Claire Deposited $\$2,500$ Into An Account That Accrues Interest Monthly. She Made No Withdrawals. After 2 Years, Claire Had $\$2,762.35$ In The Account. What Is The Annual Interest Rate?Compound Interest Formula: $V(t) =

Introduction

Claire, a savvy investor, deposited $\$2,500$ into an account that accrues interest monthly. With no withdrawals, she patiently waited for two years to see the fruits of her labor. Upon checking her account, Claire was thrilled to find that she had accumulated a whopping $\$2,762.35$. But, the burning question remains: what is the annual interest rate that made this possible? In this article, we will delve into the world of compound interest and unravel the mystery of Claire's annual interest rate.

Understanding Compound Interest

Compound interest is a powerful financial concept that allows your savings to grow exponentially over time. It's a type of interest that is calculated on both the initial principal and the accumulated interest from previous periods. The formula for compound interest is given by:

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

where:

• $V(t)$ 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 (in decimal form)
• $n$ is the number of times that interest is compounded per year
• $t$ is the time the money is invested or borrowed for, in years

Claire's Compound Interest Scenario

Let's apply the compound interest formula to Claire's situation. We know that:

• $P = \$2,500$ (initial deposit)
• $V(t) = \$2,762.35$ (final amount after 2 years)
• $t = 2$ years (time period)
• $n = 12$ (monthly compounding)

We need to find the annual interest rate, $r$. Plugging in the values, we get:

$$2,762.35 = 2,500 \left(1 + \frac{r}{12}\right)^{12 \cdot 2}$$

Solving for the Annual Interest Rate

To solve for $r$, we can start by isolating the term with $r$:

$$\left(1 + \frac{r}{12}\right)^{24} = \frac{2,762.35}{2,500}$$

Taking the 24th root of both sides:

$$1 + \frac{r}{12} = \left(\frac{2,762.35}{2,500}\right)^{\frac{1}{24}}$$

Simplifying:

$$1 + \frac{r}{12} = 1.0045$$

Subtracting 1 from both sides:

$$\frac{r}{12} = 0.0045$$

Multiplying both sides by 12:

$$r = 0.054$$

Converting the Decimal to a Percentage

To express the annual interest rate as a percentage, we can multiply by 100:

$$r = 0.054 \times 100 = 5.4\%$$

Conclusion

In this article, we used the compound interest formula to solve for the annual interest rate in Claire's account. By plugging in the values and solving for $r$, we found that the annual interest rate is approximately 5.4%. This means that Claire's investment earned an annual interest rate of 5.4% over the two-year period, resulting in a total accumulation of $\$2,762.35$.

Key Takeaways

• Compound interest is a powerful financial concept that allows your savings to grow exponentially over time.
• The compound interest formula is given by $V(t) = P \left(1 + \frac{r}{n}\right)^{nt}$.
• To solve for the annual interest rate, we can isolate the term with $r$ and take the appropriate root.
• The annual interest rate can be expressed as a percentage by multiplying by 100.

Further Reading

If you're interested in learning more about compound interest and how to calculate it, we recommend checking out the following resources:

• Compound Interest Calculator
• Compound Interest Formula
• Understanding Compound Interest

Q&A: Compound Interest and Beyond

In our previous article, we delved into the world of compound interest and solved for the annual interest rate in Claire's account. But, we know that there are many more questions to be answered. In this article, we'll tackle some of the most frequently asked questions about compound interest and provide you with a deeper understanding of this powerful financial concept.

Q: What is compound interest, and how does it work?

A: Compound interest is a type of interest that is calculated on both the initial principal and the accumulated interest from previous periods. It's a powerful financial concept that allows your savings to grow exponentially over time. The formula for compound interest is given by:

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

where:

• $V(t)$ 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 (in decimal form)
• $n$ is the number of times that interest is compounded per year
• $t$ is the time the money is invested or borrowed for, in years

Q: How often is interest compounded?

A: Interest can be compounded daily, monthly, quarterly, or annually, depending on the type of account or loan. In Claire's case, interest was compounded monthly.

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

A: Simple interest is calculated only on the initial principal, whereas compound interest is calculated on both the initial principal and the accumulated interest from previous periods. This means that compound interest grows exponentially over time, while simple interest grows linearly.

Q: How can I calculate compound interest?

A: You can use the compound interest formula to calculate the future value of an investment or loan. Alternatively, you can use a compound interest calculator or spreadsheet to make the calculation easier.

Q: What are some common applications of compound interest?

A: Compound interest is used in a variety of financial applications, including:

• Savings accounts
• Certificates of deposit (CDs)
• Bonds
• Loans
• Investments

Q: How can I maximize my compound interest earnings?

A: To maximize your compound interest earnings, consider the following strategies:

• Invest for the long-term
• Choose a high-interest rate account or loan
• Compound interest frequently (e.g., monthly or daily)
• Avoid withdrawals or early repayments

Q: What are some common mistakes to avoid when dealing with compound interest?

A: Some common mistakes to avoid when dealing with compound interest include:

• Not understanding the interest rate or compounding frequency
• Not considering the impact of inflation on interest rates
• Not taking advantage of compound interest by investing for the long-term
• Not monitoring and adjusting your investments or loans regularly

Conclusion

In this article, we've answered some of the most frequently asked questions about compound interest and provided you with a deeper understanding of this powerful financial concept. By understanding compound interest and how to calculate it, you can make informed decisions about your investments and achieve your financial goals.

Key Takeaways

• Compound interest is a type of interest that is calculated on both the initial principal and the accumulated interest from previous periods.
• The compound interest formula is given by $V(t) = P \left(1 + \frac{r}{n}\right)^{nt}$.
• To maximize your compound interest earnings, invest for the long-term, choose a high-interest rate account or loan, and compound interest frequently.
• Avoid common mistakes such as not understanding the interest rate or compounding frequency, not considering the impact of inflation on interest rates, and not taking advantage of compound interest by investing for the long-term.

Further Reading

If you're interested in learning more about compound interest and how to calculate it, we recommend checking out the following resources:

• Compound Interest Calculator
• Compound Interest Formula
• Understanding Compound Interest

By understanding compound interest and how to calculate it, you can make informed decisions about your investments and achieve your financial goals.