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\% = 8.4 ; 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\% = 0.7 ; PV = -750 ; PMT = 46.5 ; FV = 0 ; P/Y = 12 ; C/Y = 12 [/tex]; PMT: END

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

Answer :

- Calculate the monthly interest rate by dividing the annual interest rate by 12: $8.4 \div 12 = 0.7$.
- Identify the correct inputs for the TVM Solver: PV (loan amount) as negative, PMT (monthly payment) as positive, FV (future value) as 0, P/Y (payments per year) as 12, and C/Y (compounding periods per year) as 12.
- Choose the option that uses the monthly interest rate and consistent P/Y and C/Y values.
- The correct option is C: $N =; 1 \%=0.7 ; PV =-750 ; PMT =46.5 ; FV =0 ; P / Y =12 ; C / Y =12$; PMT:END. $\boxed{C}$

### Explanation
1. Understanding the Problem
We need to determine the correct inputs for the TVM Solver on a graphing calculator to find the number of payments (N) required to pay off a loan. Let's analyze the given information:

* Loan amount (PV) = $750
* Annual interest rate (APR) = 8.4%, compounded monthly
* Monthly payment (PMT) = $46.50
* Future value (FV) = $0 (since the loan will be paid off)

We need to correctly set the values for I%, PV, PMT, FV, P/Y, and C/Y in the TVM Solver.

2. Calculating the Monthly Interest Rate
First, let's consider the interest rate. The APR is 8.4%, but since the loan is compounded monthly, we need to find the monthly interest rate. This is done by dividing the APR by 12:$$\frac{8.4}{12} = 0.7$$So, the monthly interest rate is 0.7%.

3. Analyzing the Options
Now, let's analyze each option:

* **Option A:** $N =; I\%=8.4 ; PV =-750 ; PMT =46.5 ; FV =0 ; P / Y =12 ; C / Y =12$; PMT:END
* This option uses the annual interest rate (8.4%) directly, but P/Y = 12, which means it's expecting an annual rate. This is incorrect.
* **Option B:** $N=; I\%=0.7 ; P V=-750 ; P M T=46.5 ; F V=0 ; P / Y=1 ; C / Y=12$; PMT:END
* This option uses the monthly interest rate (0.7%), but P/Y = 1. This means the interest rate should be the monthly rate. However, C/Y = 12, which is inconsistent with P/Y = 1. This is incorrect.
* **Option C:** $N =; I\%=0.7 ; PV =-750 ; PMT =46.5 ; FV =0 ; P / Y =12 ; C / Y =12$; PMT:END
* This option uses the monthly interest rate (0.7%), and P/Y = 12 and C/Y = 12, which is consistent with monthly compounding and payments. PV is negative as it's money received, and PMT is positive as it's money paid out. This is the correct setup.
* **Option D:** $N =; I\%=8.4 ; PV =-750 ; PMT =46.5 ; FV =0 ; P / Y =1 ; C / Y =12$; PMT:END
* This option uses the annual interest rate (8.4%) and P/Y = 1. However, C/Y = 12, which is inconsistent. This is incorrect.

4. Conclusion
Therefore, the correct group of values is option C.

### Examples
Understanding loan calculations is crucial in personal finance. For instance, when buying a car or a house, you need to understand how the interest rate, loan term, and monthly payments affect the total amount you pay over time. Using tools like the TVM Solver helps you make informed decisions and plan your budget effectively.