Mathematical Thinking Exam 1 09 October 2017

Transcription

1 Mathematcal Thnkng Exam 1 09 October 2017 Name: Instructons: Be sure to read each problem s drectons. Wrte clearly durng the exam and fully erase or mark out anythng you do not want graded. You may use a scentfc or graphng calculator for any calculaton or dervaton unless otherwse stated so long as you show your work leadng up to that pont and then state that that s what you are dong. You must show all your work to receve full credt unless otherwse stated. Don t cheat. See below. Prohbted: Class Notes Class Handouts Study Gudes and Materals Prevous exams or exam solutons The Book Any Electroncs (excludng approved calculators) Ths ncludes, but s not lmted to, all: computers, phones, tablets, ebook readers, headphones, audo players, dctonares, and translators. Other People s Exams Page: Total Ponts: Score:

2 MATH 101 Exam 1, Page 2 of October 2017 (Some) Fnancal Formulas A = P (1 + rt) A = P (1 + ) n A = [ (1 + ) n ] 1 d d = [ ] A (1 + ) n 1 g = r a 1 + a $1 = $1 $a 1 + a 1 + a A = [ (1 + ) n ] 1 d d = [ ] P 1 (1 + ) n d = P 12t(1 rt) d = P (1 + rt) 12t Ponts earned: out of a possble 0 ponts

3 MATH 101 Exam 1, Page 3 of October 2017 Multple Choce For ths secton, you must wrte the letter correspondng to the correct answer n the blank space provded on the rght sde of the exam. Markngs on the queston tself wll not be gven any credt. 1. (3 ponts) The votng system whch does not always produce at least one wnner when there are an odd number of voters s A. Pluralty. B. Condorcet. C. Dctatorshp. D. Sequental Parwse. 2. (3 ponts) What s the name of the theorem that states there can never by a votng system of 3 + canddates that always produces a wnner, satsfes Pareto s condton and IIA, and s not a dctatorshp? A. May s B. Arrow s C. LaGrange s D. Bexley s 3. (3 ponts) In a two opton electon, whch of the followng farness crtera does Imposed Rule fal? A. Never Tes B. Far to Voters C. Far to Optons D. Monotone 4. (3 ponts) You have $5, 000 to nvest at 6% smple nterest. How much money wll you have after 3 years? A. $5, 300 B. $5, 900 C. $6, 800 D. $15, (3 ponts) What s the APY for 6% compounded weekly? A. 6.00% B. 6.09% C. 6.18% D. 7.25% 6. (3 ponts) Suppose your fourth grade teacher bought a house n 1990 for $125, 000 and sold t n Knowng that that the CPI for 1990 was and the CPI for 2016 was , how much was the house worth when t was sold? A. $68, B. $127, C. $229, D. $245, 342, (3 ponts) A 15% loan rate s most favorable (from the borrower s perspectve) f compounded A. contnuously. B. daly. C. monthly. D. annually. 8. (3 ponts) What s the APR of a credt card loan wth a 1.660% monthly nterest rate? A % B % C % D % Note: There was a typo n the correct soluton for Problem #8 where I mssed a decmal pont. Compare answer C here to the opton on the blank exam. Due to my error, everyone receves full credt for # Ponts earned: out of a possble 24 ponts

4 MATH 101 Exam 1, Page 4 of October 2017 Computaton and Technques 9. Use the followng electon for the below parts. Rank (10) (14) (13) (8) Frst A C B D Second B D A A Thrd C A D C Fourth D B C B (a) (6 ponts) Whch canddate would wn f the Hare system s used? Soluton: Frst elmnate D as t has the fewest frst place votes to gan the p.l.b. below. Rank (10) (14) (13) (8) Frst A C B A Second B A A C Thrd C B C B Now elmnate B as t has the fewest frst place votes to gan the p.l.b. below. Rank (10) (14) (13) (8) Frst A C A A Second C A C C Elmnatng C as t has the fewest frst place votes leaves A as the Hare wnner. (b) (6 ponts) Whch canddate s the pluralty runnoff wnner? Soluton: Elmnate all optons except the two wth the most frst place votes. The new preference lst ballot would then be as follows. Rank (10) (14) (13) (8) Frst B C B C Second C B C B In the majorty rule vote, B would wn 23 over 22 aganst C so B s the Pluralty Runoff wnner. Ponts earned: out of a possble 12 ponts

5 MATH 101 Exam 1, Page 5 of October Jason wants to buy a house for $205, 000 and can pay out of hs savngs, a 20% down payment on t. (a) (3 ponts) How much money does Jason need to take out as a loan to afford the house? Soluton: 0.02 $205, 000 = $41, 000 Loan = $205, 000 $41, 000 Loan = $164, 000 (b) (7 ponts) If Jason can secure a 30-year, fxed rate loan at 5.125% and makes no addtonal payments on the loan, what wll he end up payng for hs house after 30 years? Assume that he must make monthly payments (most mortgages requre ths schedule). Soluton: ( ) d = P 1 (1 + ) ( n ) /12 = $164, ( /12) 360 $ Thus to calculaton the total pad over 30 years we see: T otal = d 360 $321, Ponts earned: out of a possble 10 ponts

