Chapter 4

Find the accumulated amount \(A\) if the principal \(P\) is invested at the interest rate of \(r /\) year for \(t\) yr. $$ P=\$ 200,000, r=8 \%, t=4, \text { compounded daily } $$

Karen has been depositing $$\$ 150$$ at the end of each month in a tax-free retirement account since she was \(25 .\) Matt, who is the same age as Karen, started depositing $$\$ 250$$ at the end of each month in a taxfree retirement account when he was 35 . Assuming that both accounts have been and will be earning interest at the rate of \(5 \% /\) year compounded monthly, who will end up with the larger retirement account at the age of 65 ?

Find the effective rate corresponding to the given nominal rate. \(10 \% /\) year compounded semiannually

The Flemings secured a bank loan of $$\$ 288,000$$ to help finance the purchase of a house. The bank charges interest at a rate of \(9 \% /\) year on the unpaid balance. and interest computations are made at the end of each month. The Flemings have agreed to repay the loan in equal monthly installments over \(25 \mathrm{yr}\). What should be the size of each repayment if the loan is to be amortized at the end of the term?

Luis has $$\$ 150,000$$ in his retirement account at his present company. Because he is assuming a position with another company, Luis is planning to "roll over" his assets to a new account. Luis also plans to put $$\$3000$$/quarter into the new account until his retirement 20 yr from now. If the account earns interest at the rate of \(8 \% /\) year compounded quarterly, how much will Luis have in his account at the time of his retirement? Hint: Use the compound interest formula and the annuity formula.

Find the effective rate corresponding to the given nominal rate. \(9 \% /\) year compounded quarterly

The price of a new car is $$\$ 16,000$$. Assume that an individual makes a down payment of \(25 \%\) toward the purchase of the car and secures financing for the balance at the rate of \(10 \% /\) year compounded monthly. a. What monthly payment will she be required to make if the car is financed over a period of 36 mo? Over a period of \(48 \mathrm{mo}\) ? b. What will the interest charges be if she elects the 36 -mo plan? The 48-mo plan?

The Betzes have leased an auto for 2 yr at $$\$ 450 /$$ month. If money is worth \(9 \% /\) year compounded monthly, what is the equivalent cash payment (present value) of this annuity?

Find the effective rate corresponding to the given nominal rate. \(8 \% /\) year compounded monthly

A group of private investors purchased a condominium complex for $$\$ 2\( \)million. They made an initial down payment of \(10 \%\) and obtained financing for the balance. If the loan is to be amortized over \(15 \mathrm{yr}\) at an interest rate of \(12 \%\) /year compounded quarterly, find the required quarterly payment.

Lupé made a down payment of $$\$ 4000$$ toward the purchase of a new car. To pay the balance of the purchase price, she has secured a loan from her bank at the rate of \(12 \% /\) year compounded monthly. Under the terms of her finance agreement, she is required to make payments of $$\$ 420$$ / \mathrm{month}\( for \)36 \mathrm{mo}$. What is the cash price of the car?

Find the effective rate corresponding to the given nominal rate. \(8 \% /\) year compounded daily

The Taylors have purchased a $$\$ 270,000$$ house. They made an initial down payment of $$\$ 30,000$$ and secured a mortgage with interest charged at the rate of \(8 \%\) /year on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized over \(30 \mathrm{yr}\), what monthly payment will the Taylors be required to make? What is their equity (disregarding appreciation) after 5 yr? After 10 yr? After 20 yr?

Mike's Sporting Goods sells elliptical trainers under two payment plans: cash or installment. Under the installment plan, the customer pays $$\$ 22 /$$ month over 3 yr with interest charged on the balance at a rate of \(18 \% /\) year compounded monthly. Find the cash price for an elliptical trainer if it is equivalent to the price paid by a customer using the installment plan.

A state lottery commission pays the winner of the "Million Dollar" lottery 20 installments of $$\$ 50,000$$ /year. The commission makes the first payment of $$\$ 50,000$$ immediately and the other \(n=19\) payments at the end of each of the next 19 yr. Determine how much money the commission should have in the bank initially to guarantee the payments, assuming that the balance on deposit with the bank earns interest at the rate of \(8 \% /\) year compounded yearly. Hint: Find the present value of an annuity.

Find the present value of $$\$ 40,000$$ due in 4 yr at the given rate of interest. \(8 \% /\) year compounded quarterly

A city has $$\$ 2.5$$ million worth of school bonds that are due in \(20 \mathrm{yr}\) and has established a sinking fund to retire this debt. If the fund earns interest at the rate of \(7 \%\) /year compounded annually, what amount must be deposited annually in this fund?

