Finite Mathematics for Business Economics Life Sciences Social Sciences Barnett Ziegler Byleen Twelfth Edition

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1 Fnte Mathematcs for Busness Economcs Lfe Scences Socal Scences Barnett Zegler Byleen Twelfth Edton

2 Pearson Educaton Lmted Ednburgh Gate Harlow Essex CM20 2JE England and Assocated Companes throughout the world Vst us on the World Wde Web at: Pearson Educaton Lmted 2014 All rghts reserved. No part of ths publcaton may be reproduced, stored n a retreval system, or transmtted n any form or by any means, electronc, mechancal, photocopyng, recordng or otherwse, wthout ether the pror wrtten permsson of the publsher or a lcence permttng restrcted copyng n the Unted Kngdom ssued by the Copyrght Lcensng Agency Ltd, Saffron House, 6 10 Krby Street, London EC1N 8TS. All trademarks used heren are the property of ther respectve owners. The use of any trademark n ths text does not vest n the author or publsher any trademark ownershp rghts n such trademarks, nor does the use of such trademarks mply any afflaton wth or endorsement of ths book by such owners. ISBN 10: ISBN 13: Brtsh Lbrary Catalogung-n-Publcaton Data A catalogue record for ths book s avalable from the Brtsh Lbrary Prnted n the Unted States of Amerca

3 164 CHAPTER 3 Mathematcs of Fnance 56. A $2,000 computer can be fnanced by payng $100 per month for 2 years. What s the annual nomnal compoundng rate for ths loan? 57. The owner of a small busness has receved two offers of purchase. The frst prospectve buyer offers to pay the owner $100,000 n cash now. The second offers to pay the owner $10,000 now and monthly payments of $1,200 for 10 years. In effect, the second buyer s askng the owner for a $90,000 loan. If the owner accepts the second offer, what annual nomnal compoundng rate wll the owner receve for fnancng ths purchase? 58. At the tme they retre, a couple has $200,000 nvested n an annuty. The couple can take the entre amount n a sngle payment, or receve monthly payments of $2,000 for 15 years. If the couple elects to receve the monthly payments, what annual nomnal compoundng rate wll the couple earn on the money nvested n the annuty? Answers to Matched Problems 2. $10, PMT = $112.98/mo; total nterest = $ Payment Number Payment Interest Unpad Balance Reducton Unpad Balance 0 $1, $ $12.50 $ Totals $1, $44.21 $1, $98, Choose the 0% fnancng. 1. $13, Chapter 3 Revew Important Terms, Symbols, and Concepts 3-1 Smple Interest EXAMPLES Interest s the fee pad for the use of a sum of money P, called the prncpal. Smple nterest s gven by I = Prt I = nterest P = prncpal r = annual smple nterest rate (wrtten as a decmal) t = tme n years If a prncpal P (present value) s borrowed, then the amount A (future value) s the total of the prncpal and the nterest: A = P + Prt = P(1 + rt) 3-2 Compound and Contnuous Compound Interest Compound nterest s nterest pad on the prncpal plus renvested nterest. The future and present values are related by Ex. 1, p. 128 Ex. 2, p. 128 Ex. 3, p. 129 Ex. 4, p. 129 Ex. 5, p. 130 Ex. 1, p. 134 A = P(1 + ) n = r/m and A = amount or future value P = prncpal or present value r = annual nomnal rate (or just rate) m = number of compoundng perods per year = rate per compoundng perod n = total number of compoundng perods 164

4 CHAPTER 3 Revew 165 If a prncpal P s nvested at an annual rate r earnng contnuous compound nterest, then the amount A after t years s gven by A = Pe n The growth tme of an nvestment s the tme t takes for a gven prncpal to grow to a partcular amount. Three methods for fndng the growth tme are as follows: Ex. 2, p. 136 Ex. 3, p. 137 Ex. 4, p. 138 Ex. 5, p Use logarthms and a calculator. 2. Use graphcal approxmaton on a graphng calculator. 3. Use an equaton solver on a graphng calculator or a computer. The annual percentage yeld APY (also called the effectve rate or true nterest rate) s the smple nterest rate that would earn the same amount as a gven annual rate for whch nterest s compounded. If a prncpal s nvested at the annual rate r compounded m tmes a year, then the annual percentage yeld s gven by APY = a1 + r m m b - 1 Ex. 6, p. 140 Ex. 7, p. 141 If a prncpal s nvested at the annual rate r compounded contnuously, then the annual percentage yeld s gven by APY = e r - 1 A zero coupon bond s a bond that s sold now at a dscount and wll pay ts face value at some tme n the future when t matures. 3-3 Future Value of an Annuty; Snkng Funds An annuty s any sequence of equal perodc payments. If payments are made at the end of each tme nterval, then the annuty s called an ordnary annuty. The amount, or future value, of an annuty s the sum of all payments plus all nterest earned and s gven by Ex. 1, p. 147 Ex. 3, p. 150 Ex. 4, p. 151 (1 + ) n - 1 FV = PMT FV = future value (amount) PMT = perodc payment = rate per perod n = number of payments (perods) A balance sheet s a table that shows the nterest and balance for each payment of an annuty. An account that s establshed to accumulate funds to meet future oblgatons or debts s called a snkng fund. The snkng fund payment can be found by solvng the future value formula for PMT: Ex. 2, p. 149 PMT = FV (1 + ) n Present Value of an Annuty; Amortzaton If equal payments are made from an account untl the amount n the account s 0, the payment and the present value are related by the followng formula: Ex. 1, p. 155 Ex. 2, p (1 + ) -n PV = PMT PV = present value of all payments PMT = perodc payment = rate per perod n = number of perods 165

