est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.


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1 9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value, length, and inteest ate of a loan. The most poweful foce in the univese is.... How would you finish this quote? The woldenowned physicist Albet Einstein said,... compound inteest. Ae you supised that of all the foces that he might pick, Einstein chose this one? In this section, we will explain how inteest can eithe wok fo you o against you. As you will see, used popely, it can help you build a fotune; used impopely, it can lead you to financial uin. If you want to accumulate enough money to buy a newe ca o go on a vacation, you could deposit money in a bank account. The bank will use you money to make loans to othe customes and pay you inteest fo using you funds. Howeve, if you boow money fom the bank, say to take a college couse, then you will pay inteest to the bank. In essence, inteest is the money that one peson (a boowe) pays to anothe (a lende) to use the lende s money. Saves ean inteest; boowes pay inteest. We will discuss simple and compound inteest in this section, and discuss the cost of consume loans in Section 9.3. KEY POINT Simple inteest is a staightfowad way to compute inteest. Simple Inteest* The amount you deposit in a bank account is called the pincipal. The bank specifies an inteest ate fo that account as a pecentage of you deposit. The ate is usually expessed as an annual ate. Fo example, a bank may offe an account that has an annual inteest ate of 5%. To find the inteest that you will ean in such an account, you also need to know how long the deposit will emain in the account. The time is usually stated in yeas. Thee is a simple fomula that elates pincipal, inteest eaned, inteest ate, and time. In wods, inteest eaned = pincipal * inteest ate * time. When we compute inteest this way, it is called simple inteest. We calculate simple inte FORMULA FOR COMPUTING SIMPLE INTEREST est using the fomula I = Pt, whee I is the inteest eaned, P is the pincipal, is the inteest ate, and t is the time in yeas. *If you want some pactice with basic algeba, see Appendix A.
2 9.2 y Inteest 405 EXAMPLE 1 Calculating Simple Inteest If you deposit $500 in a bank account paying 6% annual inteest, how much inteest will the deposit ean in 4 yeas if the bank computes the inteest using simple inteest? SOLUTION: In this example: P is the pincipal, which is $500 is the annual inteest ate, which is 6% (witten as 0.06) t is the time, which is 4 (yeas) Thus, the inteest eaned is I = Pt = 500 * 0.06 * 4 = 0. In 4 yeas, this account eans $0 in inteest. Now ty Execises 5 to 8. ] KEY POINT Futue value equals pincipal plus inteest. To find the amount that will be in you account at some time in the futue, called the futue value (o sometimes called the futue amount) we add the pincipal and the inteest eaned. We will epesent futue value by A, so we can say A = pincipal + inteest = P + I. If we eplace I by Pt, we get the fomula A = P + Pt = P(1 + t). COMPUTING FUTURE VALUE USING SIMPLE INTEREST futue value of an account that pays simple inteest, use the fomula A = P(1 + t), To find the whee A is the futue value, P is the pincipal, is the annual inteest ate, and t is the time in yeas. EXAMPLE 2 Computing Futue Value Using Simple Inteest Assume that you deposit $1,000 in a bank account paying 3% annual inteest and leave the money thee fo 6 yeas. Use the simple inteest fomula to compute the futue value of this account. SOLUTION: We see that P = 1,000, = 0.03, and t = 6. Theefoe, Thus, you bank account will have $1,180 at the end of 6 yeas. ] In contast to futue value, the pincipal that you have to invest in an account now to have a specified amount in the account in the futue is called the pesent value of the account. Notice that the fomula fo computing futue value has fou unknowns. If we want, we can use this fomula fo finding the pesent value of an account povided we know the futue value, inteest ate, and time. EXAMPLE 3 P t A 1,000(1 (0.03)(6)) 1,000(1 0.18) 1,000(1.18) 1,180. Finding the Pesent Value of an Account Assume that you plan to save $2,500 to take a whitewate afting tip in Costa Rica in 2 yeas. You bank offes a cetificate of deposit (CD) that pays 4% annual inteest computed using simple inteest. How much must you put in this CD now to have the necessay money in 2 yeas?
