Sine, Cosine, and Tangent 374 11.4 – Related Trig Functions 379 11.5 – Applications of Trigonometry 384 Chapter Summary & Chapter Test APPENDIX: ANSWERS TO ODD-NUMBERED PRACTICE AND TEST PROBLEMS 397 REFERENCES 421 GLOSSARY 423 INDEX 429 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. PREFACE The primary purpose of this text is to help students understand and master the mathe- matics encountered in the allied healthcare professions. This text contains a variety of topics designed to encompass a broad range of allied health disciplines. Although all the chapters in this text may not be covered in any single course, the subject matter covered in most health-related math courses can be satisfied by picking and choosing from selected chapters and topics. With its broad approach, the text may also be con- sidered for use in the secondary classroom to help students demonstrate precalculus- level mathematics skills in career and technical education programs. CONCEPTUAL APPROACH Written using a systematic approach, this text is designed to make mathematical con- cepts easy to grasp and understand so that students can learn to perform the many mathematically based tasks utilized in their chosen healthcare profession. This text provides a variety of options to meet the needs of most allied health programs, includ- ing insurance and coding, pharmacy technician, medical assistant, and LPN programs. Not only can an instructor pick and choose chapters that focus specifically on the needs of each individual allied health program, the instructor can select specific sections from different chapters to create a tailored classroom learning approach for each course taught. This modular approach allows instructors and programs to adapt the same text for use in multiple courses across the allied health disciplines. ORGANIZATION OF TEXT Each of the 11 chapters begins with a straightforward introduction that provides a professionally based context for the topics discussed in the chapter. The chapters are broken into sections that discuss important subtopics. You may choose to have stu- dents focus on various sections as individual units or relate multiple sections to a broader chapter topic. The sections begin with bulleted objectives to help the student identify the goals of the section. Practice problems complete each section to actively apply the concepts just learned. xi Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. xii Preface The answers to all of the odd-numbered practice problem and chapter test questions are included in the Appendix to help the student practice outside of the classroom. Therefore, one option would be to assign the even-numbered problems as homework, and use the odd-numbered practice problems and chapter tests to identify areas where extra practice and in-class discussion may be necessary. A chapter summary wraps up each chapter in a concise manner by reiterating major concepts discussed. FEATURES With more than 1,500 practice problems throughout the text, including practice problems at the end of each section, users learn to actively apply important concepts while still fresh in their minds. More than 150 multipart chapter test questions provide a final review of all materials learned throughout a chapter. Answers to all odd-numbered questions (see Appendix) allow students to quiz themselves. Utilize even-numbered problems as homework, class assignments, or as test questions. More than 300 step-by-step examples throughout the text aid the instructor in teaching difficult mathematical concepts by detailing every step of a mathematical procedure to ensure that students can clearly visualize how a solution is calculated. Key terms are bolded in the text and defined in the Glossary to emphasize importance and provide a quick reference guide. Notes boxes point out useful facts and tips, whereas How to Calculate boxes provide tips and step-by-step directions for calculating the various types of problems found in the text. Important facts and information are highlighted in color throughout the text. Using Your Calculator boxes teach readers how to evaluate complex expressions on a calculator in addition to working them by hand. Chapter 9 contains a technology feature that guides users through a step-by-step process of graphing data using Microsoft Excel. End-of-chapter summaries organize all important rules and equations in one place for easy review. AVAILABLE FOR STUDENTS Student Solution Manual to Accompany Mathematics for the Health Sciences: A