Writing to Learn Explain why computing the APY for an

Designed to be solved without a calculator. Find 3.5% of 200.

Designed to be solved without a calculator. Find 2.5% of 150.

What is one-fourth of 7.25%?

What is one-twelfth of 6.5%?

78 is what percent of 120?

28 is what percent of 80?

48 is 32% of what number?

176.4 is 84% of what number?

How much does Jane have at the end of 1 year if she invests $300 at 5% simple interest?

How much does Reggie have at the end of 1 year if he invests $500 at 4.5% simple interest?

In Exercises 1–4, find the amount A accumulated after investing a principal P for t years at an interest rate r compounded annually. P = $1500, r = 7%, t = 6

In Exercises 1–4, find the amount A accumulated after investing a principal P for t years at an interest rate r compounded annually. P = $3200, r = 8%, t = 4

In Exercises 1–4, find the amount A accumulated after investing a principal P for t years at an interest rate r compounded annually. P = $12,000, r = 7.5%, t = 7

In Exercises 1–4, find the amount A accumulated after investing a principal P for t years at an interest rate r compounded annually. P = $15,500, r = 9.5%, t = 12

In Exercises 5–8, find the amount A accumulated after investing a principal P for t years at an interest rate r compounded k times per year. P = $1500, r = 7%, t = 5, k = 4

In Exercises 5–8, find the amount A accumulated after investing a principal P for t years at an interest rate r compounded k times per year. P = $3500, r = 5%, t = 10, k = 4

In Exercises 5–8, find the amount A accumulated after investing a principal P for t years at an interest rate r compounded k times per year. P = $40,500, r = 3.8%, t = 20, k = 12

In Exercises 5–8, find the amount A accumulated after investing a principal P for t years at an interest rate r compounded k times per year. P = $25,300, r = 4.5%, t = 25, k = 12

In Exercises 9–12, find the amount A accumulated after investing a principal P for t years at interest rate r compounded continuously. P = $1250, r = 5.4%, t = 6

In Exercises 9–12, find the amount A accumulated after investing a principal P for t years at interest rate r compounded continuously. P = $3350, r = 6.2%, t = 8

In Exercises 9–12, find the amount A accumulated after investing a principal P for t years at interest rate r compounded continuously. P = $21,000, r = 3.7%, t = 10

In Exercises 9–12, find the amount A accumulated after investing a principal P for t years at interest rate r compounded continuously. P = $8,875, r = 4.4%, t = 25

In Exercises 13–16, find the future value FV accumulated in an annuity after investing periodic payments R for t years at an annual interest rate r, with payments made and interest credited k times per year. R = $500, r = 7%, t = 6, k = 4

In Exercises 13–16, find the future value FV accumulated in an annuity after investing periodic payments R for t years at an annual interest rate r, with payments made and interest credited k times per year. R = $300, r = 6%, t = 12, k = 4

In Exercises 13–16, find the future value FV accumulated in an annuity after investing periodic payments R for t years at an annual interest rate r, with payments made and interest credited k times per year. R = $450, r = 5.25%, t = 10, k = 12

In Exercises 13–16, find the future value FV accumulated in an annuity after investing periodic payments R for t years at an annual interest rate r, with payments made and interest credited k times per year. R = $610, r = 6.5%, t = 25, k = 12

In Exercises 17 and 18, find the present value PV of a loan with an annual interest rate r and periodic payments R for a term of t years, with payments made and interest charged 12 times per year. r = 4.7%, R = $815.37, t = 5

In Exercises 17 and 18, find the present value PV of a loan with an annual interest rate r and periodic payments R for a term of t years, with payments made and interest charged 12 times per year. r = 6.5%, R = $1856.82, t = 30

In Exercises 19 and 20, find the periodic payment R of a loan with present value PV and an annual interest rate r for a term of t years, with payments made and interest charged 12 times per year. PV = $18,000, r = 5.4%, t = 6

In Exercises 19 and 20, find the periodic payment R of a loan with present value PV and an annual interest rate r for a term of t years, with payments made and interest charged 12 times per year. PV = $154,000, r = 7.2%, t = 15

Finding Time If John invests $2300 in a savings account with a 9% interest rate compounded quarterly, how long will it take until John’s account has a balance of $4150?

Finding Time If Joelle invests $8000 into a retirement account with a 9% interest rate compounded monthly, how long will it take until this single payment has grown in her account to $16,000?