3 406 CHAPTER 9 y Consume Mathematics Quiz Youself Redo Example 3, but now assume that you want to save $2,400 in 4 yeas and the CD has an annual inteest ate of 5%. 5 SOLUTION: We can use the fomula A = P(1 + t). We know that A = 2,500, = 4% = 0.04, and t = 2. Theefoe, We can ewite this equation as 2,500 = P(1 + (0.04)(2)). 2,500 = P(1.08). Dividing both sides of the equation by 1.08, we get P = 2, L We will ound this up to $2, to guaantee that if you put this amount in the CD now, in 2 yeas you will have the $2,500 you need fo you whitewate afting tip.* Now ty Execises 9 to 14. ] 5 Some Good Advice In Example 3, we used the ealie fomula fo computing futue value to find the pesent value athe than stating a new fomula to solve this specific poblem. You will find it easie to lean a few fomulas well and use them, togethe with simple algeba, to solve new poblems athe than tying to memoize sepaate fomulas fo evey type of poblem. KEY POINT Compounding pays inteest on peviously eaned inteest. Compound Inteest It seems fai that if money in a bank account has eaned inteest, the bank should compute the inteest due, add it to the pincipal, and then pay inteest on this new, lage amount. This is in fact the way most bank accounts wok. Inteest that is paid on pincipal plus peviously eaned inteest is called compound inteest. If the inteest is added yealy, we say that the inteest is compounded annually. If the inteest is added evey thee months, we say the inteest is compounded quately. Inteest also can be compounded monthly and daily. EXAMPLE 4 Calculating Compound Inteest the Long Way Assume that you want to eplace you sailboat with a lage one in 3 yeas. To save fo a down payment fo this puchase, you deposit $2,000 fo 3 yeas in a bank account that pays 10% annual inteest, compounded annually. How much will be in the account at the end of 3 yeas? SOLUTION: We will pefom the compound inteest calculations one yea at a time in the following table. In compounding the inteest, we will use the futue value fom the pevious yea as the new pincipal at the beginning of the yea. Notice that the quantity (1 + t) = ( * 1) = (1.10) emains the same thoughout the computations. Quiz Youself Continue Example 4 to calculate the amount in you account at the end of the fouth yea. 6 Yea Pincipal (Beginning of Yea) P Futue Value (End of Yea) P(1 + t) = P(1.10) 1 $2,000 $2,000(1.10) = $2,200 2 $2,200 $2,200(1.10) = $2,420 3 $2,420 $2,420(1.10) = $2,662 ] 6 *When calculating a deposit to accumulate a futue amount, we will always ound up to the next cent. An inteest ate of 10% would be extaodinaily high. Howeve, we will often choose ates in examples and execises to keep the computations simple.
4 9.2 y Inteest 407 PROBLEM SOLVING Veify You Answe You should always check answes to see whethe they ae easonable. In Example 4, if we had used simple inteest to find the futue value, we would have obtained A = 2,000 (1 + (0.10)(3)) = 2,000(1.30) = 2,600. The inteest we found in Example 4 is a little lage because as the inteest is added to the pincipal each yea, the bank is paying inteest on an inceasingly lage pincipal. If we wee to continue the pocess that we used in Example 4 fo a longe peiod of time, say fo 30 yeas, it would be vey tedious. In Figue 9.2 we look at the same computations in a diffeent way, keeping in mind that the amount in the account at the end of each yea is 1.10 times the amount in the account at the beginning of the yea. Amount in account is $2,000 (1.10) $2,000 (1.10) 1 Amount in account is $2,000 (1.10) (1.10) $2,000 (1.10) 2 Amount in account is $2,000 (1.10) (1.10) (1.10) $2,000 (1.10) You deposit $2,000 at the beginning of yea 1. End of yea 1 End of yea 2 End of yea 3 FIGURE % inteest being compounded annually. Quiz Youself Calculate the futue value of an account containing $3,000 fo which the annual inteest ate is 4% compounded annually fo 10 yeas. KEY POINT Knowing the pincipal, the peiodic inteest ate, and the numbe of compounding peiods, it is easy to detemine futue value. 7 If we wee to continue the patten shown in Figue 9.2 to compute the futue value of the account at the end of 30 yeas, we would see that A = 2,000(1.10) 30 L 2,000( ) L 34, * This lage amount shows how you money can gow if it is compounded ove a long peiod of time. In geneal, if we deposit a pincipal P in an account paying an annual inteest ate fo t yeas, then the futue value of the account is given by the fomula money you will have in the futue money you have now A P(1 ) t. In the example that we just calculated, P = 2,000, = 0.10, and t = 30. It is impotant to undestand that this fomula fo calculating compound inteest only woks fo the case when is the annual inteest ate and t is time being measued in yeas. Do not bothe to lean this fomula because in just a moment we will give you a simila compounding fomula that woks fo moe geneal situations. 7 Solving fo Unknowns in the Compound Inteest Fomula All banks and most othe financial institutions compound inteest moe fequently than once a yea. Fo example, many banks send savings account customes a monthly statement showing the balance in thei accounts. So fa in ou discussion of compounding, we have used a yealy inteest ate. If compounding takes place moe fequently, then the inteest ate must be adjusted accodingly. Fo example, a yealy inteest ate of % = 0. *To ensue geate accuacy, we often show calculations with eight decimal places. If you calculations do not agee with ous, it may be due to the diffeence in the way we ae ounding ou calculations.
