9.2 Interest Objectives 1. Understand the simple interest formula. 2. Use the compound interest formula to find future value. 3. Solve the compound interest formula for different unknowns, such as the present value, length, and interest rate of a loan. The most powerful force in the universe is. . . . How would you finish this quote? The world-renowned physicist Albert Einstein said, . . . compound interest. Are you surprised that of all the forces that he might pick, Einstein chose this one? In this section, we will explain how interest can either work for you—or against you. As you will see, used properly, it can help you build a fortune; used improperly, it can lead you to financial ruin. If you want to accumulate enough money to buy a newer car or go on a vacation, you could deposit money in a bank account. The bank will use your money to make loans to other customers and pay you interest for using your funds. However, if you borrow money from the bank, say to take a college course, then you will pay interest to the bank. In essence, interest is the money that one person (a borrower) pays to another (a lender) to use the lender’s money. Savers earn interest; borrowers pay interest. We will discuss simple and compound interest in this section, and discuss the cost of consumer loans in Section 9.3. KEY POINT Simple interest is a straightforward way to compute interest. Simple Interest* The amount you deposit in a bank account is called the principal. The bank specifies an interest rate for that account as a percentage of your deposit. The rate is usually expressed as an annual rate. For example, a bank may offer an account that has an annual interest rate of 5%. To find the interest that you will earn in such an account, you also need to know how long the deposit will remain in the account. The time is usually stated in years. There is a simple formula that relates principal, interest earned, interest rate, and time. In words, interest earned = principal * interest rate * time. When we compute interest this way, it is called simple interest. F O R M U L A F O R C O M P U T I N G S I M P L E I N T E R E S T We calculate simple interest 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. *If you want some practice with basic algebra, see Appendix A. Copyright © 2010 Pearson Education, Inc. 9.2 y Interest EXAMPLE 1 405 Calculating Simple Interest If you deposit $500 in a bank account paying 6% annual interest, how much interest will the deposit earn in 4 years if the bank computes the interest using simple interest? SOLUTION: In this example: P is the principal, which is $500 r is the annual interest rate, which is 6% (written as 0.06) t is the time, which is 4 (years) Thus, the interest earned is I = Prt = 500 * 0.06 * 4 = 120. In 4 years, this account earns $120 in interest. Now try Exercises 5 to 8. ] KEY POINT Future value equals principal plus interest. To find the amount that will be in your account at some time in the future, called the future value (or sometimes called the future amount) we add the principal and the interest earned. We will represent future value by A, so we can say A = principal + interest = P + I. If we replace I by Prt, we get the formula A = P + Prt = P(1 + rt). C O M P U T I N G F U T U R E V A L U E U S I N G S I M P L E I N T E R E S T To find the future value of an account that pays simple interest, use the formula A = P (1 + rt ), where A is the future value, P is the principal, r is the annual interest rate, and t is the time in years. EXAMPLE 2 Computing Future Value Using Simple Interest Assume that you deposit $1,000 in a bank account paying 3% annual interest and leave the money there for 6 years. Use the simple interest formula to compute the future value of this account. SOLUTION: We see that P = 1,000, r = 0.03, and t = 6. Therefore, P r t A 1,000(1 (0.03)(6)) 1,000(1 0.18) 1,000(1.18) 1,180. Thus, your bank account will have $1,180 at the end of 6 years. ] In contrast to future value, the principal that you have to invest in an account now to have a specified amount in the account in the future is called the present value of the account. Notice that the formula for computing future value has four unknowns. If we want, we can use this formula for finding the present value of an account provided we know the future value, interest rate, and time. EXAMPLE 3 Finding the Present Value of an Account Assume that you plan to save $2,500 to take a white-water rafting trip in Costa Rica in 2 years. Your bank offers a certificate of deposit (CD) that pays 4% annual interest computed using simple interest. How much must you put in this CD now to have the necessary money in 2 years? Copyright © 2010 Pearson Education, Inc. 406 CHAPTER 9 y Consumer Mathematics SOLUTION: We can use the formula A = P(1 + rt). We know that A = 2,500, r = 4% = 0.04, and t = 2. Therefore, 2,500 = P(1 + (0.04)(2)). We can rewrite this equation as 2,500 = P(1.08). Quiz Yourself 5 Redo Example 3, but now assume that you want to save $2,400 in 4 years and the CD has an annual interest rate of 5%. Dividing both sides of the equation by 1.08, we get P= 2,500 L 2314.814815. 