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mathematics
contemporary mathematics

Contemporary Mathematics 1st Edition OpenStax - Solutions

Suppose the constant ratio of \(A\) to \(B\) is 0.75 . Student 1 says the ratio of \(A\) to \(B\) is \(\frac{4}{3}\) Student 2 says the ratio of \(B\) to \(\mathrm{A}\) is \(\frac{3}{4}\).
Student 1 says that Standard Divisor \(=\frac{\text { House Size }}{\text { Total Population }}\). Student 2 says that Standard Divisor \(=\frac{\text { Sum of the State Populations }}{\text { Total Number of Seats Available }}\)
Student 1 says that State's Standard Divisor \(=\frac{\text { Total Population }}{\text { State's Standard Quota }}\).Student 2 says that State's Standard Quota \(=\frac{\text { State Population }}{\text { Standard Divisor }}\).
Student 1 says that State's Standard Quota \(=\) State Population \(\div \frac{\text { Total Population }}{\text { House Size }}\). Student 2 says that State's Standard Quota \(=\frac{\text { State Population }}{\text { Standard Divisor }}\)
Which of the four apportionment methods discussed in this section does not use a modified divisor?
Which of the four apportionment methods discussed in this section satisfies the quota rule?
Which of the four apportionment methods discussed in this section is biased toward states with larger populations?
Which of the four apportionment methods discussed in this section is biased toward states with smaller populations?
Which of the four apportionment methods discussed in this section begin the apportionment with a state's upper quota and adjust down?
Which of the four apportionment methods discussed in this section begin the apportionment with a state's lower quota and adjust up?
Which of the four apportionment methods discussed in this section use traditional rounding?
Does the change from a standard divisor to a modified divisor tend to change the number of seats for larger or smaller states more?
A city purchased five new firetrucks and apportioned them among the existing fire stations. Although your neighborhood fire station has the same proportion of the city's firetrucks as before the new ones were purchased, it now has one fewer.A scenario is given. Determine whether the scenario is an
The school resources officers in a county were reapportioned based on the most recent census. The number of students at Chapel Run Elementary went up while the number of students at Panther Trail Elementary went down. However, Chapel Run now has one fewer resources officer while Panther Trail has
The standard quota for States A, B, and C are 1.19, 2.73, and 5.71 respectively. State A received 1 seat, State B received 3 seats, and State \(C\) received 4 seats.A scenario is given. Determine whether the scenario is an example of a quota rule violation, the Alabama paradox, the population
A corporation that owns several hospitals purchased an additional hospital, causing the doctor to patient ratio to decrease.A scenario is given. Determine whether the scenario is an example of a quota rule violation, the Alabama paradox, the population paradox, the new-states paradox, or none of
When the city of Cocoa annexed an adjacent unincorporated community, the number of seats on the city council was increased to maintain the standard ratio of citizens to seats, but one existing community of Cocoa still lost a seat on the city council to another existing community of Cocoa when the
A reapportionment occurs because the populations of the states change, and the house size remains the same.A type of scenario is described. Indicate which paradox could arise in a scenario of this kind.
A reapportionment occurs because the house size increases, and the populations of the states remain the same.A type of scenario is described. Indicate which paradox could arise in a scenario of this kind.