5 408 CHAPTER 9 y Consume Mathematics % = 0. coesponds to a monthly inteest ate of = 0.01 = 1%. If the inteest is being % compounded quately, the quately inteest ate would then be 4 = 0. 4 = 0.03 = 3%. In ode to handle situations such as these, we will modify the fomula A = P(1 + ) t slightly. THE COMPOUND INTEREST FORMULA Assume that an account with pincipal P is paying an annual inteest ate and compounding is being done m times pe yea. If the money emains in the account fo n time peiods, then the futue value, A, of the account is given by the fomula A = P a1 + Notice that in this fomula, we have eplaced by, which is the annual ate divided by the numbe of compounding peiods pe yea, and t by n, which is the numbe of compounding peiods. m n b. m You can use the compound inteest fomula fo computing compound inteest to compae investments. Quiz Youself Saah deposits $1,000 in a CD paying 6% annual inteest fo 2 yeas. What is the futue value of he account if the inteest is compounded quately? 8 EXAMPLE 5 Undestanding How No Payments Until... Woks You have seen a home fitness cente on sale fo $3,500 and what eally makes the deal attactive is that thee is no money down and no payments due fo 6 months. Realize that although you do not have to make any payments, the deale is not loaning you the money fo 6 months fo nothing. You have boowed $3,500 and, in 6 months, you payments will be based upon that fact. Assuming that you deale is chaging an annual inteest ate of %, compounded monthly, what inteest will accumulate on you puchase ove the next 6 months? SOLUTION: To detemine the inteest that has accumulated, we will find the futue value of you loan (assuming that you make no payments) and subtact $3,500 fom that. We will use the fomula fo calculating futue value with P = 3,500, = 0., m =, and n = 6. Theefoe, monthly inteest ate n m A P 1 3, ,500(1.01) 6 3, So the accumulated inteest is $3, $3,500 = $ Now ty Execises 19 to 26. ] 8 0. numbe of months HIGHLIGHT Between the Numbes It Doesn t Hut to Ask In Example 5, you might ask youself if you would be bette off boowing the $3,500 fom anothe souce that has a lowe inteest ate and paying fo the fitness cente outight. If you have the money, sometimes a deale might give you a bette pice if you offe to pay fo an item with cash. The tick, of couse, is to be able to put money aside so that when you want to make a deal, you ae not at the mecy of someone else s money.