5 166 CHAPTER 3 Mathematcs of Fnance Amortzng a debt means that the debt s retred n a gven length of tme by equal perodc payments that nclude compound nterest. Solvng the present value formula for the payment gves us the amortzaton formula: PMT = PV 1 - (1 + ) -n An amortzaton schedule s a table that shows the nterest due and the balance reducton for each payment of a loan. The equty n a property s the dfference between the current net market value and the unpad loan balance. The unpad balance of a loan wth n remanng payments s gven by the present value formula. A strategy for solvng problems n the mathematcs of fnance s presented on page 160. Ex. 3, p. 156 Ex. 6, p. 159 Ex. 4, p. 157 Ex. 5, p. 158 Revew Exercses Work through all the problems n ths chapter revew and check your answers n the back of the book. Answers to all revew problems are there along wth secton numbers n talcs to ndcate each type of problem s dscussed. Where weaknesses show up, revew approprate sectons n the text. A In Problems 1 4, fnd the ndcated quantty, gven A = P(1 + r t) B In Problems 5 and 6, fnd the ndcated quantty, gven A = P(1 + ) n In Problems 7 and 8, fnd the ndcated quantty, gven A = Pe rt In Problems 9 and 10, fnd the ndcated quantty, gven A =?; P = $100; r = 9%; t = 6 months A = $808; P =?; r = 12%; t = 1 month A = $212; P = $200; r = 8%; t =? A = $4,120; P = $4,000; r =?; t = 6 months A =?; P = $1,200; = 0.005; n = 30 A = $5,000; P =?; = ; n = 60 A =?; P = $4,750; r = 6.8%; t = 3 years A = $36,000; P =?; r = 9.3%; t = 60 months (1 + ) n - 1 FV = PMT FV =?; PMT = $1,000; = 0.005; n = 60 FV = $8,000; PMT =?; = 0.015; n = 48 In Problems 11 and 12, fnd the ndcated quantty, gven 1 - (1 + ) -n PV = PMT 11. PV =?; PMT = $2,500; = 0.02; n = PV = $8,000; PMT =?; = ; n = 60 C 13. Solve the equaton 2,500 = 1,000(1.06) n for n to the nearest nteger usng: (A) Logarthms (B) Graphcal approxmaton technques or an equaton solver on a graphng calculator 14. Solve the equaton (1.01) n - 1 5,000 = for n to the nearest nteger usng: (A) Logarthms (B) Graphcal approxmaton technques or an equaton solver on a graphng calculator. Applcatons Fnd all dollar amounts correct to the nearest cent. When an nterest rate s requested as an answer, express the rate as a percentage, correct to two decmal places. 15. If you borrow $3,000 at 14% smple nterest for 10 months, how much wll you owe n 10 months? How much nterest wll you pay? 16. Grandparents deposted $6,000 nto a grandchld s account toward a college educaton. How much money (to the nearest dollar) wll be n the account 17 years from now f the account earns 7% compounded monthly? 17. How much should you pay for a corporate bond payng 6.6% compounded monthly n order to have $25,000 n 10 years? 18. A savngs account pays 5.4% compounded annually. Construct a balance sheet showng the nterest earned durng each year and the balance at the end of each year for 4 years f (A) A sngle depost of $400 s made at the begnnng of the frst year. (B) Four deposts of $100 are made at the end of each year. 19. One nvestment pays 13% smple nterest and another 9% compounded annually. Whch nvestment would you choose? Why? 166

