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Chapter 4: Problem 56

If the standard quota of state \(Y\) is 78.24 , then which of the following are possible apportionments to state \(Y\) under Webster's method? (a) 78 or 79 (b) 76 or 77 (c) 80 or 81 (d) 77,78,79 or 80 (e) All of the above are possible

Short Answer

Expert verified
Based on Webster's method of apportionment, the possible allocations for the quota 78.24 are 78 and 79. Therefore, only option (a) is possible.

Step by step solution

01

Understanding the Quota

The standard quota for state Y is given as 78.24. Now, let's understand possible quotas for the state using Webster's method - we will round this number to the nearest whole number. Anything above .5 gets rounded up and anything below gets rounded down.
02

Evaluate Apportionment (a)

Looking at option (a), we have 78 or 79. Since 78.24 is closer to 78 (following the usual mathematical rounding rule), it can be a possible apportionment using Webster's method. Therefore, option (a) is possible.
03

Evaluate Apportionment (b)

Moving to option (b), which is 76 or 77. These two numbers are further from the quota of 78.24, so they cannot be considered for apportionment. Therefore, option (b) is not possible.
04

Evaluate Apportionment (c)

Looking at option (c), which is 80 or 81, these numbers are too far to be considered as per Webster's method for the given quota of 78.24. Therefore, option (c) is not possible.
05

Evaluate Apportionment (d)

Analysing option (d), which are 77, 78, 79 or 80. Among these options, owing to the rounding rules in Webster's method, only 78 and 79 can be considered as potential quotas. Therefore, not all numbers in option (d) are possible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Standard Quota
The principle of a standard quota is essential when tackling apportionment problems, particularly when using Webster's Method. In such scenarios, the standard quota represents the ideal share each state or group should receive. This is derived by dividing the total number of items, such as seats in a legislature, by the total population. Then, multiply the result by the population of the respective state or group. This number helps establish a baseline for fair distribution. For instance, given the standard quota for state Y as 78.24, this figure suggests that 78.24 items should ideally be apportioned to state Y.
The standard quota is crucial because it provides a precise, mathematical expectation based on proportions. However, since you can't distribute fractional items, the actual number must be adjusted through methods like rounding to determine a suitable whole number of apportioned items.
Mathematical Rounding in Apportionment
Rounding is a technique used to make precise numbers workable, often necessary in apportionment when dealing with the standard quota. Webster's Method teaches us an adjusted form of rounding that involves rounding to the nearest whole number. In this method, the decision point is 0.5: any number equal to or greater than 0.5 is rounded up, while anything lower is rounded down. So, for example, a standard quota of 78.24 would be rounded down to 78 because 0.24 is under the 0.5 threshold.
Rounding plays a key role in determining the fair distribution of resources across groups or states. It's more than just a mathematical exercise 鈥� it ensures that each party is allocated their fair share, favoring a balance between over and under-representation. In the exercise, this means options like 78 or 79 might be possible quotas because their rounding aligns with Webster鈥檚 precision-focused approach.
Apportionment and Webster's Method
Apportionment deals with allocating a set number of items, such as legislative seats, fairly among groups based on specified quotas. It's crucial in politics, resource distribution, and statistical analyses. Webster's Method offers a way to execute apportionment by balancing precision and fairness, relying heavily on the use of rounding rules.
  • Proportional Fairness: Apportionment ensures that representation is proportionate to each group's size or population.
  • Adjusted Rounding: Webster's Method uses simplified mathematical rounding to adjust standard quotas into whole numbers, aiming for minimal disparity in representation.
  • Fair Distribution: The method prioritizes fairness by averting substantial over or under-representation, adjusting each group鈥檚 allocation within an understandable framework.
Webster's Method particularly shines in preventing anomalies seen in other methods, like giving undue advantage to certain states. Therefore, understanding apportionment using this technique lays the foundation for implementing equitable decisions based on mathematical evidence and rational interpretation.

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Most popular questions from this chapter

This exercise comes in two parts. Read Part I and answer (a) and (b), then read Part II and answer (c) and (d). Part L. A catering company contracts to provide catering services to three schools: Alexdale, with 617 students, Bromville, with 1,292 students, and Canley, with 981 students. The 30 food-service workers employed by the catering company are apportioned among the schools based on student enrollments, (a) Find the standard divisor, rounded to the nearest integer. (b) Find the apportionment of the 30 workers to the three schools under Hamilton's method. Part II. The catering company gets a contract to service one additional school - Dillwood, with 885 students. To account for the additional students, the company hires 9 additional food-service workers. [885 students represent approximately 9 workers based on the standard divisor found in (a).] (c) Find the apportionment of the 39 workers to the four schools under Hamilton's method. (d) Which paradox is illustrated by the results of (b) and (c)? Explain.

The Hamilton-Jefferson hybrid methed. The HamiltonJefferson hybrid method starts by giving each state its lower quota (as per Hamilton's method) and then apportioning the surplus seats using Jefferson's method. (a) Use the Hamilton-Jefferson hybrid method to apportion \(M=22\) seats among four states according to the following populations: \(A\) (population 18,000\(), B\) (population 18,179\(), C\) (population 40.950 ), and \(D\) (population 122,871\()\). (b) Explain why the Hamilton-Jefferson hybrid method can produce apportionments that are different from both Hamilton and Jefferson apportionments (c) Explain why the Hamilton-Jefferson hybrid method can violate the quota rule.

If the standard quota of state \(X\) is 35.41 , then which of the following are possible apportionments to state \(X\) under Webster's method? (a) 35 or 36 (b) 34,35,36 or 37 (c) 33 or 34 (d) 37 or 38 (e) All of the above are possible

Consider an apportionment problem with \(N\) states. The populations of the states are given by \(p_{1}, p_{2}, \ldots, p_{N},\) and the standard quotas are \(q_{1}, q_{2+\ldots} \ldots q_{N}\), respectively. Describe in words what each of the following quantities represents. (a) \(q_{1}+q_{2}+\cdots+q_{N}\) (b) \(\frac{p_{1}+p_{2}+\cdots+p_{N}}{q_{1}+q_{2}+\cdots+q_{N}}\) (c) \(\left(\frac{P_{N}}{P_{1}+p_{2}+\cdots++p_{N}}\right) \times 100\)

Exercises 61 and 62 are based on the following story: Mom found an open box of her children's favorite candy bars. She decides to apportion the candy bars among her three youngest children according to the number of minutes each child spent doing homework during the week. (a) Suppose that there were 10 candy bars in the box. Given that Bob did homework for a total of 54 minutes. Peter did homework for a total of 243 minutes. and Ron did homework for a total of 703 minutes. apportion the 10 candy bars among the children using Hamilton's method (b) Suppose that just before she hands out the candy bars. mom finds one extra candy bar. Using the same total minutes as in (a), apportion now the 11 candy bars among the children using Hamilton's method. (c) The results of (a) and (b) illustrate one of the para- doxes of Hamilton's method, Which one? Explain.
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