6 9.2 y Inteest 409 Doing Financial Calculations with a Calculato* When doing financial computations, often technology can speed up you wok. We will use a calculato to epoduce the solution to Example 6. On my calculato, if we pess the 2nd Finance keys, Sceen 1 comes up. The lettes TVM stand fo Time Value of Money. Then by choosing option 1, we get Sceen 2. Now we can ente the values 18 fo N, the numbe of yeas; 4.8 fo I%, HIGHLIGHT the annual inteest ate; 60,000 fo FV, the futue value; and 4 fo C/Y, the numbe of compounding peiods pe yea. Next we position the cuso ove PV (pesent value) and pess the keys Alpha Solve. The amount fo pesent value means that we must deposit $25, now to have the desied $60,000 in 18 yeas (Sceen 3). Sceen 1 Sceen 2 Sceen 3 KEY POINT We use the log function to solve fo n in the fomula A = P A1 + m B n. Example 6 illustates a diffeent way to use the compound inteest fomula. EXAMPLE 6 Finding the Pesent Value fo a College Tuition Account Upon the bith of a child, a paent wants to make a deposit into a taxfee account to use late fo the child s college education. Assume that the account has an annual inteest ate of 4.8% and that the compounding is done quately. How much must the paent deposit now so that the child will have $60,000 at age 18? SOLUTION: We will use the compound inteest fomula A = PA1 + mb n. Because we know A = 60,000, = 0.048, n = 72, and m = 4, we can find the pesent value by solving the equation 60,000 = Pa b = P fo P. Theefoe, A deposit slightly ove $25,400 now will guaantee $60,000 fo college in 18 yeas. Now ty Execises 33 and 34. ] P = 60,000 (1.0) 72 = 60, L 25, Although $60,000 may seem like a lot of money, ealize that inflation, the incease in the pice of goods and sevices, will also cause the cost of a college education to incease. We will conside the effects of inflation in the execises. So fa we have used the fomula A = PA1 + mb n to find A and P. Sometimes we want to find o n. To do this, we need to intoduce some new techniques. If you want to solve fo n in the fomula A = PA1 + mb n, you need to be able to solve an equation of the fom a x = b, whee a and b ae fixed numbes. A popety of logaithmic functions enables you to solve such equations. Many calculatos have a key labeled eithe log o log x, which stands fo the common logaithmic function. Pessing this key *Fo this example, I am using a TI83 calculato, but many othe calculatos have simila featues fo doing financial calculations. On the TI83 plus and TI84, pess the key and then choose option 1 to get sceen 1. APPS
7 410 CHAPTER 9 y Consume Mathematics eveses the opeation of aising 10 to a powe. Fo example, suppose that you compute 10 5 = 100,000 on you calculato. If you next pess the log key, the display will show 5. If you ente 1,000, which is 10 aised to the thid powe, and pess the log key, the display will show 3. Pactice finding the log of powes of 10 such as 100 and 1,000,000. If you ente 23 and then pess the log key, the display will show The intepetation of this esult is that = 23.* The log function has an impotant popety that will help us solve equations of the fom a x = b. EXPONENT PROPERTY OF THE LOG FUNCTION log y x = x log y To undestand this popety, you should use you calculato to veify the following: log 4 5 = 5 log 4 log 6 3 = 3 log 6 Example 7 illustates how to use the exponent popety to solve equations. EXAMPLE 7 Solve 3 x = 20. Solving an Equation Using the Exponent Popety of the Log Function SOLUTION: We illustate the steps equied to solve this equation. Quiz Youself 9 Step 1 Take the log of both sides of the equation. log 3 x = log 20 Step 2 Use the exponent popety of the log function. x log 3 = log 20 Step 3 Divide both sides by log 3. log 20 x = log 3 Step 4 Use a calculato to evaluate the ight side of the equation (you calculato may give a slightly diffeent answe). x = Solve 6 x = 15. Now ty Execises 35 to 42. ] 9 In Example 8, we use the exponent popety of the log function to find the time it takes an investment to gow to a cetain amount. EXAMPLE 8 Saving fo Equipment fo a Business Maa wants to buy lighting equipment fom he cousin to stat a dance studio. He will sell his equipment fo $2,800. She pesently has $2,500 and found an investment that will pay he 9% annual inteest, compounded monthly. In how many months will Maa be able to pay he cousin fo the equipment? SOLUTION: We know that the futue value that Maa must pay he cousin is A = 2,800. She pesently has $2,500 and the monthly inteest ate is m = 0.09 = We must solve the compound inteest fomula A = PA1 + mb n fo n, which epesents the numbe of months of the compounding. Substituting fo A, P, and, we get the equation 2,800 = 2,500a b n. m *We will not discuss what it means to aise 10 to a powe such as