1.08 We will round this up to $2,314.82 to guarantee that if you put this amount in the CD now, in 2 years you will have the $2,500 you need for your white-water rafting trip.* Now try Exercises 9 to 14. ] 5 › Some Good Advice In Example 3, we used the earlier formula for computing future value to find the present value rather than stating a new formula to solve this specific problem. You will find it easier to learn a few formulas well and use them, together with simple algebra, to solve new problems rather than trying to memorize separate formulas for every type of problem. Compound Interest KEY POINT Compounding pays interest on previously earned interest. It seems fair that if money in a bank account has earned interest, the bank should compute the interest due, add it to the principal, and then pay interest on this new, larger amount. This is in fact the way most bank accounts work. Interest that is paid on principal plus previously earned interest is called compound interest. If the interest is added yearly, we say that the interest is compounded annually. If the interest is added every three months, we say the interest is compounded quarterly. Interest also can be compounded monthly and daily. EXAMPLE 4 Calculating Compound Interest the Long Way Assume that you want to replace your sailboat with a larger one in 3 years. To save for a down payment for this purchase, you deposit $2,000 for 3 years in a bank account that pays 10% annual interest,† compounded annually. How much will be in the account at the end of 3 years? S O LU T I O N : We will perform the compound interest calculations one year at a time in the following table. In compounding the interest, we will use the future value from the previous year as the new principal at the beginning of the year. Notice that the quantity (1 + rt) = (1 + 0.10 * 1) = (1.10) remains the same throughout the computations. Quiz Yourself 6 Continue Example 4 to calculate the amount in your account at the end of the fourth year. Year Principal (Beginning of Year) P Future Value (End of Year) P(1 + rt) = 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 future amount, we will always round up to the next cent. †An interest rate of 10% would be extraordinarily high. However, we will often choose rates in examples and exercises to keep the computations simple. Copyright © 2010 Pearson Education, Inc. 9.2 y Interest 407 PROBLEM SOLVING Verify Your Answer You should always check answers to see whether they are reasonable. In Example 4, if we had used simple interest to find the future value, we would have obtained A = 2,000 (1 + (0.10)(3)) = 2,000(1.30) = 2,600. The interest we found in Example 4 is a little larger because as the interest is added to the principal each year, the bank is paying interest on an increasingly larger principal. If we were to continue the process that we used in Example 4 for a longer period of time, say for 30 years, it would be very tedious. In Figure 9.2 we look at the same computations in a different way, keeping in mind that the amount in the account at the end of each year is 1.10 times the amount in the account at the beginning of the year. 0 You deposit $2,000 at the beginning of year 1. 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)3 1 2 3 End of year 1 End of year 2 End of year 3 FIGURE 9.2 10% interest being compounded annually. If we were to continue the pattern shown in Figure 9.2 to compute the future value of the account at the end of 30 years, we would see that A = 2,000(1.10)30 L 2,000(17.44940227) L 34,898.80. * This large amount shows how your money can grow if it is compounded over a long period of time. In general, if we deposit a principal P in an account paying an annual interest rate r for t years, then the future value of the account is given by the formula Quiz Yourself 7 Calculate the future value of an account containing $3,000 for which the annual interest rate is 4% compounded annually for 10 years. KEY POINT Knowing the principal, the periodic interest rate, and the number of compounding periods, it is easy to determine future value. money you will have in the future money you have now A P(1 r)t. In the example that we just calculated, P = 2,000, r = 0.10, and t = 30. It is important to understand that this formula for calculating compound interest only works for the case when r is the annual interest rate and t is time being measured in years. Do not bother to learn this formula because in just a moment we will give you a similar compounding formula that works for more general situations. 