Complete the following using compound future value. (Use the Table provided.) Note: Do not round intermediate calculations. Round your final answers to the nearest cent.
| Time | Principal | Rate | Compounded | Amount | Interest |
|---|---|---|---|---|---|
| 3 years | $17,100 | 6 | % | Quarterly | |
Answer (2)
Calculate the quarterly interest rate: i = 4 0.06 = 0.015 .
Calculate the total number of compounding periods: N = 4 × 3 = 12 .
Calculate the amount using the compound interest formula: A = 17100 ( 1 + 0.015 ) 12 = 20445.07 .
Calculate the interest earned: I n t eres t = 20445.07 − 17100 = 3345.07 . The amount is $20 , 445.07 and the interest is $3 , 345.07 .
Explanation
Understanding the Problem We are given a principal amount of $17 , 100 that is invested for 3 years at an annual interest rate of 6%, compounded quarterly. We need to find the amount after 3 years and the interest earned.
Calculating Quarterly Interest Rate First, we need to calculate the quarterly interest rate. The annual interest rate is 6%, so the quarterly interest rate is 4 6% = 4 0.06 = 0.015 .
Calculating Total Compounding Periods Next, we need to calculate the total number of compounding periods. Since the interest is compounded quarterly for 3 years, the total number of compounding periods is 4 × 3 = 12 .
Calculating the Amount Now, we can use the compound interest formula to calculate the amount (A) after 3 years: A = P ( 1 + i ) N , where P is the principal amount, i is the quarterly interest rate, and N is the total number of compounding periods. Plugging in the values, we get: A = 17100 ( 1 + 0.015 ) 12 = 17100 ( 1.015 ) 12 ≈ 17100 × 1.195618 ≈ 20445.07 .
Calculating the Interest Earned Finally, we can calculate the interest earned by subtracting the principal amount from the amount after 3 years: I n t eres t = A − P = 20445.07 − 17100 = 3345.07 .
Final Answer Therefore, the amount after 3 years is $20 , 445.07 and the interest earned is $3 , 345.07 .
Examples
Compound interest is a powerful tool for growing wealth over time. For example, if you invest $10 , 000 in a retirement account that earns an average annual return of 7% compounded annually, after 30 years, your investment would grow to approximately $76 , 122.55 . This demonstrates the importance of starting to invest early and taking advantage of the power of compounding. Understanding compound interest can help you make informed decisions about your savings and investments, enabling you to reach your financial goals more effectively. The formula for compound interest, A = P ( 1 + i ) N , allows you to calculate the future value of an investment based on the principal, interest rate, and compounding frequency.
After 3 years, the total amount will be approximately $20,445.07, with earned interest of about $3,345.07 from an initial investment of $17,100 at a 6% annual interest rate compounded quarterly.
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