How Much Would $200 Invested At 5% Interest Compounded Monthly Be Worth After 9 Years? Round The Answer To The Nearest Cent.

Investing money and watching it grow over time is a cornerstone of financial planning. Understanding how different interest rates and compounding periods affect your investment is crucial. In this article, we will dive into the concept of compound interest and calculate the future value of a hypothetical investment. Let's explore the formula and the step-by-step process involved in determining the worth of an initial investment of $200 at a 5% interest rate compounded monthly over a period of 9 years.

Understanding the Compound Interest Formula

The compound interest formula is a powerful tool for calculating the future value of an investment or loan, taking into account the effects of compounding. It allows us to determine how much an initial principal amount will grow over time, considering the interest rate, compounding frequency, and the investment period. The formula is expressed as follows:

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

Where:

• $A(t)$ represents the future value of the investment after time t.
• $P$ is the principal amount (the initial investment).
• $r$ denotes the annual interest rate (expressed as a decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ represents the number of years the money is invested or borrowed for.

Each component of this formula plays a vital role in determining the final amount. The principal is the foundation, the interest rate dictates the growth percentage, the compounding frequency accelerates the earnings, and the investment period allows the magic of compounding to work over time. By understanding these elements, investors can make informed decisions about their financial future.

Applying the Formula to Our Scenario

In our specific scenario, we are looking to calculate the future value of an investment of $200 with an annual interest rate of 5%, compounded monthly over a period of 9 years. This means we have the following values:

• $P = $200$ (the initial investment)
• $r = 0.05$ (the annual interest rate, expressed as a decimal)
• $n = 12$ (the number of times interest is compounded per year, since it's monthly)
• $t = 9$ (the number of years)

Now, we will plug these values into the compound interest formula to determine the future value, $A(t)$. Substituting the given values into the formula, we get:

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

This equation represents the mathematical model for our investment's growth. It captures the essence of how the principal will increase over time, influenced by the interest rate and the compounding frequency. The next step involves simplifying this equation to arrive at the final answer.

Step-by-Step Calculation

To accurately calculate the future value, we need to follow the order of operations, which dictates the sequence in which we perform mathematical calculations. Let's break down the calculation step by step:

1. Divide the annual interest rate by the number of compounding periods per year: $\frac{0.05}{12} \approx 0.00416667$. This gives us the periodic interest rate, which is the interest rate applied to each compounding period.
2. Add 1 to the result: $1 + 0.00416667 \approx 1.00416667$. This represents the growth factor for each period.
3. Multiply the number of compounding periods per year by the number of years: $12 \cdot 9 = 108$. This is the total number of compounding periods over the investment term.
4. Raise the result from step 2 to the power of the result from step 3: $(1.00416667)^{108} \approx 1.56568$. This calculates the total growth factor over the entire investment period.
5. Multiply the principal amount by the result from step 4: $200 \cdot 1.56568 \approx 313.136$. This gives us the future value of the investment before rounding.

By following these steps meticulously, we ensure an accurate calculation of the investment's growth. Each step builds upon the previous one, ultimately leading to the final future value.

Rounding to the Nearest Cent

The question asks us to round the answer to the nearest cent, which is two decimal places. Our calculated future value is approximately $313.136. To round this to the nearest cent, we look at the third decimal place. Since it is 6 (which is 5 or greater), we round up the second decimal place.

Therefore, the future value rounded to the nearest cent is $313.14.

Rounding is a crucial step in financial calculations as it ensures that the answer is presented in a practical and easily understandable format. In this case, rounding to the nearest cent provides a precise value that is relevant in real-world financial scenarios.

The Final Answer

Based on our calculations, an investment of $200 at a 5% interest rate compounded monthly would be worth approximately $313.14 after 9 years. This result demonstrates the power of compound interest in growing an investment over time.

Compound interest is often called the