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Questions and Answers
What is apportionment primarily concerned with in mathematical contexts?
What is apportionment primarily concerned with in mathematical contexts?
Which method of apportionment allocates resources based on the largest fractional parts of quotas?
Which method of apportionment allocates resources based on the largest fractional parts of quotas?
How is the standard divisor calculated?
How is the standard divisor calculated?
What does the lower quota represent in apportionment?
What does the lower quota represent in apportionment?
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If the total population is 1,000 and there are 10 seats to apportion, what would be the standard divisor?
If the total population is 1,000 and there are 10 seats to apportion, what would be the standard divisor?
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What is a key limitation of apportionment methods?
What is a key limitation of apportionment methods?
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Which method involves modifying the standard divisor to achieve more equitable results?
Which method involves modifying the standard divisor to achieve more equitable results?
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If a group has a population of 250 and the standard divisor is 100, what is its quota?
If a group has a population of 250 and the standard divisor is 100, what is its quota?
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What is the total number of assignments for the three departments combined?
What is the total number of assignments for the three departments combined?
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Which department received the highest number of assignments?
Which department received the highest number of assignments?
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In the example provided, how many total teachers are to be assigned to Cabuyao Elementary School?
In the example provided, how many total teachers are to be assigned to Cabuyao Elementary School?
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Which grade has the highest enrollment in Cabuyao Elementary School?
Which grade has the highest enrollment in Cabuyao Elementary School?
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What method is suggested for assigning teachers to grades in the example?
What method is suggested for assigning teachers to grades in the example?
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What is the total enrollment of students across all grades in Cabuyao Elementary School?
What is the total enrollment of students across all grades in Cabuyao Elementary School?
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What is a significant outcome of using voting methods in decision-making?
What is a significant outcome of using voting methods in decision-making?
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What is the purpose of Jefferson’s Method in apportionment?
What is the purpose of Jefferson’s Method in apportionment?
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What is the result of assigning 10 assignments to Department A, 15 to Department B, and 25 to Department C?
What is the result of assigning 10 assignments to Department A, 15 to Department B, and 25 to Department C?
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How is the standard divisor calculated in Jefferson’s Method?
How is the standard divisor calculated in Jefferson’s Method?
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What is the initial action taken after calculating the standard divisor?
What is the initial action taken after calculating the standard divisor?
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What does it signify if the sum of rounded-down quotas is less than the available resources?
What does it signify if the sum of rounded-down quotas is less than the available resources?
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In the example provided, how many seats were assigned to the College of Science?
In the example provided, how many seats were assigned to the College of Science?
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What happens to the calculated quotas in Jefferson's Method after rounding down?
What happens to the calculated quotas in Jefferson's Method after rounding down?
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What is the total enrollment of the three colleges in the allocation problem?
What is the total enrollment of the three colleges in the allocation problem?
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Which college has the highest enrollment among the three listed?
Which college has the highest enrollment among the three listed?
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What is the total number of seats allocated among the colleges?
What is the total number of seats allocated among the colleges?
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Which department receives the highest number of project assignments when using Webster’s Method?
Which department receives the highest number of project assignments when using Webster’s Method?
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What is the calculated standard divisor for distributing the 50 assignments?
What is the calculated standard divisor for distributing the 50 assignments?
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Using Webster's Method, what rounding rule is applied to quotas?
Using Webster's Method, what rounding rule is applied to quotas?
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If the total proposals submitted were increased to 700 but the total assignments remain 50, what would the new standard divisor be?
If the total proposals submitted were increased to 700 but the total assignments remain 50, what would the new standard divisor be?
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If quotas lead to over-allocation in Webster's Method, what should be adjusted?
If quotas lead to over-allocation in Webster's Method, what should be adjusted?
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What is the rounded quota for Department A if calculated quotas are not adjusted?
What is the rounded quota for Department A if calculated quotas are not adjusted?
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Which statement about the allocation of assignments to departments is true?
Which statement about the allocation of assignments to departments is true?
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What is the primary goal of the fairness criteria in voting methods?
What is the primary goal of the fairness criteria in voting methods?
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Which voting method ensures that the option with the most votes ultimately wins?
Which voting method ensures that the option with the most votes ultimately wins?
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In the Borda Count Method, what does each option receive based on voter rankings?
In the Borda Count Method, what does each option receive based on voter rankings?
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What does the Condorcet Criterion state about the winning option?
What does the Condorcet Criterion state about the winning option?
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What is a drawback of the Plurality Method?
What is a drawback of the Plurality Method?
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Which statement accurately describes Approval Voting?
Which statement accurately describes Approval Voting?
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Which of the following best describes the outcome of a Condorcet Method election?
Which of the following best describes the outcome of a Condorcet Method election?
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Which criterion ensures that adding or removing an option will not change the original winner?
Which criterion ensures that adding or removing an option will not change the original winner?
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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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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.