7 Solving for Unknowns in the Compound Interest Formula All banks and most other financial institutions compound interest more frequently than once a year. For example, many banks send savings account customers a monthly statement showing the balance in their accounts. So far in our discussion of compounding, we have used a yearly interest rate. If compounding takes place more frequently, then the interest rate must be adjusted accordingly. For example, a yearly interest rate of 12% = 0.12 *To ensure greater accuracy, we often show calculations with eight decimal places. If your calculations do not agree with ours, it may be due to the difference in the way we are rounding our calculations. Copyright © 2010 Pearson Education, Inc. 408 CHAPTER 9 y Consumer Mathematics 0.12 corresponds to a monthly interest rate of 12% 12 = 12 = 0.01 = 1%. If the interest is being 0.12 compounded quarterly, the quarterly interest rate would then be 12% 4 = 4 = 0.03 = 3%. In order to handle situations such as these, we will modify the formula A = P(1 + r)t slightly. T H E C O M P O U N D I N T E R E S T F O R M U L A Assume that an account with principal P is paying an annual interest rate r and compounding is being done m times per year. If the money remains in the account for n time periods, then the future value, A, of the account is given by the formula A = P a1 + r n b. m r Notice that in this formula, we have replaced r by m , which is the annual rate divided by the number of compounding periods per year, and t by n, which is the number of compounding periods. You can use the compound interest formula for computing compound interest to compare investments. Understanding How “No Payments Until . . .” Works EXAMPLE 5 You have seen a home fitness center on sale for $3,500 and what really makes the deal attractive is that there is no money down and no payments due for 6 months. Realize that although you do not have to make any payments, the dealer is not loaning you the money for 6 months for nothing. You have borrowed $3,500 and, in 6 months, your payments will be based upon that fact. Assuming that your dealer is charging an annual interest rate of 12%, compounded monthly, what interest will accumulate on your purchase over the next 6 months? SOLUTION: To determine the interest that has accumulated, we will find the future value of Quiz Yourself 8 Sarah deposits $1,000 in a CD paying 6% annual interest for 2 years. What is the future value of her account if the interest is compounded quarterly? your “loan” (assuming that you make no payments) and subtract $3,500 from that. We will use the formula for calculating future value with P = 3,500, r = 0.12, m = 12, and n = 6. Therefore, monthly interest rate AP 1 r m n 3,500 1 number of months 0.12 12 3,500(1.01) 3,715.33. 6 6 So the accumulated interest is $3,715.33 - $3,500 = $215.33. Now try Exercises 19 to 26. ] 8 ¶ ¶ ¶ HIGHLIGHT Between the Numbers—It Doesn’t Hurt to Ask In Example 5, you might ask yourself if you would be better off borrowing the $3,500 from another source that has a lower interest rate and paying for the fitness center outright. If you have the money, sometimes a dealer might give you a better price if you offer to pay for an item with cash. The trick, of course, is to be able to put money aside so that when you want to make a deal, you are not at the mercy of someone else’s money. Copyright © 2010 Pearson Education, Inc. 9.2 y Interest 409 HIGHLIGHT ¶ ¶ ¶ Doing Financial Calculations with a Calculator* When doing financial computations, often technology can speed up your work. We will use a calculator to reproduce the solution to Example 6. On my calculator, if we press the 2nd Finance keys, Screen 1 comes up. The letters TVM stand for “Time Value of Money.” Then by choosing option 1, we get Screen 2. Now we can enter the values 18 for N, the number of years; 4.8 for I%, the annual interest rate; 60,000 for FV, the future value; and 4 for C/Y, the number of compounding periods per year. Next we position