A reapportionment occurs because an additional state is added to the union, the populations of the original states remain the same, and the house size is increased to correspond to the population of the new state.A type of scenario is described. Indicate which paradox could arise in a scenario of
Refer back to Table 11.1. Did any single candidate secure the majority of popular votes? Candidate (Party Label) Popular Vote Total Al Gore (Democrat) 50,999,897 George W. Bush (Republican) 50,456,002 Ralph Nader (Green) 2,882,955 448,895 Patrick J. Buchanan (Reform/Independent) Table 11.1 Results
Refer again to Table 11.1. In the 2000 U.S. presidential election, which candidate had a plurality of popular votes? Candidate (Party Label) Popular Vote Total 50,999,897 Al Gore (Democrat) George W. Bush (Republican) 50,456,002 Ralph Nader (Green) 2,882,955 448,895 Patrick J. Buchanan
A condominium association elects a new president every two years by a two-round system of voting. If none of the candidates receive a majority, the association charter states that the top two candidates will be eligible to participate in a runoff election. In a particular year, five residents were
The five members of the Chionilis family-Annette, Rene, Seema, Titus, and Galen-have decided to get takeout for dinner. They are trying to decide on a restaurant. The options are Rainbow China, Dough Boys Pizza, Taco City, or Caribbean Flavor. They will use majority election with runoffs where the
Refer to the table below containing voters' preference rankings to answer the following questions.1. How many voters ranked the options in the following order: Option A in fourth place, Option B in second place, Option C in first place, and Option D in third place?2. How many ballots in total were
Let's review the kindergarten class color preferences again, and this time determine which color would be selected based on these results using the ranked-choice method. Number of Ballots 46 Red 4 7 26 25 Blue 1 2 1 4 Green 65 6 5 5 2 Yellow 5 4 6 3 Purple 4 3 4 1 Pink 3 1 3 160
Let's review the ballots from the kindergarten class again, as shown in the table below. This time, let's determine the Borda score received by the color purple. Number of Ballots 4 6 4 7 Red 26 25 Blue 1 2 1 4 Green 6 5 5 2 Yellow 5 4 6 3 Purple 4 3 4 1 Pink 3 1 - 3 6
Use the table below, which displays a sample preference summary, to answer the questions that follow.1. Use the ranked-choice voting method to determine the winner of the election.2. Use the Borda count method to determine the winner of the election. Number of Ballots 95 90 110 115 Option A 4 4 1 1
Consider the summary of ranked ballots shown in the table below. Determine the winner of an election using the pairwise comparison method.1. Construct a pairwise comparison matrix for the sample summary of ranked ballots in the table above.2. Use the pairwise comparison method to determine a
Suppose that three people are playing the game Rock, Paper, Scissors. On the count of three, each person shows a hand signal for rock, paper, or scissors. Each hand signal beats another hand signal. The group keeps having a tie because Person A always picks rock, Person B always picks paper which
1. Suppose there is an election with five candidates-A, B, C, D, and E-and that Candidate \(C\) is a Condorcet candidate. How many points did Candidate \(\mathrm{C}\) win?2. What is the greatest number of points that any one of the other candidates could win?3. Is it possible for Candidate
Suppose that Person A/B/C were just about to give up on their game of Rock, Paper, Scissors when they were joined by Person D who reminded them that their updated version, Rock, Paper, Scissors, Lizard, Spock was a far superior game with the added rules that Lizard eats Paper, Paper disproves
The eight members of the Chionilis family-Annette, Rene, Seema, Titus, Galen, Elena, Max and Demitri-have another decision to make. Approval voting worked out nicely the last time. They are going to use it again, but this time, Annette, Rene, Seema, and Galen are feeling a little indecisive. They
Because the vote counts for George W. Bush and Al Gore differed by only 537 votes, many Democrats blamed Ralph Nader and the Green Party for their loss. Let's consider how the election results might have differed if the approval voting method had been used.Use Table 11.3 and the following
In the future, humans have explored distant solar systems and found three habitable planets which could be colonized. Since it will take all available resources to colonize one planet, humans must agree on the planet. Planet A has the most comfortable climate and most plentiful resources, but it is