8 9.2 y Inteest 411 We solve this equation by the following steps: Quiz Youself 10 Do Example 8 again, but now assume that the inteest ate is 6%. Solving fo n, we get the equation 1. = (1.0075) n Divide both sides of the equation by 2,500 and simplify. log(1.) = log(1.0075) n Take the log of both sides. log(1.) = n log(1.0075) n = log (1.) log (1.0075) L Use the exponent popety of the log function. This means that Maa will have the money she needs by the end of the 16th month. ] 10 The last situation that we will conside is how to solve the compound inteest equation A = PA1 + mb n fo. To do this, we have to be able to solve an equation of the fom x a = b, whee a and b ae fixed numbes. We show how to solve such an equation in Example 9. EXAMPLE 9 Negotiating a Basketball Contact Kobe is negotiating a new basketball contact with the Lakes and expects to etie afte playing one moe yea. In ode to educe his cuent taxes, his agent has ageed to defe a bonus of $1.4 million to be paid as $1.68 million in 2 yeas. If the Lakes invest the $1.4 million now, what ate of investment would they need to have $1.68 million to pay Kobe in 2 yeas? Assume that you want to find an annual inteest ate that is compounded monthly. SOLUTION: To solve this compound inteest poblem, we again use the fomula A = PA1 + mb n. We know that A = 1.68, P = 1.4, m =, and n = 24. Substituting fo A, P, m, and n, we get the equation 1.68 = 1.4a1 + b 24. Dividing both sides of the equation by 1.4 gives us 1.2 = A1 +. We can get id of 1 the exponent 24 if we aise both sides of the equation to the powe. This gives us the equation (1.2) 1/24 = aa /24 b b = 1 +. Subtacting 1 fom both sides of the equation, we get * B 24 = (1.2)1/241 = = Now, multiplying this equation by, we find the annual inteest ate,, to be ( ) L Thus, the Lakes need to find an investment that pays an annual inteest ate of about 9.15% compounded monthly. Now ty Execises 43 to 46. ] Some Good Advice Be caeful to distinguish between the situations in Examples 8 and 9. In Example 8, we used the log function to solve an equation of the fom a x = b. In Example 9, we solved an equation of the fom x a 1 = b by aising both sides of the equation to the powe. 24 a *In algeba, (a x ) y = a xy. That is why AA1 + B 24 B 1/24 = A1 + B (24)(1/24) = A1 + B 1 = 1 +.
9 4 CHAPTER 9 y Consume Mathematics Execises 9.2 Looking Back* These execises follow the geneal outline of the topics pesented in this section and will give you a good oveview of the mateial that you have just studied. 1. How did we find the pesent value in Example 3? 2. Why did we divide the yealy inteest ate of 0. by in Example 5? 3. What popety of the log function did we use to solve the equation 3 x = 20 in Example 7? 4. What was ou ecommendation in the Between the Numbes Highlight following Example 5? Shapening You Skills In Execises 5 8, use the simple inteest fomula I = Pt and elementay algeba to find the missing quantities in the table below. I P t 5. $1,000 8% 3 yeas 6. $196 7% 2 yeas 7. $700 $3,500 4 yeas 8. $1,920 $8,000 6% In Execises 9 14, use the futue value fomula A = P(1 + t) and elementay algeba to find the missing quantities in the table below. A P t 9. $2,500 8% 3 yeas 10. $1,600 4% 5 yeas 11. $1,770 6% 3 yeas. $2,332 3% 2 yeas 13. $1,400 $1,250 2 yeas 14. $966 $840 5% In Execises 15 18, you ae given an annual inteest ate and the compounding peiod. Find the inteest ate pe compounding peiod %; monthly 16. 8%; quately 17. %; daily %; daily In Execises 19 26, you ae given the pincipal, the annual inteest ate, and the compounding peiod. Use the fomula fo computing futue value using compound inteest to detemine the value of the account at the end of the specified time peiod. 