To determine the present value [tex]\( P \)[/tex] that must be invested to have a future value [tex]\( A \)[/tex] at a given simple interest rate [tex]\( r \)[/tex] after time [tex]\( t \)[/tex], we use the formula for future value in simple interest, which is:
[tex]\[ A = P(1 + rt) \][/tex]
Here, [tex]\( A \)[/tex] is the future value, [tex]\( P \)[/tex] is the present value, [tex]\( r \)[/tex] is the interest rate per period (as a decimal), and [tex]\( t \)[/tex] is the time in years.
Given:
- [tex]\( A = \$6500 \)[/tex]
- [tex]\( r = 4.5\% = 0.045 \)[/tex] (converted from percent to decimal)
- [tex]\( t = 6 \)[/tex] years
To find the present value [tex]\( P \)[/tex], we rearrange the formula to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{A}{1 + rt} \][/tex]
Substitute the known values into the formula:
[tex]\[ P = \frac{6500}{1 + 0.045 \times 6} \][/tex]
Calculate the denominator:
[tex]\[ 1 + 0.045 \times 6 = 1 + 0.27 = 1.27 \][/tex]
Now, divide the future value by this result:
[tex]\[ P = \frac{6500}{1.27} \][/tex]
Perform the division:
[tex]\[ P \approx 5118.11 \][/tex]
Thus, the present value [tex]\( P \)[/tex] that must be invested is approximately \[tex]$5118.11 (rounded to the nearest cent).
Therefore, the present value \( P \) that must be invested is \$[/tex]5118.11.