Compre- hensive Approach, ISBN: 1-4354-4111-7 The solution manual provides step-by-step solutions to all of the odd-numbered practice problems and chapter test questions in the text. The manual allows students to further Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Preface xiii understand how to correctly work a math problem by showing every step of the solution process so they can learn from their mistakes, whether in the classroom or at home. AVAILABLE FOR INSTRUCTORS Instructor Resources to Accompany Mathematics for the Health Sciences: A Comprehen- sive Approach, ISBN 1-4354-4112-5 This valuable classroom learning tool contains a full Instructor Solution Manual with in-depth, step-by-step solutions to all practice problems and chapter test questions in the book. After assigning even-numbered problems for homework, instructors have the option to first refer the student to the step-by-step examples in the core text to illustrate how to properly solve a problem. If the student continues to struggle, use the solution manual to help the student understand how to calculate a correct answer for each as- signed problem. PowerPoint presentations corresponding to each chapter are provided for use during in-class lectures and as handouts. The slides allow students to follow along as you work through examples using these new math concepts, or you can hand out the presenta- tions for students to use for test review and in-class note taking. A computerized test bank with approximately 500 additional mathematical problems in multiple-choice and short-answer formats provides you with additional test materials. Modify or create your own test questions to add a personalized touch or simply use the test questions offered in any combination you so choose. This free electronic resource is an excellent companion for any instructor looking to enhance a student’s overall learning experience. REVIEWERS Delmar Cengage Learning would like to extend its appreciation to the following re- viewers for their expertise and valuable suggestions during the development process: Keith Kuchar Mathematics Professor College of DuPage Glen Ellyn, IL Thalea Longhurst, M.S. Health Science & Technology Education Specialist Utah State Office of Education Salt Lake City, UT Mary Marlin Instructor of Developmental Mathematics West Virginia Northern Community College Wheeling, WV Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. xiv Preface Mary L. Phillips, B.S.Ed. Assistant Instructor/Coordinator Lancaster General College of Nursing and Health Sciences Lancaster, PA Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. CHAPTER 1 Mathematical Essentials INTRODUCTION This chapter reviews basic operations with integers, fractions, percents, and decimals. These topics lay the foundation for the rest of this book. Being proficient with integers, fractions, percents, and decimals is essential to doing well in the remaining chapters. In addition to integers, fractions, percents, and decimals, the order of operations and Roman numerals are studied. 1.1 – OPERATIONS WITH INTEGERS OBJECTIVES The goal of this section is for the student to: ✓ add, subtract, multiply, and divide integers. ✓ understand division involving zero. Adding Integers When adding integers, keep in mind that the negative sign means the opposite. Thus, the opposite of 3 is −3. When two opposite numbers are added, the sum is zero. Math- ematically, the opposite of −3 would be written as −(−3); but this must equal positive 3. Therefore, −(−3) = 3. One way to think about these types of numbers is to associate 1 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 2 Chapter 1 Mathematical Essentials them with money. For example, the opposite of having $3 is owing $3. When evaluating the sum of two numbers, think of money. For example, 3 + (−3) = 0. ↑ ↑ have owe If you have $3 and you owe $3, after you pay the person, you have no money left. ? How to Calculate Sign Rules for Adding Two Integers When adding two integers with the same sign, add the two integers and the sign of the answer will have the same sign as the two integers being added. When adding two integers with different signs, the sign of the answer will have the sign attached to the larger integer. EXAMPLE 1-1: Add the following integers. 1) −5 + 3 If you owe 5 and have 3, after you pay 3, you still owe 2. Therefore, the answer is −2. Notice the sign of the larger integer, 5, is negative. Thus, the answer will also be negative. 