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Sagot :
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]
### 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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