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 700 B 8, 000 C 820 D 1, 500 E 4, 000 F 550 100 representatives are apportioned among these six states. Calculation: Jefferson’s Apportionment Method: The total population is 700 + 8 , 000 + 820 + 1 , 500 + 4 , 000 + 550 = 15 , 570 The number of representatives to be apportioned is 100. Standard divisor for the apportionment = Total population Number of representives to be apportioned = 15 , 570 100 = 155.70 Let’s calculate the standard quota for each state. State Population Standard Quota A 700 700 155.7 ≈ 4.496 B 8, 000 8 , 000 155.7 ≈ 51.38 C 820 820 155.7 ≈ 5.27 D 1, 500 1 , 500 155.7 ≈ 9.63 E 4, 000 4 , 000 155.7 ≈ 25.69 F 550 550 155.7 ≈ 3.53 Total Let’s begin with the modified divisor 152. (Note that the modified divisor is always smaller than the standard divisor, 155.7). The following table illustrates the modified quota for the modified divisor 152 for each state. State Population Standard Quota Modified Quota Round Modified Quota Down A 700 4.496 700 152 ≈ 4.61 4 B 8, 000 51.38 8 , 000 152 ≈ 52.63 52 C 820 5.27 820 152 ≈ 5.39 5 D 1, 500 9.63 1 , 500 152 ≈ 9.87 9 E 4, 000 25.69 4 , 000 152 ≈ 26.32 26 F 550 3.53 550 152 ≈ 3.62 3 Total 99 The total of round modified quota is 99. Since it is slightly smaller, we need to reduce the modified divisor. If the modified divisor becomes smaller, the modified quota becomes larger. Let’s try a modified divisor of 150. (Note that the modified divisor is always smaller than the standard divisor, 155.7). The following table illustrates the modified quota for the modified divisor 150 for each state. State Population Standard Quota Modified Quota Round Modified Quota Down A 700 4.496 700 150 ≈ 4.67 4 B 8, 000 51.38 8 , 000 150 ≈ 53.33 53 C 820 5.27 820 150 ≈ 5.47 5 D 1, 500 9.63 1 , 500 150 ≈ 10 10 E 4, 000 25.69 4 , 000 150 ≈ 26.67 26 F 550 3.53 550 150 ≈ 3.67 3 Total 101 The total of round modified quota is 101. 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 150.5. (Note that the modified divisor is always smaller than the standard divisor, 155.7). The following table illustrates the modified quota for the modified divisor 150.5 for each state. State Population Standard Quota Modified Quota Round Modified Quota Down A 700 4.496 700 150.5 ≈ 4.65 4 B 8, 000 51.38 8 , 000 150.5 ≈ 53.16 53

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Chapter 10.3, Problem 34E

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The method (Jefferson’s apportionment method or Adam’s apportionment method) that violates quota rule in apportioning 100 representatives among six states A, B, C, D, E and F.

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Chapter 10.3, Problem 33E

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The method (Jefferson’s apportionment method or Adam’s apportionment method) that violates quota rule in apportioning 100 representatives among six states A, B, C, D, E and F.

Solution Summary: The author explains how Jefferson's apportionment method violates the quota rule by aggregating 100 representatives among six states.

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The method (Jefferson’s apportionment method or Adam’s apportionment method) that violates quota rule in apportioning 100 representatives among six states A, B, C, D, E and F.

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