How Much Would $\$ 200$ Invested At $5 \%$ Interest Compounded Monthly Be Worth After 9 Years? Round Your Answer To The Nearest Cent. The Formula To Use Is: \[ A(t) = P\left(1+\frac{r}{n}\right)^{n T} \] A. $\$

How Much Would $200 Invested at 5% Interest Compounded Monthly Be Worth After 9 Years?

Understanding the Power of Compound Interest

Compound interest is a powerful financial concept that allows your savings to grow exponentially over time. When you invest money at a fixed interest rate, the interest earned is added to the principal amount, and the interest is then calculated on the new, higher balance. This process is repeated at regular intervals, resulting in a snowball effect that can lead to significant growth in your investment.

The Formula for Compound Interest

The formula for compound interest is given by:

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

Where:

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

Plugging in the Values

In this case, we want to find the future value of an investment of $200 at a 5% interest rate compounded monthly for 9 years. We can plug in the values as follows:

• $P = 200$
• $r = 0.05$ (5% interest rate in decimal form)
• $n = 12$ (monthly compounding)
• $t = 9$ years

Calculating the Future Value

Now, let's calculate the future value of the investment using the formula:

${ A(9) = 200\left(1+\frac{0.05}{12}\right)^{12 \times 9} }$

${ A(9) = 200\left(1+0.0041667\right)^{108} }$

${ A(9) = 200\left(1.0041667\right)^{108} }$

Using a calculator to evaluate the expression, we get:

${ A(9) \approx 200 \times 1.5493 }$

${ A(9) \approx 309.86 }$

Rounding to the Nearest Cent

Rounding the result to the nearest cent, we get:

${ A(9) \approx 309.86 }$

Conclusion

In conclusion, an investment of $200 at a 5% interest rate compounded monthly for 9 years would be worth approximately $309.86. This represents a growth of $109.86, or a 54.93% increase in the initial investment. The power of compound interest is evident in this example, and it highlights the importance of starting to save and invest early in life to take advantage of this powerful financial concept.

The Impact of Compounding Frequency

The frequency of compounding can have a significant impact on the growth of an investment. In this case, we assumed monthly compounding, which resulted in a higher growth rate compared to annual compounding. To illustrate this, let's recalculate the future value assuming annual compounding:

• $P = 200$
• $r = 0.05$ (5% interest rate in decimal form)
• $n = 1$ (annual compounding)
• $t = 9$ years

Using the formula, we get:

${ A(9) = 200\left(1+\frac{0.05}{1}\right)^{1 \times 9} }$

${ A(9) = 200\left(1+0.05\right)^{9} }$

${ A(9) = 200\left(1.05\right)^{9} }$

Using a calculator to evaluate the expression, we get:

${ A(9) \approx 200 \times 1.5513 }$

${ A(9) \approx 310.26 }$

Comparison of Results

Comparing the results, we can see that the investment with monthly compounding ($309.86) grew slightly less than the investment with annual compounding ($310.26). This highlights the importance of considering the compounding frequency when investing.

The Importance of Starting Early

The power of compound interest is not just about the interest rate or the compounding frequency, but also about the time the money is invested for. Starting to save and invest early in life can have a significant impact on the growth of your investment. To illustrate this, let's consider an example where the investment is made at a younger age:

• $P = 200$
• $r = 0.05$ (5% interest rate in decimal form)
• $n = 12$ (monthly compounding)
• $t = 30$ years

Using the formula, we get:

${ A(30) = 200\left(1+\frac{0.05}{12}\right)^{12 \times 30} }$

${ A(30) = 200\left(1+0.0041667\right)^{360} }$

${ A(30) = 200\left(1.0041667\right)^{360} }$

Using a calculator to evaluate the expression, we get:

${ A(30) \approx 200 \times 7.0399 }$

${ A(30) \approx 1,407.98 }$

Conclusion

In conclusion, the power of compound interest is a powerful financial concept that can help your savings grow exponentially over time. By starting to save and invest early in life, you can take advantage of this concept and achieve significant growth in your investment. The frequency of compounding and the time the money is invested for can also have a significant impact on the growth of your investment.
Frequently Asked Questions About Compound Interest

Q: What is compound interest?

A: Compound interest is the interest earned on both the principal amount and any accrued interest over time. It's a powerful financial concept that can help your savings grow exponentially.

Q: How does compound interest work?

A: Compound interest works by adding the interest earned to the principal amount, and then calculating the interest on the new, higher balance. This process is repeated at regular intervals, resulting in a snowball effect that can lead to significant growth in your investment.

Q: What are the key factors that affect compound interest?

A: The key factors that affect compound interest are:

• Principal amount (initial investment)
• Interest rate (in decimal form)
• Compounding frequency (how often interest is compounded per year)
• Time the money is invested for (in years)

Q: How often should I compound my interest?

A: The frequency of compounding can have a significant impact on the growth of your investment. Monthly compounding is generally more effective than annual compounding, but it depends on your individual financial goals and circumstances.

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

A: Simple interest is calculated only on the principal amount, whereas compound interest is calculated on both the principal amount and any accrued interest. Compound interest is generally more effective than simple interest, especially over longer periods of time.

Q: Can I use compound interest to my advantage?

A: Yes, you can use compound interest to your advantage by starting to save and invest early in life, taking advantage of high-yield savings accounts or certificates of deposit (CDs), and considering tax-advantaged retirement accounts such as 401(k) or IRA.

Q: How can I calculate compound interest?

A: You can calculate compound interest using the formula:

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

Where:

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

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

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

• Not starting to save and invest early in life
• Not taking advantage of high-yield savings accounts or CDs
• Not considering tax-advantaged retirement accounts
• Not understanding the compounding frequency and its impact on the growth of your investment

Q: Can I use compound interest to pay off debt?

A: Yes, you can use compound interest to pay off debt by taking advantage of high-yield savings accounts or CDs and using the interest earned to pay off your debt. However, it's generally more effective to focus on paying off high-interest debt first and then using compound interest to grow your savings.

Q: How can I maximize the benefits of compound interest?

A: To maximize the benefits of compound interest, you can:

• Start to save and invest early in life
• Take advantage of high-yield savings accounts or CDs
• Consider tax-advantaged retirement accounts
• Understand the compounding frequency and its impact on the growth of your investment
• Avoid common mistakes such as not starting to save and invest early in life and not taking advantage of high-yield savings accounts or CDs.