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Chapter 8: Problem 6

\(P=\$ 18,000, r=7.5 \%, t=18\) months

Short Answer

Expert verified
The interest earned on $18,000 over a period of 18 months at an annual interest rate of 7.5% is \$2025.

Step by step solution

01

Convert Percentages to Decimals

Convert the annual interest rate from percentage to decimal form by dividing it by 100. Hence, \(r = 7.5/100 = 0.075\).
02

Convert Time to Years

Convert the time from months to years by dividing it by 12. Therefore, \(t = 18/12 = 1.5\) years.
03

Substituting the Values into the Formula

Substitute the values of \(P, r, t\) into the simple interest formula \(I = P \cdot r \cdot t\). Hence, \(I = 18000 \cdot 0.075 \cdot 1.5\).
04

Calcuate the Interest

Do the multiplication to calculate the interest. \(I = 18000 \cdot 0.075 \cdot 1.5 = \$2025\).

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Key Concepts

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

Percentage Conversion
When working with interest rates in financial mathematics, it's crucial to convert percentages into decimals. This is because most mathematical formulas in finance use decimals for interest rates. For example, an interest rate of 7.5% needs to be converted to its decimal form before use. To convert any percentage to a decimal:
  • Divide the percentage value by 100.
  • For 7.5%, the conversion is done as follows: \( 7.5 \div 100 = 0.075 \).
This simple conversion ensures that the calculations remain accurate and in line with mathematical standards.
Time Conversion
In financial calculations, time is often measured in different units such as months or years. To ensure uniformity and correct calculations, it's essential to convert these units when necessary. Typically, if interest is calculated annually, the time should also be in years. Here's how you can convert time from months to years:
  • Divide the number of months by 12 (since there are 12 months in a year).
  • In our case, 18 months equals \( 18 \div 12 = 1.5 \) years.
Converting time accurately is vital for ensuring that your financial mathematics calculations are correct and meaningful.
Financial Mathematics
Financial mathematics involves applying mathematical methods to solve problems related to finance, including issues such as interest calculations. Simple interest is a fundamental concept, used to calculate the interest on a sum of money over a period. The formula for simple interest is:\[ I = P \cdot r \cdot t \]Where:
  • \(I\) is the interest.
  • \(P\) is the principal amount.
  • \(r\) is the rate of interest as a decimal.
  • \(t\) is the time in years.
This formula helps in understanding how money grows over time at a given interest rate, making it powerful in financial planning and analysis.
Mathematical Formulas
Mathematical formulas provide the framework for solving complex problems systematically. In the realm of financial mathematics, they simplify the computation and help achieve accurate results. For instance, the simple interest formula \( I = P \cdot r \cdot t \) provides a straightforward way to calculate interest:
  • The principal amount \(P = \\(18,000\)
  • The converted rate \(r = 0.075\)
  • The period \(t = 1.5 \) years
Putting these values into the formula gives us:\[ I = 18000 \cdot 0.075 \cdot 1.5 = \\)2025 \]This equation is a powerful tool, showing how formulas streamline the process of finding solutions to financial problems.

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Most popular questions from this chapter

Suppose that you decide to buy a car for \(\$ 29,635\), including taxes and license fees. You saved \(\$ 9000\) for a down payment and can get a five-year car loan at \(6.62 \%\). Find the monthly payment and the total interest for the loan.

The unpaid balance of an installment loan is equal to the present value of the remaining payments. The unpaid balance, \(P\), is given by $$ P=P M T \frac{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}{\left(\frac{r}{n}\right)}, $$

What is the future value of a loan and how is it determined?

How much should you deposit at the end of each month in an IRA that pays \(8 \%\) compounded monthly to earn \(\$ 60,000\) per year from interest alone, while leaving the principal untouched, when you retire in 30 years?

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 \(\operatorname{Loan} A\). 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.
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