2) −2 + (−4) If you owe 2 and then owe 4 more, in total, you owe 6. Therefore, the answer is −6. Notice these two integers have the same sign (a negative sign). Thus, the answer will also have the same sign (negative). 3) 5 + (−3) If you have 5 and owe 3, after you pay 3, you have 2 left. Therefore, the answer is 2. Notice the sign of the larger integer, 5, is positive. Thus, the answer will be positive. 4) A patient’s temperature was 101 degrees. The temperature then fell by 2 degrees and later rose by 1 degree. What was the patient’s final temperature? 101 + (−2) + 1 = 100 degrees fall rise Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 3 Subtracting Integers As discussed earlier, the negative sign means the opposite. Therefore, −3 can be thought of as “the opposite of having $3,” which means owing $3. Therefore, subtrac- tion can be thought of the same way as addition. 3 − 3 can be represented as 3 + (−3) ? How to Calculate Subtracting Integers a − b = a + (−b) EXAMPLE 1-2: Evaluate the following. 1) −5 − 4 If you owe 5 and owe 4 more, in total, you owe 9. Therefore, the answer is −9. Notice the sign of each integer is negative. Thus, the answer must be negative. 2) 6 − 8 If you have 6 and owe 8, after paying 6, you owe 2. Therefore, the answer is −2. Notice the sign of the larger integer, 8, is negative. Thus, the answer must be negative. 3) −4 + 3 If you owe 4 and have 3, after paying 3, you owe 1. Therefore, the answer is −1. Also notice the sign of the larger integer, 4, is negative. This means the answer must also be negative. 4) −8 − (−5) Notice that −(−5) is equivalent to +5. Rewriting gives −8 + 5 = −3. 5) −7 − 4 − (−2) First, write −(−2) as +2, which becomes −7 − 4 + 2. Adding gives an answer of −9. Multiplying Integers Recall that multiplication is repeated addition. For example, (3)(4) = 12, because this actually means add 4 three times. (3)(4) = 4 + 4 + 4 = 12 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 4 Chapter 1 Mathematical Essentials With this in mind, now consider multiplication of positive and negative numbers. Most students already have it ingrained in their mind that a positive multiplied by a negative equals a negative, but why? This can be answered by analyzing (3)(−4). (3)(−4) = −12 because this actually means add −4 three times. (3)(−4) = (−4) + (−4) + (−4) = −12 Before considering multiplying a negative by a negative, let us take a closer look at the number −3. This can also be represented by (−3). But −(3) also means the opposite of 3. Therefore, (−3) is equivalent to −(3). With this in mind, rewrite (−3)( −4) as follows: (−3)( −4) = − (3)(−4) = −(−12) = 12. (−12) In conclusion, a negative multiplied by a negative equals a positive. ? How to Calculate Sign Rules for Multiplying Two Integers A positive multiplied by a positive equals a positive: (+)(+) = +. A negative multiplied by a negative equals a positive: (−)(−) = +. A negative multiplied by a positive equals a negative: (−)(+) = −. A positive multiplied by a negative equals a negative: (+)(−) = −. EXAMPLE 1-3: Multiply the following. 1) (5)(−6) A positive multiplied by a negative equals a negative. Therefore, the answer is −30. 2) (−7)( −5) A negative multiplied by a negative equals a positive. Therefore, the answer is 35. 3) (−2)(8) A negative multiplied by a positive equals a negative. Therefore, the answer is −16. 4) (−2)( −4)( −3) A negative multiplied by a negative equals a positive. Therefore, (−2)(−4) = 8, but (8)(−3) equals a negative. Therefore, the answer is −24. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 5 5) Product is another word that means multiply. If a patient’s temperature decreased 1 degree every hour for 3 hours, express the change in the patient’s temperature as a product of two integers. Because the patient’s temperature decreased 1 degree every hour, this would be expressed as −1. Because this occurred every hour for 3 hours, the product would be (−1)(3) = −3 degrees. NOTE The negative result implies an overall decrease of 3 degrees. Dividing Integers Division and multiplication are related. For example, 6 = 6 ÷ 3 = 2. But we can also 3 look at 6 ÷ 3 as a multiplication problem. To see this, write 6 ÷ 3 in long division format. N ) 3 6 The question is: What number N multiplied by 3 equals 6? Clearly, the number is 2. It is a fact that 6 ÷ (−3) = −2. To see why, we again look at this as a long division problem. −3)6 The question is: What number multiplied by −3 equals positive 6? As discussed earlier in the study of multiplication, the answer is −2 because (−2)(−3) = 6. Therefore, −2 −3) 6. What about −6 ÷ (−3)? Again, look at this in long division format. −3)− 6 Here, the question is: What number multiplied by −3 equals −6? The answer is posi- tive 2 because (2)(−3) = −6. Therefore, −6 ÷ (−3) = 2. