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Troy took out a loan for [tex]$\$1850$[/tex] at a [tex]$9.6\%$[/tex] APR, compounded monthly, to buy a television. If he will make monthly payments of [tex]$\[tex]$102.50$[/tex][/tex] to pay off the loan, which of these groups of values plugged into the TVM Solver of a graphing calculator could be used to calculate the number of payments he will have to make?

A. [tex]N = \, \text{ } \, I\%=0.8 \, ; \, PV=-1850 \, ; \, PMT=102.5 \, ; \, FV=0 \, ; \, P/Y=12 \, ; \, C/Y=12 \, ; \, PMT:\,END[/tex]

B. [tex]N = \, \text{ } \, I\%=9.6 \, ; \, PV=-1850 \, ; \, PMT=102.5 \, ; \, FV=0 \, ; \, P/Y=1 \, ; \, C/Y=12 \, ; \, PMT:\,END[/tex]

C. [tex]N = \, \text{ } \, I\%=0.8 \, ; \, PV=-1850 \, ; \, PMT=102.5 \, ; \, FV=0 \, ; \, P/Y=1 \, ; \, C/Y=12 \, ; \, PMT:\,END[/tex]

D. [tex]N = \, \text{ } \, I\%=9.6 \, ; \, PV=-1850 \, ; \, PMT=102.5 \, ; \, FV=0 \, ; \, P/Y=12 \, ; \, C/Y=12 \, ; \, PMT:\,END[/tex]



Answer :

GinnyAnswer
To determine the correct group of values to use in the TVM Solver to calculate the number of payments Troy will have to make to pay off his loan, we need to carefully analyze the given parameters for the problem:

1. Annual Percentage Rate (APR):
- The loan has an APR of [tex]\(9.6\% \)[/tex], which is an annual interest rate.

2. Compounding Monthly:
- Since the interest is compounded monthly, the monthly interest rate ([tex]\(i\%\)[/tex]) will be the annual rate divided by 12.
- Monthly interest rate [tex]\(= \frac{9.6\%}{12} = 0.8\%\)[/tex].

3. Present Value (PV):
- The present value of the loan is [tex]\(\$1850\)[/tex].
- Since this amount is a loan taken out (i.e., an outgoing payment), it will be considered as [tex]\(-1850\)[/tex] in the TVM equations.

4. Monthly Payment (PMT):
- The monthly payment Troy makes is [tex]\(\$102.50\)[/tex].

5. Future Value (FV):
- The future value of the loan, when it is fully paid off, should be [tex]\(\$0\)[/tex].

6. Payments per Year (P/Y) and Compounding Periods per Year (C/Y):
- Since the payments are monthly, there will be 12 payments per year.
- Since interest is compounded monthly, there will also be 12 compounding periods per year.

Given the above analysis, let's compare the options provided:

- Option A:
- [tex]\(N=\)[/tex]
- [tex]\(i\% = 0.8\)[/tex]
- [tex]\(PV = -1850\)[/tex]
- [tex]\(PMT = 102.5\)[/tex]
- [tex]\(FV = 0\)[/tex]
- [tex]\(P/Y = 12\)[/tex]
- [tex]\(C/Y = 12\)[/tex]
- [tex]\(PMT : END\)[/tex]

- Option B:
- [tex]\(N=\)[/tex]
- [tex]\(i\% = 9.6\)[/tex]
- [tex]\(PV = -1850\)[/tex]
- [tex]\(PMT = 102.5\)[/tex]
- [tex]\(FV = 0\)[/tex]
- [tex]\(P/Y = 1\)[/tex]
- [tex]\(C/Y = 12\)[/tex]
- [tex]\(PMT : END\)[/tex]

- Option C:
- [tex]\(N=\)[/tex]
- [tex]\(i\% = 0.8\)[/tex]
- [tex]\(PV = -1850\)[/tex]
- [tex]\(PMT = 102.5\)[/tex]
- [tex]\(FV = 0\)[/tex]
- [tex]\(P/Y = 1\)[/tex]
- [tex]\(C/Y = 12\)[/tex]
- [tex]\(PMT : END\)[/tex]

- Option D:
- [tex]\(N=\)[/tex]
- [tex]\(i\% = 9.6\)[/tex]
- [tex]\(PV = -1850\)[/tex]
- [tex]\(PMT = 102.5\)[/tex]
- [tex]\(FV = 0\)[/tex]
- [tex]\(P/Y = 12\)[/tex]
- [tex]\(C/Y = 12\)[/tex]
- [tex]\(PMT : END\)[/tex]

Analysis:
- Option A is correct because it accurately reflects the monthly interest rate ([tex]\(0.8\%\)[/tex]), the present value ([tex]\(-1850\)[/tex]), the payment amount ([tex]\(102.5\)[/tex]), the future value ([tex]\(0\)[/tex]), and both the payments per year and compounding periods per year are 12.
- Option D correctly uses the annual interest rate ([tex]\(9.6\%\)[/tex]), but while it adjusts payments per year correctly, it doesn't directly align with all our breakdowns since the [tex]\(TVM\)[/tex] solver typically requires specifying monthly units.

Given that the TVM solver components must work harmoniously for a typical financial calculator setup, the most reliable option here would ensure precise use of monthly break-ups:

Thus, Option A ([tex]\(i\% = 0.8\)[/tex], PV = -1850, PMT = 102.5, FV = 0, P/Y = 12, C/Y = 12, PMT:END\)) is the correct choice for solving the loan payment problem.

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