To determine the present value [tex]\( P \)[/tex] you must invest to have a future value [tex]\( A \)[/tex] with a simple interest rate [tex]\( r \)[/tex] after time [tex]\( t \)[/tex], follow these steps:
1. Identify the given values:
- Future Value ([tex]\( A \)[/tex]): [tex]$5000.00
- Interest Rate (\( r \)): 13.0%
- Time (\( t \)): 13 weeks
2. Convert the interest rate from percentage to decimal:
- \( r = 13.0\% = 0.13 \)
3. Convert the time from weeks to years:
- There are approximately 52 weeks in a year.
- \( t = \frac{13\text{ weeks}}{52\text{ weeks/year}} = \frac{13}{52} = 0.25 \text{ years} \)
4. Use the simple interest formula to find the present value \( P \):
The future value formula for simple interest is:
\[
A = P(1 + rt)
\]
Rearrange this formula to solve for \( P \):
\[
P = \frac{A}{1 + rt}
\]
5. Plug in the known values to find \( P \):
\[
P = \frac{5000.00}{1 + (0.13 \times 0.25)}
\]
Calculate the denominator first:
\[
1 + (0.13 \times 0.25) = 1 + 0.0325 = 1.0325
\]
Now, divide the future value by this amount:
\[
P = \frac{5000.00}{1.0325} \approx 4842.6150121065375
\]
6. Round the present value to the nearest cent:
\[
P \approx 4842.62
\]
Therefore, to have a future value of $[/tex]5000.00 after 13 weeks at a simple interest rate of 13%, you need to invest approximately [tex]\( \boxed{4842.62} \)[/tex].
