- The initial principal is \(\$11,960,000\).
- The amount after year 1 (at \(3\%\)) is \(\$11,960,000 \times 1.03 = \$12,318,800.00\).
- The amount after year 2 (at \(4\%\)) is \(\$12,318,800.00 \times 1.04 = \$12,811,552.00\).
- The final amount after year 3 (at \(5.5\%\)) is \(\$12,811,552.00 \times 1.055 = \boxed{\$13,516,187.36}\).
Explanation
- Problem Analysis and Initial Data
To determine the final amount after three years, we start with an initial principal amount of \(P_0 = \$11,960,000\). The interest is compounded annually at different rates for each year: \(3\%\) for the first year, \(4\%\) for the second year, and \(5 \frac{1}{2}\%\) (or \(5.5\%\)) for the third year. We need to calculate the amount at the end of each year sequentially and round the result to two decimal places where specified.
- Calculating the Amount After the First Year
The amount at the end of the first year, \(A_1\), is calculated by applying the \(3\%\) interest rate to the initial principal:
\[A_1 = P_0 \times (1 + 0.03) = 11,960,000 \times 1.03 = 12,318,800.00\]
Thus, the amount after the first year is \(\$12,318,800.00\).
- Calculating the Amount After the Second Year
Next, we apply the \(4\%\) interest rate to the amount at the end of the first year to find the amount at the end of the second year, \(A_2\):
\[A_2 = A_1 \times (1 + 0.04) = 12,318,800.00 \times 1.04 = 12,811,552.00\]
Rounding to two decimal places, we get \(\$12,811,552.00\).
- Calculating the Final Amount After the Third Year
Finally, we apply the \(5.5\%\) interest rate to the amount at the end of the second year to find the final amount, \(A_3\):
\[A_3 = A_2 \times (1 + 0.055) = 12,811,552.00 \times 1.055 = 13,516,187.36\]
Rounding to two decimal places, the final amount is \(\$13,516,187.36\). The total accrued amount after three years is \(\boxed{\$13,516,187.36}\).
Examples
In finance, calculating interest over varying periods is essential for managing savings accounts or investment portfolios. For instance, if you invest money in a savings account where the interest rates are tied to fluctuating market conditions, you would perform similar step-by-step calculations. This process helps you project future wealth and understand how compounding interest, even with changing rates, contributes to the growth of your capital over time. This foundational knowledge is crucial for making informed financial decisions.