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Sagot :
Alright, let's break down the problem step-by-step:
1. Price of the boat including tax: \[tex]$5,275.00 2. Down payment: \$[/tex]500
3. Time financed over: 36 months
4. Annual interest rate: 15%
### Step-by-Step Solution:
#### Step 1: Calculate the amount financed
The amount financed is the price of the boat minus the down payment:
[tex]\[ \text{Amount Financed} = \$5,275.00 - \$500 = \$4,775.00 \][/tex]
#### Step 2: Convert the annual interest rate to a monthly interest rate
The monthly interest rate is calculated by dividing the annual interest rate by 12:
[tex]\[ \text{Monthly Interest Rate} = \frac{15\%}{12} = \frac{0.15}{12} \approx 0.0125 \][/tex]
#### Step 3: Calculate the monthly payment using the formula for an installment loan
The formula for the monthly payment for an installment loan is:
[tex]\[ c = \frac{\text{Amount Financed} \times \text{Monthly Interest Rate}}{1 - (1 + \text{Monthly Interest Rate})^{-\text{Number of Months}}} \][/tex]
Plugging in the values:
[tex]\[ c = \frac{4775 \times 0.0125}{1 - (1 + 0.0125)^{-36}} \approx 165.53 \][/tex]
So, the monthly payment [tex]\( c \)[/tex] is [tex]\( \$165.53 \)[/tex].
#### Step 4: Calculate the total of the payments
Total of payments is the monthly payment multiplied by the number of months:
[tex]\[ \text{Total Payments} = 36 \times 165.53 \approx 5958.97 \][/tex]
#### Step 5: Verify the total of the payments divided by the number of payments equals the monthly payment
[tex]\[ \frac{5958.97}{36} \approx 165.53 \][/tex]
### Answers:
- To the nearest penny, [tex]\( c = \$165.53 \)[/tex]
- Total of payments = amount financed + [tex]\(c = 4775 + 165.53 = \$5958.97\)[/tex]
- Total of payments divided by number of payments = monthly payment = [tex]\( \frac{5958.97}{36} = \$165.53 \)[/tex]
So the detailed answers are:
- Monthly Payments (principal plus interest):
[tex]\[ c = \$165.53 \][/tex]
- Total of Payments:
[tex]\[ \text{Total of payments} = 5958.97 \][/tex]
- Verification of Monthly Payment:
[tex]\[ \frac{5958.97}{36} = \text{Monthly Payment} = \$165.53 \][/tex]
1. Price of the boat including tax: \[tex]$5,275.00 2. Down payment: \$[/tex]500
3. Time financed over: 36 months
4. Annual interest rate: 15%
### Step-by-Step Solution:
#### Step 1: Calculate the amount financed
The amount financed is the price of the boat minus the down payment:
[tex]\[ \text{Amount Financed} = \$5,275.00 - \$500 = \$4,775.00 \][/tex]
#### Step 2: Convert the annual interest rate to a monthly interest rate
The monthly interest rate is calculated by dividing the annual interest rate by 12:
[tex]\[ \text{Monthly Interest Rate} = \frac{15\%}{12} = \frac{0.15}{12} \approx 0.0125 \][/tex]
#### Step 3: Calculate the monthly payment using the formula for an installment loan
The formula for the monthly payment for an installment loan is:
[tex]\[ c = \frac{\text{Amount Financed} \times \text{Monthly Interest Rate}}{1 - (1 + \text{Monthly Interest Rate})^{-\text{Number of Months}}} \][/tex]
Plugging in the values:
[tex]\[ c = \frac{4775 \times 0.0125}{1 - (1 + 0.0125)^{-36}} \approx 165.53 \][/tex]
So, the monthly payment [tex]\( c \)[/tex] is [tex]\( \$165.53 \)[/tex].
#### Step 4: Calculate the total of the payments
Total of payments is the monthly payment multiplied by the number of months:
[tex]\[ \text{Total Payments} = 36 \times 165.53 \approx 5958.97 \][/tex]
#### Step 5: Verify the total of the payments divided by the number of payments equals the monthly payment
[tex]\[ \frac{5958.97}{36} \approx 165.53 \][/tex]
### Answers:
- To the nearest penny, [tex]\( c = \$165.53 \)[/tex]
- Total of payments = amount financed + [tex]\(c = 4775 + 165.53 = \$5958.97\)[/tex]
- Total of payments divided by number of payments = monthly payment = [tex]\( \frac{5958.97}{36} = \$165.53 \)[/tex]
So the detailed answers are:
- Monthly Payments (principal plus interest):
[tex]\[ c = \$165.53 \][/tex]
- Total of Payments:
[tex]\[ \text{Total of payments} = 5958.97 \][/tex]
- Verification of Monthly Payment:
[tex]\[ \frac{5958.97}{36} = \text{Monthly Payment} = \$165.53 \][/tex]
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