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 6 Chapter 1 Mathematical Essentials ? How to Calculate Sign Rules for Dividing Two Integers A positive divided by a positive equals a positive: (+) ÷ (+) = +. A negative divided by a negative equals a positive: (−) ÷ (−) = +. A negative divided by a positive equals a negative: (−) ÷ (+) = −. A positive divided by a negative equals a negative: (+) ÷ (−) = −. EXAMPLE 1-4: Divide the following. −20 1) −5 The answer is 4 because a negative divided by a negative is a positive. 2) −12 ÷ 2 The answer is − 6 because a negative divided by a positive is a negative. 15 3) −3 The answer is −5 because a positive divided by a negative is a negative. NOTE is equivalent to −15 , which is equivalent to −. All three are equal to −5. 15 15 −3 3 3 In other words, the negative sign can be placed in the top (numerator), the bottom (denominator), or in front. Understanding this concept can be especially helpful when adding or subtracting fractions, which is studied in the next section. 4) The temperature decreased 2 degrees every hour. The total decline was 8 degrees. For how many hours was the temperature falling at this rate? A decline of 2 degrees is expressed as −2. Because the temperature decreased a total of 8 degrees, this is expressed as −8. Therefore, the number of hours the temperature decreased at this rate is −8 = 4 hours. −2 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 7 Division Involving Zero Why does zero divided by any nonzero number equal zero? To answer this, we begin by analyzing the specific case 0. Remember from previous discussions that 0 , as a 6 6 long division problem, would be written as 6)0. To evaluate this, you must determine what number multiplied by 6 equals zero. The answer is zero because any number multiplied by zero is equal to zero. Therefore, Zero divided by any nonzero number is equal to zero. Next, we analyze why any number divided by zero is undefined. The specific case of 6 0 is used. In long division format, this is written as 0)6. To evaluate this, you must deter- mine what number multiplied by zero equals 6? There is no answer because any num- ber multiplied by zero equals zero. Hence, it is undefined. Therefore, Any nonzero number divided by zero is undefined. NOTE 0 is often referred to as an indeterminate form and is studied thoroughly in 0 higher level mathematics. PRACTICE PROBLEMS: Section 1.1 Add or subtract. 1. 3 − 8 6. −18 + 12 11. 15 − (−3) 2. −4 − 6 7. 24 − 30 12. −4 + 3 − (−5) 3. −7 + 4 8. 11 − 21 13. 3 − 8 + 2 4. −6 − (−5) 9. −20 − 30 14. −7 − (−8) − 12 5. 8 − (−3) 10. 16 − (−7) 15. −38 − (−21) Continues Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 8 Chapter 1 Mathematical Essentials PRACTICE PROBLEMS: Section 1.1 (continued) 16. −28 − 28 19. −7 − (−2) 22. −3 − (−1) 17. −45 + 50 20. 6 − 8 23. 15 − 22 + 10 18. 40 − (−20) 21. −5 − 10 24. −8 − (−8) − (−12) − 5 Multiply. 25. 5 × (−4) 30. 7 × (−8) 35. 8(−3) 26. −8(−9) 31. −5(−5) 36. (−5)(6) 27. (−6)7 32. −8 × 8 37. 4 × (−5) × (−2) 28. −2 × −20 33. (−2)(−7) 38. −2(−5)(−3) 29. 9 × 9 34. (−5)(4) Divide. −70 −30 39. 10 ÷ (−5) 46. 52. 10 3 40. −12 ÷ (−6) 40 0 47. 53. 41. −20 ÷ 4 −5 12 −30 −64 4 42. 48. 54. −15 −8 0 −45 −20 −6 43. 49. 55. 9 −2 0 60 −12 0 44. 50. 56. −12 −4 −8 −32 25 45. 51. −4 −5 57. A patient’s temperature was 103 degrees. The temperature then fell by 4 degrees and later rose by 2 degrees. What was the patient’s ﬁnal temperature? 58. The temperature fell 4 degrees every hour. The total decline was 20 degrees. For how many hours was the temperature decreasing at this rate? 59. If the outside temperature declined by 4 degrees every hour for 5 hours, express the change in temperature as a product of two integers. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 9 1.2 – FRACTIONS OBJECTIVES The goal of this section is for the student to: ✓ add, subtract, multiply, and divide fractions. ✓ work with unit rates. ✓ convert between improper fractions and mixed numbers. ✓ add, subtract, multiply, and divide mixed numbers and improper fractions. ✓ simplify complex fractions. In fractions, the top number is called the numerator, and the bottom number is called the denominator. A common denominator is necessary to add or subtract fractions. However, when multiplying or dividing fractions, a common denominator is not necessary. Common denominators are necessary when adding or subtracting fractions. Common denominators are not necessary when multiplying or dividing fractions. Multiplying Fractions ? How to Calculate Multiplying Fractions a b × c d = a ×c b×d For example, 3 5 3× 5 15 × = =. 