Trust Officer Megan is the trust officer for an estate. If she invests $15,000 into an account that carries an interest rate of 8% compounded monthly, how long will it be until the account has a value of $45,000 for Megan’s client?

Chief Financial Officer Willis is the financial officer of a private university with the responsibility for managing an endowment. If he invests $1.5 million at an interest rate of 8% compounded quarterly, how long will it be until the account exceeds $3.75 million?

Finding the Interest Rate What interest rate compounded daily (365 days/year) is required for a $22,000 investment to grow to $36,500 in 5 years?

Finding the Interest Rate What interest rate compounded monthly is required for an $8500 investment to triple in 5 years?

Pension Officer Jack is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $22 million in 6 years. What interest rate compounded annually does he need for this investment?

Bank President The president of a bank has $18 million in his bank’s investment portfolio that he wants to grow to $25 million in 8 years. What interest rate compounded annually does he need for this investment?

Doubling Your Money Determine how much time is required for an investment to double in value if interest is earned at the rate of 5.75% compounded quarterly.

Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 6.25% compounded monthly.

In Exercises 31–34, complete the table about continuous compounding. $12,500 9% ? ?

In Exercises 31–34, complete the table about continuous compounding. $32,500 8% ? ?

In Exercises 31–34, complete the table about continuous compounding. $ 9,500 ? 4 years ?

In Exercises 31–34, complete the table about continuous compounding. $16,800 ? 6 years ?

In Exercises 35–40, complete the table about doubling time of an investment. 4% Quarterly ?

In Exercises 35–40, complete the table about doubling time of an investment. 8% Quarterly ?

In Exercises 35–40, complete the table about doubling time of an investment. 7% Annually ?

In Exercises 35–40, complete the table about doubling time of an investment. 7% Quarterly ?

In Exercises 35–40, complete the table about doubling time of an investment. 7% Monthly ?

In Exercises 35–40, complete the table about doubling time of an investment. 7% Continuously ?

In Exercises 41–44, find the annual percentage yield (APY) for the investment. $3000 at 6% compounded quarterly

In Exercises 41–44, find the annual percentage yield (APY) for the investment. $8000 at 5.75% compounded daily

In Exercises 41–44, find the annual percentage yield (APY) for the investment. P dollars at 6.3% compounded continuously

In Exercises 41–44, find the annual percentage yield (APY) for the investment. P dollars at 4.7% compounded monthly

Comparing Investments Which investment is more attractive, 5% compounded monthly or 5.1% compounded quarterly?

Comparing Investments Which investment is more attractive, \(5 \frac{1}{8} \%\) compounded annually or 5% compounded continuously?

In Exercises 47–50, payments are made and interest is credited at the end of each month. An IRA Account Amy contributes $50 per month into the Lincoln National Bond Fund that earns 7.26% annual interest. What is the value of Amy’s investment after 25 years?

In Exercises 47–50, payments are made and interest is credited at the end of each month. An IRA Account Andrew contributes $50 per month into the Hoffbrau Fund that earns 15.5% annual interest. What is the value of his investment after 20 years?

In Exercises 47–50, payments are made and interest is credited at the end of each month. An Investment Annuity Jolinda contributes to the Celebrity Retirement Fund that earns 12.4% annual interest. What should her monthly payments be if she wants to accumulate $250,000 in 20 years?

In Exercises 47–50, payments are made and interest is credited at the end of each month. An Investment Annuity Diego contributes to a Commercial National money market account that earns 4.5% annual interest. What should his monthly payments be if he wants to accumulate $120,000 in 30 years?

Car Loan Payment What is Kim’s monthly payment for a 4-year $9000 car loan with an APR of 7.95% from Century Bank?

Car Loan Payment What is Ericka’s monthly payment for a 3-year $4500 car loan with an APR of 10.25% from County Savings Bank?

House Mortgage Payment Gendo obtains a 30-year $86,000 house loan with an APR of 8.75% from National City Bank. What is her monthly payment?

House Mortgage Payment Roberta obtains a 25-year $100,000 house loan with an APR of 9.25% from NBD Bank. What is her monthly payment?

Mortgage Payment Planning An $86,000 mortgage for 30 years at 12% APR requires monthly payments of $884.61. Suppose you decided to make monthly payments of $1050.00. (a) When would the mortgage be completely paid? (b) How much do you save with the greater payments compared with the original plan?