It's final exams week and seven college students are hungry. They must get food, but from which drive thru? Their preferences are listed in the table below. The majority have listed McDonald's as their top choice. Let's calculate what the results of the election will be using various voting
Those seven college students are hungry again! Their preferences haven't changed, as shown below. Let's calculate if the results change when we use different voting methods.1. Which restaurant is the winner using the pairwise comparison voting method?2. Which restaurant is the winner using the
A survey asked a random sample of 100 people in the United States to rank their priorities for spending their tax refund. The options were (V) go on vacation, (S) put into savings, (D) pay off debt, or (T) other. The pairwise comparison matrix for the results is in Figure 11.11. Determine whether
Let's return to the survey about tax refund spending from Example 11.18. We know that the Condorcet criterion applies because Option D, "Pay off debt," is a Condorcet candidate, which wins every pairwise match up.Use the information in the ballot summary from the table below to find the winner and
The local animal shelter is having a vote-by-donation charity event. For a \(\$ 10\) donation, an individual can complete a ranked ballot indicating their favorite large dog breed: standard poodle, golden retriever, Labrador retriever, or bulldog. Use the summary of ballots below to answer each
Earlier, we discovered that the summary of ranked ballots shown in the table below results in the pairwise comparison matrix in Figure 11.12. Use this information to answer the questions.1. Determine the winner of the election by the pairwise comparison method.2. Suppose that the 95 voters in the
Suppose that 30 students in a class are going to vote on whether to have apple, blueberry, or cherry pie. Use the summary of ranked ballots in below to answer each question.1. Determine the winner of the election by plurality.2. Which candidate would win a plurality election if cherry pie were
The NBC sitcom The Office ran for nine years and has been one of the most popular streamed television shows of all time. One of the trademarks of the show was that characters would often break the fourth wall to communicate with the audience just by staring directly into the camera. In fact, there
The following table provides a comparison of the number of faculty members in each department at a particular college to the student head count in that department and the number of class sections in that department in the Spring semester. Use this information to answer the questions.1. Determine
Refer to the information given in Example 11.24 .1. If there are 32 class sections each semester in the Fine Art department, and the same ratio is used to determine the number of faculty members, how many faculty members would you expect to see in the Fine Art department?2. If the Health Sciences
Table 11.4 contains a list of the five U.S. states with the greatest number of representatives in the U.S. House of Representatives, along with the population of that state in 2021. Use the information in the table to answer the questions.1. What is the ratio of State Population to Representative
As of this writing, the Census.gov website U.S. Population clock showed a population of 332,693,997. There are 435 seats in the U.S. House of Representatives. Find the standard divisor rounded to the nearest tenth.
The public schools in a certain county have been allotted 349 school resource officers to be distributed among 327 public schools attended by approximately 271,500 students.1. Identify the states, seats, house size, state population, and total population in this apportionment scenario.2. Describe
Example 27 outlined that the Census.gov website U.S. Population clock showed a population of \(330,147,881\), there are 435 seats in the U.S. House of Representatives, and the standard divisor was \(758,960.6\) people per representative. The state of California has a population of approximately
The science department of a high school has received a grant for 34 laptops. They plan to apportion them among their six classrooms based on each classroom's student capacity. Use the values in the table below to find the standard quota for each classroom. Room Students A 30 B 25 C 28 D 32 E FL 24
The board of trustees of a college has recently approved the installation of 70 new emergency blue lights in three parking lots. The number of lights in each lot will be proportionate to the size of the parking lot, which isto be measured in acres. The total number of acres is 34 ; so the standard
In 2015, the U.S. Air Force had a fleet of approximately 281 A-10C Thunderbolt II aircraft. Suppose that the Air Force administration wanted to distribute 27 aircrafts across six bases based on the number of qualified pilots stationed at those bases. Use the information in the table below to answer
