Apportionment (MAT101: Lesson 1)
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Questions and Answers

What is apportionment primarily concerned with in mathematical contexts?

  • Increasing the population of each group.
  • Distributing a set number of resources fairly among groups. (correct)
  • Maximizing the total resources available.
  • Eliminating the need for quotas in resource allocation.

Which method of apportionment allocates resources based on the largest fractional parts of quotas?

  • Webster’s Method
  • Hamilton’s Method (correct)
  • Standard Divisor Method
  • Jefferson’s Method

How is the standard divisor calculated?

  • By dividing total population by total resources. (correct)
  • By dividing total resources by total population.
  • By adding total resources to total population.
  • By multiplying total resources by total population.

What does the lower quota represent in apportionment?

<p>The largest whole number less than or equal to the quota. (C)</p> Signup and view all the answers

If the total population is 1,000 and there are 10 seats to apportion, what would be the standard divisor?

<p>$100 (B)</p> Signup and view all the answers

What is a key limitation of apportionment methods?

<p>They may lead to unfair results under certain conditions. (A)</p> Signup and view all the answers

Which method involves modifying the standard divisor to achieve more equitable results?

<p>Jefferson’s Method (D)</p> Signup and view all the answers

If a group has a population of 250 and the standard divisor is 100, what is its quota?

<p>2 (D)</p> Signup and view all the answers

What is the total number of assignments for the three departments combined?

<p>50 assignments (B)</p> Signup and view all the answers

Which department received the highest number of assignments?

<p>Department C (D)</p> Signup and view all the answers

In the example provided, how many total teachers are to be assigned to Cabuyao Elementary School?

<p>40 teachers (A)</p> Signup and view all the answers

Which grade has the highest enrollment in Cabuyao Elementary School?

<p>Grade 5 (D)</p> Signup and view all the answers

What method is suggested for assigning teachers to grades in the example?

<p>Hamilton’s method (B)</p> Signup and view all the answers

What is the total enrollment of students across all grades in Cabuyao Elementary School?

<p>700 students (B)</p> Signup and view all the answers

What is a significant outcome of using voting methods in decision-making?

<p>Expressing preferences among alternatives (B)</p> Signup and view all the answers

What is the purpose of Jefferson’s Method in apportionment?

<p>To modify the divisor downward until the total matches resources (A)</p> Signup and view all the answers

What is the result of assigning 10 assignments to Department A, 15 to Department B, and 25 to Department C?

<p>Total matches available assignments (D)</p> Signup and view all the answers

How is the standard divisor calculated in Jefferson’s Method?

<p>Total Enrollment divided by available seats (A)</p> Signup and view all the answers

What is the initial action taken after calculating the standard divisor?

<p>Compute quotas based on the standard divisor (A)</p> Signup and view all the answers

What does it signify if the sum of rounded-down quotas is less than the available resources?

<p>The divisor needs to be adjusted downward (D)</p> Signup and view all the answers

In the example provided, how many seats were assigned to the College of Science?

<p>3 (B)</p> Signup and view all the answers

What happens to the calculated quotas in Jefferson's Method after rounding down?

<p>They can lead to an insufficient total compared to available resources (B)</p> Signup and view all the answers

What is the total enrollment of the three colleges in the allocation problem?

<p>6,000 students (A)</p> Signup and view all the answers

Which college has the highest enrollment among the three listed?

<p>College of Business (D)</p> Signup and view all the answers

What is the total number of seats allocated among the colleges?

<p>20 (A)</p> Signup and view all the answers

Which department receives the highest number of project assignments when using Webster’s Method?

<p>Department C (C)</p> Signup and view all the answers

What is the calculated standard divisor for distributing the 50 assignments?

<p>12 (A)</p> Signup and view all the answers

Using Webster's Method, what rounding rule is applied to quotas?

<p>Round to the nearest whole number (A)</p> Signup and view all the answers

If the total proposals submitted were increased to 700 but the total assignments remain 50, what would the new standard divisor be?

<p>14 (C)</p> Signup and view all the answers

If quotas lead to over-allocation in Webster's Method, what should be adjusted?

