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Chapter 3: Problem 4

Find the interest earned on a \(\$ 50,000\) deposited for six years at 4\(\frac{1}{8} \%\) interest, compounded continuously.

Short Answer

Expert verified
The interest earned on a sum of \$50,000 after six years at 4\frac{1}{8} \% interest, compounded continuously, will be \$ 13,775.58.

Step by step solution

01

Convert Interest Rate to Decimal

The presented interest rate is given as \(4\frac{1}{8} \% \). To convert this into decimal form, which is preferred for calculations, simply divide the fraction by 100. So, \(4\frac{1}{8} \% = 0.04125\).
02

Apply Continuous Compound Interest Formula

The formula for continuous compound interest is \(A = P e^{rt}\). Substitute the given values into the formula: principal (P) = \$50,000, interest rate (r) = 0.04125, and time (t) = 6 years. This amounts to \(A = 50000 * e^{0.04125*6}\).
03

Calculate the Total Amount

By performing the calculation of the earlier step, the total amount A that will be present after 6 years will be \$63,775.58.
04

Find the Interest Earned

The interest earned will be the total amount at the end minus the initial deposit, thus Interest Earned = \(A - P = \$63,775.58 - \$50,000 = \$13,775.58\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Compounding
Continuous compounding is a concept where interest is calculated and added to the principal balance of an investment continuously, or at every possible instant, rather than on a discrete, periodic schedule. This produces a situation where the interest itself earns interest at the highest possible frequency. Mathematically, when interest is compounded continuously, the formula used is \(A = P e^{rt}\), where:
  • \(A\) is the amount of money accumulated after time t, including interest.
  • \(P\) is the principal amount (the initial amount of money).
  • \(r\) is the annual interest rate (in decimal form).
  • \(t\) is the time the money is invested for, also called the investment period.
The exponential function \(e\) plays a crucial role in continuous compounding, allowing interest to be compounded at infinitesimal intervals over time.
Interest Rate Conversion
Interest rate conversion is important for calculations involving different compounding methods or when adjusting the percentage rate to make it usable in formulas. For instance, when provided with an interest rate like \(4\frac{1}{8} \%\), it must be converted into decimal form. This is achieved by dividing the percentage by 100, so \(4\frac{1}{8} \%\) becomes \(0.04125\).
  • First, transform fractions into a single decimal. In this example, \(\frac{1}{8}\) is \(0.125\).
  • Add this to the whole number part of the percentage (4 + 0.125 = 4.125).
  • Finally, divide by 100 to convert into decimal form (4.125/100 = 0.04125).
Accurate interest rate conversion ensures valid inputs for further financial calculations, such as those in compound interest formulas.
Financial Algebra
Financial algebra blends mathematical equations and financial concepts to solve problems related to finance. This could involve calculating interest, loan payments, or investments. The compound interest formula \(A = P e^{rt}\) is a prime example representing how algebraic expressions can model real-world financial calculations.Financial algebra involves steps like:
  • Identifying the variables: principal amount (\(P\)), interest rate (\(r\)), time (\(t\)), and total amount (\(A\)).
  • Substituting values into formulas to find unknowns.
  • Simplifying calculations using arithmetic and algebraic manipulations.
Understanding financial algebra is essential to navigate a range of fiscal situations, from personal finance to corporate investments.
Exponential Functions
Exponential functions describe processes that grow proportionally to their current value, a common occurrence in finance with continuous compounding interest. The function takes the form \(y = a e^{bx}\) where \(e\) is the base of natural logarithms, approximately equal to 2.71828. In the context of continuous compounding, it models the growth of an investment over time.
  • The \(e^{rt}\) part of the formula \(A = P e^{rt}\) shows how the investment grows exponentially over time, dictated by the rate of interest (\(r\)) and the period (\(t\)).
  • The function suggests that the longer the time or higher the rate, the more rapid and substantial the growth.
Recognizing the nature of exponential functions helps in predicting not only financial growth but also in understanding compounding's vast potential impact on investments.

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Most popular questions from this chapter

Ridgewood Savings Bank charges a \(\$ 27\) per check overdraft protection fee. On July \(8,\) Nancy had \(\$ 1,400\) in her account. Over the next four days, the following checks arrived for payment at her bank: July \(9, \$ 1,380.15,\) July \(10, \$ 670\) and \(\$ 95.67 ;\) July \(11, \$ 130 ;\) and July \(12, \$ 87.60 .\) How much will she pay in overdraft protection fees? How much will she owe the bank after July 12\(?\)

On May 29, Rocky had an opening balance of x dollars in an account that pays 3% interest, compounded daily. He deposits y dollars. Express his ending balance on May 30 algebraically.

Joby had \(\$ 421.56\) in her checking account when she deposited \(g\) twenty- dollar bills and \(k\) quarters. Write an expression that represents the amount of money in her account after the deposit.

Pierre deposits \(\$ 9,000\) in a certificate of deposit that pays 8\(\%\) interest, compounded semiannually. How much interest does the account earn in the first six months? What is the balance after six months?

Caroline is opening a CD to save for college. She is considering a 3 -year \(\mathrm{CD}\) or a 3\(\frac{1}{2}\) -year CD since she starts college around that time. She needs to be able to have the money to make tuition payments on time, and she does not want to have to withdraw money early from the CD and face a penalty. She has \(\$ 19,400\) to deposit. a. How much interest would she earn at 4.2\(\%\) compounded monthly for three years? Round to the nearest cent. b. How much interest would she earn at 4.2\(\%\) compounded monthly for 3\(\frac{1}{2}\) years? Round to the nearest cent. c. Caroline decides on a college after opening the 3\(\frac{1}{2}\) -year \(\mathrm{CD},\) and the college needs the first tuition payment a month before the \(\mathrm{CD}\) matures. Caroline must withdraw money from the CD early, after 3 years and 5 months. She faces two penalties. First, the interest rate for the last five months of the CD was lowered to 2\(\%\) . Additionally, there was a \(\$ 250\) penalty. Find the interest on the last five months of the CD. Round to the nearest cent. d. Find the total interest on the 3\(\frac{1}{2}\) year CD after 3 years and 5 months. e. The interest is reduced by subtracting the \(\$ 250\) penalty. What does the account earn for the 3 years and 5 months? f. Find the balance on the CD after she withdraws \(\$ 12,000\) after 3 years and five months. g. The final month of the CD receives 2\(\%\) interest. What is the final month's interest? Round to the nearest. What is the final month's interest? Round to the nearest cent. h. What is the total interest for the 3\(\frac{1}{2}\) year \(\mathrm{CD} ?\) i. Would Caroline have been better off with the 3 -year CD? Explain?
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