Select The Correct Answer.Charlie Wants To Make A One-time Investment Into An Account That Earns $6\%$ Interest Compounded Semiannually. To Earn \$338,893 After 26 Years, Approximately How Much Money Must He Invest?$S = P(1 +

Select the Correct Answer: Calculating Investment Amount for Compounded Interest

Understanding Compounded Interest

Compounded interest is a type of interest that is calculated on both the initial principal and the accumulated interest from previous periods. In this case, Charlie wants to make a one-time investment into an account that earns a $6\%$ interest rate compounded semiannually. This means that the interest is applied twice a year, resulting in a higher total amount compared to simple interest.

The Formula for Compounded Interest

The formula for compounded interest is given by:

$$S = P\left(1 + \frac{r}{n}\right)^{nt}$$

where:

• $S$ is the future value of the investment (the amount Charlie wants to earn)
• $P$ is the principal amount (the initial amount Charlie invests)
• $r$ is the annual interest rate (in decimal form)
• $n$ is the number of times the interest is compounded per year
• $t$ is the time the money is invested for, in years

Given Values and the Goal

In this problem, we are given the following values:

• $S = \$338,893$ (the amount Charlie wants to earn)
• $r = 0.06$ (the annual interest rate, in decimal form)
• $n = 2$ (the interest is compounded semiannually, so twice a year)
• $t = 26$ years (the time the money is invested for)

We need to find the principal amount $P$ that Charlie must invest to earn $\$338,893$ after 26 years.

Solving for the Principal Amount

To solve for the principal amount $P$, we can rearrange the formula for compounded interest as follows:

$$P = \frac{S}{\left(1 + \frac{r}{n}\right)^{nt}}$$

Substituting the given values, we get:

$$P = \frac{\$338,893}{\left(1 + \frac{0.06}{2}\right)^{2 \cdot 26}}$$

Calculating the Principal Amount

Now, we can calculate the principal amount $P$ using the formula above:

$$P = \frac{\$338,893}{\left(1 + 0.03\right)^{52}}$$

$$P = \frac{\$338,893}{\left(1.03\right)^{52}}$$

Using a calculator to evaluate the expression, we get:

$$P \approx \$100,000$$

Therefore, Charlie must invest approximately $\$100,000$ to earn $\$338,893$ after 26 years, assuming a $6\%$ interest rate compounded semiannually.

Conclusion

In this problem, we used the formula for compounded interest to calculate the principal amount that Charlie must invest to earn a certain amount after a given period of time. We found that Charlie must invest approximately $\$100,000$ to earn $\$338,893$ after 26 years, assuming a $6\%$ interest rate compounded semiannually. This problem demonstrates the importance of understanding compounded interest and how it can be used to calculate investment amounts.

Discussion and Further Exploration

This problem can be extended to explore other scenarios, such as:

• What if the interest rate is higher or lower?
• What if the interest is compounded more or less frequently?
• What if the time period is shorter or longer?

These questions can be answered by modifying the formula for compounded interest and recalculating the principal amount. This can help to develop a deeper understanding of compounded interest and its applications in finance.

Real-World Applications

Compounded interest is a fundamental concept in finance and is used in a variety of real-world applications, such as:

• Calculating investment returns
• Determining loan interest rates
• Evaluating the effectiveness of savings plans

Understanding compounded interest is essential for making informed financial decisions and achieving long-term financial goals.

Additional Resources

For further reading and exploration, the following resources are recommended:

• [1] Investopedia: Compounded Interest
• [2] Khan Academy: Compounded Interest
• [3] Math Is Fun: Compounded Interest

These resources provide a comprehensive overview of compounded interest and its applications in finance. They can be used as a starting point for further exploration and learning.
Q&A: Compounded Interest and Investment Calculations

Frequently Asked Questions

Compounded interest is a fundamental concept in finance that can be used to calculate investment returns, determine loan interest rates, and evaluate the effectiveness of savings plans. However, it can be a complex topic, and many people have questions about how it works. In this article, we will answer some of the most frequently asked questions about compounded interest and investment calculations.

Q: What is compounded interest?

A: Compounded interest is a type of interest that is calculated on both the initial principal and the accumulated interest from previous periods. This means that the interest is applied multiple times, resulting in a higher total amount compared to simple interest.

Q: How is compounded interest calculated?

A: The formula for compounded interest is given by:

$$S = P\left(1 + \frac{r}{n}\right)^{nt}$$

where:

• $S$ is the future value of the investment
• $P$ is the principal amount (the initial amount invested)
• $r$ is the annual interest rate (in decimal form)
• $n$ is the number of times the interest is compounded per year
• $t$ is the time the money is invested for, in years

Q: What is the difference between simple interest and compounded interest?

A: Simple interest is calculated only on the initial principal, while compounded interest is calculated on both the initial principal and the accumulated interest from previous periods. This means that compounded interest results in a higher total amount compared to simple interest.

Q: How often is interest compounded?

A: Interest can be compounded daily, monthly, quarterly, semiannually, or annually, depending on the specific investment or loan. The more frequently the interest is compounded, the higher the total amount will be.

Q: What is the effect of time on compounded interest?

A: The longer the time period, the higher the total amount will be. This is because the interest is applied multiple times, resulting in a higher total amount.

Q: How can I use compounded interest to calculate investment returns?

A: To calculate investment returns using compounded interest, you can use the formula:

$$S = P\left(1 + \frac{r}{n}\right)^{nt}$$

where:

• $S$ is the future value of the investment
• $P$ is the principal amount (the initial amount invested)
• $r$ is the annual interest rate (in decimal form)
• $n$ is the number of times the interest is compounded per year
• $t$ is the time the money is invested for, in years

Q: How can I use compounded interest to determine loan interest rates?

A: To determine loan interest rates using compounded interest, you can use the formula:

$$S = P\left(1 + \frac{r}{n}\right)^{nt}$$

where:

• $S$ is the total amount owed
• $P$ is the principal amount (the initial amount borrowed)
• $r$ is the annual interest rate (in decimal form)
• $n$ is the number of times the interest is compounded per year
• $t$ is the time the loan is for, in years

Q: What are some common mistakes to avoid when using compounded interest?

A: Some common mistakes to avoid when using compounded interest include:

• Not considering the effect of time on compounded interest
• Not using the correct formula for compounded interest
• Not taking into account the frequency of compounding
• Not considering the impact of inflation on the investment or loan

Conclusion

Compounded interest is a powerful tool for calculating investment returns, determining loan interest rates, and evaluating the effectiveness of savings plans. By understanding how compounded interest works and using the correct formula, you can make informed financial decisions and achieve your long-term financial goals.

Additional Resources

For further reading and exploration, the following resources are recommended:

• [1] Investopedia: Compounded Interest
• [2] Khan Academy: Compounded Interest
• [3] Math Is Fun: Compounded Interest

These resources provide a comprehensive overview of compounded interest and its applications in finance. They can be used as a starting point for further exploration and learning.