Quiz

Huntington-Hill Method The Huntington-Hill method of apportionment is the currently used method of apportionment in the United States Congress. It involves what is called the geometric mean between two numbers. Geometric Mean of two numbers 𝑛 𝑎𝑛𝑑 𝑚 is 𝑛∙𝑚 .

Huntington-Hill Method Find the standard divisor. Find the standard quotas. For each standard quota, find the geometric mean between its upper and lower quotas. If the standard quota is greater than the geometric mean round up. If the standard quota is less than the geometric mean round down. Find a modified divisor that perfectly apportions the seats.

Example #1 The United States of Canines has a senate with 100 seats to give out to four states. Find the standard divisor and standard quotas. Find the geometric means for the upper and lower quotas and determine an initial apportionment. Find a modified divisor that gives a perfect apportionment.

Example #1 Solution Standard Divisor: 100,000 100 =1,000 𝑝𝑢𝑝𝑠/𝑠𝑒𝑎𝑡 Standard Quotas: A: 86915 1000 =86.9 B: 4.325 C: 5.4 D:3.36 Geometric Means: 𝐴: 86∙87 =86.50 B: 4∙5 =4.472 C: 5∙6 =5.477 D: 3∙4 =3.464 Initial Apportionment: A: 87 B: 4 C: 5 D: 3 Total: 99

Example #1 Solution We are one seat short, so we reduce our divisor. Modified Divisor: 990 Quotas: A: 87.793 B: 4.369 C: 5.455 D: 3.394 Notice we have to recalculate our geometric mean for A as it went up a number. The rest remain the same.

Example #1 Solution Quotas: A: 87.793 B: 4.369 C: 5.455 D: 3.394 Geometric Means: A: 87∙88 =87.499 B: 4.472 C: 5.477 D: 3.464 Apportionment: A: 88 B: 4 C: 6 D: 3 Total: 100. WOOOHOOO!

Example #2 Five classes being taught and apportioned based on how many students sign up. Algebra: 36 Geometry: 61 Calculus: 3

Example #2 Solution Standard Divisor: 100 5 =20 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠/𝑠𝑒𝑎𝑡 Standard Quotas: Algebra: 1.8 Geometry: 3.05 Calc: .15 Geometric Means: Algebra: 1∙2 =1.414… Geometry: 3∙4 =3.464… Calc: 0∙1 =0.000

Example #2 Solution If the standard quota is higher than the geometric mean, we round up! Standard Quotas: Algebra: 1.8 Geometry: 3.05 Calc: .15 Geometric Means: Algebra: 1.414 Geometry: 3.464 Calc: 0 Round Up Round Down Round Up Initial Apportionment: Algebra: 2 Geometry: 3 Calc: 1

Example #2 Solution Our apportionment is over five classes so we must make our standard divisor bigger. Modified Divisor: 21 Algebra: 1.7 Geometry:2.9 Calc: 0.14 Geometric Means: A: 1.414 G: 2.45 C: 0 Apportionment: A: 2 G: 3 C: 1 Notice our geometric mean for Geometry changed.

Example #2 Solution Modified Divisor: 25 Algebra: 1.44 G: 2.44 C: .12 Geometric Means: A: 1.414 G: 2.45 C: 0 Round Up Down Up Apportionment: A: 2 G: 2 C: 1