The Air Force administration wants to distribute 27 aircrafts across six bases based on the number of qualified pilots stationed at those bases. The standard quotas for each base are listed in the table below. Use this information to answer the question1. Give the lower quota for each Air Force
Suppose that the Hawaii State Department of Education has a budget for 616 schools and is doing a research study to determine the equitable number of schools to have in each of the five counties based on the residents under 19 years old, This data is provided in the table below. Using the Hamilton
Several apportionment methods have been used to allocate 125 seats to ten states and the results are shown in the table below. Determine which apportionments do not satisfy the quota rule and justify your answer. State A State B State C State D State E State F State G Standard Quota 41.26 16.00
Suppose the population of a state is 50 and the standard divisor is 12.5 .1. Find the state's standard quota.2. Increase the standard divisor by 2 units and use the modified divisor to determine the modified quota for the state.3. Decrease the modified divisor from part 2 by 1.5 units and use the
Suppose that the Hawaii State Department of Education has a budget for 616 schools and is doing a research study to determine the equitable number of schools to have in each of five counties based on the residents under the age of 19. With the data in the table below, apply Jefferson's method to
As in earlier examples, suppose that the Hawaii State Department of Education has a budget for 616 schools and is doing a research study to determine the equitable number of schools to have in each of the five counties based on the residents under the age of 19 . Use the data in the following table
Use the data in the table below to apportion 616 schools to Hawaiian counties. This time, use Webster's method. Hawaii Honolulu Kalawao Kauai Maui Total Residents under Age 19 46,310 224,230 20 16,560 38,450 325,570
Let's use the results from Example 11.34, Example 11.37, Example 11.38, and Example 11.39 to compare the four apportionment methods we have discussed. The following table summarizes the results of the results of the Hamilton, Jefferson, Adams and Webster methods when applied to the apportionment of
The following table displays the effect of reducing the size of the divisor. Observe the effect this has on the modified quotas of smaller states versus larger states and use the table answer each question.1. When the divisor decreases from 10,500 to 10,000 , how many representatives are gained by
The 1880 census recorded the population of Alabama as \(1,513,401\) and that of the U.S. as \(62,979,766\).1. Calculate the standard divisor and standard quota for the State of Alabama based on a house size of 299.2. Calculate the standard divisor and standard quota for the State of Alabama based
Suppose that States A and B each have a population of 6 , while State \(C\) has a population of 2.1. Use the Hamilton method to apportion 10 seats.2. Use the Hamilton method to apportion 11 seats.3. Does this example demonstrate the Alabama paradox? If so, how?
Suppose that 18 respirators are to be apportioned to three hospitals based on their capacities. The Hamilton method is used to allocate the respirators in 2020, then to reallocate based on new capacities in 2021. The results are shown in the table below. How does this demonstrate the population
The country of Costaguana has three states: Azuera with a population of 894,000; Punta Mala with a population of 696,000; and Esmeralda with a population of 215,000. There are 38 seats in the Congress of Costaguana. The apportionment of the seats is determined by Hamilton's method to be: 19 for
The country of San Lorenzo has grown from two states to three. The house size of the congress has been increased by eight and the seats have been reapportioned to accommodate the new state of Oscuridad. The constitution mandates the use of the Hamilton method of apportionment. Use this information
The country of Gulliversia has two states: Lilliput with a population of 700,000 and Brobdingnag with a population of 937,000. The constitution of Gulliversia requires that the 90 congressional seats be apportioned by Hamilton's method. Lilliput has received 38 seats while Brobdingnag has received
Oklahoma was admitted as the 46th state on November 16, 1907. Before Oklahoma joined the union, the U.S. population was approximately \(75,030,000\) and the House of Representatives had 386 seats. The new state had a population of approximately 970,000 . Use this information to estimate the
