Problem 7 - Prealgebra

Question

A These problems are similar to the examples found in this section. They should be set up and solved in the same way. (Problems 1-12 involve simple interest.) Bank Loan A bank lends one of its customers \(\$ 2,000\) at \(8 \%\) for 1 year. If the customer pays the loan back at the end of the year, how much does he pay the bank?

Step-by-Step Solution

Answer

The customer pays $2160 at the end of the year.

Step 1: Understanding Simple Interest

Simple interest is calculated using the formula \( I = P \times r \times t \), where \( I \) is the interest, \( P \) is the principal amount (initial loan), \( r \) is the rate of interest, and \( t \) is the time duration of the loan in years.

Step 2: Identify Given Values

Here, the principal \( P = 2000 \) dollars, annual interest rate \( r = 8\% = 0.08 \), and the time \( t = 1 \) year.

Step 3: Calculate the Interest

Plug the values into the simple interest formula: \( I = 2000 \times 0.08 \times 1 \).

Key Concepts

Interest Calculation

Interest calculation is a fundamental concept in financial mathematics, crucial for determining how much extra one will pay or earn over time on a certain principal amount. In this context, simple interest is the focus. Unlike compound interest, which takes interest on both the principal and previously earned interest, simple interest only applies to the principal. This makes calculations straightforward and easy.To calculate simple interest, you use the formula \( I = P \times r \times t \), where:

• \( I \) is the interest earned or paid,
• \( P \) is the principal amount (the initial sum of money),
• \( r \) is the annual interest rate (converted into decimal form, so 8% becomes 0.08), and
• \( t \) is the time period the money is lent or borrowed, in years.

Understanding each component helps in leveraging financial products effectively. For example, in the bank loan scenario from the exercise, knowing these variables allows for precise computation of interest owed, enhancing financial literacy.

Financial Mathematics

Financial mathematics explores the application of mathematical concepts to solve problems in finance, such as calculating interest, understanding investments, and managing debts. It's crucial because it helps individuals and organizations make informed financial decisions. In simple interest scenarios like the bank loan example, this kind of math demystifies how borrowing and repayments work. Here’s how financial mathematics helps:

• Determining how much extra money will be paid or received over time, enabling better budget planning.
• Assisting in the comparison of different investment or loan options by providing a clear picture of costs or returns.
• The clarity from these calculations aids in making strategic decisions to ensure profitability and sustainability in financial undertakings.

Mastering concepts like simple interest is foundational, as it applies to savings accounts, loans, and more. This ensures one’s financial health through educated decisions.

Prealgebra Concepts

Prealgebra serves as a stepping stone into more complex fields of mathematics. It includes basic arithmetic operations, understanding percentages, decimals, and fractions - all of which are integral to interest calculations. In the context of simple interest:

• Understanding percentages becomes pivotal, as interest rates are often given in percentage form and need conversion to decimals. For example, converting 8% to 0.08 is a prealgebra skill.
• Basic multiplication of the principal, the rate, and the time period forms the core operation in simple interest calculations.
• Grasping these areas lays the groundwork for more advanced topics in mathematics, finance, and economics.

By solidifying prealgebra skills, students gain confidence and competence in handling more complex financial scenarios in both personal and academic contexts.