Questions: Suppose you found a CD that pays 2.7% interest compounded monthly for 5 years. If you deposit 11,000 now, how much will you have in the account in 5 years? (Round to the nearest cent.) What was the interest earned? (Round to the nearest cent.) Now suppose that you would like to have 20,000 in the account in 5 years. How much would you need to deposit now? (Round to the nearest cent.)

Solution Steps

1. To find the future value of the investment, use the formula for compound interest: \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (initial deposit), \( r \) is the annual interest rate (decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time in years.
2. To find the interest earned, subtract the principal from the future value.
3. To find the present value needed to achieve a future value of $20,000, rearrange the compound interest formula to solve for \( P \).

To find the future value \( A \) of the investment, we use the compound interest formula:

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

Substituting the given values:

• \( P = 11000 \)
• \( r = 0.027 \)
• \( n = 12 \)
• \( t = 5 \)

\[ A = 11000 \left(1 + \frac{0.027}{12}\right)^{12 \times 5} \approx 12588.0 \]

The interest earned is the difference between the future value and the principal:

\[ \text{Interest Earned} = A - P = 12588.0 - 11000 = 1588.0 \]

To find the present value \( P \) needed to achieve a future value of $20,000, we rearrange the compound interest formula:

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

Substituting the desired future value:

• \( A = 20000 \)

\[ P = \frac{20000}{\left(1 + \frac{0.027}{12}\right)^{12 \times 5}} \approx 17476.97 \]

Final Answer