The incorporated town of Orange Grove consists of two subdivisions: The Oaks with 1,254 residents and The Villages with 10,746 residents. A council with 100 members supervises the municipality's operations. The council votes to annex an unincorporated subdivision called The Lakes with a population
A company with an office in each of four cities must distribute 145 new Chromebooks to the four offices. It is determined that Office \(A\) will receive 42 , Office \(B\) will receive 17 , Office \(C\) will receive 35 , and Office \(D\) will receive 51.At the last minute, it is discovered that
A county with three towns has 30 garbage trucks to apportion. Attenborough receives 6 trucks, Breckenridge receives 8 trucks, and Cabbotsville receives 16 trucks, in proportion to their populations. When one additional truck is purchased, the reallocation results in 5 trucks for Attenborough, 9 for
A group of 60 dentists work for a company that runs five offices. The dentists have been apportioned to the offices by the number of patients. Office A receives 14 dentists, Office B receives 11, Office C receives 11, Office \(D\) receives 12 , and Office \(E\) receives 12 . When a new dentist
A company with locations in three cities plans to give 200 achievement awards that shall be apportioned to the three cities by population. City A with 9,150 employees is allotted 61 awards, City B with 6,040 employees is allotted 40 awards, and City C with 14,810 employees is allotted 99 awards to
A soccer club must apportion soccer balls to the teams among four age brackets based on the number of teams in each bracket. Bracket U8 receives 32 balls, U12 receives 46 balls, U15 receives 29 balls, and U18 receives 25 balls. Mid-season, the balls are reapportioned. U8 has decreased by 10
READ (Reading Education Assistance Dogs) trains and certifies therapy animals to serve in classrooms and help children develop a love of reading in a low-stress environment. Suppose that 20 therapy teams are apportioned to three schools based on their populations. When the population of School A
A charity organization has 851 volunteers in Country A and 3449 volunteers in Country B. There are 43 lead organizers that must be apportioned to the two locations. Country A receives 9 while Country B receives 34.When the operations are expanded to Country \(C\) with 725 volunteers, 7 new lead
A country has two states. There are 22 seats in the legislature. State A has 4 seats, and State B has 18 seats. When State \(\mathrm{C}\) joins the union, the number of representatives is increased by five. Under the new apportionment, State A receives 3 seats, State B receives 19 seats, and State
In the Garunga Solar System, there are four inhabited planets. Three of the planets are members of the United Association of Garungan Planets (UAGP). The UAGP has 67 seats. Planet Angluertka has 40 seats, planet Bangluertka has 27 seats, and planet Clangluertka has 13 seats. When planet
When the number of seats changed from 147 to 148 , the standard quotas changed from A 44.24, B 17.35 , C 37.12 , and D 48.29 to A 45.54, B 17.47, C 37.37, and D 48.62 .a. How did the increase in seats impact the apportionment?b. Is this apportionment an example of a paradox? Justify your answer.Use
When the number of seats changed from 126 to 127 , the standard quotas changed from \(A 9.57, B 29.49, C\) 33.89, D 28.43, and E 24.61 to A 9.65, B 29.72, C 34.16, D 38.66, and E 24.81.a. How did the increase in seats impact the apportionment?b. Is this apportionment an example of a paradox?
When the number of seats changed from 25 to 26 , the standard quotas changed from \(A 2.21, B 5.25, C\) 11.27, and D 6.27 to A \(2.30, B 5.46\), C 11.72 , and D 6.52 .a. How did the increase in seats impact the apportionment?b. Is this apportionment an example of a paradox? Justify your answer.Use
When the number of seats changed from 25 to 26 , the standard quotas changed from A 2.43, B 5.42, and C 8.15 to A 2.46, B 5.47 , and C 8.07 .a. How did the increase in seats impact the apportionment?b. Is this apportionment an example of a paradox? Justify your answer.Use the Hamilton method of
The house size is 18 . When the population of State \(A\) increases by 11.76 percent, State \(B\) increases by 16.22 percent, and State C increases by 12.18 percent, the standard quotas change from A 2.46, B 6.00, and C 9.53 to A 2.41, B 6.25, and C 9.34 .a. How did the change in populations impact
The house size is 100 . When the population of State \(A\) increases by 20 percent, State \(B\) increases by 10 percent, State \(C\) increases by 30 percent, and the populations of States \(D, E\), and \(F\) remain the same. The standard quotas change from A 12.50, B 25.00, C 9.38, D 18.75, E 12.50