The Johnsons have accumulated a nest egg of $$\$ 40,000$$ that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $$\$ 2400 /$$ month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $$\$ 3000$$. If local mortgage rates are \(7.5 \% /\) year compounded monthly for a conventional 30 -yr mortgage, what is the price range of houses that they should consider?

Find the present value of $$\$ 40,000$$ due in 4 yr at the given rate of interest. \(7 \% /\) year compounded monthly

Carl is the beneficiary of a $$\$ 20,000$$ trust fund set up for him by his grandparents. Under the terms of the trust, he is to receive the money over a 5 -yr period in equal installments at the end of each year. If the fund earns interest at the rate of \(9 \%\) /year compounded annually, what amount will he receive each year?

Find the present value of $$\$ 40,000$$ due in 4 yr at the given rate of interest. \(9 \%\) year compounded daily

Lowell Corporation wishes to establish a sinking fund to retire a $$\$ 200,000$$ debt that is due in 10 yr. If the investment will earn interest at the rate of \(9 \% /\) year compounded quarterly, find the amount of the quarterly deposit that must be made in order to accumulate the required sum.

Find the accumulated amount after 4 yr if $$\$ 5000$$ is invested at \(8 \% /\) year compounded continuously.

The management of Gibraltar Brokerage Services anticipates a capital expenditure of $$\$ 20,000$$ in 3 yr for the purchase of new computers and has decided to set up a sinking fund to finance this purchase. If the fund earns interest at the rate of \(10 \% /\) year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.

Lauren plans to deposit $$\$ 5000$$ into a bank account at the beginning of next month and $$\$ 200 / \mathrm{month}$$ into the same account at the end of that month and at the end of each subsequent month for the next 5 yr. If her bank pays interest at the rate of \(6 \% /\) year compounded monthly, how much will Lauren have in her account at the end of 5 yr? (Assume she makes no withdrawals during the 5 -yr period.)

Find the accumulated amount after 6 yr if $$\$ 6500$$ is invested at \(7 \% /\) year compounded continuously.

Andrea, a self-employed individual, wishes to accumulate a retirement fund of $$\$ 250,000$$. How much should she deposit each month into her retirement account, which pays interest at the rate of \(8.5 \% /\) year compounded monthly, to reach her goal upon retirement 25 yr from now?

Joe plans to deposit $$\$ 200$$ at the end of each month into a bank account for a period of 2 yr, after which he plans to deposit $$\$ 300$$ at the end of each month into the same account for another 3 yr. If the bank pays interest at the rate of \(6 \%\) /year compounded monthly, how much will Joe have in his account by the end of 5 yr? (Assume no withdrawals are made during the 5-yr period.)

Joe secured a loan of $$\$ 12,0003$$ yr ago from a bank for use toward his college expenses. The bank charged interest at the rate of \(4 \% /\) year compounded monthly on his loan. Now that he has graduated from college, Joe wishes to repay the loan by amortizing it through monthly payments over \(10 \mathrm{yr}\) at the same interest rate. Find the size of the monthly payments he will be required to make.

From age 25 to age 40 , Jessica deposited $$\$ 200$$ at the end of each month into a tax-free retirement account. She made no withdrawals or further contributions until age \(65 .\) Alex made deposits of $$\$ 300$$ into his tax- free retirement account from age 40 to age \(65 .\) If both accounts earned interest at the rate of \(5 \% /\) year compounded monthly, who ends up with a bigger nest egg upon reaching the age of 65 ? Hint: Use both the annuity formula and the compound interest formula.

Use logarithms to solve each problem. How long will it take $$\$ 12,000$$ to grow to $$\$ 15,000$$ if the investment earns interest at the rate of \(8 \% /\) year compounded monthly?

Robin wishes to accumulate a sum of $$\$ 450,000$$ in a retirement account by the time of her retirement 30 yr from now. If she wishes to do this through monthly payments into the account that earn interest at the rate of \(10 \% /\) year compounded monthly, what should be the size of each payment?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The future value of an annuity can be found by adding together all the payments that are paid into the account.

Use logarithms to solve each problem. How long will it take an investment of $$\$ 2000$$ to double if the investment earns interest at the rate of \(9 \% /\) year compounded monthly?

Yumi's grandparents presented her with a gift of $$\$ 20,000$$ when she was 10 yr old to be used for her college education. Over the next \(7 \mathrm{yr}\), until she turned 17 , Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 5.5\%lyear compounded monthly. Upon turning 17 , Yumi now plans to withdraw her funds in equal annual installments over the next 4 yr, starting at age 18 . If the college fund is expected to earn interest at the rate of \(6 \% /\) year, compounded annually, what will be the size of each installment?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the future value of an annuity consisting of \(n\) payments of \(R\) dollars each-paid at the end of each investment period into an account that earns interest at the rate of \(i\) per period - is \(S\) dollars, then $$ R=\frac{i S}{(1+i)^{n}-1} $$

Use logarithms to solve each problem. How long will it take an investment of $$\$ 5000$$ to triple if the investment earns interest at the rate of \(8 \% /\) year compounded daily?

Martin has deposited $$\$ 375$$ in his IRA at the end of each quarter for the past 20 yr. His investment has earned interest at the rate of \(8 \% /\) year compounded quarterly over this period. Now, at age 60 , he is considering retirement. What quarterly payment will he receive over the next 15 yr? (Assume that the money is earning interest at the same rate and that payments are made at the end of each quarter.) If he continues working and makes quarterly payments of the same amount in his IRA until age 65, what quarterly payment will he receive from his fund upon retirement over the following \(10 \mathrm{yr}\) ?

Use logarithms to solve each problem. Find the interest rate needed for an investment of $$\$ 5000$$ to grow to an amount of $$\$ 6000$$ in \(3 \mathrm{yr}\) if interest is compounded continuously.

Darla purchased a new car during a special sales promotion by the manufacturer. She secured a loan from the manufacturer in the amount of $$\$ 16,000$$ at a rate of \(7.9 \% /\) year compounded monthly. Her bank is now charging \(11.5 \% /\) year compounded monthly for new car loans. Assuming that each loan would be amortized by 36 equal monthly installments, determine the amount of interest she would have paid at the end of 3 yr for each loan. How much less will she have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank?

Dan is contemplating trading in his car for a new one. He can afford a monthly payment of at most $$\$ 400 .$$ If the prevailing interest rate is \(7.2 \% /\) year compounded monthly for a 48 -mo loan, what is the most expensive car that Dan can afford, assuming that he will receive $$\$ 8000$$ for the trade-in?

Use logarithms to solve each problem. How long will it take an investment of $$\$ 6000$$ to grow to $$\$ 7000$$ if the investment earns interest at the rate of \(7 \frac{1}{2} \%\) compounded continuously?

Paula is considering the purchase of a new car. She has narrowed her search to two cars that are equally appealing to her. Car A costs $$\$ 28,000$$, and car B costs $$\$ 28,200$$. The manufacturer of car A is offering \(0 \%\) financing for 48 months with zero down, while the manufacturer of car \(B\) is offering a rebate of $$\$ 2000$$ at the time of purchase plus financing at the rate of \(3 \%\) /year compounded monthly over 48 mo with zero down. If Paula has decided to buy the car with the lower net cost to her, which car should she purchase?

Use logarithms to solve each problem. How long will it take an investment of $$\$ 8000$$ to double if the investment earns interest at the rate of \(8 \%\) compounded continuously?

The Sandersons are planning to refinance their home. The outstanding principal on their original loan is $$\$ 100,000$$ and was to be amortized in 240 equal monthly installments at an interest rate of \(10 \% /\) year compounded monthly. The new loan they expect to secure is to be amortized over the same period at an interest rate of 7.8\%/year compounded monthly. How much less can they expect to pay over the life of the loan in interest payments by refinancing the loan at this time?

Mitchell has been given the option of either paying his $$\$ 300$$ bill now or settling it for $$\$ 306$$ after 1 mo ( 30 days). If he chooses to pay after 1 mo, find the simple interest rate at which he would be charged.

Since he was 22 years old, Ben has been depositing $$\$ 200$$ at the end of each month into a taxfree retirement account earning interest at the rate of \(6.5 \%\) /year compounded monthly. Larry, who is the same age as Ben, decided to open a tax-free retirement account 5 yr after Ben opened his. If Larry's account earns interest at the same rate as Ben's, determine how much Larry should deposit each month into his account so that both men will have the same amount of money in their accounts at age 65 .

Jennifer was awarded damages of $$\$ 150,000$$ in a successful lawsuit she brought against her employer 5 yr ago. Interest (simple) on the judgment accrues at the rate of \(12 \% /\) year from the date of filing. If the case were settled today, how much would Jennifer receive in the final judgment?

Two years ago, Paul borrowed $$\$ 10,000$$ from his sister Gerri to start a business. Paul agreed to pay Gerri interest for the loan at the rate of \(6 \% /\) year, compounded continuously. Paul will now begin repaying the amount he owes by amortizing the loan (plus the interest that has accrued over the past 2 yr) through monthly payments over the next 5 yr at an interest rate of \(5 \% /\) year compounded monthly. Find the size of the monthly payments Paul will be required to make.

To help finance the purchase of a new house, the Abdullahs have decided to apply for a shortterm loan (a bridge loan) in the amount of $$\$ 120,000$$ for a term of 3 mo. If the bank charges simple interest at the rate of \(12 \% /\) year, how much will the Abdullahs owe the bank at the end of the term?