Mortgage Payment Planning Suppose you make payments of $884.61 for the $86,000 mortgage in Exercise 53 for 10 years and then make payments of $1050 until the loan is paid. (a) When will the mortgage be completely paid under these circumstances? (b) How much do you save with the greater payments compared with the original plan?

Writing to Learn Explain why computing the APY for an investment does not depend on the actual amount being invested. Give a formula for the APY on a $1 investment at annual rate r compounded k times a year. How do you extend the result to a $1000 investment?

Writing to Learn Give reasons why banks might not announce their APY on a loan they would make to you at a given APR. What is the bank’s APY on a loan that they make at 4.5% APR?

Group Activity Work in groups of three or four. Consider population growth of humans or other animals, bacterial growth, radioactive decay, and compounded interest. Explain how these problem situations are similar and how they are different. Give examples to support your point of view.

Simple Interest Versus Compounding Annually Steve purchases a $1000 certificate of deposit and will earn 6% each year. The interest will be mailed to him, so he will not earn interest on his interest. (a) Writing to Learn Explain why after t years, the total amount of interest he receives from his investment plus the original $1000 is given by \(f(t)=1000(1+0.06 t)\) (b) Steve invests another $1000 at 6% compounded annually. Make a table that compares the value of the two investments for t = 1, 2,..., 10 years.

True or False If $100 is invested at 5% annual interest for 1 year, there is no limit to the final value of the investment if it is compounded sufficiently often. Justify your answer.

True or False The total interest paid on a 15-year mortgage is less than half of the total interest paid on a 30-year mortgage with the same loan amount and APR. Justify your answer.

In Exercises 63–66, you may use a graphing calculator to solve the problem. Multiple Choice What is the total value after 6 years of an initial investment of $2250 that earns 7% interest compounded quarterly? (A) $3376.64 (B) $3412.00 (C) $3424.41 (D) $3472.16 (E) $3472.27

In Exercises 63–66, you may use a graphing calculator to solve the problem. Multiple Choice The annual percentage yield of an account paying 6% compounded monthly is (A) 6.03%. (B) 6.12%. (C) 6.17%. (D) 6.20%. (E) 6.24%.

In Exercises 63–66, you may use a graphing calculator to solve the problem. Multiple Choice Mary Jo deposits $300 each month into her retirement account that pays 4.5% APR (0.375% per month). Use the formula \(F V=R\left((1+i)^{n}-1\right) / i\) to find the value of her annuity after 20 years. (A) $71,625.00 (B) $72,000.00 (C) $72,375.20 (D) $73,453.62 (E) $116,437.31

In Exercises 63–66, you may use a graphing calculator to solve the problem. Multiple Choice To finance their home, Mr. and Mrs. Dass have agreed to a $120,000 mortgage loan at 7.25% APR. Use the formula \(P V=R\left(1-(1+i)^{-n}\right) / i\) to determine their monthly payments if the loan has a term of 15 years. (A) $1095.44 (B) $1145.44 (C) $1195.44 (D) $1245.44 (E) $1295.44

Loan Payoff Use the information about Carlos’s truck loan in Example 9 to make a spreadsheet of the payment schedule. The first few lines of the spreadsheet should look like the following table: To create the spreadsheet successfully, however, you need to use formulas for many of the cells, as shown in boldface type in the following sample: Continue the spreadsheet using copy-and-paste techniques, and determine the amount of the 48th and final payment so that the final balance is $0.00.

Writing to Learn Loan Payoff Which of the following graphs is an accurate graph of the loan balance as a function of time, based on Carlos’s truck loan in Example 9 and Exercise 67? Explain your choice based on increasing or decreasing behavior and other analytical characteristics. Would you expect the graph of loan balance versus time for a 30-year mortgage loan at twice the interest rate to have the same shape or a different shape as the one for the truck loan? Explain.

The function \(f(x)=100 \frac{(1+0.08 / 12)^{x}-1}{0.08 / 12}\) describes the future value of a certain annuity. (a) What is the annual interest rate? (b) How many payments per year are there? (c) What is the amount of each payment?

The function \(f(x)=200 \frac{1-(1+0.08 / 12)^{-x}}{0.08 / 12}\) describes the present value of a certain annuity. (a) What is the annual interest rate? (b) How many payments per year are there? (c) What is the amount of each payment?