Math for General Education – Test 2 Review

Apportionment

1. Overview of Rhode Island’s House of Representatives

• Total seats: 75 seats distributed among five counties.
• County populations:
  • Bristol: 49,875
  • Kent: 166,158
  • Newport: 82,888
  • Providence: 626,667
  • Washington: 126,979

1. Population and Apportionment Calculations
   a. Total Population:

   • Calculated as the sum of all county populations:
     $ext{Total Population} = 49875 + 166158 + 82888 + 626667 + 126979 = 1000567$
     b. Divisor:
   • The divisor is found by dividing the total population by total seats:
     $ext{Divisor} = \frac{1000567}{75} ext{ (exact value needed)}$
     c. Quota for Newport:
   • Calculated as:
     $ext{Quota}<em>{Newport} = \frac{Population</em>{Newport}}{Divisor}$
   • Quota to 5 decimal places required.
     d. Quota for Kent County:
   • Calculated similarly:
     $ext{Quota}<em>{Kent} = \frac{Population</em>{Kent}}{Divisor}$
   • Quota to 5 decimal places required.

2. Apportionment using Hamilton’s Method
   a. Seats assigned to Bristol County:
   b. Seats assigned to Providence County:

   • Total of 75 seats distributed among counties based on quotas.

3. Apportionment using Jefferson’s Method
   a. Seats assigned to Bristol County:
   b. Seats assigned to Providence County:
   c. Seats assigned to Kent County:

4. Apportionment using Huntington-Hill Method
   a. Geometric mean for Washington County:

   • Required down to 3 decimal places based on initial divisor.
     b. Final seats for Bristol County:
     c. Final seats for Providence County:

Additional Apportionment Questions

1. Geometric Mean Calculation

   • Example for a county with a quota of 4.87563:
     a. Geometric mean:
     $4.47214$
     b. Initial allocation:
     $5$

2. Historical Apportionment Methods
   a. Vetoed method by President Washington: (Refer to page 75 for details)
   b. Rhode Island Paradox: Reasonably could be named based on specific problematic cases.
   c. New States Paradox: Inspired by a specific state.
   d. Apportionment method following Quota Rule: (Refer to page 79)
   e. Political Districting based on Affiliation:

   • Term for districts drawn based on political advantages: (Refer to p. 89)

Graph Theory

1. True/False Statement

   • Graph Theory applications: True, it is applicable in laying fiber optic lines. (Refer to p. 117)

2. Historical Context of Graph Theory

   • Mathematician Pioneering Graph Theory: (Mentioned on p. 119)

3. Details from City Graph
   a. Number of vertices and edges: Inquiry for provided graph representation.
   b. Graph connectedness: Determined based on vertex connectivity.
   c. Degree of vertex representing LA:
   d. Path vs Circuit:

   • Flying sequence from Seattle to Dallas to Atlanta: Path vs Circuit.
     e. Circuit verification:
   • Sequence from LA to Chicago to Dallas to LA checked.
     f. Cost of Seattle to LA Circuit:
   • Total cost: $415$

4. Definitions in Graph Theory
   a. Euler Path:

   • Complete the definition based on material (referenced on p. 127).
     b. Path Characteristics: Questions on return to starting vertex: Yes/No.
     c. Euler Circuit Definition:
     d. Circuit Characteristics: Questions on circuits returning: Yes/No.
     e. Existence of Euler path/circuit in graph: Explained based on specific characteristics (Example 7, p. 129).
     f. Fleury’s Algorithm: Find an Euler Circuit based on provided graph (Example 10, p. 130).

5. Hamiltonian Concepts
   a. Hamiltonian Path Definition:

   • Complete the definition from source (p. 132).
     b. Hamiltonian Circuit Definition:
   • Complete the definition from source (p. 132).
     c. Hamiltonian Circuit Found: Inquiry into existence on a provided graph (Example 14, p. 133).

6. Finding Hamiltonian Circuits
   a. Nearest Neighbor Algorithm findings starting with vertex A.
   b. Repeated Nearest Neighbor Algorithm findings.
   c. Sorted Edges Algorithm findings. (Refer to Examples 17, 18, 19, p. 136-138)
   d. Non-Greedy Algorithm Identification: Questions about algorithms mentioned:

   • Identify which is NOT greedy:
     a. Nearest Neighbor
     b. Repeated Nearest Neighbor
     c. Sorted Edges
   • Reference material pages 136-138 for clarity.