practiceexam2solutions

Math 130 Midterm 2 Spring 2024 You have 50 minutes to complete the test. This exam contains 3 two-sided pages and 4 questions. If you have any questions, raise your hand and wait for the proctor to come to your seat. Write your final solutions on the test papers. Solutions presented with no supporting work may receive no credit. If you need extra space, use the extra page at the back. Clearly indicate on the question when you use paper at the back, and clearly indicate at the back which question you are answering. You may use the the backs of the pages and / or the last page of the exam to write your scratch-work. Please ask the proctor if you need more paper. Recall that you are allowed to use the results from the book and the class. When doing so, please be sure to write down the statements of the Theorems you are using. YOU MAY NOT USE NOTES, CELL PHONES, OR LAPTOPS AT ANY TIME DURING THE TEST PERIOD. When going to the bathroom, please leave your phone on the table. You’ve got this! PLEASE PRINT YOUR NAME HERE IN BLOCK CAPITALS: PLEASE WRITE YOUR BNUMBER HERE: Page 1

Math 130 1. (20 points) (a) (10 points) Do Hamilton State A B C D E Total Population 399 765 84 1111 406 2765 Number of Seats: 95 Standard divisor: 29.10526 Exact Quota 13.70886 26.28391 2.88608 38.17179 13.94937 95 Lower Quota 13 26 2 38 13 92 Fractional part 0.70886 0.28391 0.88608 0.17179 0.94937 3 Surplus 1 0 1 0 1 3 Total 14 26 3 38 14 95 (b) (5 points) Do Adam once State A B C D E Total Population 399 765 84 1111 406 2765 Number of Seats: 95 Standard divisor: 29.10526 Exact Quota 13.70886 26.28391 2.88608 38.17179 13.94937 95 Adam Quota 14 27 3 39 14 97 (c) (5 points) Do we lower / raise the divisor? We need to lower the quotas, we can do this by raising the divisor. Page 2

4. (25 points) Sealed bids. Person A B C D Item 1 31000 30000 29000 33000 Item 2 27000 29000 28000 27000 Item 3 14000 17000 11000 16000 Total Value 72000 76000 68000 76000 Fair Share 18000 19000 17000 19000 Allocation 0 46000 0 33000 Di ff erence 18000 -27000 17000 -14000 Total Surplus = 6000 Surplus Share 1500 1500 1500 1500 Summary Person A B C D Item(s) 2, 3 1 Item’s Value 0 46000 0 33000 Cash 19500 -25500 18500 -12500 Net Total 19500 20500 18500 20500 5. (25 points) Alice, Bob, and Claire want to divide a $42 cake. Half of the cake is vanilla, and half of the cake is chocolate. They decide to use the cut and choose method. Alice likes vanilla and chocolate the same amount. Bob likes vanilla 6 times as much as chocolate. Claire likes chocolate twice as much as vanilla. They order themselves: Alice first, Bob second, Claire third. (a) (5 points) Draw a diagram which indicates a possible cut Alice may make in the first round. Diagram not included due to latex complications. Alice could cut the cake in such a way that the chocolate part on top is parted at a 2:1 ratio and the vanilla part is parted at a 1:2 ratio. (b) (10 points) For the cut Alice made, find the value of each of the two parts from Bob’s perspective. Bob thinks the vanilla half is worth $36 and the chocolate half is worth $6. So the part with more vanilla is worth $ 2 3 6 + $ 1 3 36 = $4 + $12 = $16 . The part with more chocolate is worth $ 2 3 36 + $ 1 3 6 = $24 + $2 = $26 . (c) (10 points) Suppose that Bob takes the part worth more to him (pick one for him if there is a tie). Indicate possible further cuts that could be made in the last round (they don’t need to be reasonable choices of cut) and which parts Claire would take in this case. Diagram not included due to latex complications. In this stage Alice and Bob must each cut their piece into 3 pieces. For example they could each split their part into 2 very small parts and one large part containing all the chocolate. In this case Claire would choose to take the large part from each of them. Page 4

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