Math in Society Edition 2.3 Contents Problem Solving . 1 Extension: Taxes . 30 David Lippman Voting Theory . 35 David Lippman Weighted Voting . 59 David Lippman Apportionment . 75 Mike Kenyon, David Lippman Fair Division . 93 David Lippman Graph Theory . 117 David Lippman Scheduling . 155 David Lippman Growth Models . 173 David Lippman Finance . 197 David Lippman Statistics . 227 David Lippman, Jeff Eldridge, onlinestatbook.com Describing Data . 247 David Lippman, Jeff Eldridge, onlinestatbook.com Probability . 279 David Lippman, Jeff Eldridge, onlinestatbook.com Sets . 319 David Lippman Historical Counting Systems . 333 Lawrence Morales, David Lippman Fractals . 367 David Lippman Cryptography . 387 David Lippman, Melonie Rasmussen Solutions to Selected Exercises . 407 David Lippman Pierce College Ft Steilacoom Copyright © 2013 David Lippman This book was edited by David Lippman, Pierce College Ft Steilacoom Development of this book was supported, in part, by the Transition Math Project and the Open Course Library Project. Statistics, Describing Data, and Probability contain portions derived from works by: Jeff Eldridge, Edmonds Community College (used under CC-BY-SA license) www.onlinestatbook.com (used under public domain declaration) Apportionment is largely based on work by: Mike Kenyon, Green River Community College (used under CC-BY-SA license) Historical Counting Systems derived from work by: Lawrence Morales, Seattle Central Community College (used under CC-BY-SA license) Cryptography contains portions taken from Precalculus: An investigation of functions by: David Lippman and Melonie Rasmussen (used under CC-BY-SA license) Front cover photo: Jan Tik, http://www.flickr.com/photos/jantik/, CC-BY 2.0 This text is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. You are free: to Share — to copy, distribute, display, and perform the work to Remix — to make derivative works Under the following conditions: Attribution. You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). Share Alike. If you alter, transform, or build upon this work, you may distribute the resulting work only under the same, similar or a compatible license. With the understanding that: Waiver. Any of the above conditions can be waived if you get permission from the copyright holder. Other Rights. In no way are any of the following rights affected by the license: • Your fair dealing or fair use rights; • Apart from the remix rights granted under this license, the author's moral rights; • Rights other persons may have either in the work itself or in how the work is used, such as publicity or privacy rights. • Notice — For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to this web page: http://creativecommons.org/licenses/by-sa/3.0/us/ About the Author/Editor David Lippman received his master’s degree in mathematics from Western Washington University and has been teaching at Pierce College since Fall 2000. David has been a long time advocate of open learning, open materials, and basically any idea that will reduce the cost of education for students. It started by supporting the college’s calculator rental program, and running a book loan scholarship program. Eventually the frustration with the escalating costs of commercial text books and the online homework systems that charged for access led to action. First, David developed IMathAS, open source online math homework software that runs WAMAP.org and MyOpenMath.com. Through this platform, he became an integral part of a vibrant sharing and learning community of teachers from around Washington State that support and contribute to WAMAP. These pioneering efforts, supported by dozens of other dedicated faculty and financial support from the Transition Math Project, have led to a system used by thousands of students every quarter, saving hundreds of thousands of dollars over comparable commercial offerings. David continued further and wrote the first edition of this textbook, Math in Society, after being frustrated by students having to pay $100+ for a textbook for a terminal course. Together with Melonie Rasmussen, he co-authored PreCalculus: An Investigation of Functions in 2010. Acknowledgements David would like to thank the following for their generous support and feedback. • Jeff Eldridge, Lawrence Morales, and Mike Kenyon, who were kind enough to license me use of their works. • The community of WAMAP users and developers for creating some of the homework content used in the online homework sets. • Pierce College students in David’s online Math 107 classes for helping correct typos and identifying portions of the text that needed improving, along with other users of the text. • The Open Course Library Project for providing the support needed to produce a full course package for this book. Preface The traditional high school and college mathematics sequence leading from algebra up through calculus could leave one with the impression that mathematics is all about algebraic manipulations. This book is an exploration of the wide world of mathematics, of which algebra is only one small piece. The topics were chosen because they provide glimpses into other ways of thinking mathematically, and because they have interesting applications to everyday life. Together, they highlight algorithmic, graphical, algebraic, statistical, and analytic approaches to solving problems. This book is available online for free, in both Word and PDF format. You are free to change the wording, add materials and sections or take them away. I welcome feedback, comments and suggestions for future development. If you add a section, chapter or problems, I would love to hear from you and possibly add your materials so everyone can benefit. New in This Edition Edition 2 has been heavily revised to introduce a new layout that emphasizes core concepts and definitions, and examples. Based on experience using the first edition for three years as the primarily learning materials in a fully online course, concepts that were causing students confusion were clarified, and additional examples were added. New “Try it Now” problems were introduced, which give students the opportunity to test out their understanding in a zero-stakes format. Edition 2.0 also added four new chapters. Edition 2.1 was a typo and clarification update on the first 14 chapters, and added 2 additional new chapters. No page or exercise numbers changed on the first 14 chapters. Edition 2.2 was a typo revision. A couple new exploration exercises were added. Edition 2.3 is a typo revision. Supplements The Washington Open Course Library (OCL) project helped fund the creation of a full course package for this book, which contains the following features: • Suggested syllabus for a fully online course • Possible syllabi for an on-campus course • Online homework for most chapters (algorithmically generated, free response) • Online quizzes for most chapters (algorithmically generated, free response) • Written assignments and discussion forum assignments for most chapters The course shell was built for the IMathAS online homework platform, and is available for Washington State faculty at www.wamap.org and mirrored for others at www.myopenmath.com. The course shell was designed to follow Quality Matters (QM) guidelines, but has not yet been formally reviewed. Problem Solving 1 Problem Solving In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you will need (and won’t need) to answer the question. In this chapter, we will review several basic but powerful algebraic ideas: percents, rates, and proportions. We will then focus on the problem solving process, and explore how to use these ideas to solve problems where we don’t have perfect information. Percents In the 2004 vice-presidential debates, Edwards's claimed that US forces have suffered "90% of the coalition casualties" in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies "have taken almost 50 percent" of the casualties1. Who is correct? How can we make sense of these numbers? Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is 40 equivalent to the fraction or the decimal 0.40. Notice that 80 out of 200 and 10 out of 100 80 10 40 25 are also 40%, since = = . 200 25 100 Example 1 243 people out of 400 state that they like dogs. What percent is this? 243 60.75 =0.6075 = . This is 60.75%. 400 100 Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right. Example 2 1 Write each as a percent: a) b) 0.02 c) 2.35 4 1 a) = 0.25 = 25% b) 0.02 = 2% c) 2.35 = 235% 4 1 http://www.factcheck.org/cheney_edwards_mangle_facts.html © David Lippman Creative Commons BY-SA 2 Percents If we have a part that is some percent of a whole, then part percent = , or equivalently, part= percent ⋅ whole whole To do the calculations, we write the percent as a decimal. Example 3 The sales tax in a town is 9.4%. How much tax will you pay on a $140 purchase? Here, $140 is the whole, and we want to find 9.4% of $140.