Excursions in Modern Mathematics Peter Tannenbaum Chapter 1 The Mathematics of Voting Slides prepared by Beth Kirby and Carl Lee University of Kentucky MA 111 Fall 2011 Voting UK. Ballots and Schedules Plurality Borda Plurality with Elimination Pairwise Comparisons Ballots and Schedules.

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• Excursions in Modern Mathematics introduces readers to the power and beauty of math. By developing an appreciation for the aesthetics and applicability of mathematics, readers who previously felt math was an “unknowable” subject can approach it with a new perspective.

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Excursions in Modern Mathematics introduces you to the power of math by exploring applications like social choice and management science, showing that math is more than a set of formulas. Ideal for an applied liberal arts math course, Tannenbaum’s text is known for its clear, accessible writing style and its unique exercise sets that build in complexity from basic to more challenging. The Eighth Edition offers more real data and applications to connect with today’s readesr, expanded coverage of applications like growth, and revised exercise sets.

Sample questions asked in the 8th edition of Excursions in Modern Mathematics:

While the number of smokers for the general adult population is decreasing, it is not decreasing equally across all subpopulations (and for some groups it is actually increasing).For the 18-to-24 age group, the number of smokers was 8million in 1965 and 6.3 million in 2009. ( Source: “Trendsin Tobacco Use,” American Lung Association, 2011.) Assuming the number of smokers in the 18-to-24 age groupcontinues decreasing according to a negative linear growthmodel, (a) predict the number of smokers in the 18-to-24 agegroup in 2015 (round your answer to the nearest thousand). (b) predict in what year the number of smokers in the18-to-24 age group will reach 5 million.

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The city of Lightsville currently has 137 streetlights. As partof an urban renewal program, the city council has decidedto install and have operational 2 additional streetlights atthe end of each week for the next 52 weeks. Each streetlightcosts $1 to operate for 1 week. (a) How many streetlights will the city have at the end of38 weeks? (b) How many streetlights will the city have at the end of N weeks? (Assume N ? 52.) (c) What is the cost of operating the original 137 lights for52 weeks? (d) What is the additional cost for operating the newlyinstalled lights for the 52-week period during whichthey are being installed?

Consider the data set {-4, 6, 8, -5.2, 10.4, 10, 12.6, -13} (a) Find the average A of the data set. (b) Find the median M of the data set. (c) Consider the data set {-4, 6, 8, -5.2, 10.4, 10, 12.6} having one less data point than the original set. Find the average and the median of this data set.

Consider the data set {-5, 7, 4, 8, 2, 8, -3, -6}. (a) Find the first quartile Q 1 of the data set. (b) Find the third quartile Q 3 of the data set. (c) Consider the data set {-5, 7, 4, 8, 2, 8, -3, -6, 2} obtained by adding one more data point to the original data set. Find the first and third quartiles of this data set.

The world population reached 6 billion people in 1999 and 7billion in 2012. ( Source: Negative Population Growth, www .npg.org .) Assuming a linear growth model for the worldpopulation, (a) predict the year when the world population wouldreach 8 billion. (b) predict the world population in 2020.

Explain each of the following statements regarding the median score in one of the SAT sections: (a) If the number of test-takers N is odd, then the median score must end in 0. (b) If the number of test-takers N is even, then the median score can end in 0 or 5, but the chances that it will end in 5 are very low.

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• Step 1 of 4

  (a)

  A ballot in which the voters are asked to rank the candidates inorder of preference is called a preference ballot.

  A logical way to organize the ballots, is to group together theidentical ballots. The ballots for an election with 21 votes andfive candidates are organized into piles as shown below:

• Step 2 of 4

  Write the preference schedule for the election.

  Number of Voters	5	5	3	3	3	2
  1st choice	A	C	A	B	D	D
  2nd choice	B	E	D	E	C	C
  3rd choice	C	D	B	A	B	B
  4th choice	D	A	C	C	E	A
  5th choice	E	B	E	D	A	E

  Table 1

• Step 3 of 4

  (b)

  The candidate with the most first-place votes wins. This methodis called the plurality Method.

  Thus, in an election conducted under the plurality method, onedoes not need each voter to rank the candidates.

  From the preference schedule for the election given in Table 1,the following can be ascertained:

  The candidate A gets eight first-place votes.

  The candidate B gets three first-place votes.

  The candidate C gets five first-place votes.

  The candidate D gets five first-place votes.

  The candidate E gets zero first-place votes.

  Therefore, by the plurality Method, the candidategets the highest first-place votes.

• Step 4 of 4

  (c)

  To calculate the candidates with the fewest last-place votes,use the preference schedule for the election given in Table 1.

  It is observed that at the last place (5^th choice),candidates A, B, D, and E appear.

  From the table, the following can be observed:

  The candidate A gets three last-place votes.

  The candidate B gets five last-place votes.

  The candidate C gets zero last-place votes.

  The candidate D gets three last-place votes.

  The candidate E gets 10 last-place votes.

  Thus, the candidategets the fewest last-place votes.