19. $5,000, 5%, yealy; 5 yeas 20. $7,500, 7%, yealy; 6 yeas 21. $4,000, 8%, quately; 2 yeas 22. $8,000, 4%, quately; 3 yeas 23. $20,000, 8%, monthly; 2 yeas 24. $10,000, 6%, monthly; 5 yeas 25. $4,000, 10%, daily; 2 yeas 26. $6,000, 4%, daily; 3 yeas Savings institutions often state two ates in thei advetising. One is the nominal yield, which you can think of as an annual simple inteest ate. The othe is called the effective annual yield, which is the actual inteest ate that the account eans due to the compounding. If $1,000 is in an account that pays a nominal yield of 9% and if the compounding is done monthly, then afte 1 yea, the account would contain $1,093.80, which coesponds to a simple inteest ate of 9.38%. We would say that this account has an effective annual yield of 9.38%. In Execises 27 30, find the effective annual yield fo each account. 27. nominal yield, 7.5%; compounded monthly 28. nominal yield, 10%; compounded twice a yea 29. nominal yield, 6%; compounded quately 30. nominal yield, 8%; compounded daily In Execises 31 and 32, you ae given an annual inteest ate and the compounding peiod fo two investments. Decide which is the bette investment % compounded yealy; 4.95% compounded quately % compounded monthly; 4.70% compounded daily In Execises 33 and 34, Ann and Tom want to establish a fund fo thei gandson s college education. What lump sum must they deposit in each account in ode to have $30,000 in the fund at the end of 15 yeas? 33. Saving fo college. 6% annual inteest ate, compounded quately 34. Saving fo college. 7.5% annual inteest ate, compounded monthly In Execises 35 42, solve each equation x = x = 37. (1.05) x = (1.15) x = x 3 = x 2 = x 4 = x 4 = 25 In Execises 43 46, use the compound inteest fomula A = P(1 + ) t and the given infomation to solve fo eithe t o. (We ae assuming that n = 1.) 43. A = $2,500, P = $2,000, t = A = $400, P = $20, t = A = $1,500, P = $1,000, = 4% 46. A = $2,500, P = $1,000, = 6% *Befoe doing these execises, you may find it useful to eview the note How to Succeed at Mathematics on page xix. We will assume thee ae 365 days in a yea.
10 9.2 y Execises 413 Applying What You ve Leaned 47. Buying an entetainment system. You have puchased a home entetainment system fo $3,600 and have ageed to pay off the system in 36 monthly payments of $136 each. a. What will be the total sum of you payments? b. What will be the total amount of inteest that you have paid? 48. Buying a ca. You have puchased a used ca fo $6,000 and have ageed to pay off the ca in 24 monthly payments of $325 each. a. What will be the total sum of you payments? b. What will be the total amount of inteest that you have paid? Often, though govenmentsuppoted pogams, students may obtain bagain inteest ates such as 6% o 8% to attend college. Fequently, payments ae not due and inteest does not accumulate until you stop attending college. In Execises 49 and 50, calculate the amount of inteest due 1 month afte you must begin payments. 49. Boowing fo college. You have boowed $10,000 at an annual inteest ate of 8%. 50. Boowing fo college. You have boowed $15,000 at an annual inteest ate of 6%. In Execises 51 54, we will assume that the lende is using simple inteest to compute the inteest on the loan. 51. Boowing fo a tip. You plan to take a tip to the Gand Canyon in 2 yeas. You want to buy a cetificate of deposit fo $1,200 that you will cash in fo you tip. What annual inteest ate must you obtain on the cetificate if you need $1,500 fo you tip? 52. Paying inteest on late taxes. Jonathan wants to defe payment of his $4,500 tax bill fo 4 months. If he must pay an annual inteest ate of 15% fo doing this, what will his total payment be? 53. Boowing fom a pawn shop. Sanjay has boowed $400 on his fathe s watch fom the Main Steet Pawn Shop. He has ageed to pay off the loan with $425 one month late. What is the annual inteest ate that he is being chaged? 54. Boowing fom a bail bondsman. If a peson accused of a cime does not have sufficient esouces, he may have a bail bondsman post bail to be eleased until a tial is held. Assume that a bondsman chages a $50 fee plus 8% of the amount of the bail. If a bondsman posts $20,000 fo a tial that takes place in 2 months, what is the inteest ate being chaged by the bondsman? (Teat the $50 fee plus the 8% as inteest on a $20,000 loan fo two months.) The computations fo dealing with inflation ae the same as fo detemining futue value. If an item sells fo $100 today and thee is an annual inflation ate of 4% fo 10 yeas, the item would then cost 100(1.04) 10 = $ The Bueau of Labo Statistics maintains an index called the consume pice index (CPI), which is a measue of inflation. The accompanying table shows the CPI fo seveal ecent yeas. The CPI of fo 2007 means that the pice of cetain basic items such as clothing, food, enegy, automobiles, etc. that would have cost $100 in 1982 to 1984, which ae the base yeas fo the index, would now cost $ Yea CPI Pecent Incease In Execises 55 58, you ae given a yea and the pice of an item. Use the pecent incease in the CPI as the ate of inflation fo the next 10 yeas to calculate the pice of that item 10 yeas late. 55. Inflation. 