Math 1030Q Sprig 2013 Exam #2 Review 1. Solve for x: 2 + 9.2 8x = 2.32 Solutio: 2 + 9.2 8x = 2.32 9.2 8x = 2.32 2 log 9.2 8x = log.32 8x log 9.2 = log.32 8x log 9.2 log.32 = 8 log 9.2 8 log 9.2 x =.06418 2. Solve for y: 3y 7 13 = 1540 Solutio: 3y 7 13 = 1540 y 7 13 = 513.33333 y 7 = 1.61619 y = 8.61619
3. Which will be worth more i 10 years: $10,000 ivested at 8.2% simple iterest, or $10,000 ivested at 5% iterest, compouded mothly? Solutio: For simple iterest: F = 10000.08210 = $18,200 For compoud iterest: F = 10000.05 10 = 100001.00416667 0 = 100001.64701015 = $16,470.10... so you ear more with simple iterest. 4. Suppose a fried leds you $100, ad you agree to pay him back $1 i 18 moths. If we assume that this is simple iterest, the what is the iterest rate? Solutio: Note that 18 moths is t = 1.5 years. The solvig for r, F = P rt 1 = 100 1.5r 1. = 1.5r 0. = 1.5r 0.08 = r = 8%
5. For a accout with a aual iterest rate of 6%, fid the aual percetage yield APY if iterest is compouded: a quarterly? Solutio: APY =.06 4 1 4 = 1.015 4 1 = 1.06136355 1.0614 = 6.14% b mothly? Solutio: APY =.06 1 = 1.005 1 = 1.06167781 1.0617 = 6.17% c daily? Solutio: APY =.06 1 = 1.00016438 1 = 1.06182993 1.0618 = 6.18%
6. A bak advertises a Certificate of Deposit CD with 4.8% iterest, compouded mothly. If I ivest $3,500 today, how log will it take for my ivestmet to grow to $4,200? Solutio: Usig the compoud iterest formula ad solvig for t, F = P r t 4200 = 3500 0.048 t 1.2 = 0.048 t log1.2 = log 0.048 t log1.2 = t log 0.048 log1.2 = t log1.004 log1.2 = t = 3.806 years log1.004 7. Reba would like to make the $2,150 dow paymet o a ew car i 6 moths. If she has $2,000 i her savigs accout, ad iterest is compouded daily, what iterest rate would she eed to ear to have eough? Solutio: Usig the compoud iterest formula t must be i years, ot moths: 2150 = 2000 r 2150 = 2000 r 1.075 = r 2 1.075 2 = r 1.00039636 = r 0.00039636 = r 0.00039636 = r 0.14466994 = r 14.47% 6 2 2 2
8. Whe Jed was bor, his gradfather deposited $1,982 ito a savigs accout for his gradso, uder the coditio that obody touches it util Jed turs 21. If this accout ears 3.9% iterest compouded semi-aually twice per year, the how much will Jed have o his 21st birthday? Solutio: Usig the compoud iterest formula ad solvig for F, F = P r t = 1982 0.039 2 = 19821.0195 42 2 21 = 19822.25042 = $4,460.33 9. May years later, Jed s graddaughter is bor, ad he would like to do somethig similar for her. He would like her to have exactly $10,000 i the accout o her 21st birthday. If the accout ears 4.1% compouded aually, how much would Jed eed to deposit o the day she is bor? Solutio: 10000 = P.041 1 21 1 10000 = P 1.041 21 10000 = P 2.32522680 10000 2.32522680 = P 2.32522680 2.32522680 P = $4,300.66
10. It s ever too early to start savig for retiremet! Suppose you fid a savigs accout that will pay 5% iterest compouded mothly. If, startig o your ext birthday, you deposit $85 per moth, ad cotiue this util your 65th birthday, how much will you have i your accout? Solutio: This depeds o your curret age, obviously, so let s assume you do this startig whe you tur 21. The o your 65th birthday you ve bee makig deposits for t = 65 21 = 44 years. Usig the systematic savigs formula ad solvig for F, r t F = D 1 r 0.05 44 1 = 85 0.05 1.0041667 528 1 = 85 0.0041667 8.98386 1 = 85 0.0041667 = 851916.58 = $162,870.69 11. Let s say you d like to retire with, oh I do t kow, $1 millio. Give the same accout from #10, how much would you eed to deposit every moth for this to happe? Solutio: Assume you start o your 21st birthday..05 44 1 1000000 = D.05 1.00416667 528 1 1000000 = D.00416667 7.98387327 1000000 = D.00416667 1000000 = D1916.8052 1000000 1916.8052 = D1916.8052 1916.8052 D = $521.89