8 7 8 ×7 56 Reducing Fractions The fraction 14 can be reduced as follows: 21 14 7×2 7 2 2 2 = = × = 1× =. 21 7×3 7 3 3 3 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 10 Chapter 1 Mathematical Essentials In this example, the 7s reduce to a value of 1. Students often say the 7s “cancel” and work the problem as illustrated below. However, it helps to keep in mind that what is actually happening is the 7s reduce to a value of 1. 14 7 ×2 2 = = 21 7 ×3 3 In the example below, 5 cross “cancels” with 30. That is, 5 divides into 5 one time, and 5 divides into 30 six times. This is why there is a small 1 by the 5 and a small 6 by the 30. Likewise, 4 cross “cancels” with 12 because 4 divides into 4 one time and 4 divides into 12 three times. 5 4 51 41 1×1 1 × = × = = 12 30 12 3 30 6 3× 6 18 Instead of using the symbol for multiplication (×), parentheses are often used. The ex- ample below illustrates this idea. 2⎛ 3 ⎞ 2 1 ⎛ 31 ⎞ 1(1) 1 ⎜ ⎟ = ⎜ ⎟ = = 9 ⎝ 10 ⎠ 9 3 ⎝ 10 5 ⎠ 3 (5) 15 One situation that often throws students is multiplying a whole number by a fraction. The best way to handle this is to think of the whole number as a fraction (over 1). For example, 3 12 3 12 3 3 3× 3 9 12 × = × = × = = = 9. 4 1 4 1 41 1×1 1 EXAMPLE 1-5: Work the following problems. 15 6 1) × Reduce 15 with 25, and 20 with 6. Then simplify. 20 25 15 3 63 3× 3 9 × = = 20 10 25 5 10 × 5 50 7 ⎛ 10 ⎞ 2) ⎜ ⎟ Reduce 7 with 21, and 18 with 10. Then simplify. 18 ⎝ 21⎠ 7 1 ⎛ 10 5 ⎞ 1(5) 5 ⎜ ⎟ = = 18 9 ⎝ 213 ⎠ 9 (3) 27 3) Find half of 6. 1 Half of 6 is expressed as × 6. To evaluate this, first write 6 as a fraction and 2 then simplify. 1 63 1× 3 3 × = = =3 21 1 1×1 1 4) A healthcare technician took a 120-fluid ounce (fl oz) container of hydrogen peroxide and divided it in half. The technician then split the Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 11 remaining half by one-fourth. How much hydrogen peroxide did this technician have? 1 1 120 120 × × = = 15 fl oz 2 4 8 Dividing Fractions ? How to Calculate To divide fractions, invert the second fraction and multiply. a b ÷ c d = a b × d c = a×d b×c For example, 2 4 2 9 21 93 1× 3 3 ÷ = × = × = = 15 9 15 4 15 5 42 5×2 10 ↑ invert second fraction and multiply EXAMPLE 1-6: Simplify and express the answer in reduced form. 3 9 1) ÷ 7 14 a) Invert and multiply. 3 14 × 7 9 b) Reduce. 31 14 2 2 × = 71 93 3 5 ⎛ ⎞ 2) ÷ ⎜ − 15 ⎟ 6 ⎝ 36 ⎠ a) Invert and multiply. 5 ⎛ 36 ⎞ ⎜− ⎟ 6 ⎝ 15 ⎠ b) Reduce. 5 1 ⎛ 36 6 ⎞ 6 ⎜− ⎟ = − = −2 6 1 ⎝ 15 3 ⎠ 3 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 12 Chapter 1 Mathematical Essentials 1 3) ÷ 6 × 18 4 a) Invert 6 and change to multiplication. 1 1 × × 18 4 6 b) Rewrite 18 as 18. 1 1 1 18 × × 4 6 1 c) Reduce. 1 1 18 3 3 × × = 4 61 1 4 2 1 4) ×9÷ 3 4 a) Invert 1 and change to multiplication. 4 2 4 ×9× 3 1 b) Rewrite 9 as 9. 1 2 9 4 × × 3 1 1 c) Reduce. 2 93 4 2 × 3× 4 × × = = 24 31 1 1 1×1×1 Adding and Subtracting Fractions with Common Denominators ? How to Calculate Adding or Subtracting Fractions with Common Denominators a c + = b c a+b c or a c b − = c a−b c When adding or subtracting fractions, if the denominators are equivalent, simply add or subtract the numerators (top numbers) and place this value over the common de- nominator. For example, 1 3 1+ 3 4 + = =. 7 7 7 7 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 13 An example of subtraction is 3 5 3− 5 −2 2 − = = =−. 11 11 11 11 11 Adding and Subtracting Fractions with Uncommon Denominators If the denominators are not equivalent, the fractions must first be rewritten so they have common denominators. An example of adding fractions with unlike denominators is 1 3 +. 6 8 Because the denominators are not the same, you must first determine the least common denominator, or LCD. ? How to Calculate The LCD of two distinct numbers can be determined by finding the least common multiple of the two numbers. The LCD is the smallest number that both denominators divide into evenly (without a remainder). To determine the LCD, begin writing down the multiples of each denomi- nator. The LCD will be the first common number. The procedure below illustrates this idea. Multiples of 6: 6 12 18 24 30 Multiples of 8: 8 16 24 32 40 The LCD is 24 because 24 is the first common number. Therefore, each fraction must be expressed as an equivalent fraction with a denominator of 24. To accomplish this, 1 6 4 3 must be multiplied by , and 3 must be multiplied by. 4 8 3 ⎛ 4⎞ 1 3 ⎛ 3⎞ 4 9 4+9 13 ⎜ ⎟ + ⎜ ⎟ = + = = ⎝ 4⎠ 6 8 ⎝ 3⎠ 24 24 24 24 EXAMPLE 1-7: Evaluate the following. 