the cursor over PV (present value) and press the keys Alpha Solve . The amount -25418.75939 for present value means that we must deposit $25,418.76 now to have the desired $60,000 in 18 years (Screen 3). Screen 2 Screen 1 Screen 3 Example 6 illustrates a different way to use the compound interest formula. EXAMPLE 6 Finding the Present Value for a College Tuition Account Upon the birth of a child, a parent wants to make a deposit into a tax-free account to use later for the child’s college education. Assume that the account has an annual interest rate of 4.8% and that the compounding is done quarterly. How much must the parent deposit now so that the child will have $60,000 at age 18? r n S O L U T I O N : We will use the compound interest formula A = P A 1 + m B . Because we know A = 60,000, r = 0.048, n = 72, and m = 4, we can find the present value by solving the equation 60,000 = Pa1 + 0.048 72 b = P11 + 0.012272 4 for P. Therefore, P= 60,000 60,000 = L 25,418.76. 72 2.360461386 (1.012) A deposit slightly over $25,400 now will guarantee $60,000 for college in 18 years. Now try Exercises 33 and 34. ] Although $60,000 may seem like a lot of money, realize that inflation, the increase in the price of goods and services, will also cause the cost of a college education to increase. We will consider the effects of inflation in the exercises. r n So far we have used the formula A = P A 1 + m B to find A and P. Sometimes we want to find r or n. To do this, we need to introduce some new techniques. KEY POINT n We use the log function to solve for n in the formula n A = P A 1 + mr B . If you want to solve for n in the formula A = P A 1 + mr B , you need to be able to solve an equation of the form ax = b, where a and b are fixed numbers. A property of logarithmic functions enables you to solve such equations. Many calculators have a key labeled either “log” or “log x,” which stands for the common logarithmic function. Pressing this key *For this example, I am using a TI-83 calculator, but many other calculators have similar features for doing financial calculations. On the TI-83 plus and TI-84, press the APPS key and then choose option 1 to get screen 1. Copyright © 2010 Pearson Education, Inc. 410 CHAPTER 9 y Consumer Mathematics reverses the operation of raising 10 to a power. For example, suppose that you compute 105 = 100,000 on your calculator. If you next press the log key, the display will show 5. If you enter 1,000, which is 10 raised to the third power, and press the log key, the display will show 3. Practice finding the log of powers of 10 such as 100 and 1,000,000. If you enter 23 and then press the log key, the display will show 1.361727836. The interpretation of this result is that 101.361727836 = 23.* The log function has an important property that will help us solve equations of the form ax = b. EXPONENT PROPERTY OF THE LOG FUNCTION log y x = x log y To understand this property, you should use your calculator to verify the following: log 45 = 5 log 4 log 63 = 3 log 6 Example 7 illustrates how to use the exponent property to solve equations. EXAMPLE 7 Solving an Equation Using the Exponent Property of the Log Function Solve 3x = 20. SOLUTION: We illustrate the steps required to solve this equation. Quiz Yourself Solve 6x = 15. log 3x = log 20 Step 1 Take the log of both sides of the equation. Step 2 Use the exponent property of the log function. Step 3 Divide both sides by log 3. x= Step 4 Use a calculator to evaluate the right side of the equation (your calculator may give a slightly different answer). x = 2.726833028 9 Now try Exercises 35 to 42. ] x log 3 = log 20 log 20 log 3 9 In Example 8, we use the exponent property of the log function to find the time it takes an investment to grow to a certain amount. EXAMPLE 8 Saving for Equipment for a Business Mara wants to buy lighting equipment from her cousin to start a dance studio. He will sell his equipment for $2,800. She presently has $2,500 and found an investment that will pay her 9% annual interest, compounded monthly. In how many months will Mara be able to pay her cousin for the equipment? SOLUTION: We know that the future value that Mara must pay her cousin is A = 2,800. She presently has $2,500 and the monthly interest rate is mr = 0.09 12 = 0.0075 . We must solve n the compound interest formula A = P A 1 + mr B for n, which represents the number of months of the compounding. Substituting for A, P, and mr , we get the equation 2,800 = 2,500 a1 + 0.09 n b . 