Problem 69 - College Algebra with Modeling and Visualization

Question

\(\$ 1600\) at \(10.4 \%\) compounded monthly for 2.5 years

Step-by-Step Solution

Answer

The amount after 2.5 years is approximately $2084.45.

Step 1: Understand the formula for compound interest

The formula for calculating compound interest is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (initial money), \( r \) is the annual interest rate (decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time the money is invested or borrowed for, in years.

Step 2: Assign values to the variables

From the problem, we have: - Principal amount \( P = 1600 \) dollars.- Annual interest rate \( r = 10.4\% = 0.104 \).- Compounded monthly \( n = 12 \).- Time period \( t = 2.5 \) years.

Step 3: Substitute values into the formula

Substituting the known values into the formula leads to: \[ A = 1600 \left(1 + \frac{0.104}{12}\right)^{12 \times 2.5} \].Start by calculating the term \( \frac{0.104}{12} \).

Step 4: Calculate interest rate per period

Find \( \frac{0.104}{12} \) which equals approximately \( 0.00867 \). This is the monthly interest rate.

Step 5: Calculate the number of compounding periods

Multiply the number of years \( t \) by the number of compounding periods in a year \( n \), which is \( 12 \times 2.5 = 30 \) periods.

Step 6: Compute the future value

Substitute the results from the steps to compute:\[ A = 1600 \left(1 + 0.00867 \right)^{30} \].Calculate \( 1.00867^{30} \) which gives approximately \( 1.30278 \).Thus, the future value \( A \) is \( 1600 \times 1.30278 \approx 2084.45 \) dollars.

Key Concepts

Algebraic Formulas

Algebraic formulas play a crucial role in calculating compound interest, as they allow us to systematically solve problems and perform calculations. When it comes to finance, one key formula is the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] This formula helps determine the future value of an investment or loan after interest has compounded over a set period. Each variable in the formula has a specific role:

• A is the total amount after the interest has compounded, including both the principal and interest.
• P stands for the principal, which is the original amount of money.
• r symbolizes the annual interest rate in decimal form.
• n denotes the frequency at which the interest is compounded annually.
• t represents the time in years over which the interest is applied.

It's like a mathematical recipe. Substitute the known values into their respective spots in the formula, apply the steps needed, and you will find the future value of your investment or debt.

Financial Mathematics

Financial mathematics is a field that uses mathematical tools to solve problems related to finance. One of the popular concepts in this field is compound interest. This concept helps us understand how investments grow over time when interest is applied not just on the principal, but also on accumulated interest.
By using the compound interest formula, financial mathematics provides insights into how different factors affect savings or investments, including:

• The initial principal amount.
• The interest rate (how much you earn on your money).
• The frequency of compounding (how often interest is added to the total).
• The duration of the investment (the total time your money is invested).

Understanding how to use these components effectively can aid in making well-informed financial decisions, whether for personal savings, investments, or loans.

Exponential Functions

Exponential functions are foundational in describing how quantities grow or decay over time, particularly in the realm of finance. The compound interest formula embodies this concept, showing how money grows at an exponential rate due to regular compounding periods. In the formula, the expression \( \left(1 + \frac{r}{n}\right)^{nt} \) involves exponential growth. The base of the exponential function \(1 + \frac{r}{n}\) reflects how much the investment grows in one compounding period. Raising this figure to the power of \(nt\) results in the total growth over multiple compounding periods.
Key features of exponential functions in compound interest are:

• Rapid growth: As time and the number of compounding periods increase, the total amount grows faster.
• Compounding effect: The more frequent the compounding, the higher the future value.

Understanding exponential growth can help individuals predict future financial states better, strategize investments, and comprehend the impact of different compounding intervals on their financial plans.