2 1 1) + 9 3 2 1 ⎛ 3⎞ 2 3 5 The LCD is 9. 9 + ⎜ ⎟ 3 ⎝ 3⎠ = 9 + 9 = 9 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 14 Chapter 1 Mathematical Essentials 5 3 2) − 6 8 ⎛ 4⎞ 5 3 ⎛ 3⎞ 20 9 11 The LCD is 24. ⎜ 4 ⎟ 6 − 8 ⎜ 3 ⎟ = − = ⎝ ⎠ ⎝ ⎠ 24 24 24 2 1 3) + 5 3 ⎛ 3⎞ 2 1 ⎛ 5⎞ 6 5 11 The LCD is 15. ⎜ 3 ⎟ 5 + ⎜ ⎟ = + = ⎝ ⎠ 3 ⎝ 5⎠ 15 15 15 1 2 4) − 3 9 ⎛ 3⎞ 1 2 3 2 1 The LCD is 9. ⎜ 3 ⎟ 3 − 9 = − = ⎝ ⎠ 9 9 9 2 1 5) − 3 8 ⎛ 8⎞ 2 1 ⎛ 3⎞ 16 3 13 The LCD is 24. ⎜ 8 ⎟ 3 − 8 ⎜ 3 ⎟ = − = ⎝ ⎠ ⎝ ⎠ 24 24 24 Unit Rates When units in the numerator and denominator are distinct, the fraction is called a rate. However, in many applications, we want rates to be written as unit rates. Unit rates have a 1 in the denominator. For example, if a car travels 280 miles per 8 gallons of gasoline, we usually want to know how many miles the car can travel on 1 gallon of gasoline. 280 mi That is, write as a fraction where the denominator is 1. To accomplish this, 8 gal 280 mi 35 mi divide 280 by 8 to get 35. Therefore, =. Because 35 mi has a 1 in the de- 8 gal 1 gal 1 gal nominator, it is a unit rate. EXAMPLE 1-8: A person receives $36 for every 4 hours of work. Write this as a unit rate and interpret this result. $36 $ 36 9 $9 36 dollars every four hours can be expressed as. Simplifying gives 1 =. In 4 hr 4 hr 1 hr words, this is saying, “Nine dollars per hour,” which is the person’s hourly wage. EXAMPLE 1-9: A car travels 180 miles in 3 hours. Write this as a unit rate and interpret the results. How far would this car travel in 8 hours? 180 60 mi 60 mi 180 miles in 3 hours can be expressed as 180 mi. Simplifying gives =. In 3 hr 31 hr 1 hr words, this is saying, “Sixty miles per hour,” which gives the velocity at which the car is traveling. Because this car is traveling 60 miles each hour, in 8 hours it will travel 8 × 60 = 480 miles. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 15 EXAMPLE 1-10: Three hundred seventy-five milliliters of an intravenous (IV) solution drips into a patient every 3 hours. Write this as a unit rate. Assuming the instruments continue to function at the same rate, determine how many milliliters will drip into this patient in 8 hours. 375 mL 375 125 mL 125 mL 375 mL every 3 hours can be written as. Simplifying gives =. 3 hr 31 hr 1 hr Because 125 mL is dripping into this patient every hour, in 8 hours there will be a 125 mL total of 8 hours × = 1,000 mL that will drip into this patient. 1 hour Improper Fractions and Mixed Numbers Up to this point, the primary type of fraction discussed is this chapter is the proper fraction. Proper fractions have a denominator that is greater than the numerator. If the numerator is greater than the denominator, such as 3 , the fraction is called an improper 2 fraction. All improper fractions are greater than 1. Therefore, all improper fractions can 3 be expressed as a mixed number. An example of a mixed number is 5. This leads to 4 the following question: How do we write an improper fraction as a mixed number? The answer is, divide the denominator into the numerator and keep track of the re- mainder. For example, to write 7 as a mixed number, we see that 2 divides into 7 three 2 7 1 times with a remainder of 1. Thus, = 3. To write a mixed number as an improper 2 2 fraction, do the following: 1 2 × 3+1 6 +1 7 3 = = =. 2 2 2 2 ? How to Calculate To add or subtract mixed numbers: 1. convert each mixed number into an improper fraction. 2. add or subtract as explained earlier. 3. convert the answer to a mixed number. For example, 1 1 11 7 ⎛ 2 ⎞ 11 ⎛ 5 ⎞ 7 22 35 57 7 2 +3 = + =⎜ ⎟ +⎜ ⎟ = + = =5 5 2 5 2 ⎝ 2⎠ 5 ⎝ 5⎠ 2 10 10 10 10. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 16 Chapter 1 Mathematical Essentials EXAMPLE 1-11: Evaluate parts 1 and 2. 1 1 1) 5 − 2 4 3 21 7 a) Write as improper fractions: −. 4 3 b) Because the LCD is 12, convert each fraction so they have a denominator of 12 and then subtract. ⎛ 3 ⎞ 21 7 ⎛ 4 ⎞ 63 28 35 ⎜ ⎟ − ⎜ ⎟ = − = ⎝ 3⎠ 4 3 ⎝ 4⎠ 12 12 12 c) Convert to a mixed number. 35 11 =2 12 12 5 1 2) 2 + 1 8 2 21 3 a) Write as improper fractions: +. 8 2 b) Because the LCD is 8, convert the second fraction so its denominator is 8 and then add. 21 3 ⎛ 4 ⎞ 21 12 33 + ⎜ ⎟ = + = 8 2 ⎝ 4⎠ 8 8 8 3) Convert to a mixed number. 