<p>The divisor (A)</p> Signup and view all the answers

What is the rounded quota for Department A if calculated quotas are not adjusted?

<p>10 (D)</p> Signup and view all the answers

Which statement about the allocation of assignments to departments is true?

<p>Department A received the fewest assignments (B), Department B received more than Department A (D)</p> Signup and view all the answers

What is the primary goal of the fairness criteria in voting methods?

<p>To evaluate the effectiveness of different voting methods. (C)</p> Signup and view all the answers

Which voting method ensures that the option with the most votes ultimately wins?

<p>Plurality Method (A)</p> Signup and view all the answers

In the Borda Count Method, what does each option receive based on voter rankings?

<p>Points assigned according to their rank. (B)</p> Signup and view all the answers

What does the Condorcet Criterion state about the winning option?

<p>It must defeat all other options in head-to-head comparisons. (D)</p> Signup and view all the answers

What is a drawback of the Plurality Method?

<p>It can result in a winner lacking majority support. (D)</p> Signup and view all the answers

Which statement accurately describes Approval Voting?

<p>Voters can vote for multiple candidates they approve of. (A)</p> Signup and view all the answers

Which of the following best describes the outcome of a Condorcet Method election?

<p>The candidate who defeats all opponents in pairwise contests wins. (B)</p> Signup and view all the answers

Which criterion ensures that adding or removing an option will not change the original winner?

<p>Independence of Irrelevant Alternatives (A)</p> Signup and view all the answers

Flashcards

Apportionment

The practice of distributing a fixed number of resources (like seats or funds) among different groups in proportion to their respective quotas.

Standard Divisor

The ratio of the total population (or relevant measure) to the total number of resources available.

Quota

The calculated share of resources for each group, determined by dividing the group's population by the standard divisor.

Lower Quota

The largest whole number less than or equal to the calculated quota.

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Upper Quota

The smallest whole number greater than or equal to the calculated quota.

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Hamilton's Method

A method of apportionment where resources are allocated based on quotas, and any remaining resources are distributed to the groups with the largest fractional parts of their quotas.

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Apportionment in Real-World Problems

The process of applying a specific apportionment method (like Hamilton's) to real-world problems involving allocation of resources.

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Evaluating Apportionment Methods

The process of evaluating the fairness and limitations of different apportionment methods, considering factors like accuracy, bias, and ease of implementation.

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Apportionment Methods

A method used to distribute resources fairly based on population or other relevant factors. It ensures that each group receives a proportional share of the resources.

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Jefferson's Method

In Jefferson's Method, the divisor is adjusted downward until the sum of the rounded-down quotas equals the total number of resources.

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Compute and Round Down Quotas

In Jefferson's Method, quotas are computed by dividing each group's population (or relevant factor) by the divisor. These are then rounded down to the nearest whole number.

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Adjust Divisor

If the sum of the rounded-down quotas is less than the total number of resources, the divisor is adjusted downward, typically by a small increment. This process is repeated until the sum of the rounded-down quotas matches the total number of resources

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Assign Seats

After the divisor is adjusted and quotas are calculated, the rounded-down quotas represent the final allocation of the resources.

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Fair Allocation

The allocation process ensures that each group receives a proportional share of the resources, taking into account their relative size or importance

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Voting

A method of decision-making where individuals express their preferences among choices.

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Final Apportionment

The final number of resources allocated to each group after adjustments are made.

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Ranked Voting

A method of voting where each individual's preference is ranked in order of priority.

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Plurality Voting

A voting system where the candidate with the most votes wins, regardless of how many voters they get.

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Quota Calculation

A quota represents the proportion of resources a specific unit is entitled to receive based on the standard divisor.

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Webster's Method

Webster's method utilizes standard rounding to allocate resources. If rounding leads to an over-allocation, the divisor is increased; if under-allocation, the divisor is decreased.

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Initial Quota Allocation

The initial quota allocation for each department is calculated by dividing the number of proposals submitted by the standard divisor.

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Rounding Quotas

Quotas are rounded to the nearest whole number, ensuring whole units are allocated.

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Divisor Adjustment

In case of over-allocation or under-allocation due to rounding, the standard divisor needs to be adjusted upward or downward respectively.