2004, fastfood meal, $ Inflation. 2006, automobile, $17, Inflation. 2007, gallon of gasoline, $ Inflation. 2005, athletic shoes, $ Inflation. Fom 1992 to 1995, Albania expeienced a yealy inflation ate of 226%. Detemine the pice of the fastfood meal in Execise 55 afte 5 yeas at a 226% inflation ate. 60. Inflation. The inflation ate in Hungay duing the mid1990s was about 28%. Detemine the pice of the athletic shoes in Execise 58 afte 10 yeas at a 28% inflation ate. 61. Compaing investments. Jocelyn puchased 100 shaes of Jet Blue stock fo $23.75 pe shae. Eight months late she sold the stock at $24.50 pe shae. a. What annual ate, calculated using simple inteest, did she ean on this tansaction? b. What annual ate would she have to ean in a savings account compounded monthly to ean the same money on he investment? 62. Compaing investments. Dominick puchased a bond fo $2,400 to peseve a wildlife sanctuay and 10 months late he sold it fo $2,580. a. What annual ate, calculated using simple inteest, did he ean on this tansaction? b. What annual ate would he have to ean in a savings account compounded monthly to ean the same money on his investment? 63. Investment eanings. Emily puchased a bond valued at $20,000 fo highway constuction fo $9,420. If the bond pays 7.5% annual inteest compounded monthly, how long must she hold it until it eaches its full face value?
11 414 CHAPTER 9 y Consume Mathematics 64. Investment eanings. Lucas puchased a bond with a face value of $10,000 fo $4,200 to build a new spots stadium. If the bond pays 6.5% annual inteest compounded monthly, how long must he hold it until it eaches its full face value? Communicating Mathematics 65. What fomula do we use to compute simple inteest? 66. What is the diffeence between simple inteest and compound inteest? 67. What is the meaning of each vaiable in the compound inteest fomula A = PA1 + m B n? 68. Explain the elationship between the fomulas A = P(1 + ) t and A = PA1 + m B n. 69. Unde what cicumstances will A = P(1 + ) t and A = PA1 + m B n give you the same answes to a compound inteest poblem? 70. Explain the diffeence in the techniques that you have to use to solve a poblem like Example 8 vesus a poblem like Example 9. Using Technology to Investigate Mathematics 71. Get a tutoial fom you instucto that explains in moe detail how to use a calculato to solve finance poblems. Use you calculato to epoduce some of the examples in this section. You instucto also has Excel speadsheets available fo doing financial computations; use them to epoduce some of the computations in this section.* 72. Thee ae many good inteactive financial calculatos available on the Intenet. Find seveal and use them to veify some of the computations that we did in this section. Fo Exta Cedit Some banks advetise that money in thei accounts is compounded continuously. To get an undestanding of what this means, apply the compound inteest fomula using a vey lage numbe of compounding peiods pe yea. In Execises 73 and 74, divide the yea into 100,000 compounding peiods pe yea. Apply the compound inteest fomula fo finding futue value to appoximate what the effective annual yield would be if the compounding wee done continuously fo the stated nominal yield. 73. nominal yield, 10% 74. nominal yield, % If the pincipal P is invested in an account that pays an annual inteest ate of % and the compounding is done continuously, then the futue value, A, that will be in the account afte t yeas is given by the fomula A = Pe t. The numbe e is appoximately Use the fomula fo continuous compounding to find the effective annual yield if the compounding in Execise 73 is done continuously. 76. Use the fomula fo continuous compounding to find the effective annual yield if the compounding in Execise 74 is done continuously.
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