33 1 =4 8 8 EXAMPLE 1-12: Divide and give your answer as a mixed number. 3 3 5 ÷3 4 8 1) Write as improper fractions. 23 27 ÷ 4 8 2) Invert and multiply. 23 82 46 × = 41 27 27 3) Convert to a mixed number. 46 19 =1 27 27 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 17 1 1 EXAMPLE 1-13: Terry lost 3 pounds and then lost 2 more pounds. How much 2 4 weight did Terry lose altogether? 1 1 7 9 7 ⎛ 2⎞ 9 14 9 23 3 3 +2 = + = ⎜ ⎟ + = + = =5 2 4 2 4 2 ⎝ 2⎠ 4 4 4 4 4 3 Therefore, Terry lost a total of 5 pounds. 4 EXAMPLE 1-14: A healthcare professional was asked to administer 600 units of a medication to a patient. However, all that was in stock were bottles containing 125 units. How many bottles should be administered? Give your answer as a mixed number. 600 24 × 25 24 4 600 ÷ 125 = = = = 4 bottles should be administered. 125 5 × 25 5 5 Simplifying Complex Fractions A complex fraction is a fraction that has a fraction(s) in the numerator or the denomina- tor, or in both. Complex fractions arise in a variety of applications within the health science disciplines. To simplify complex fractions, remember that the fraction a can b also be written as a ÷ b. EXAMPLE 1-15: Completely simplify the following complex fractions. 3 4 1) 3 8 3 3 31 82 2 This can be written as ÷ = × = = 2. 4 8 41 31 1 3 4 2) 6 3 3 6 3 1 31 1 1 This means ÷ 6, which is the same as ÷ = × = × =. 4 4 1 4 6 4 62 8 ⎛ 6 +12 ⎞ ⎜⎝ 4 + 8 ⎟⎠ 3) 1 2 18 12 Simplifying the numerator gives. 1 2 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 18 Chapter 1 Mathematical Essentials Rewriting and simplifying gives 18 1 18 2 18 21 18 ÷ = × = × = = 3. 12 2 12 1 12 6 1 6 PRACTICE PROBLEMS: Section 1.2 Add or subtract. Reduce your answer when necessary. 5 2 7 3 2 1 1. + 9. − 16. −2 8 8 12 8 3 6 5 4 1 2 3 1 2. − 10. + 17. 3 + 7 7 9 3 4 2 7 5 11 1 3 5 3. − 11. − 18. 5 − 4 8 6 16 6 8 6 1 3 5 3 1 3 4. + 12. + 19. 10 − 2 12 8 6 4 3 5 3 1 1 1 1 1 5. + 13. 2 − 1 20. 6 − 3 8 4 2 3 4 2 3 1 7 3 5 6. + 14. 3 + 2 21. 1 + 2 10 6 12 8 6 2 1 4 ⎛ 2⎞ 3 7. + 15. 5 − ⎜ −2 ⎟ ⎜⎝ 3⎟⎠ 22. 5 − 3 3 6 9 4 3 1 8. − 5 6 Multiply. 2 1 5 3 3 23. × 29. × 35. 2 × 4 3 5 9 10 4 5 2 1 ⎛ 6⎞ 5 1 24. × 30. ⎜− ⎟ 36. ×2 8 15 4 ⎜⎝ 8 ⎟⎠ 6 3 3 21 9 4 1 4 25. × 31. × 37. 2 × 3 7 30 16 27 2 5 6 5 3 1 3 5 26. − × 32. × 38. 2 × 1 8 12 5 4 4 8 2 3 1 1 5 5 27. × 33. 3 × 2 39. 1 ×2 9 5 2 4 12 6 5 4 2 ⎛ 3⎞ 3 2 28. × 34. 1 ⎜ −2 ⎟ 40. 3 × 2 16 5 3 ⎜⎝ 5⎟⎠ 8 5 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 19 Divide. 2 1 7 21 5 3 41. ÷ 47. ÷ 53. 1 ÷ 3 3 12 36 8 2 3 9 15 21 1 1 42. ÷ 48. ÷ 54. −1 ÷ 5 15 24 48 2 4 5 ⎛ 3⎞ 3 1 5 43. ÷ ⎜− ⎟ 49. 3 ÷ 2 55. 3 ÷ 2 6 ⎜⎝ 8 ⎟⎠ 4 3 6 5 10 3 3 44. ÷ 50. 9 ÷ 56. −9 ÷ 4 9 3 4 8 7 3 5 5 1 45. ÷ 51. ÷ 10 57. 5 ÷ 1 8 4 8 8 6 1 1 3 1 1 2 46. ÷ 52. 3 ÷ 1 58. 2 ÷ 6 2 4 4 4 4 3 Write as a unit rate. $1, 800 24, 000 cells 16 g 59. 61. 63. 4 wk 4 mm2 80 L $100 120 mL 9 ft 60. 62. 64. 10 sq ft 8 kg 2 sec 65. Express $30 for 50 shirts as a unit rate. 66. Express $3 for 12 test tubes as a unit rate. 67. Express $1,000 for 5 days’ work as a unit rate. 68. Express $2.79 for 36 ounces as a unit rate. 69. A person makes $3,200 for 4 weeks of work. How much does this person make per week? If this person works 50 weeks in a year, how much will he or she make per year? 70. A person makes $6,000 for 5 weeks of work. How much does this person make per week? If this person works 50 weeks in a year, how much will he or she make per year? 71. Three hundred milliliters of an IV solution drips into a patient every 2 hours. Write this as a unit rate. Assuming the instruments continue to function at the same rate, determine how many milliliters will drip into this patient in 6 hours. 72. Four hundred milliliters of an IV solution drips into a patient every 4 hours. Write this as a unit rate. Assuming the instruments continue to function at the same rate, determine how many milliliters will drip into this patient in 10 hours. 73. Five hundred twenty-ﬁve milliliters of an IV solution drips into a patient every 3 hours. Write this as a unit rate. Assuming the instruments continue to function at the same rate, determine how many milliliters will drip into this patient in 5 hours. 74. Sixty-eight patients visited a hospital emergency department during an 8-hour shift. On average, how many patients visited the emergency department per hour during this particular shift? Continues Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 20 Chapter 1 Mathematical Essentials PRACTICE PROBLEMS: Section 1.2 (continued) 75. One hundred twenty patients visited a hospital emergency department during a 10-hour shift. On average, how many patients visited the emergency department per hour during this particular shift? 