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Final Allocation

After adjusting the divisor and recalculating quotas, they again are rounded to the nearest whole number, ensuring a final allocation that matches the total available resources.

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Plurality Method

A method for choosing a winner by counting the number of votes for each alternative, the one with the most votes wins.

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Borda Count Method

A method for choosing a winner by ranking all alternatives and assigning points based on ranks. Higher ranking equals more points, the candidate with the most total points wins.

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Condorcet Method

A method for choosing a winner by comparing alternatives in head-to-head pairwise contests. The candidate who wins against all others is the winner.

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Approval Voting

A method for choosing a winner where voters can approve of multiple candidates. The candidate with the most votes wins.

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Fairness Criteria

A set of criteria used to evaluate voting methods, ensuring fairness and transparency.

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Outcome

The option chosen based on the voting method used.

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Majority Rule

The option that is preferred by more than half of the voters should win.

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Study Notes

Apportionment (MAT101: Lesson 1)

  • Apportionment is the mathematical practice of distributing resources (seats, goods, funds) proportionally among groups based on criteria like population, contribution, or need.
  • It's vital for fair allocation of resources.

Objectives

  • Define apportionment and its significance in mathematical contexts.
  • Explain apportionment methods (Hamilton, Jefferson, Webster).
  • Apply methods to solve real-world allocation problems.
  • Evaluate the fairness and limitations of apportionment methods.

Key Concepts

  • Standard Divisor: The ratio of total population to the number of resources. (Total Population / Total Resources)
  • Quota: Each group's share of resources calculated by dividing the group's population by the standard divisor. (Group Population / Standard Divisor)
  • Lower Quota: The largest whole number less than or equal to the quota.
  • Upper Quota: The smallest whole number greater than or equal to the quota.

Hamilton's Method

  • Allocates resources based on quotas, distributing remaining resources to groups with the largest fractional parts of their quotas.
  • Steps:
    • Calculate the standard divisor.
    • Compute quotas for each group.
    • Assign each group its lower quota.
    • Distribute remaining resources to groups with largest fractional parts.

Jefferson's Method

  • Modifies the divisor downward until the sum of rounded-down quotas equals the total number of resources.
  • Steps:
    • Start with the standard divisor.
    • Compute quotas and round each down to the nearest whole number.
    • If the total is less than the available resources, adjust the divisor downward and recalculate.

Webster's Method

  • Uses standard rounding to allocate resources. If rounding leads to over-allocation, adjust the divisor upward; if under-allocation, adjust downward.
  • Steps:
    • Compute quotas.
    • Round quotas to the nearest whole number.
    • Adjust the divisor if necessary and recalculate.

Voting Methods (MAT101: Lesson 2)

Objectives

  • Define voting and its significance in decision-making.
  • Explain different voting methods and their applications.
  • Analyze advantages and limitations of each voting method.
  • Solve problems involving various voting scenarios.

Key Concepts

  • Voter Preferences: Individuals express their choices.
  • Outcome: The result of the voting process.
  • Fairness Criteria: Principles to evaluate voting methods.
    • Majority Rule: The option preferred by more than half of voters wins.
    • Condorcet Criterion: Preferred option in head-to-head comparisons wins.
    • Independence of Irrelevant Alternatives: Adding/removing options shouldn't change outcome.

Plurality Method

  • Candidate with most votes wins.
  • Can result in a winner without majority support.

Borda Count Method

  • Voters rank all options.
  • Options with highest total points score win.

Condorcet Method

  • Winner must defeat every other option in head-to-head comparisons.
  • May not always produce a result.

Approval Voting

  • Voters vote for any number of candidates.
  • Candidate with most votes wins.

Instant Runoff Voting (IRV)

  • Voters rank candidates.
  • Eliminates candidate with fewest votes, redistributes their votes until one candidate has a majority.

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Related Documents

Lesson 12: Apportionment & Voting Methods PDF

Description

Explore the concept of apportionment in mathematics, its significance, and various methods such as Hamilton, Jefferson, and Webster. This lesson covers definitions, applications, and evaluates the fairness of these methods in resource allocation. Engage with real-world scenarios to enhance your understanding of equitable distribution.

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