. Maggie borrows $7,000 from the bak at 8% iterest compouded mothly. a If she makes a $400 paymet at the ed of the first moth, how much does she owe? Solutio: This is the remaiig balace etry that would be at the ed of the first row of a amortizatio schedule. It would read Paymet Iterest Paid Pricipal Paid Remaiig Balace 0.08 $400 7000 = 46.67 400-46.67 = 353.33 7000-353.33 = $6,646.67 b If she cotiues payig $400 mothly, how log will it take to pay off the loa? Solutio: Usig the loa formula ad solvig for t, 1 r P = R 7000 = 400 r t 1 0.08 0.08 17.5 = 1 1.006667 t 0.006667 0.116667 = 1 1.006667 t 0.883333 = 1.006667 t log0.883333 = log 1.006667 t log0.883333 = t log1.006667 log0.883333 = t = 1.556 years log1.006667 t
13. Adrew takes out a $18,500 studet loa to pay for graduate school. If the iterest rate is 6.3% compouded quarterly, how large would his quarterly paymets be i order to pay off this loa i 10 years? Solutio: Usig the loa formula ad solvig for R, 1 r t P = R 18500 = R r 1.063 4.063 4 4 10 1 1.01575 40 18500 = R.01575.46478687 18500 = R.10575 18500 R = 29.51027746 = $626.90 14. Fray ad Zooey are ready to buy their first house. They determie that they ca pay $1100 per moth towards a mortgage. If the 20 year mortgage available to them charges 7.8% iterest compouded mothly, a how large of a loa ca they afford? Solutio: Usig the loa formula ad solvig for P, 1 r t P = R = 1100 r 1 0.078 0.078 20 1 1.0065 240 = 1100 0.0065 1 0.2111995 = 1100 0.0065 0.7888045 = 1100 = $133,489.31 0.0065
b create a amortizatio schedule for the first 3 moths of the loa. Solutio: The rows detailig the first three mothly paymets would read Paymet Iterest Paid Pricipal Paid Remaiig Bal. $133,489.31 1100 0.078 133489.31 = $867.68 1100 867.68 = $232.32 $133,256.99 1100 0.078 133256.99 = $866.17 1100 867.68 = $233.83 $133,023.16 1100 0.078 133023.16 = $864.65 1100 864.65 = $235.35 $132,787.81 15. Whe rollig two dice, what is the probability that you: a Roll a 5? Solutio: You could have 1,4, 2,3, 3,2, 4,1 as possible rolls. There are 36 total possibilities, so P 5 = 4 36 = 1 9. b Roll a umber higher tha 9? Solutio: You could have 4,6, 5,5, 6,4, 5,6, 6,5, 6,6 as possible roll, so P 9 = 6 36 = 1 6. c Do t roll a 3? Solutio: We kow we could have 1,2 or 2,1 as our possible outcomes for rollig a 3, so to fid P ot 3, we have P ot 3 = 1 P 3 = 1 2 = 34 = 17. 36 36 18 d Roll a umber that is at least 5? Solutio: We could cosider all outcomes that result i 5 or higher, but there are a lot of those, so it will be faster to otice that the oly outcomes which are ot 5 or higher are the oes which are less tha 5, i.e. 1,1, 1,2, 1,3, 2,1, 2,2, ad 3,1. So P 5 = 1 P 4 = 1 6 = 30 = 15. 36 36 18 e Roll a eve umber or a umber larger tha 3? Solutio: Use the additio rule. Clearly half of the possible outcomes are eve ad half are odd, so P eve = 18 33. P > 3 =, sice every outcome except 36 36 for 1,1, 2,1, ad 1,2 is larger tha 3. Of the 18 eve outcomes, oly 1,1 is ot larger tha 3, so P eve AND > 3 = 17, ad thus our fial aswer is 36 P eve OR > 3 = 18 + 33 17 = 34 = 17. 36 36 36 36 18
16. Accordig to the America Medical Associatio, i 1996 there were 737,764 physicias i the Uited Sates, 157,387 of whom were female. There were 133,005 physicias uder 35 years of age, 47,348 of whom were female. What is the probability that a radomly chose physicia i 1996 was female or uder the age of 35? Solutio: P female OR 35 = P female + P 35 P female AND 35 = 157387 737764 + 133005 737764 47348 737764 = 0.21333 + 0.18028 0.064178 = 0.329432 = 32.9% 17. A recet poll at a uiversity shows that, i a vote for the ew mascot: 60% of studets would approve of a giraffe, 42% would approve of a hippo, ad 17% would approve of both. If we select a studet at radom, what s the probability that he or she would approve of either the giraffe or the hippo? Solutio: Notice that P either A or B = 1 P either A or B... The P G or H = P G + P H P G ad H = 0.60 + 0.42 0.17 = 0.85, so, P either G or H = 1 0.85 = 0.15 = 15%.