6 Revew Exercses A $10,000 retrement account s left to earn nterest at 7% compounded daly. How much money wll be n the account 40 years from now when the owner reaches 65? (Use a 365-day year and round answer to the nearest dollar.) 21. A couple wshes to have $40,000 n 6 years for the down payment on a house. At what rate of nterest compounded contnuously must $25,000 be nvested now to accomplsh ths goal? 22. Whch s the better nvestment and why: 9% compounded quarterly or 9.25% compounded annually? 23. What s the value of an ordnary annuty at the end of 8 years f $200 per month s deposted nto an account earnng 7.2% compounded monthly? How much of ths value s nterest? 24. A credt card company charges a 22% annual rate for overdue accounts. How much nterest wll be owed on a $635 account 1 month overdue? 25. What wll a $23,000 car cost (to the nearest dollar) 5 years from now f the nflaton rate over that perod averages 5% compounded annually? 26. What would the $23,000 car n Problem 25 have cost (to the nearest dollar) 5 years ago f the nflaton rate over that perod had averaged 5% compounded annually? 27. A loan of $2,500 was repad at the end of 10 months wth a check for $2, What annual rate of nterest was charged? 28. You want to purchase an automoble for $21,600. The dealer offers you 0% fnancng for 48 months or a $3,000 rebate. You can obtan 4.8% fnancng for 48 months at the local bank. Whch opton should you choose? Explan. 29. Fnd the annual percentage yeld on a CD earnng 6.25% f nterest s compounded (A) monthly. (B) contnuously. 30. You have $5,000 toward the purchase of a boat that wll cost $6,000. How long wll t take the $5,000 to grow to $6,000 f t s nvested at 9% compounded quarterly? (Round up to the next-hgher quarter f not exact.) 31. How long wll t take money to double f t s nvested at 6% compounded monthly? 9% compounded monthly? (Round up to the next-hgher month f not exact.) 32. Startng on hs 21st brthday, and contnung on every brthday up to and ncludng hs 65th, John deposts $2,000 a year nto an IRA. How much (to the nearest dollar) wll be n the account on John s 65th brthday, f the account earns: (A) 7% compounded annually? (B) 11% compounded annually? 33. If you just sold a stock for $17, (net) that cost you $12, (net) 3 years ago, what annual compound rate of return dd you make on your nvestment? 34. The table shows the fees for refund antcpaton loans (RALs) offered by an onlne tax preparaton frm. Fnd the annual rate of nterest for each of the followng loans. (A) A $400 RAL pad back n 15 days (B) A $1,800 RAL pad back n 21 days RAL Amount RAL Fee $10 $500 $29.00 $501 $1,000 $39.00 $1,001 $1,500 $49.00 $1,501 $2,000 $69.00 $2,001 $5,000 $ Lncoln Beneft Lfe offered an annuty that pays 5.5% compounded monthly. What equal monthly depost should be made nto ths annuty n order to have $50,000 n 5 years? 36. A person wants to establsh an annuty for retrement purposes. He wants to make quarterly deposts for 20 years so that he can then make quarterly wthdrawals of $5,000 for 10 years. The annuty earns 7.32% nterest compounded quarterly. (A) How much wll have to be n the account at the tme he retres? (B) How much should be deposted each quarter for 20 years n order to accumulate the requred amount? (C) What s the total amount of nterest earned durng the 30-year perod? 37. If you borrow $4,000 from an onlne lendng frm for the purchase of a computer and agree to repay t n 48 equal nstallments at 0.9% nterest per month on the unpad balance, how much are your monthly payments? How much total nterest wll be pad? 38. A company decdes to establsh a snkng fund to replace a pece of equpment n 6 years at an estmated cost of $50,000. To accomplsh ths, they decde to make fxed monthly payments nto an account that pays 6.12% compounded monthly. How much should each payment be? 39. How long wll t take money to double f t s nvested at 7.5% compounded daly? 7.5% compounded annually? 40. A student receves a student loan for $8,000 at 5.5% nterest compounded monthly to help her fnsh the last 1.5 years of college. Startng 1 year after fnshng college, the student must amortze the loan n the next 5 years by makng equal monthly payments. What wll the payments be and what total nterest wll the student pay? 41. If you nvest $5,650 n an account payng 8.65% compounded contnuously, how much money wll be n the account at the end of 10 years? 42. A company makes a payment of $1,200 each month nto a snkng fund that earns 6% compounded monthly. Use graphcal approxmaton technques on a graphng calculator to determne when the fund wll be worth $100, A couple has a $50,000, 20-year mortgage at 9% compounded monthly. Use graphcal approxmaton technques on a graphng calculator to determne when the unpad balance wll drop below $10, A loan company advertses n the paper that you wll pay only 8 a day for each $100 borrowed. What annual rate of nterest are they chargng? (Use a 360-day year.) 45. Construct the amortzaton schedule for a $1,000 debt that s to be amortzed n four equal quarterly payments at 2.5% nterest per quarter on the unpad balance. 167

