Calculate The Interest Earned And The Amount To Be Paid After 6 Years On A Debt Of $80,000.00 At An Interest Rate Of 4% Compounded Monthly.

Jun 20, 2025 by ADMIN 140 views

Understanding Compound Interest A Comprehensive Guide to Problem 11

Calculating the future value of a debt and the interest accrued over time is a fundamental concept in finance. In Problem 11, we are presented with a scenario involving a debt of $80,000.00 that needs to be paid after 6 years, with an interest rate of 4% compounded monthly. This problem allows us to explore the power of compound interest and how it impacts the total amount to be repaid. This article aims to break down the problem step by step, providing a clear understanding of the concepts and calculations involved. By the end of this article, you will be equipped with the knowledge to tackle similar financial problems and make informed decisions about your own finances. Understanding compound interest is essential for anyone dealing with loans, investments, or any financial planning scenario. The ability to accurately calculate interest earned and the final amount due is crucial for both personal and professional financial management.

Breaking Down the Problem The Key Components

To effectively solve Problem 11, it's crucial to first identify and understand the key components. The principal amount, which is the initial debt, is $80,000.00. The interest rate is 4% per year, but since the interest is compounded monthly, we need to convert this annual rate to a monthly rate by dividing it by 12. This gives us a monthly interest rate of 4%/12 = 0.003333 (approximately). The time period is 6 years, which we need to convert to months since the interest is compounded monthly. This gives us a total of 6 years * 12 months/year = 72 months. With these components in hand, we can now proceed to calculate the amount to be paid after 6 years and the total interest earned. The formula for compound interest is a cornerstone of financial calculations. It allows us to determine how an initial amount grows over time, taking into account the effect of interest accumulating not only on the principal but also on the previously earned interest. This compounding effect is what makes compound interest such a powerful tool for wealth accumulation, but it also means that debts can grow significantly over time if not managed properly. Understanding the variables in the formula and how they interact is essential for accurate financial planning.

Calculating the Future Value Using the Compound Interest Formula

The formula for compound interest is: A = P (1 + r/n)^(nt), where:

A is the future value of the investment/loan, including interest.
P is the principal investment amount (the initial deposit or loan amount).
r is the annual interest rate (as a decimal).
n is the number of times that interest is compounded per year.
t is the number of years the money is invested or borrowed for.

In our case:

P = $80,000.00
r = 4% = 0.04
n = 12 (compounded monthly)
t = 6 years

Plugging these values into the formula, we get:

A = 80000 (1 + 0.04/12)^(12*6) A = 80000 (1 + 0.003333)^(72) A = 80000 (1.003333)^(72) A ≈ 80000 * 1.27024 A ≈ $101,619.20

Therefore, the amount to be paid after 6 years is approximately $101,619.20. The compound interest formula is a versatile tool that can be applied to a wide range of financial scenarios. Whether you are calculating the future value of an investment, the total cost of a loan, or the impact of inflation on your savings, understanding this formula is essential. By manipulating the variables, you can explore different scenarios and make informed decisions about your financial future. For example, you can change the interest rate, the compounding frequency, or the time period to see how these factors affect the final outcome. This ability to model different scenarios is a key benefit of understanding the compound interest formula.

Determining the Interest Earned The Final Calculation

Now that we have the future value, we can calculate the interest earned by subtracting the principal amount from the future value:

Interest Earned = Future Value - Principal Interest Earned = $101,619.20 - $80,000.00 Interest Earned = $21,619.20

Thus, the interest earned after 6 years is approximately $21,619.20. This calculation highlights the impact of compound interest over time. The interest earned represents the cost of borrowing the money, and it is a significant amount in this case. Understanding how interest accumulates is crucial for managing debt effectively. By comparing the interest earned to the principal amount, you can get a sense of the true cost of borrowing and make informed decisions about your financial obligations. In this scenario, the interest earned is more than 25% of the original principal, which underscores the importance of considering the long-term impact of interest rates and compounding frequency.

Analyzing the Options The Correct Answer

Looking at the options provided:

a. 21660 b. 21445 c. 24863 d. 26996

The closest answer to our calculated interest earned of $21,619.20 is option a. 21660. This slight difference could be due to rounding during the calculations. The process of analyzing the options and selecting the closest answer is a crucial skill in problem-solving. In many real-world scenarios, you may not have an exact match to your calculated result, and you need to be able to identify the most reasonable option. This requires understanding the context of the problem, the potential sources of error, and the relative magnitude of the differences between the options. In this case, the slight difference between our calculated result and option a is likely due to rounding, and option a is clearly the most plausible answer.

Key Takeaways and Practical Applications

Compound interest is a powerful force in finance, impacting both investments and debts.
Understanding the compound interest formula is crucial for calculating future values and interest earned.
Breaking down problems into key components makes them easier to solve.
Analyzing options and selecting the closest answer is an important problem-solving skill.

This problem illustrates the importance of understanding compound interest in managing debt. By knowing how interest accumulates over time, you can make informed decisions about borrowing and repayment strategies. Compound interest is not just a theoretical concept; it has practical implications for everyday financial decisions. Whether you are taking out a loan, investing for retirement, or saving for a down payment, understanding how compound interest works can help you achieve your financial goals. By applying the principles learned in this problem, you can take control of your finances and make informed decisions that will benefit you in the long run.

Conclusion Mastering Financial Calculations

Problem 11 provides a valuable lesson in understanding compound interest and its impact on debt. By calculating the future value and interest earned, we can gain insights into the financial implications of borrowing money. This knowledge empowers us to make informed decisions and manage our finances effectively. Mastering financial calculations is an essential skill for navigating the complexities of the modern financial world. By understanding concepts such as compound interest, present value, and future value, you can make informed decisions about your investments, loans, and savings. This article has provided a step-by-step guide to solving Problem 11, but the principles and techniques discussed can be applied to a wide range of financial scenarios. By continuing to develop your financial literacy, you can build a solid foundation for your financial future and achieve your long-term goals.

This detailed explanation not only solves the problem but also provides a comprehensive understanding of the underlying concepts. Understanding these concepts is crucial for making informed financial decisions in various real-life scenarios.