Explanation Approach: The following are the steps to be followed in Jefferson’s Apportionment Method: Assume a modified divisor for the given apportionment. Note that the modified divisor is smaller than the standard divisor in this method. Calculate the modified quota for each state using the assumed modified divisor and round the modified quotas down. Assign the rounded value to each state. Repeat the above steps till the sum of all the assignments equals the total number of apportionments. The following are the steps to be followed in Adam’s Apportionment Method: Assume a modified divisor for the given apportionment. Note that the modified divisor is greater than the standard divisor in this method. Calculate the modified quota for each state using the assumed modified divisor and round the modified quotas up. Assign the rounded value to each state. Repeat the above steps till the sum of all the assignments equals the total number of apportionments. The standard quota or the number of representatives/seats allocated for a state is calculated by the formula Population of the state Total population × Number of representatives/seats . If the standard quota is rounded down to the nearest integer, we call it as lower quota. If the standard quota is rounded up to the nearest integer, we call it as upper quota. Quota rule states that a state/party can have only two allocations, either lower quota or upper quota. Given Information: The population of the six states are given in the following table: State Population A 1, 400 B 2, 000 C 1, 500 D 9, 000 E 4, 300 F 500 500 representatives are apportioned among these six states. Calculation: Jefferson’s Apportionment Method: The total population is 1 , 400 + 2 , 000 + 1 , 500 + 9 , 000 + 4 , 300 + 500 = 18 , 700 The number of representatives to be apportioned is 500. Standard divisor for the apportionment = Total population Number of representives to be apportioned = 18 , 700 500 = 37.4 Let’s calculate the standard quota for each state. State Population Standard Quota A 1, 400 1 , 400 37.4 ≈ 37.43 B 2, 000 2 , 000 37.4 ≈ 53.48 C 1, 500 1 , 500 37.4 ≈ 40.11 D 9, 000 9 , 000 37.4 ≈ 240.64 E 4, 300 4 , 300 37.4 ≈ 114.97 F 500 500 37.4 ≈ 13.37 Let’s begin with the modified divisor 37. (Note that the modified divisor is always smaller than the standard divisor, 37.4). The following table illustrates the modified quota for the modified divisor 37 for each state. State Population Standard Quota Modified Quota Round Modified Quota Down A 1, 400 37.43 1 , 400 37 ≈ 37.84 37 B 2, 000 53.48 2 , 000 37 ≈ 54.05 54 C 1, 500 40.11 1 , 500 37 ≈ 40.54 40 D 9, 000 240.64 9 , 000 37 ≈ 243.24 243 E 4, 300 114.97 4 , 300 37 ≈ 116.22 116 F 500 13.37 500 37 ≈ 13.51 13 Total 503 The total of round modified quota is 503. Since it is slightly larger, we need to increase the modified divisor. If the modified divisor becomes larger, the modified quota becomes smaller. Let’s try a modified divisor of 37.07. (Note that the modified divisor is always smaller than the standard divisor, 37.4). The following table illustrates the modified quota for the modified divisor 37.07 for each state. State Population Standard Quota (Natural Quota) Modified Quota Round Modified Quota Down A 1, 400 37.43 1 , 400 37.07 ≈ 37.77 37 B 2, 000 53.48 2 , 000 37.07 ≈ 53.95 53 C 1, 500 40.11 1 , 500 37.07 ≈ 40.46 40 D 9, 000 240.64 9 , 000 37.07 ≈ 242.78 242 E 4, 300 114.97 4 , 300 37.07 ≈ 115.997 115 F 500 13.37 500 37.07 ≈ 13.49 13 Total

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Chapter 10.3, Problem 35E

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The method (Jefferson’s apportionment method or Adam’s apportionment method) that violates quota rule in apportioning 500 representatives among six states A, B, C, D, E and F.

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Chapter 10.3, Problem 34E

Chapter 10.3, Problem 36E

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The method (Jefferson’s apportionment method or Adam’s apportionment method) that violates quota rule in apportioning 500 representatives among six states A, B, C, D, E and F.

Solution Summary: The author explains Jefferson's apportionment method, which violates the quota rule, and the steps to follow in Adam’s Apportionment Method.

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The method (Jefferson’s apportionment method or Adam’s apportionment method) that violates quota rule in apportioning 500 representatives among six states A, B, C, D, E and F.

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