7 168 CHAPTER 3 Mathematcs of Fnance 46. You can afford monthly deposts of only $300 nto an account that pays 7.98% compounded monthly. How long wll t be untl you wll have $9,000 to purchase a used car? (Round to the next-hgher month f not exact.) 47. A company establshes a snkng fund for plant retoolng n 6 years at an estmated cost of $850,000. How much should be nvested semannually nto an account payng 8.76% compounded semannually? How much nterest wll the account earn n the 6 years? 48. What s the annual nomnal rate compounded monthly for a CD that has an annual percentage yeld of 6.48%? 49. If you buy a 13-week T-bll wth a maturty value of $5,000 for $4, from the U.S. Treasury Department, what annual nterest rate wll you earn? 50. In order to save enough money for the down payment on a condomnum, a young couple deposts $200 each month nto an account that pays 7.02% nterest compounded monthly. If the couple needs $10,000 for a down payment, how many deposts wll the couple have to make? 51. A busness borrows $80,000 at 9.42% nterest compounded monthly for 8 years. (A) What s the monthly payment? (B) What s the unpad balance at the end of the frst year? (C) How much nterest was pad durng the frst year? 52. You unexpectedly nhert $10,000 just after you have made the 72nd monthly payment on a 30-year mortgage of $60,000 at 8.2% compounded monthly. Dscuss the relatve merts of usng the nhertance to reduce the prncpal of the loan or to buy a certfcate of depost payng 7% compounded monthly. 53. Your parents are consderng a $75,000, 30-year mortgage to purchase a new home. The bank at whch they have done busness for many years offers a rate of 7.54% compounded monthly. A compettor s offerng 6.87% compounded monthly. Would t be worthwhle for your parents to swtch banks? Explan. 54. How much should a $5,000 face value zero coupon bond, maturng n 5 years, be sold for now, f ts rate of return s to be 5.6% compounded annually? 55. If you pay $5,695 for a $10,000 face value zero coupon bond that matures n 10 years, what s your annual compound rate of return? 56. If an nvestor wants to earn an annual nterest rate of 6.4% on a 26-week T-bll wth a maturty value of $5,000, how much should the nvestor pay for the T-bll? 57. Two years ago you borrowed $10,000 at 12% nterest compounded monthly, whch was to be amortzed over 5 years. Now you have acqured some addtonal funds and decde that you want to pay off ths loan. What s the unpad balance after makng equal monthly payments for 2 years? 58. What annual nomnal rate compounded monthly has the same annual percentage yeld as 7.28% compounded quarterly? 59. (A) A man deposts $2,000 n an IRA on hs 21st brthday and on each subsequent brthday up to, and ncludng, hs 29th (nne deposts n all). The account earns 8% compounded annually. If he leaves the money n the account wthout makng any more deposts, how much wll he have on hs 65th brthday, assumng the account contnues to earn the same rate of nterest? (B) How much would be n the account (to the nearest dollar) on hs 65th brthday f he had started the deposts on hs 30th brthday and contnued makng deposts on each brthday untl (and ncludng) hs 65th brthday? 60. A promssory note wll pay $27,000 at maturty 10 years from now. How much money should you be wllng to pay now f money s worth 5.5% compounded contnuously? 61. In a new housng development, the houses are sellng for $100,000 and requre a 20% down payment. The buyer s gven a choce of 30-year or 15-year fnancng, both at 7.68% compounded monthly. (A) What s the monthly payment for the 30-year choce? For the 15-year choce? (B) What s the unpad balance after 10 years for the 30-year choce? For the 15-year choce? 62. A loan company wll loan up to 60% of the equty n a home. A famly purchased ther home 8 years ago for $83,000. The home was fnanced by payng 20% down and sgnng a 30-year mortgage at 8.4% for the balance. Equal monthly payments were made to amortze the loan over the 30-year perod. The market value of the house s now $95,000. After makng the 96th payment, the famly appled to the loan company for the maxmum loan. How much (to the nearest dollar) wll the famly receve? 63. A $600 stereo s fnanced for 6 months by makng monthly payments of $110. What s the annual nomnal compoundng rate for ths loan? 64. A person deposts $2,000 each year for 25 years nto an IRA. When she retres mmedately after makng the 25th depost, the IRA s worth $220,000. (A) Fnd the nterest rate earned by the IRA over the 25-year perod leadng up to retrement. (B) Assume that the IRA contnues to earn the nterest rate found n part (A). How long can the retree wthdraw $30,000 per year? How long can she wthdraw $24,000 per year? 168