76. A person burns 375 calories every 5 hours. Write this as a unit rate. How many calories will this person burn in 24 hours? 77. A person burns 360 calories every 4 hours. Write this as a unit rate. How many calories will this person burn in 24 hours? 78. A person burns 570 calories every 6 hours. Write this as a unit rate. How many calories will this person burn in 24 hours? Simplify the complex fractions. 5 3 ⎛ 3⎞ ⎜ 1− ⎟ 8 10 ⎝ 4⎠ 79. 85. 89. 15 1 8 5 2 ⎛ 5 1⎞ 3 9 ⎜ + ⎟ 80. ⎝ 9 5⎠ 6 2 90. 86. 3 3 1 1 2 4 + 2 81. 91. 6 7 2 ⎛ 1 5⎞ 3 1 ⎜ + ⎟ − 8 ⎝ 4 6⎠ 5 4 82. 87. 2 12 1 1 3 − ⎛5 7 ⎞ 5 7 ⎜ − ⎟ 92. 5 5 1 ⎝ 6 12⎠ + 88. 7 10 12 4 83. 10 3 1 4 84. 1 2 93. A technician took one fourth of the amount of solution in a container. The technician then split this amount by one-fourth again. How much of the original solution did this technician have left? 94. A healthcare professional split 12 ounces of a powdered medicine into four equal amounts. How much did each amount weigh? 1 3 95. Tim lost 2 pounds and then lost 1 more pounds. How much weight did Tim lose altogether? 3 4 96. A pharmacy received 125 ﬂ oz of a bulk order of particular medication. If 50 smaller containers were made from this bulk order, how many ﬂuid ounces were in each smaller order? Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Chapter 1 Mathematical Essentials 21 97. A healthcare professional was asked to administer 1,000 units of a medication. However, all that is in stock are bottles containing 150 units. How many bottles should be administered? Give your answer as a mixed number. 98. A healthcare professional was asked to administer 1,500 units of a medication. However, all that is in stock are bottles containing 175 units. How many bottles should be administered? Give your answer as a mixed number. 1.3 – ORDER OF OPERATIONS OBJECTIVE The goal of this section is for the student to: ✓ evaluate expressions using the order of operations. The order of operations is a systematic way of calculating the value of a mathematical expression such as (3 + 3)2 ÷ 4 + 5. The order of operations provides the procedure to follow when evaluating mathe- matical expressions. Mathematical expressions must be evaluated by following this procedure. Calculators and computers are therefore programmed to follow this same process. ? How to Calculate Order of Operations 1. 2. 3. Simplify parentheses and brackets (in general grouping symbols). Evaluate exponents. Multiply and divide as symbols occur from left to right. 4. Add and subtract (as symbols occur from left to right). Many individuals remember the order of operations by thinking of the acronym PEMDAS. P stands for parentheses, E for exponents, M for multiplication, D for divi- sion, A for addition, and S for subtraction. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 22 Chapter 1 Mathematical Essentials EXAMPLE 1-16: Work parts 1 through 4. 1) Evaluate: 2 + 3 × 4 − 6 ÷ 2 + 4 × 5. a) Simplify grouping symbols. There are no grouping symbols, so move to step b. b) Evaluate exponents. There are no exponents so move to step c. c) Multiply and divide from left to right. 2 + 3 × 4 − 6 ÷ 2 + 4 × 5 = 2 + 12 − 3 + 20 12 3 20 d) Add and subtract, which gives 2 + 12 − 3 + 20 = 31. 2) Evaluate: (2 + 4)2 ÷ 4 × 2 a) Simplify the parenthesis: (6)2 ÷ 4 × 2. b) Evaluate exponents: 36 ÷ 4 × 2. c) Multiply and divide from left to right. 36 ÷ 4 × 2 gives 9 × 2 = 18. 9 Therefore, the answer is 18. 3) Evaluate: 5 + (4 + 6) ÷ (2 + 3) − 1. a) Simplify parentheses: 5 + (10) ÷ (5) − 1. b) There are no exponents, so move to step c. c) Multiply and divide from left to right. 5 + (10) ÷ (5) − 1 = 5 + 2 − 1 2 d) Add and subtract to get the answer of 6. 4) Compare the result of 4 × 4 ÷ 4 × 4 with the result of 4 × 4 ÷ (4 × 4). a) To evaluate 4 × 4 ÷ 4 × 4, multiply and divide as they occur from left to right: 4 × 4 ÷ 4 × 4 = 16 ÷ 4 × 4 = 4 × 4 = 16. 16 4 b) To evaluate 4 × 4 ÷ (4 × 4), first simplify the parentheses; then multiply and divide as they occur from left to right: 4 × 4 ÷ (4 × 4) = 4 × 4 ÷ 16 = 4 × 4 ÷ 16 = 16 ÷ 16 = 1. 16 16 NOTE Understanding when to use parentheses, and when not to, is important when using the calculator to evaluate mathematical expressions. Using the calculator is discussed in Section 2.10. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

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