HON DISCRETE CHAPTER FOUR REVIEW
Identify the states, populations, seats, and what would the standard divisor represents:
- Mr. Smith is planning to give candy to students based on the numerical value of their quarter grades.
- WSFCS has seen a rise in students and is planning to hire 100 new teachers for either elementary, middle, or high schools. WSFCS plans to apportion the teachers based on the number of students each school level.
- The subway has 5 main subway lines (Green, Red, Yellow, Orange, and Blue) and the city is considering a more efficient way to organize public transportation. The plan is to re-distribute the 60 subway trains to the different lines based on the amount of profit that each line is currently making the city.
STANDARD APPORTIONMENT METHOD PRACTICE PROBLEMS:
1) Find apportionment according to Hamilton’s, Jefferson’s, Adam’s, and Webster’s.
State / A / B / C / 200 SEATSPopulation / 4400 / 8300 / 10,300
2) Find apportionment according to Hamilton’s, Jefferson’s, Adam’s, and Webster’s.
State / A / B / C / D / E / 200 SEATSPopulation / 4400 / 8300 / 5300 / 9700 / 10,300
3) Find apportionment according to Hamilton’s, Jefferson’s, Adam’s, and Webster’s.
State / A / B / C /D
/ E / 150 SEATSPopulation / 1700 / 2300 / 3100 / 2700 / 1900
APPORTIONMENT WORD PROBLEMS:
1)The 6 main departments at a local college need to split the 50 scholarships that the university offers to give to incoming freshmen. The admission department decides to assign scholarships to the departments based on their current number of majors which is given in the table below.
Political Science / History / Math / Science / English / Foreign Language112 / 78 / 45 / 57 / 68 / 60
- Identify the seats, populations, and states in this problem.
- What is the standard divisor?
- What does the standard divisor represent in this problem? (Hint: Use part a answers)
- What is the standard quota for each state?
- What is the lower quota for each state?
- What is the upper quota for each state?
- What are HAMILTON’S AND JEFFERSON’S APPORTIONMENTS to this problem?
2)There are 5 main interstates in a state. The state legislature plans to assign work crews to the 5 main interstates based on the average number of cars on the interstate each day. The interstates combined have an average of 231,000 cars each day. The STANDARD QUOTA for each interstate is given in the table below.
Interstate 1 / Interstate 40 / Interstate 85 / Interstate 485 / Interstate 77165.4 / 220.7 / 285.4 / 180.9 / 197.6
- Identify the seats, populations, and states in this problem.
- What is the standard divisor?
- What does the standard divisor represent in this problem? (Hint: Use part a answers)
- What is the average number of cars for each interstate?
- How many work crews are there in the entire state?
- What is the lower quota for each interstate?
- What is the upper quota for each interstate?
- What are WEBSTER’S AND ADAM’S APPORTIONMENTS to this problem?
HON DISCRETE CHAPTER FIVE REVIEW
#1:In each graph:
a)Find the Edge Set
b)Identify any multiple edge
c)Identify any loops
d)Identify any bridges
e)Find the degree of each vertex
#2: If possible, draw each of the following graphs.
1)The graph has 3 bridges and 4 vertices.
2)The graph has 4 odd vertices and 2 even vertices.
3)Disconnected in which each component has a bridge.
4)Has at least 3 multiple edges, 2 loops, 5 vertices, and 10 total edges.
5)Disconnected graph with 2 even vertices and 3 odd vertices.
6)Euler Path and odd vertices are not adjacent.
#3: For each of the following graphs
a)Identify any components.
b)Does the graph have a circuit starting and ending at A? Find one if it exists.
c) Is there a path from A to F? Find One.
#4: Does the Graph have an Euler Path, Euler Circuit, or Neither?
a)Determine by degree of vertices if an Euler Circuit or Path Exists
b)If one exists, then find it from a starting vertex.
#5: For the below graph,
a)Find a circuit of length 1
b)Find a circuit of length 4
c)Find a circuit of length 5
d)Find a circuit of length 6
e)Find a path of length 3
f)Find a path of length 8
g)Find a path of length 11
HON DISCRETE CHAPTER SIX REVIEW
#1: For each of the following situations: Be able to find how many vertices, edges, degree of vertex, and number of distinct circuits exist in the complete graph.
1) 12 vertices
2) degree of each vertex is 5
3) 18 vertices
4) 21edges
5) Distinct Hamilton circuits = 720
6) 4 vertices
7) 78 edges
#2: Determine if the graph has a Hamilton Path and/or circuit. If so, find one.
#3:For the weighted complete graphs/tables below: find the shortest Hamilton circuit for
A. Nearest Neighbor
B. Repetitive Nearest Neighbor
C. Cheapest Link
D. Brute Force – How many circuits exist in the graph?
A / $9 / $13 / $7 / $4 / $6
B / $9 / $5 / $15 / $8 / $12
C / $13 / $5 / $18 / $11 / $17
D / $7 / $15 / $18 / $3 / $5
E / $4 / $8 / $11 / $3 / $2
F / $6 / $12 / $17 / $5 / $2
#2 / A / B / C / D / E
A / 19 / 27 / 33 / 25
B / 19 / 26 / 13 / 21
C / 27 / 26 / 35 / 31
D / 33 / 13 / 35 / 18
E / 25 / 21 / 31 / 18
GRAPH #3:
GRAPH #4: