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Chapter 8: Problem 42

You deposit \(\$ 6000\) in an account that pays \(5.25 \%\) interest compounded semiannually. After 10 years, the interest rate is increased to \(5.4 \%\) compounded quarterly. What will be the value of the account after 18 years?

Short Answer

Expert verified
The final amount present in the account after 18 years can be calculated using two stages of compound interest. First, by the interest compounded semiannually for the first 10 years and then for 8 years with the increased rate of interest compounded quarterly.

Step by step solution

01

Calculate Initial Compound Interest

Calculate the compound interest generated over the first 10 years using the formula described. Here, \(P = \$6000\), \(r = 0.0525\) (which is \(5.25 \%\) converted to decimal), \(n = 2\) (since interest is compounded semiannually), and \(t = 10\) years.\nTherefore, the value of the account after 10 years, \(A1\) is given by the formula: \( A1 = P(1 + r/n)^{nt} = \$6000(1 + 0.0525/2)^{2*10}\)
02

Calculate Compound Interest After Rate Change

Now calculate the compound interest generated over the next 8 years. Details have changed, the principal amount will be the amount in the account after 10 years, \(P = A1\) (amount calculated in step 1). Interest rate changes to \(r=0.054\), it will now compound quarterly hence, \(n=4\) and time is the remaining 8 years, \(t = 8\) years.\nSo, the value of the account after 18 years, \(A2\) would be given by: \( A2 = A1(1 + r/n)^{nt}\)
03

Calculate Final Amount in Account

Calculate the final amount present in the account after 18 years by evaluating \(A2\).

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Most popular questions from this chapter

In Exercises 25-30, round to the nearest dollar. Suppose that you earned a bachelor's degree and now you're teaching high school. The school district offers teachers the opportunity to take a year off to earn a master's degree. To achieve this goal, you deposit \(\$ 2000\) at the end of each year in an annuity that pays \(7.5 \%\) compounded annually. a. How much will you have saved at the end of 5 years? b. Find the interest.

Suppose your credit card has a balance of \(\$ 3600\) and an annual interest rate of \(16.5 \%\). You decide to pay off the balance over two years. If there are no further purchases charged to the card, a. How much must you pay each month? b. How much total interest will you pay?

Exercises 19 and 20 refer to the stock tables for Goodyear (the tire d. How many shares of this company's stock were traded company) and Dow Chemical given below. In each exercise, use yesterday? the stock table to answer the following questions. Where necessary, e. What were the high and low prices for a share yesterday? round dollar amounts to the nearest cent. f. What was the price at which a share last traded when the stock a. What were the high and low prices for a share for the past exchange closed yesterday? b. If you owned 700 shares of this stock last year, what dividend g. What was the change in price for a share of stock from the did you receive? h. Compute the company's annual earnings per share using c. What is the annual return for the dividends alone? How does Annual earnings per share this compare to a bank offering a \(3 \%\) interest rate? $$ =\frac{\text { Yesterday's closing price per share }}{P E \text { ratio }} . $$ $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { 52-Week High } & \text { 52-Week Low } & \text { Stock } & \text { SYM } & \text { Div } & \text { Yld \% } & \text { PE } & \text { Vol 100s } & \text { Hi } & \text { Lo } & \text { Close } & \text { Net Chg } \\ \hline 56.75 & 37.95 & \begin{array}{c} \text { Dow } \\ \text { Chemical } \end{array} & \text { DOW } & 1.34 & 3.0 & 12 & 23997 & 44.75 & 44.35 & 44.69 & +0.16 \\ \hline \end{array} $$

In terms of paying less in interest, which is more economical for a $$ 150,000\( mortgage: a 30 -year fixed-rate at \)8 \%\( or a 20 -year fixed-rate at \)7.5 \%$ ? How much is saved in interest?

In Exercises 1-10, a. Find the value of each annuity. Round to the nearest dollar b. Find the interest.$$ \begin{array}{|l|l|l|} \hline \begin{array}{l} \$ 1000 \text { at the end of } \\ \text { every three months } \end{array} & \begin{array}{l} 6.25 \% \text { compounded } \\ \text { quarterly } \end{array} & 6 \text { years } \\ \hline \end{array} $$
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