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Homer took out a 6-month loan for [tex][tex]$\$[/tex]700[tex]$[/tex] at an appliance store to be paid back with monthly payments at a [tex]$[/tex]20.4 \%$[/tex] APR, compounded monthly. If the loan offers no payments for the first 3 months, which of these groups of values plugged into the TVM Solver of a graphing calculator will give him the correct answer for the amount of the monthly payment over the last 3 months of the loan?

A. [tex]N=3 ; I \%=20.4 ; PV=-700 ; PMT= ; FV=0 ; P/Y=12 ; C/Y=12 ; PMT:END[/tex]

B. [tex]N=3 ; I \%=20.4 ; PV=-736.31 ; PMT= ; FV=0 ; P/Y=12 ; C/Y=12 ; PMT:END[/tex]

C. [tex]N=0.25 ; I \%=20.4 ; PV=-700 ; PMT= ; FV=0 ; P/Y=12 ; C/Y=12 ; PMT:END[/tex]

D. [tex]N=0.25 ; I \%=20.4 ; PV=-736.31 ; PMT= ; FV=0 ; P/Y=12 ; C/Y=12 ; PMT:END[/tex]

Sagot :

GinnyAnswer
To solve the problem of determining which group of values will give Homer the correct answer for the amount of the monthly payment over the last 3 months of his loan, let's follow the steps outlined below:

### Step 1: Understand the Terms and Context
- Principal (loan amount): [tex]$700$[/tex]
- Annual Percentage Rate (APR): [tex]$20.4\%$[/tex]
- Loan duration: 6 months
- Payments start after 3 months (no payments for the first 3 months)

### Step 2: Convert APR to Monthly Interest Rate
Since the APR is compounded monthly, we need the monthly interest rate:
[tex]\[ \text{Monthly Interest Rate} = \frac{20.4\%}{12} = \frac{20.4}{100 \times 12} = 0.017 \][/tex]

### Step 3: Calculate the Amount After First 3 Months
After the first 3 months with no payments, the amount of the loan increases due to the interest compounding monthly:
[tex]\[ \text{Amount After 3 Months} = \$700 \times (1 + 0.017)^3 \][/tex]

After calculating:
[tex]\[ \text{Amount After 3 Months} = 700 \times 1.017^3 \approx 736.31 \][/tex]

### Step 4: Use the TVM Solver for Monthly Payments
For the remaining 3 months, Homer needs to pay off this increased amount [tex]\(736.31\)[/tex].

The key values to input into the TVM (Time Value of Money) solver are:
- [tex]\( N = 3 \)[/tex] (Number of payments is 3)
- [tex]\( I\% = 20.4 \)[/tex] (Annual Interest Rate)
- [tex]\( PV = -736.31 \)[/tex] (Present Value after first 3 months, negative because it's an outflow from Homer's perspective)
- [tex]\( PMT = \)[/tex] (This is the value we are solving for)
- [tex]\( FV = 0 \)[/tex] (Future Value at the end of the loan, which should be zero)
- [tex]\( P/Y = 12 \)[/tex] (Payments Per Year)
- [tex]\( C/Y = 12 \)[/tex] (Compounding Periods Per Year)
- [tex]\( PMT: END \)[/tex] (Payments are assumed to be made at the end of each period)

### Step 5: Verification
Given these key values:
[tex]\[ \text{Group B: } N = 3, I\% = 20.4, PV = -736.31, \text{PMT = }, FV = 0, P/Y = 12, C/Y = 12, PMT: END \][/tex]

Thus, the correct choice that would give Homer the amount of the monthly payment over the last 3 months is:
[tex]\[ \boxed{B. \, N=3; I\%=20.4; PV=-736.31; PMT=; FV=0; P/Y=12; C/Y=12; PMT: END} \][/tex]
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