state districts?

Question

a state has 4 districts with the populations shown below. The house of representatives has 20 seats that are to be apportioned using the Jefferson method. Find the first critical multiplier for district A.

District
A 87,000
B 56,000
C. 72,000
D. 35,000

A. 1.0057 B. 1.16 C. 1.348 D. 1.696

Answer 1

Your questions are so interesting.

I'm not sure what a critical multiplier is, but I found an algorithm for the Jefferson Method at http://www.ctl.ua.edu/math103/apportionment/appmeth.htm

1) Find the standard divisor: average number of people per seat over the entire population.

(87 + 56 + 72 + 45)/20 = 13,000 people per seat.

2) Calculate each district's standard quota: population / standard divisor.

A: 87/13 = 6.308
B: 56/13 = 4.308
C: 72/13 = 5.538
D: 35/13 = 2.692

3) Initially assign each district its lower quota (standard quota rounded down):
A: 6
B: 4
C: 5
D: 2

4) Check to see if the right number has been apportioned: 6 + 4 + 5 + 2 = 17, so there are 20 - 17 = 3 seats left.

4a) If not, then try to find a standard divisor that will result in all seats being apportioned.

Well 13,000 didn't work, so try 12,000.

A: 87/12 = 7.25
B: 56/12 = 4.667
C: 72/12 = 6
D: 35/12 = 2.917

7 + 4 + 6 + 2 = 19, so still not quite... try 11,500:

A: 87/11.5 = 7.565
B: 56/11.5 = 4.870
C: 72/11.5 = 6.261
D: 35/11.5 = 3.043

7 + 4 + 6 + 3 = 20, so that's the correct apportionment.

So now I have to try to guess what the critical multiplier is....

I'm at a loss...can you post a definition of a critical multiplier? I can't find a definition on the net.

Answer 2

Well, I looked up the Jefferson Method for ya (see below). There doesn't seem to be a "critical multiplier" listed.

What it basically says is to take the total population and divide by the seats available to get D = 260,000 / 20 = 13,000, then decrease D by a factor d such that, when each state is given it's # of seats as (population of state / ( D - d )), the total correct # of seats is met.

In this case, maybe instead of subtracting d, we'll just multiply by a factor M, replacing (D-d) with (DM) in the equation.

With M = 1.0, we get:
A 87000/13000 (round down) = 6 seats
B 56000/13000 = 4 seats
C 72000/13000 = 5 seats
D 35000/13000 = 2 seats
Total = 17, so we need to lower M, probably so MD = 35000/3 = 11666.666 (thus giving district D 3 seats)

Now we try again:
A 87000/11666.66 = 7
B 56000/11666.66 = 4
C 72000/11666.66 = 6
D 35000/11666.66 = 3
Total = 20 seats, and we're good!

MD = 11666.66
M(12000) = 11666.66
M = 35/36 = 0.97222222 Hmm, this doesn't match any of your answers... maybe we should use D/M ?? so invert = 1.0286...

Hmm, that didn't work either. Not only that, but the answer would be the same for all 4 districts, which makes the question a little weird.

Sorry, that's all I've got.

Well, one more try... maybe the critical multiplier for district A is the ratio of population essentially "unused", say....

87000 / ( 7 seats * 11666.66 people/seat) =
87000 / 81666 = 1.065... bah! Also not an answer.