12 *We will not discuss what it means to raise 10 to a power such as 1.361727836. Copyright © 2010 Pearson Education, Inc. 9.2 y Interest 411 We solve this equation by the following steps: 1.12 = (1.0075)n log(1.12) = log(1.0075)n log(1.12) = n log(1.0075) Quiz Yourself 10 Do Example 8 again, but now assume that the interest rate is 6%. Divide both sides of the equation by 2,500 and simplify. Take the log of both sides. Use the exponent property of the log function. Solving for n, we get the equation n= log (1.12) L 15.17. log (1.0075) This means that Mara will have the money she needs by the end of the 16th month. ] 10 The last situation that we will consider is how to solve the compound interest equation n A = P A 1 + mr B for r. To do this, we have to be able to solve an equation of the form x a = b, where a and b are fixed numbers. We show how to solve such an equation in Example 9. EXAMPLE 9 Negotiating a Basketball Contract Kobe is negotiating a new basketball contract with the Lakers and expects to retire after playing one more year. In order to reduce his current taxes, his agent has agreed to defer a bonus of $1.4 million to be paid as $1.68 million in 2 years. If the Lakers invest the $1.4 million now, what rate of investment would they need to have $1.68 million to pay Kobe in 2 years? Assume that you want to find an annual interest rate that is compounded monthly. S O L U T I O N : nTo solve this compound interest problem, we again use the formula A = P A 1 + mr B . We know that A = 1.68, P = 1.4, m = 12, and n = 24. Substituting for A, P, m, and n, we get the equation 1.68 = 1.4 a1 + r 24 b . 12 24 Dividing both sides of the equation by 1.4 gives us 1.2 = A 1 + 12r B . We can get rid of 1 the exponent 24 if we raise both sides of the equation to the 24 power. This gives us the equation (1.2)1/24 = a a1 + r 24 1/24 r * b b =1+ . 12 12 Subtracting 1 from both sides of the equation, we get r = (1.2)1/24 - 1 = 1.00762566 - 1 = 0.00762566. 12 Now, multiplying this equation by 12, we find the annual interest rate, r, to be 12(0.00762566) L 0.0915. Thus, the Lakers need to find an investment that pays an annual interest rate of about 9.15% compounded monthly. Now try Exercises 43 to 46. ] › Some Good Advice Be careful to distinguish between the situations in Examples 8 and 9. In Example 8, we used the log function to solve an equation of the form ax = b. In Example 9, we solved an equation of the form x a = b by raising both sides of the equation to the 1a power. *In algebra, (a x ) y = a xy. That is why A A 1 + 12r B 24 B 1/24 = A 1 + 12r B (24)(1/24) = A 1 + 12r B 1 = 1 + 12r . Copyright © 2010 Pearson Education, Inc. 412 CHAPTER 9 y Consumer Mathematics 9.2 Exercises Looking Back* 22. $8,000, 4%, quarterly; 3 years These exercises follow the general outline of the topics presented in this section and will give you a good overview of the material that you have just studied. 23. $20,000, 8%, monthly; 2 years 1. How did we find the present value in Example 3? 24. $10,000, 6%, monthly; 5 years 25. $4,000, 10%, daily; 2 years 26. $6,000, 4%, daily; 3 years 2. Why did we divide the yearly interest rate of 0.12 by 12 in Example 5? 3. What property of the log function did we use to solve the equation 3x = 20 in Example 7? 4. What was our recommendation in the “Between the Numbers” Highlight following Example 5? Sharpening Your Skills In Exercises 5–8, use the simple interest formula I = Prt and elementary algebra to find the missing quantities in the table below. Savings institutions often state two rates in their advertising. One is the nominal yield, which you can think of as an annual simple interest rate. The other is called the effective annual yield, which is the actual interest rate that the account earns 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 after 1 year, the account would contain $1,093.80, which corresponds to a simple interest rate of 9.38%. We would say that this account has an effective annual yield of 9.38%. In Exercises 27–30, find the effective annual yield for each account. 27. nominal yield, 7.5%; compounded monthly 28. nominal yield, 10%; compounded twice a year I 5. P r t $1,000 8% 3 years 7% 2 years 6. $196 7. $700 $3,500 8. $1,920 $8,000 29. nominal yield, 6%; compounded quarterly 30. nominal yield, 8%; compounded daily In Exercises 31 and 32, you are given an annual interest rate and the compounding period for two investments. Decide which is the better investment. 4 years 6% In Exercises 9–14, use the future value formula A = P(1 + rt) and elementary algebra to find the missing quantities in the table below. A P r t 9. $2,500 8% 3 years 10. $1,600 4% 5 years 11. $1,770 6% 3 years 12. $2,332 3% 2 years 13. $1,400 $1,250 14. $966 $840 33. Saving for college. 