Question

Suppose you run up a debt of \(\$ 300\) on a credit card that charges an annual rate of 12 percent, compounded annually. How much will you owe at the end of two years? Assume no additional charges or payments are made.

Short answer

You will owe $376.32 at the end of two years.

Step-by-step solution

Understand the Compounded Interest Formula

The formula to calculate the amount owed after a certain period with compounded interest is \( A = P(1 + r)^n \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (initial amount), \( r \) is the annual interest rate (decimal), and \( n \) is the number of years the money is invested or borrowed for.

Identify Given Values

From the exercise, we know the principal \( P = 300 \), the annual interest rate \( r = 0.12 \) (or 12% converted to a decimal), and the time period \( n = 2 \) years.

Apply the Compounded Interest Formula

Substitute the given values into the formula: \[ A = 300 \times (1 + 0.12)^2 \]

Calculate Inside the Parentheses

First calculate the expression inside the parentheses: \( 1 + 0.12 = 1.12 \).

Compute the Power of the Parentheses Result

Raise 1.12 to the power of 2: \( (1.12)^2 = 1.2544 \).

Find the Final Amount

Multiply the principal amount by the result from Step 5: \( A = 300 \times 1.2544 = 376.32 \)

Key concepts

Interest Rates

Interest rates play a crucial role in the world of finance and can determine how much you owe on a loan or gain from an investment. An interest rate is the percentage at which interest is paid by a borrower or earned by a lender over a particular period of time. It's usually expressed annually. The interest rate can vary: it might be simple or compounded.

With compounded interest, the interest that accumulates over a period is added back to the principal amount, increasing the total sum for future interest calculations. Here's how it works:

• The principal amount is the starting sum (like the $300 in our exercise).
• The rate of interest is given per year; 12% becomes 0.12 when converted into a decimal.
• Compounding means the interest is calculated not only on the initial principal but also on the accumulated interest.

These points help demonstrate how an interest rate can significantly affect the calculations for savings or debts.

Credit Card Debt

Credit card debt is a common financial challenge faced by many individuals. It occurs when a person borrows money via a credit card and agrees to pay back not only this principal amount (the amount borrowed or spent) but also interest on it if not paid within a stipulated period, typically referred to as a grace period.

Understanding how interest on credit card debt works is crucial:

• The interest compounds, meaning that interest is charged on top of interest if the balance is not paid off completely.
• If you only make minimum payments, the interest can greatly increase how long it takes to pay off the debt.
• The annual percentage rate (APR) on credit cards can vary widely, and higher rates mean owning more if not managed properly.

By being aware of how credit card interest accumulates, you can avoid falling into cycles of debt that seem hard to overcome.

Financial Mathematics

Financial mathematics is a field involving the application of mathematical methods to financial problems. It helps in determining the values of investments, calculating interest rates, and understanding financial risk.

In the context of compounded interest, we use mathematical formulas to predict future amounts owed or gained. Specifically, the formula \( A = P(1 + r)^n \) is vital. This formula can answer questions like how much you'll owe after a certain time if you don't make payments on a debt.

• \( P \) stands for principal or initial investment.
• \( r \) is the rate of interest annually expressed as a decimal.
• \( n \) is the number of years the amount is borrowed or invested for.
• \( A \) is the amount including interest after \( n \) years.

With a concrete understanding of financial mathematics, managing your finances becomes a methodical process. By learning this, you can make decisions that are informed and potentially more profitable or cost-saving.