Calculate Accumulated Amount With 3% Interest Compounded Quarterly

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Calculate the accumulated amount after 18 years when $1,999 is deposited into an account with a 3% interest rate, compounded quarterly (4 times per year). Round your answer to the nearest cent.

In this article, we'll explore the concept of compound interest and delve into how to calculate the accumulated amount after a certain period. We'll use a specific example to illustrate the process, providing a step-by-step guide that you can apply to various financial scenarios. Let's consider a scenario where $1,999 is deposited into an account with a 3% interest rate, compounded quarterly (4 times per year). Our goal is to find the accumulated amount after 18 years. Compound interest is a powerful financial tool that allows your money to grow exponentially over time. Unlike simple interest, which is calculated only on the principal amount, compound interest is calculated on the principal plus the accumulated interest from previous periods. This means that you earn interest not only on your initial investment but also on the interest that has already been added to your account.

The Formula for Compound Interest

To calculate the accumulated amount with compound interest, we use the following formula:

A=P(1+rk)ktA = P(1 + \frac{r}{k})^{kt}

Where:

  • A is the accumulated amount (the total amount after interest is applied).
  • P is the principal amount (the initial deposit).
  • r is the annual interest rate (as a decimal).
  • k is the number of times the interest is compounded per year.
  • t is the number of years the money is invested.

This formula is the cornerstone of compound interest calculations, and understanding each component is crucial for accurate financial planning. The principal amount (P) is the initial sum of money you invest, while the annual interest rate (r) represents the percentage of the principal that will be added as interest each year. The number of times the interest is compounded per year (k) determines how frequently the interest is calculated and added to the account. Finally, the number of years (t) represents the investment horizon, or the length of time the money is invested.

Applying the Formula to Our Example

Now, let's apply this formula to our example. We have:

  • P = $1,999
  • r = 3% = 0.03 (as a decimal)
  • k = 4 (compounded quarterly)
  • t = 18 years

Plugging these values into the formula, we get:

A=1999(1+0.034)4imes18A = 1999(1 + \frac{0.03}{4})^{4 imes 18}

This equation represents the accumulated amount after 18 years, taking into account the quarterly compounding of interest. The next step is to simplify and solve the equation, which will give us the final accumulated amount. It's important to follow the order of operations (PEMDAS/BODMAS) to ensure an accurate calculation. First, we'll divide the annual interest rate by the number of compounding periods per year. Then, we'll add 1 to the result. Next, we'll raise the expression in parentheses to the power of the total number of compounding periods (number of compounding periods per year multiplied by the number of years). Finally, we'll multiply the result by the principal amount.

Step-by-Step Calculation

Let's break down the calculation step by step:

  1. Calculate the interest rate per quarter: 0.034=0.0075\frac{0.03}{4} = 0.0075
  2. Add 1 to the result: 1+0.0075=1.00751 + 0.0075 = 1.0075
  3. Calculate the total number of compounding periods: 4imes18=724 imes 18 = 72
  4. Raise 1.0075 to the power of 72: (1.0075)72≈1.709139(1.0075)^{72} \approx 1.709139
  5. Multiply the result by the principal amount: 1999imes1.709139≈3416.5688611999 imes 1.709139 \approx 3416.568861

Therefore, the accumulated amount after 18 years is approximately $3,416.57. This step-by-step calculation illustrates how the compound interest formula works in practice. By breaking down the formula into smaller steps, it becomes easier to understand the impact of each variable on the final accumulated amount. The interest rate per quarter, the total number of compounding periods, and the power to which the expression in parentheses is raised all play crucial roles in determining the final result. It's also important to note the use of approximation in step 4, as the result of raising 1.0075 to the power of 72 is a decimal number with many digits. Rounding to a reasonable number of decimal places is necessary for practical purposes.

Rounding to the Nearest Cent

As instructed, we need to round our answer to the nearest cent (hundredth). So, the accumulated amount is $3,416.57.

Rounding to the nearest cent is a common practice in financial calculations, as it ensures that the final result is accurate and practical. In this case, rounding to the nearest cent means considering the digit in the thousandths place (the third digit after the decimal point). If this digit is 5 or greater, we round up the digit in the hundredths place (the second digit after the decimal point). If the digit in the thousandths place is less than 5, we leave the digit in the hundredths place as it is. In our example, the accumulated amount before rounding was approximately $3,416.568861. The digit in the thousandths place is 8, which is greater than 5, so we round up the digit in the hundredths place (6) to 7. This gives us the final rounded accumulated amount of $3,416.57.

Conclusion

In conclusion, by depositing $1,999 into an account with a 3% interest rate, compounded quarterly, the accumulated amount after 18 years is approximately $3,416.57. This example demonstrates the power of compound interest and how it can help your money grow over time. Understanding the principles of compound interest is essential for making informed financial decisions. Whether you're saving for retirement, investing in stocks or bonds, or simply trying to grow your savings, compound interest can play a significant role in helping you achieve your financial goals. By understanding the formula for compound interest and how each variable affects the final accumulated amount, you can make better decisions about where to invest your money and how long to invest it for. Additionally, it's important to consider the frequency of compounding, as more frequent compounding (e.g., daily or monthly) can lead to higher returns over time compared to less frequent compounding (e.g., annually). Finally, remember that the longer you leave your money invested, the more time it has to grow through the power of compound interest.

  • Compound interest
  • Accumulated amount
  • Interest rate
  • Compounded quarterly
  • Financial calculations
  • Investment growth
  • Principal amount
  • Time value of money
  • Financial planning
  • Interest compounding frequency