6% annual interest rate, compounded quarterly 34. Saving for college. 7.5% annual interest rate, compounded monthly 5% In Exercises 35–42, solve each equation. In Exercises 15–18, you are given an annual interest rate and the compounding period. Find the interest rate per compounding period. 16. 8%; quarterly 17. 12%; daily† 18. 10%; daily 32. 4.75% compounded monthly; 4.70% compounded daily In Exercises 33 and 34, Ann and Tom want to establish a fund for their grandson’s college education. What lump sum must they deposit in each account in order to have $30,000 in the fund at the end of 15 years? 2 years 15. 18%; monthly 31. 5% compounded yearly; 4.95% compounded quarterly 35. 3x = 10 36. 2x = 12 37. (1.05)x = 2 38. (1.15)x = 3 = 10 40. x2 = 10 41. x4 = 10 42. x4 = 25 39. In Exercises 19–26, you are given the principal, the annual interest rate, and the compounding period. Use the formula for computing future value using compound interest to determine the value of the account at the end of the specified time period. x3 In Exercises 43–46, use the compound interest formula A = P(1 + r)t and the given information to solve for either t or r. (We are assuming that n = 1.) 43. A = $2,500, P = $2,000, t = 5 19. $5,000, 5%, yearly; 5 years 44. A = $400, P = $20, t = 35 20. $7,500, 7%, yearly; 6 years 45. A = $1,500, P = $1,000, r = 4% 21. $4,000, 8%, quarterly; 2 years 46. A = $2,500, P = $1,000, r = 6% *Before doing these exercises, you may find it useful to review the note How to Succeed at Mathematics on page xix. †We will assume there are 365 days in a year. Copyright © 2010 Pearson Education, Inc. 9.2 y Exercises Applying What You’ve Learned 47. Buying an entertainment system. You have purchased a home entertainment system for $3,600 and have agreed to pay off the system in 36 monthly payments of $136 each. a. What will be the total sum of your payments? b. What will be the total amount of interest that you have paid? 48. Buying a car. You have purchased a used car for $6,000 and have agreed to pay off the car in 24 monthly payments of $325 each. a. What will be the total sum of your payments? b. What will be the total amount of interest that you have paid? Often, through government-supported programs, students may obtain “bargain” interest rates such as 6% or 8% to attend college. Frequently, payments are not due and interest does not accumulate until you stop attending college. In Exercises 49 and 50, calculate the amount of interest due 1 month after you must begin payments. 49. Borrowing for college. You have borrowed $10,000 at an annual interest rate of 8%. 413 trial is held. Assume that a bondsman charges a $50 fee plus 8% of the amount of the bail. If a bondsman posts $20,000 for a trial that takes place in 2 months, what is the interest rate being charged by the bondsman? (Treat the $50 fee plus the 8% as interest on a $20,000 loan for two months.) The computations for dealing with inflation are the same as for determining future value. If an item sells for $100 today and there is an annual inflation rate of 4% for 10 years, the item would then cost 100(1.04)10 = $148.02. The Bureau of Labor Statistics maintains an index called the consumer price index (CPI), which is a measure of inflation. The accompanying table shows the CPI for several recent years. The CPI of 207.3 for 2007 means that the price of certain basic items such as clothing, food, energy, automobiles, etc. that would have cost $100 in 1982 to 1984, which are the base years for the index, would now cost $207.30. Year 2002 2003 2004 2005 2006 2007 CPI 179.9 184.0 188.9 195.3 201.6 207.3 2.3 2.7 3.4 3.2 2.8 Percent Increase 50. Borrowing for college. You have borrowed $15,000 at an annual interest rate of 6%. In Exercises 55–58, you are given a year and the price of an item. Use the percent increase in the CPI as the rate of inflation for the next 10 years to calculate the price of that item 10 years later. In Exercises 51–54, we will assume that the lender is using simple interest to compute the interest on the loan. 56. Inflation. 2006, automobile, $17,650 51. Borrowing for a trip. You plan to take a trip to the Grand Canyon in 2 years. You want to buy a certificate of deposit for $1,200 that you will cash in for your trip. What annual interest rate must you obtain on the certificate if you need $1,500 for your trip? 55. Inflation. 2004, fast-food meal, $4.65 57. Inflation. 2007, gallon of gasoline, $3.25 58. Inflation. 