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Chapter 8: Problem 18

In Exercises 15-20, the principal \(P\) is borrowed and the loan's future value, \(A\), at time \(t\) is given. Determine the loan's simple interest rate, \(r\), to the nearest tenth of a percent. \(P=\$ 10,000, A-\$ 14,060, t-2\) years

Short Answer

Expert verified
The simple interest rate of the loan is 20.3%.

Step by step solution

01

Rearrange the Formula to Solve for r

We start by rearranging the formula \(A = P(1 + rt)\) to solve for r. We can do this by first subtracting P from both sides of the equation. This will give us \(A - P = Prt\). Then, dividing both sides of the equation by \(Pt\), the formula will be readjusted for r as follows: \(r = (A - P) / (Pt)\).
02

Substitute the Given Values

Now that we have the formula for r, we are going to substitute the given values into that formula. So \(P = \$10000\), \(A = \$14060\), and \(t = 2\) years. Substituting these into the formula gives \(r = (14060 - 10000) / (10000 * 2)\).
03

Solve for r and Convert to Percentage

Doing the calculations from the previous step now gives \(r = 0.203\). Since we need to find the interest rate as a percentage, we now need to multiply the resulting decimal by 100. Doing so, we get \(r = 0.203 * 100 = 20.3 %\).

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Key Concepts

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

Principal Amount
The principal amount, often denoted as \(P\), is the initial sum of money borrowed or invested. In the context of loans, it is the initial amount the borrower receives from the lender. This number is critical as it forms the basis for the calculation of simple interest.
  • It represents the core amount without any accrual of interest or other fees.
  • In our example, the principal amount is \( \$10,000 \), which is the original loan amount the borrower took.
  • The principal is essential for determining future values and calculating the interest.
Understanding the principal is the first step in comprehending how loans and investments generate earnings or costs.
Future Value
The future value, represented as \(A\), of a loan or investment is the total amount that will be paid back or accumulated at the end of the investment period. It includes both the principal and the interest accrued.
  • In simple interest scenarios, future value is computed over a specific timeframe and is a sum of the principal and the interest.
  • For our loan example, the future value amount is \( \$14,060 \), which is what the borrower will owe after 2 years.
  • Future value calculations help borrowers and investors understand the total cost or benefit of their financial decisions.
Assigning importance to future value allows one to plan financials efficiently, ensuring all costs or returns can be anticipated.
Interest Rate Calculation
Calculating the interest rate is essential to understand how much cost the borrower will incur or how much profit an investor will make. In simple interest loans, the interest rate \(r\) can be found using the formula \( r = \frac{A - P}{Pt} \).
  • This formula derives from the equation \( A = P(1 + rt) \), where \(A\) is the future value, \(P\) is the principal, \(t\) is the time, and \(r\) is the rate.
  • For example, substituting our known values \(P = 10,000\), \(A = 14,060\), and \(t = 2\), we compute \( r = \frac{14,060 - 10,000}{10,000 \times 2} = 0.203 \), or \(20.3\%\).
  • The rate is crucial for analyzing the cost-effectiveness of loans.
Proper calculation of interest rates enables informed decisions regarding borrowing and investing.
Loan Terms
Loan terms refer to the specific conditions and duration under which a loan is granted. These components encompass the excess amount paid beyond the principal, ensuring lenders are compensated for their risk.
  • Timeframe: It specifies the length of time over which the loan is to be repaid or the interest is to be calculated. In our scenario, the loan term is 2 years.
  • Interest: Indicates whether the interest is simple or compound, affecting how the future value is calculated.
  • Clear understanding of terms prevents misunderstandings and ensures that borrowers know their obligations.
Grasping loan terms is essential for both lenders and borrowers to ensure efficient and clear transactions, usually laid out in a formal agreement.

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Most popular questions from this chapter

In Exercises 11-18, a. Determine the periodic deposit. Round up to the nearest dollar. b. How much of the financial goal comes from deposits and how much comes from interest? \(\$ ?\) at the end of each month \(4.5 \%\) compounded monthly 10 years \(\$ 200,000\)

In Exercises 1-10, use $$ P M T=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} . $$ Round answers to the nearest dollar. Suppose that you decide to borrow \(\$ 40,000\) for a new car. You can select one of the following loans, each requiring regular monthly payments: Installment Loan A: three-year loan at \(6.1 \%\) Installment Loan B: five-year loan at \(7.2 \%\). a. Find the monthly payments and the total interest for Loan A. b. Find the monthly payments and the total interest for Loan B. c. Compare the monthly payments and the total interest for the two loans.

What is a mortgage?

Exercises 19 and 20 refer to the stock tables for Goodyear (the tire company) and Dow Chemical given below. In each exercise, use the stock table to answer the following questions. Where necessary, round dollar amounts to the nearest cent. a. What were the high and low prices for a share for the past 52 weeks? b. If you owned 700 shares of this stock last year, what dividend did you receive? c. What is the annual return for the dividends alone? How does this compare to a bank offering a \(3 \%\) interest rate? d. How many shares of this company's stock were traded yesterday? e. What were the high and low prices for a share yesterday? f. What was the price at which a share last traded when the stock exchange closed yesterday? g. What was the change in price for a share of stock from the market close two days ago to yesterday's market close? h. Compute the company's annual earnings per share using Annual earnings per share $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { 52-Week High } & \text { 52-Week Low } & \text { Stock } & \text { SYM } & \text { Div } & \text { Yld \% } & \text { PE } & \text { Vol 100s } & \text { Hi } & \text { Lo } & \text { Close } & \text { Net Chg } \\ \hline 73.25 & 45.44 & \text { Goodyear } & \text { GT } & 1.20 & 2.2 & 17 & 5915 & 56.38 & 54.38 & 55.50 & +1.25 \\ \hline \end{array} $$

Each group should have a newspaper with current stock quotations. Choose nine stocks that group members think would make good investments. Imagine that you invest \(\$ 10,000\) in each of these nine investments. Check the value of your stock each day over the next five weeks and then sell the nine stocks after five weeks. What is the group's profit or loss over the five-week period? Compare this figure with the profit or loss of other groups in your class for this activity.
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