Question

Consider a nation of 12,400,000 represented in a legislative body of 400 representatives. There are six states A, B, C, D, E, and F with the following population distribution

A B C    D    E F

840,000 2,560,000 520,000    6,810,000 1,220,000    450,000

a) Find the standard divisor (SD) representing number of citizens per representative

b) Find the quota for each state, as well as the lower and upper quotas

c) Show there is an apportionment problem since neither of these quotas give the required number of 400 representatives.

BONUS: Give the solution to this problem (the apportionment of representatives) according to Hamilton's method.

Answer 1

a) Standard Divisor is (Total population / No of representative seats) = 12,400,000/400 = 31000 citizens per reprentative

b) Standard quota for each state is achieved by dividing population of that state with the standard divisor. Lower quota is the rounded down value of the standard quota. Upper quota is the rounded up value of the standard quota.

States	A	B	C	D	E	F	Total		No of reprentatives	400
Population	840000	2560000	520000	6810000	1220000	450000	12400000		Standard Divisor	31000
Standard quota	27.09677	82.58065	16.77419	219.6774	39.35484	14.51613	400
Lower quota	27	82	16	219	39	14	397
Upper quota	28	83	17	220	40	15	403

c) We can clearly see the  apportionment problem as none of the two options of lower quota or upper quota give the seats count to 400. We will try to solve it using Hamilton's method

Per Hamilton method, we will allocate the Initial seats as per lower quota. After assigning that, we will assign the additional remaining seats to the states whose decimal parts of the quotas were largest until the desired result is achieved

Hamilton Approach
States	A	B	C	D	E	F	Total
Population	840000	2560000	520000	6810000	1220000	450000	12400000
Standard quota	27.09677	82.58065	16.77419	219.6774	39.35484	14.51613	400
Initial	27	82	16	219	39	14	397
Final	27	83	17	220	39	14	400

Explanation - Initial quota is achieved by trimming down the decimal points. As total seats after initial quota are 397, which are 3 short of 400, we will allocate those 3 seats to the states having largest decimal parts.

So additional seats will be assigned to C (0.77419), D (0.6774) and B (0.58065) in the same order.