Questions: Calculate, to the nearest cent, the present value of an investment that will be worth 1,000 at the stated interest rate after the stated amount of time. Hin 6 years, at 5.8% per year, compounded weekly (assume 52 weeks per year)

Solution Steps

To find the present value of an investment, we use the formula for present value with compound interest:

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

where:

• \( FV \) is the future value of the investment (\$1,000 in this case),
• \( r \) is the annual interest rate (5.8% or 0.058),
• \( n \) is the number of compounding periods per year (52 for weekly),
• \( t \) is the number of years (6 years).

We will plug these values into the formula to calculate the present value.

We are given the following values:

• Future Value (\( FV \)) = \$1,000
• Annual interest rate (\( r \)) = 5.8% = 0.058
• Compounding periods per year (\( n \)) = 52 (weekly)
• Time in years (\( t \)) = 6

The present value (\( PV \)) can be calculated using the formula:

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

Substituting the known values into the formula:

\[ PV = \frac{1000}{\left(1 + \frac{0.058}{52}\right)^{52 \times 6}} \]

Calculating the expression step-by-step:

1. Calculate \( \frac{r}{n} \): \[ \frac{0.058}{52} \approx 0.0011153846 \]

2. Calculate \( 1 + \frac{r}{n} \): \[ 1 + 0.0011153846 \approx 1.0011153846 \]

3. Calculate \( nt \): \[ 52 \times 6 = 312 \]

4. Raise \( 1 + \frac{r}{n} \) to the power of \( nt \): \[ (1.0011153846)^{312} \approx 1.233 \]

5. Finally, calculate \( PV \): \[ PV = \frac{1000}{1.233} \approx 810.65 \]

After rounding to the nearest cent, we find:

\[ PV \approx 706.24 \]

Final Answer