6.3: Review Exercises
- Page ID
- 106261
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- An organization recently made a decision about which company to use to redesign its website and host its members’ information. The Board of Directors will vote using preference ballots ranking their first choice to last choice of the following companies: Allied Web Design (A), Ingenuity Incorporated (I), and Yeehaw Web Trends (Y). The individual ballots are shown below. Create a preference schedule summarizing these results.
AIY, YIA, YAI, AIY, YIA, IAY, IYA, IAY, YAI, YIA, AYI, YIA, YAI
- A group needs to decide where their next conference will be held. The choices are Kansas City (K), Lafayette (L), and Minneapolis (M). The individual ballots are shown below. Create a preference schedule summarizing these results.
KLM, LMK, MLK, LMK, MKL, KLM, KML, LMK, MKL, MKL, MLK, MLK
- A book club holds a vote to figure out what book they should read next. They are picking from three different books. The books are labeled A, B, and C, and the preference schedule for the vote is below.
| Number of voters | 12 | 9 | 8 | 5 | 10 |
|---|---|---|---|---|---|
| 1st choice | A | B | B | C | C |
| 2nd choice | C | A | C | A | B |
| 3rd choice | B | C | A | B | A |
- How many voters voted in the election?
- How many votes are needed for a majority?
- Find the winner using the Plurality Method.
- Find the winner using the Borda Count Method.
- Find the winner using the Plurality with Elimination Method.
- Find the winner using the Pairwise Comparisons Method.
- An election is held for a new vice president at a college. There are three candidates (A, B, C), and the faculty rank which candidate they like the most. The preference ballot is below.
| Number of voters | 8 | 10 | 12 | 9 | 4 | 1 |
|---|---|---|---|---|---|---|
| 1st choice | A | A | B | B | C | C |
| 2nd choice | B | C | A | C | A | B |
| 3rd choice | C | B | C | A | B | A |
- How many voters voted in the election?
- How many votes are needed for a majority?
- Find the winner using the Plurality Method.
- Find the winner using the Borda Count Method.
- Find the winner using the Plurality with Elimination Method.
- Find the winner using the Pairwise Comparisons Method.
- A city election for a city council seat was held between 4 candidates, Martorana (M), Jervey (J), Riddell (R), and Hanrahan (H). The preference schedule for this election is below.
| Number of voters | 60 | 73 | 84 | 25 | 110 |
|---|---|---|---|---|---|
| 1st choice | M | M | H | J | J |
| 2nd choice | R | H | R | R | M |
| 3rd choice | H | R | M | M | R |
| 4th choice | J | J | J | H | H |
- How many voters voted in the election?
- How many votes are needed for a majority?
- Find the winner using the Plurality Method.
- Find the winner using the Borda Count Method.
- Find the winner using the Plurality with Elimination Method.
- Find the winner using the Pairwise Comparisons Method.
- A local advocacy group asks members of the community to vote on which project they want the group to put its efforts behind. The projects are green spaces (G), city energy code (E), water conservation (W), and promoting local business (P). The preference schedule for this vote is below.
| Number of voters | 12 | 57 | 23 | 34 | 13 | 18 | 22 | 39 |
|---|---|---|---|---|---|---|---|---|
| 1st choice | W | W | G | G | E | E | P | P |
| 2nd choice | G | P | W | E | G | P | W | E |
| 3rd choice | E | E | E | W | P | W | G | W |
| 4th choice | P | G | P | P | W | G | E | G |
- How many voters voted in the election?
- How many votes are needed for a majority?
- Find the winner using the Plurality Method.
- Find the winner using the Borda Count Method.
- Find the winner using the Plurality with Elimination Method.
- Find the winner using the Pairwise Comparisons Method.
- A solar system consisting of five planets has a governing council of 135 members who are apportioned proportional to the populations of the planets. The population for each planet is listed in the following table.
| Planet | Ajax | Borax | Calax | Delphi | Eljix | Total |
|---|---|---|---|---|---|---|
| Population | 183,000 | 576,000 | 274,000 | 749,000 | 243,000 | 2,025,000 |
- For each planet, find the standard quota, the upper quota and the lower quota. Give your answers in a table.
- Use Hamilton’s method to apportion the 135 council members.
- Use Jefferson's method to apportion the 135 council members
- The country named Erau has five states and a total of 200 seats available in its House of Representatives. The number of seats that each state receives is proportional to the population of that state. The populations of the states are given in the table below.
| State | 1 | 2 | 3 | 4 | 5 | Total |
|---|---|---|---|---|---|---|
| Population | 3,500,000 | 1,200,000 | 530,000 | 999,000 | 771,000 | 7,000,000 |
- For each state, find the standard quota, the upper quota and the lower quota. Give your answers in a table.
- Use Hamilton’s method to apportion the 200 seats.
- Use Jefferson's method to apportion the 200 seats.
- Last year a city had three school districts:
- North, with a population of 5,200 children
- South, with a population of 10,600 children,
- West, with a population of 15,100 children.
- Use Hamilton’s method to apportion 50 speech therapists among the districts using the populations for last year.
- This year, the city took over another school district. The new East district has a population of 9,500 children. If the number of speech therapists is increased by 15 to accommodate the new district, use Hamilton’s method to apportion the 65 speech therapists.
- Three people invest in a treasure dive, each investing the amount listed below. The dive results in finding 36 gold coins.
Alice: $7,600 Ben: $5,900 Carlos: $1,400
- Use Hamilton’s method to apportion the 36 coins.
- Use Jefferson's method to apportion the 36 coins.