Ultimate Final Exam Guide: Graph Theory, Voting Methods & Apportionment

Document 1: Created-Midterm Study Sheet Keywords: simple interest, compound interest, present value, percent change, APY calculation, installment loan payment, finance charge, down payment, mortgage points, ordinary annuity, sinking fund, modular arithmetic, base-n systems, Roman numerals, golden ratio, Egyptian numeration, ancient numeration systems, Mayan numeration, Babylonian numeration, Greek numeration, Chinese numeration Detailed Description: This study sheet is a comprehensive resource for core mathematical finance concepts and ancient numeration systems. It explains and gives step‑by‑step formulas for calculating simple interest, compound interest, present value, percent change, APY, installment loan payments, finance charges, down payments, points on mortgages, ordinary annuities, and sinking funds. Each topic is accompanied by clear worked examples to illustrate how to apply the formulas in real situations (e.g., calculating interest on loans or determining future value of annuities). It also includes a section on modular arithmetic with examples like clock calculations, as well as explanations of base‑n systems and their conversions. Additional sections dive into historical numeration methods from different cultures—Egyptian, Babylonian, Mayan, Greek, and Chinese—explaining the symbols, bases, and usage. It even touches on Roman numerals and the Golden Ratio with practical examples. Overall, this sheet combines modern financial math with ancient mathematical history in a concise yet detailed reference. Document 2: Final Exam – Graph Theory, Voting, and Apportionment Methods Keywords: graph, vertex, edge, loop, connected graph, disconnected graph, degree of vertex, bridge, path, circuit, Euler path, Euler circuit, Hamilton path, Hamilton circuit, Fleury’s algorithm, Kruskal’s algorithm, spanning tree, voting methods, plurality method, plurality with elimination, Borda count, pairwise comparison method, voting criteria, majority criterion, head-to-head criterion, monotonicity, irrelevant alternatives, apportionment methods, standard divisor, standard quota, lower quota, upper quota, Hamilton’s method, Jefferson’s method, Adams’s method, Webster’s method, apportionment paradoxes, new-states paradox, Alabama paradox, population paradox, functions, slope, linear equation, exponential functions Detailed Description: This study guide thoroughly covers three major topics: graph theory, voting methods, and apportionment techniques. In the graph theory section, it defines fundamental concepts such as vertices, edges, loops, connected and disconnected graphs, degrees of vertices, and bridges. It explains special paths and circuits, including Euler paths/circuits and Hamilton paths/circuits, with conditions for their existence. It also introduces algorithms like Fleury’s for tracing Euler circuits and Kruskal’s for building minimum spanning trees. The voting methods section details various systems—plurality, plurality with elimination, Borda count, and pairwise comparisons—along with voting fairness criteria such as majority, head-to-head (Condorcet), monotonicity, and irrelevant alternatives. It emphasizes how to apply these criteria and includes tips to avoid common counting mistakes. The apportionment section covers core definitions like standard divisor/quota and explains methods (Hamilton’s, Jefferson’s, Adams’s, and Webster’s) with their rounding rules. It also explains well-known paradoxes (new-states, Alabama, and population paradoxes) and why they occur. Finally, there’s a brief review of functions and graphs, including slope calculation, linear and exponential equations, and how to interpret intercepts. This sheet is an in‑depth, exam‑ready reference blending discrete mathematics and real‑world voting/apportionment applications.