The house size is 17 . When the population of State \(A\) increases by 12.73 percent, State \(B\) increases by 15.63 percent, and State C increases by 12.90 percent, the standard quotas change from A 2.53, B 5.90, and C 8.57 to A 2.51, B 5.99, and C 8.50 .a. How did the change in populations impact
The house size is 38 seats. When the population of State A increases by 6.60 percent, State B increases by 1.44 percent, and State C increases by 1.40 percent, the standard quotas change from A 18.82, B 14.65, and C 4.53 to A 19.59, B 14.29, and C 4.41 .a. How did the change in populations impact
The house size is 24 seats. When the population of State A increases by 28 percent, State B increases by 26 percent, and State C increases by 15 percent, the standard quotas change from A 3.38, B 6.32, and C 14.30 to \(A\) 3.63, B 6.67, and C 13.71 .a. How did the change in populations impact the
The house size was 60 . There were three states with standard quotas of A 4.18, B 15.38, and C 40.44. A fourth state was annexed, and the house size was increased to 65 . The new standard quotas are A 4.17, B 15.33, C 40.32, and D 5.18.a. How did the additional state impact the apportionment?b. Is
The house size was 50.There were three states with standard quotas of A 9.41, B 24.42, and C 16.17. A fourth state was annexed, and the house size was increased to 66 . The new standard quotas are A 9.36, B 24.30, C 16.09 , and 16.25 .a. How did the additional state impact the apportionment?b. Is
The house size was 27.There were three states with standard quotas of A 6.39, B 11.40, and C 9.21. A fourth state was annexed, and the house size was increased to 35 . The new standard quotas are A 6.38, B 11.37, C 9.19 , and 8.06 .a. How did the additional state impact the apportionment?b. Is this
The house size was 100 . There were three states with standard quotas of A 26.09, B 30.43 , and \(C 43.48\). A fourth state was annexed, and the house size was increased to 122.The new standard quotas are A 26.14, B 30.50, C 43.57 , and 21.78 .a. How did the additional state impact the
Consider States A, B, and C.a. Determine the apportionment for States A, B, and C with the original house size using the Hamilton method.b. Determine the apportionment for States A, B, and C with the updated house size using the Hamilton method.c. Does the change in the house size and use of the
Consider States \(D, E, F\), and G.a. Determine the apportionment for States D, E, F, and G with the original house size using the Hamilton method.b. Determine the apportionment for States D, E, F, and G with the updated house size using the Hamilton method.c. Does the change in the house size and
Consider States H, I, J, K, and L.a. Determine the apportionment for States \(\mathrm{H}, \mathrm{I}, \mathrm{J}, \mathrm{K}\), and \(\mathrm{L}\) with the original house size using the Hamilton method.b. Determine the apportionment for States H, I, J, K, and L with the updated house size using the
Consider States P, Q and R.a. Determine the apportionment for States \(P, Q\), and \(R\) with the original house size using the Hamilton method.b. Determine the apportionment for States P, Q, and R with the updated house size using the Hamilton method.c. Does the change in the house size and use of
Calculate the population growth rates \(d, e\), and \(f\) for States D, E, and F. Give answer as a percentage rounded to one decimal place.Use the information in the table below. State Original Population Updated Population Population Growth Rate A 889 958 7.76% House Size B 674 692 2.67% 42 C 237
Calculate the population growth rates \(p, q, r\), and \(s\) for States \(P, Q, R\), and \(S\). Give answer as a percentage rounded to one decimal place.Use the information in the table below. State Original Population Updated Population Population Growth Rate A 889 958 7.76% House Size B 674 692
Calculate the population growth rates \(x, y\), and \(z\) for States \(X, Y\), and \(Z\). Give answer as a percentage rounded to one decimal place.Use the information in the table below. State Original Population Updated Population Population Growth Rate A 889 958 7.76% House Size B 674 692 2.67%
Consider States A, B, and C.a. Determine the Hamilton apportionment for States A, B, and C with the original population.b. Determine the Hamilton apportionment for States A, B, and C with the updated population.c. Does the increase in population of States A, B, and C from the original population to
Consider States \(D, E\), and \(F\).a. Determine the Hamilton apportionment for States \(\mathrm{D}, \mathrm{E}\), and \(\mathrm{F}\) with the original population.b. Determine the Hamilton apportionment for States D, E, and F with the updated population.c. Does the increase in population and the

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