Math 1101

Final Review

Fall 2002

Number of Voters	12	21	8	10	2
1^st Choice	B	A	D	C	A
2^nd Choice	D	D	C	A	C
3^rd Choice	A	C	A	D	D
4^th Choice	C	B	B	B	B

1. Given the above voting schedule determine the winner via the following methods:
1. Plurality method
2. Borda count method
3. Pairwise comparison method.
4. Plurality-with-elimination method.

2. Determine who came in second using
1. The extend plurality method.
2. The recursive Borda count method.
3. Is there a condorcet candidate? Why or why not?
4. know the meanings of the following criterion: the majority criterion, the condorcet criterion, the monotonicity criterion, and the independence-of-irrelevantoalternatives criterion.

Weighted Voting:

5. Given the voting system: {15: 5, 4, 3, 2, 1}
1. Who is the dictator?
6. Given the voting system: {15:10, 7, 5, 3, 3}
1. Who is the dictator:
2. Are there any dummies?
3. What is the power index of each voter using the Banzhaf power index?
4. How many Banzhaf coalitions are there?
5. What is the power index of each voter using the Shapley-Shubik power index?
6. How many Shapley-Shubik sequential coalitions are there?
7. Given the voting system: {q: 10, 8, 6, 4, 3}, find q if a 2/3 majority is needed to pass a referendum?
8. Given the above voting system, what is the minimum quota?
9. which of the following is not a possible Shapley-Shubik power index for a system with 6 players. 1/3, 1/5, 1/7, 1/8.
10. Henry likes chocolate cake 5 times as much as he likes strawberry cake. If a cake which is half and half is worth $24, how much is the chocolate half worth to him?
11. If the cake is cut in half so that one piece is 1/3 chocolate and 2/3 strawberry, how much is this piece worth to Henry.
12. Explain the divider-chooser method.
13. Know the lone divider and lone chooser methods.
14. Using the method of sealed bids determine who gets which item and how much cash they receive or pay out.

The Bids are:

	Mavis	Donald	Douglas
Car	15000	10,000	5,000
Yacht	15,000	25,000	20,000
Jewelry	10,000	5,000	18,000
Total Value

Apportionment:

	Population In millions	Lower Quota	Standard Quota	Upper Quota	Apportionment
Minnesota	492
Wisconsin	536
Iowa	293
Kansas	269
Nebraska	171

15. A panel is to be created consisting of representation of the following states, and the panel is to consist of 100 members.
1. What is the standard divisor?
2. Find the standard quota for each state.
3. Fill in the lower and upper quotas.
4. What will be the apportionment using the Hamilton method?
5. What will be the apportionment using the Jefferson method?
6. What will be the apportionment using the Adams method?
7. What will be the apportionment using Webster’s method?
16. Which of the above methods favor larger states?
17. Which favor smaller states?
18. Which violate the quota rule? The Alabama paradox? Population paradox? New-States paradox?
19. Know the terminology:
1. Adjacent edges and vertices,
2. Bridges
3. Circuit
4. path
5. Euler circuit
6. Euler path
7. How to eulerize a graph
8. Connected graphs
9. Loop
10. Multiple edges
11. Unicursal tracings.
20. problem 25 page 183. If the graphs do not have Euler paths, eulerize them.
21. problem 62 on page 190.
22. Draw a complete graph with 6 vertices.
23. How many edges does a complete graph with 10 vertices have?
24. How many Hamilton circuits will a complete graph with 10 vertices have?
25. Traveling salesman problems

Use the following algorithms to find a path for a traveling salesman visiting each of the following cities:

	Alexandria	Minneapolis	Rochester	Brainerd	Bemidji	Mankato
Alexandria	*	131	214	86	133	165
Minneapolis	131	*	83	125	214	120
Rochester	214	83	*	208	297	83
Brainerd	86	125	208	*	97	179
Bemidji	133	214	297	97	*	266
Mankato	165	120	83	179	266	*

1. Nearest-neighbor algorithm.
2. Repetitive-nearest-neighbor algorithm
3. Cheapest-Link algorithm
4. Kruskal’s algorithm to find the minimum spanning network between the above cities.

26. Create a project digraph for the following information:
1. Then determine a descending time priority list and make a schedule accordingly.
2. Determine the critical values of each vertex.
3. Create a critical path list.
4. Find a schedule using the critical path priority list.

Task	Length of Task	Tasks that must be completed Before you can start
A	7
B	5
C	6
D	8
E	5	A ,B
F	7	E, C
G	5	D, F
H	4	E
I	4	F
J	3	G,I
K	1	H, L
L	4	I
M	3	L, J
N	6	K
O	3	L

27. Find the 10^th Fibonacci number.
28. What is the golden ratio?
29. What is the golden ratio to the 10^th power?
30. Calculate gnomons
31. Use chapter 10 reviews already provided…
32. Find the value of $40,000 in 8 years if interest is 8% nominal compounded quarterly .
33. How much needs to be invested now at 9% compounded semi-annually to have $40,000 in 5 years?
34. James invests $200 at the beginning of each month from January 1^st 2003 until December 1^st 2013. On January 1^st 2014 he starts investing $500 per month until he retires on December 31^st 2039. What will the value of this annuity be on the day he retires if interest is 9% compounded monthly during this time?
35. Find the effective interest rate on 7% compounded monthly to the nearest hundredth of a percent.
36. You buy a house for $150,000. The loan is for 30 years at 7.2% compounded monthly. What is the house payment before insurance and taxes? Create the first 3 rows of an amortization table for the loan.