6 MATH 101 Exam 1, Page 6 of October 2017 (c) (7 ponts) In ths scenaro, what would be hs monthly payments? Soluton: From the prevous part, d $ (d) (7 ponts) Assume that Jason decdes he can actually afford to take out a 15-year, fxed rate loan at 6.875%. How much would he save by takng the 15-year loan compared to the 30-year loan from the prevous part or would he actually pay more n total? (Agan assume monthly payments.) Soluton: ( ) d = P 1 (1 + ) ( n ) /12 = $164, ( /12) 180 $ Thus to calculaton the total pad over 15 years we see: T otal = d 180 $263, Thus the total saved s the dfference of these two totals. Saved = $321, $263, = $57, Ponts earned: out of a possble 14 ponts

7 MATH 101 Exam 1, Page 7 of October 2017 Concepts 11. (10 ponts) Produce a preference lst ballot of 13 voters showng that the pluralty votng system can elect as a wnner, the opton that appears last on the majorty of ballots (7 of the 13). Be sure to justfy your ballot wth complete sentences and do actually calculate the wnner. Soluton: Rank (4) (3) (3) (3) 1 A B C C 2 B A A B 3 C C B A Clearly C wns the pluralty vote wth the frst place vote counts beng Addtonally, C s ranked last on seven of the 13 ballots whch accounts for approxmately 53.85% > 50%. Ponts earned: out of a possble 10 ponts

8 MATH 101 Exam 1, Page 8 of October (10 ponts) Suppose you decde you want to have two mllon dollars saved by the tme you retre n 40 years to be able to lve comfortably. If you can get a 10% return n the stock market (bg assumpton), how much do you have to save (nvest n the stock market) each month to reach ths goal? Soluton: $2, 000, 000 = ( (1 +.1/12) 480 ) 1 d.1/12 $16, = d(( ) 480 1) d $ (3 ponts (bonus)) How much money n 40-years would you need f n fact you wanted 2 mllon dollars n today s money? Soluton: I forgot to nclude an expected average nflaton rate for the tme perod or an expected CPI. Everyone earns the full ponts for ths extra credt problem. Ponts earned: out of a possble 10 ponts

9 MATH 101 Exam 1, Page 9 of October 2017 Short Response 14. A votng system s sad to satsfy the Majorty Crteron f an opton ranked frst by a majorty of voters s always among the wnners. For each of the followng votng systems, explan n a few sentences why the votng system satsfes or doesn t satsfy the Majorty Crteron. (a) (5 ponts) The Pluralty System Soluton: The Pluralty Votng System declares a wnner by lookng at whchever canddate or canddates receved the most frst place votes. Snce the opton or optons wth the most frst place votes s automatcally the wnner or wnners of a Pluralty system, ths votng system satsfes the Majorty Crteron. (b) (5 ponts) The Borda Count System Soluton: The Borda Count System s gong to fal the Majorty Crteron. Suppose n a vote wth three optons and eleven voters, sx voters lst opton A frst and the other fve lst opton A last. Then the Borda score for A wll be 2 6 = 12. However f those same sx voters lst opton B second and the same other frst lst opton B frst, then the Borda score for B wll be = 16. As 16 > 12, opton A can t be among the wnners even though t s lsted frst on a majorty of the voters lsts. In essence, Borda Count fals ths farness crtera because of the large amount of weght gven to second place relatve to frst. Ponts earned: out of a possble 10 ponts

10 MATH 101 Exam 1, Page 10 of October (10 ponts) Explan what the real (effectve) growth rate tells a person. Complete the followng problem and reference t n your explanaton when descrbng each of the factors. Nyk wants to make an nvestment that wll have real growth rate of 12%. If nflaton s current at 2.7%, what annual nterest rate much she earn to get on her nvestment to accomplsh ths goal? Soluton: The real growth rate represents the growth of an nvestment takng nto account for nflaton. Snce after a perod of tme, the prncpal nvested s worth less n terms of purchasng power (as well as any accrued nterest), the real growth rate offers a more consumer purchasng power vew of an nvestment. To address the example, Nyk wants her nterest rate to generate wealth that grows at 12% = g takng nto account nflaton. Assumng nflaton s constant at 2.7% = a, we are tryng to determne what advertsed nterest rate (r) must she fnd to acheve her goal. g = r a 1 + a.12 = r = r = r Thus Nyk needs to fnd an nvestment wth nterest rate r 15.02%. Ponts earned: out of a possble 10 ponts