To find the equivalent replacement payment due today for an original scheduled payment of [tex]$1400 that was due nine months ago with an interest rate of 8.5%, follow these steps:
1. Determine the original payment:
- The original payment that was due nine months ago is $[/tex]1400.
2. Convert the interest rate to a decimal:
- The interest rate provided is 8.5%. To use this in our calculations, we need to convert it to a decimal. Dividing 8.5 by 100 gives 0.085.
3. Convert the time period to years:
- The time period is given in months (nine months). To use this in our interest formula, convert months to years by dividing by 12. Therefore, nine months is equivalent to [tex]\( \frac{9}{12} \)[/tex] or 0.75 years.
4. Calculate the replacement payment using the present value formula for simple interest:
- The formula for the future value of a payment using simple interest is:
[tex]\[
\text{Replacement Payment} = \text{Original Payment} \times \left(1 + \text{Rate} \times \text{Time Period}\right)
\][/tex]
- Substituting in our values:
[tex]\[
\text{Replacement Payment} = 1400 \times \left(1 + 0.085 \times 0.75\right)
\][/tex]
- Calculate the term inside the parentheses first:
[tex]\[
0.085 \times 0.75 = 0.06375
\][/tex]
- Adding 1 to this result gives:
[tex]\[
1 + 0.06375 = 1.06375
\][/tex]
- Now multiply this by the original payment amount:
[tex]\[
1400 \times 1.06375 = 1489.25
\][/tex]
Thus, the equivalent replacement payment due today is $1489.25.
