High School

Kendra took out a loan for [tex]\$750[/tex] at an [tex]8.4\%[/tex] APR, compounded monthly, to buy a stereo. If she will make monthly payments of [tex]\$46.50[/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 she will have to make?

A. [tex]N=?; I\%=0.7; PV=-750; PMT=46.5; FV=0; P/Y=12; C/Y=12[/tex]; PMT:END

B. [tex]N=?; I\%=0.7; PV=-750; PMT=46.5; FV=0; P/Y=1; C/Y=12[/tex]; PMT:END

C. [tex]N=?; I\%=8.4; PV=-750; PMT=46.5; FV=0; P/Y=1; C/Y=12[/tex]; PMT:END

D. [tex]N=?; I\%=8.4; PV=-750; PMT=46.5; FV=0; P/Y=12; C/Y=12[/tex]; PMT:END

Answer :

To determine which group of values can be used in the TVM Solver to calculate the number of payments Kendra will need to make for her loan, we need to break down the given options and identify the correct one.

Kendra's loan details are:
- Loan amount (Present Value, PV) = [tex]$750 (borrowed, so it's typically negative in TVM calculations)
- Annual Percentage Rate (APR) = 8.4%
- Monthly payment (PMT) = $[/tex]46.50
- The goal is to calculate the number of payments necessary to pay off the loan, which means we should end with a Future Value (FV) of $0.
- Compounded monthly, so payments per year (P/Y) = 12 and compounding periods per year (C/Y) = 12.

Step-by-Step Analysis:

1. Interest Rate Per Period:
- Since the APR is split across 12 months, the monthly interest rate needs to be calculated by dividing the annual rate by the number of compounding periods per year.
- Interest rate per month = [tex]\( \frac{8.4\%}{12} = 0.7\% \)[/tex]

2. Match the Information to the TVM Solver Options:
- Option A:
- [tex]\( I\% = 0.7\)[/tex], which matches our calculated monthly interest rate.
- [tex]\( PV = -750 \)[/tex] (borrowed amount, hence negative)
- [tex]\( PMT = 46.5 \)[/tex] (monthly payment)
- [tex]\( FV = 0 \)[/tex] (goal is to fully repay the loan)
- [tex]\( P/Y = 12 \)[/tex], [tex]\( C/Y = 12 \)[/tex] (monthly payments and monthly compounding)
- Setting payments at the end of the month, typically PMT is set to END, which is common.

3. Verifying Other Options:
- Option B has [tex]\( I\% = 0.7 \)[/tex] correctly but [tex]\( P/Y = 1 \)[/tex], which is incorrect as payments are monthly, not yearly.
- Option C has [tex]\( I\% = 8.4 \)[/tex], which is incorrect since it represents the annual rate and not the monthly one.
- Option D has the correct [tex]\( P/Y \)[/tex] and [tex]\( C/Y \)[/tex] but uses the annual interest rate [tex]\( I\% = 8.4 \)[/tex], which doesn't match the monthly rate needed.

Thus, using the given details, Option A is the correct set of values to plug into the TVM Solver to calculate the number of payments Kendra will need to make.

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