2005, athletic shoes, $96 59. Inflation. From 1992 to 1995, Albania experienced a yearly inflation rate of 226%. Determine the price of the fast-food meal in Exercise 55 after 5 years at a 226% inflation rate. 60. Inflation. The inflation rate in Hungary during the mid-1990s was about 28%. Determine the price of the athletic shoes in Exercise 58 after 10 years at a 28% inflation rate. 61. Comparing investments. Jocelyn purchased 100 shares of Jet Blue stock for $23.75 per share. Eight months later she sold the stock at $24.50 per share. a. What annual rate, calculated using simple interest, did she earn on this transaction? 52. Paying interest on late taxes. Jonathan wants to defer payment of his $4,500 tax bill for 4 months. If he must pay an annual interest rate of 15% for doing this, what will his total payment be? b. What annual rate would she have to earn in a savings account compounded monthly to earn the same money on her investment? 62. Comparing investments. Dominick purchased a bond for $2,400 to preserve a wildlife sanctuary and 10 months later he sold it for $2,580. a. What annual rate, calculated using simple interest, did he earn on this transaction? 53. Borrowing from a pawn shop. Sanjay has borrowed $400 on his father’s watch from the Main Street Pawn Shop. He has agreed to pay off the loan with $425 one month later. What is the annual interest rate that he is being charged? b. What annual rate would he have to earn in a savings account compounded monthly to earn the same money on his investment? 54. Borrowing from a bail bondsman. If a person accused of a crime does not have sufficient resources, he may have a bail bondsman post bail to be released until a 63. Investment earnings. Emily purchased a bond valued at $20,000 for highway construction for $9,420. If the bond pays 7.5% annual interest compounded monthly, how long must she hold it until it reaches its full face value? Copyright © 2010 Pearson Education, Inc. 414 CHAPTER 9 y Consumer Mathematics 64. Investment earnings. Lucas purchased a bond with a face value of $10,000 for $4,200 to build a new sports stadium. If the bond pays 6.5% annual interest compounded monthly, how long must he hold it until it reaches its full face value? Communicating Mathematics 65. What formula do we use to compute simple interest? 66. What is the difference between simple interest and compound interest? 67. What is the meaning of each variable in the compound interest n formula A = P A 1 + mr B ? 68. Explain the relationship between the formulas A = P(1 + r)t n and A = P A 1 + mr B . 69. Under what circumstances will A = P(1 + r)t and A = n P A 1 + mr B give you the same answers to a compound interest problem? 70. Explain the difference in the techniques that you have to use to solve a problem like Example 8 versus a problem like Example 9. 72. There are many good interactive financial calculators available on the Internet. Find several and use them to verify some of the computations that we did in this section. For Extra Credit Some banks advertise that money in their accounts is compounded continuously. To get an understanding of what this means, apply the compound interest formula using a very large number of compounding periods per year. In Exercises 73 and 74, divide the year into 100,000 compounding periods per year. Apply the compound interest formula for finding future value to approximate what the effective annual yield would be if the compounding were done continuously for the stated nominal yield. 73. nominal yield, 10% 74. nominal yield, 12% If the principal P is invested in an account that pays an annual interest rate of r% and the compounding is done continuously, then the future value, A, that will be in the account after t years is given by the formula A = Pert. Using Technology to Investigate Mathematics The number e is approximately 2.718281828. 71. Get a tutorial from your instructor that explains in more detail how to use a calculator to solve finance problems. Use your calculator to reproduce some of the examples in this section. Your instructor also has Excel spreadsheets available for doing financial computations; use them to reproduce some of the computations in this section.* 75. Use the formula for continuous compounding to find the effective annual yield if the compounding in Exercise 73 is done continuously. 76. Use the formula for continuous compounding to find the effective annual yield if the compounding in Exercise 74 is done continuously. Copyright © 2010 Pearson Education, Inc.
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