Questions: Find the present value for the following future amount. 9810 at 2.5% compounded semiannually for 11 years The present value is (Do not round until the final answer. Then round to the nearest cent as needed.)

Solution Steps

To find the present value of a future amount compounded semiannually, we use the present value formula for compound interest:

\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]

where:

• \( PV \) is the present value,
• \( FV \) is the future value (\$9810 in this case),
• \( r \) is the annual interest rate (2.5% or 0.025),
• \( n \) is the number of compounding periods per year (2 for semiannual),
• \( t \) is the number of years (11 years).

We are given the future value \( FV = 9810 \), the annual interest rate \( r = 0.025 \), the number of compounding periods per year \( n = 2 \) (since it is compounded semiannually), and the time in years \( t = 11 \).

To find the present value, we use the formula for compound interest:

\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]

Substituting the given values into the formula:

\[ PV = \frac{9810}{(1 + \frac{0.025}{2})^{2 \times 11}} \]

First, calculate the base of the exponent:

\[ 1 + \frac{0.025}{2} = 1.0125 \]

Next, calculate the exponent:

\[ 2 \times 11 = 22 \]

Now, raise the base to the power of the exponent:

\[ 1.0125^{22} \approx 1.314 \]

Finally, calculate the present value:

\[ PV = \frac{9810}{1.314} \approx 7464.1147 \]

Round the present value to the nearest cent:

\[ PV \approx 7464.11 \]

Final Answer