Complete the following using compound future value. (Use the Table 12.1 provided.) Note: Round your answers to the nearest cent.
| Time | Principal | Rate | Compounded | Amount |
| :------ | :-------- | :-------- | :--------- | :----- |
| 12 years | $15,000 | 3 1/2 % | Annually | |
Answer (2)
Convert the annual interest rate to decimal form: r = 100 3.5 = 0.035 .
Apply the compound interest formula: A = P ( 1 + r ) n , where P = 15000 and n = 12 .
Substitute the values into the formula: A = 15000 ( 1 + 0.035 ) 12 .
Calculate the future value and round to the nearest cent: A ≈ 22666.03 .
Explanation
Understanding the Problem We are given a principal amount of $15,000 that is invested for 12 years at an annual interest rate of 3.5%, compounded annually. We need to find the amount (future value) after 12 years.
Identifying the Formula To solve this, we will use the compound interest formula: A = P ( 1 + r ) n where:
A 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 (as a decimal)
n is the number of years the money is invested or borrowed for
Converting Interest Rate First, convert the interest rate to a decimal: r = 100 3.5 = 0.035
Substituting Values Now, substitute the given values into the formula: A = 15000 ( 1 + 0.035 ) 12 A = 15000 ( 1.035 ) 12
Calculating the Power Calculate ( 1.035 ) 12 :
( 1.035 ) 12 ≈ 1.5110686
Finding the Future Value Multiply the result by the principal amount: A = 15000 × 1.5110686 ≈ 22666.029
Rounding the Answer Round the final amount to the nearest cent: A ≈ 22666.03
Final Answer Therefore, the amount after 12 years is approximately $22,666.03.
Examples
Compound interest is a powerful tool for growing wealth over time. For example, if you invest $10,000 in a retirement account with an average annual return of 7%, compounded annually, after 30 years, your investment would grow significantly. Understanding compound interest helps in making informed financial decisions, such as planning for retirement, saving for a down payment on a house, or investing in the stock market. The longer the investment period and the higher the interest rate, the greater the impact of compounding.
The future value of an investment of $15,000 at an interest rate of 3.5% compounded annually for 12 years is approximately $22,666.03. This calculation uses the compound interest formula, taking into account the principal amount, interest rate converted to decimal, and the number of years. Thus, the final amount reflects the growth over the investment period.
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