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Chapter 3: Problem 3

Sally deposits \(\$ 4,000\) in a certificate of deposit that pays 6\(\frac{3}{4} \%\) simple interest. What is her balance after one year?

Short Answer

Expert verified
Sally's balance after one year will be $\$4,270.

Step by step solution

01

Calculate the interest rate as a decimal

The interest rate is given as a fraction: 6 and 3/4 percent. First, add 6 to 3/4 to get the percentage in a decimal format: 6.75 . Then, divide this by 100 to convert it to a decimal: 0.0675
02

Calculate the interest

Apply the equation I = Prt , where I is the interest, P is the principal (\$4,000), r is the interest rate (0.0675), and t is the time (1 year). This yields the result I = 4000 * 0.0675 * 1 = $270.
03

Calculate the balance after one year

Add the interest to the initial deposit: \$4,000 + \$270 = \$4,270

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Key Concepts

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

Understanding Certificates of Deposit
A Certificate of Deposit (CD) is a financial product commonly offered by banks. It's a type of savings account with a set interest rate and fixed maturity date. When you open a CD, you deposit money for a specified term during which you can鈥檛 access the funds without a penalty. In return, the bank pays you interest on your deposit.
  • Fixed Term: CD terms can range from a few months to several years. You agree to leave your money in the account until the term ends.
  • Interest Rates: CDs often offer a predictable and typically higher interest rate than regular savings accounts, making them an attractive option for those looking to earn more from their savings.
  • Security: Your investment is secure as long as it remains within the limits of FDIC insurance in the United States.
After the specified term, you receive your initial deposit plus the accrued interest. Understanding how interest is calculated, such as simple interest, is crucial when investing in CDs.
Interest Rate Conversion Explained
Converting interest rates into a usable format is an important step in financial calculations. Rates are often presented as percentages, but we need them as decimals for mathematical equations.Let鈥檚 break down the process:
  • Convert fractions to decimals: If an interest rate includes a fraction, convert it into a decimal. For example, 6 3/4% becomes 6.75%.
  • Change percentage to decimal: Divide the percentage by 100. So, 6.75% becomes 0.0675 in decimal form.
  • Use in calculations: The decimal form allows integration into formulas, like calculating simple interest using the formula \(I = Prt\).
This method ensures accuracy in calculations involving interest-bearing instruments like CDs.
Financial Algebra Simplified
Financial algebra involves using mathematical operations and formulas to solve financial problems, helping calculate values like interest or future balances.In the context of simple interest:
  • Formula: Simple interest is calculated with \(I = Prt\), where \(I\) is the interest earned, \(P\) is the principal amount, \(r\) is the interest rate in decimal form, and \(t\) is the time in years.
  • Understanding Units: Each part of the formula requires specific units (money in dollars, time in years) to maintain consistency.
  • Application: Once you've calculated the interest, you can determine the total balance by adding it to the initial principal.
Financial algebra is an invaluable tool in personal finance, enabling you to make informed decisions about various financial products and understand their benefits. It bridges the gap between theoretical percentages and real-world applications, such as understanding how a CD can grow your savings over time.

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Most popular questions from this chapter

Mike deposits \(\$ 5,000\) in a three-year \(\mathrm{CD}\) account that yields 3.5\(\%\) interest, compounded weekly. What is his ending balance at the end of three years?

Mason discovered that when he recorded a deposit of \(\$ 75\) two weeks ago, he mistakenly subtracted it from the running total in his check register. He decided that he would write a new entry after his most recent entry and add \(\$ 75 .\) Will this correct his mistake? Explain.

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?

Olivia cashed a check for \(\$ 113 .\) The teller gave her four twenty-dollar bills, \(x\) ten-dollar bills, and three one-dollar bills. Find the value of \(x .\)

Investigate the difference between compounding annually and simple interest for parts a-j. a. Find the simple interest for a one-year \(\mathrm{CD}\) for \(\$ 5,000\) at a 6\(\%\) b. Find the interest for a one-year CD for \(\$ 5,000\) at an interest rate of \(6 \%,\) compounded annually. c. Compare the results from parts a and b. d. Find the simple interest for a three-year \(\mathrm{CD}\) for \(\$ 5,000\) at an enterest rate of 6\(\% .\) e. Find the interest for a three-year CD for \(\$ 5,000\) at an interest rate f. Compare the results from parts d and e. g. Find the simple interest for a six-year \(\mathrm{CD}\) for \(\$ 5,000\) at an interest rate of 4\(\% .\) h. Find the interest for a six-year CD for \(\$ 5,000\) at an interest rate of 4\(\%\) , componded annually. i. Compare the results from parts \(\mathrm{g}\) and \(\mathrm{h} .\) j. Is interest compounded annually the same as simple interest? Explain.
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