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Sagot :
Let's break down the problem step by step to determine what percent of the total amount paid will be interest.
### Step 1: Calculate the total initial cost
List Price:
[tex]\[ \$21,450 \][/tex]
Sales Tax Rate:
[tex]\[ 6.88\% \][/tex]
[tex]\[ \text{Sales Tax} = 0.0688 \times \$21,450 = \$1,475.76 \][/tex]
Vehicle Registration Fee:
[tex]\[ \$1,089 \][/tex]
Documentation Fee:
[tex]\[ \$124 \][/tex]
Total Initial Cost:
[tex]\[ \text{Total Initial Cost} = \text{List Price} + \text{Sales Tax} + \text{Registration Fee} + \text{Documentation Fee} \][/tex]
[tex]\[ \text{Total Initial Cost} = \$21,450 + \$1,475.76 + \$1,089 + \$124 \][/tex]
[tex]\[ \text{Total Initial Cost} = \$24,138.76 \][/tex]
### Step 2: Calculate trade-in value reduction
Trade-in Value of 2004 Dodge Neon:
[tex]\[ \$6,591 \][/tex]
[tex]\[ 80\% \text{ of Trade-in Value} = 0.80 \times \$6,591 = \$5,272.80 \][/tex]
Financed Amount:
[tex]\[ \text{Financed Amount} = \text{Total Initial Cost} - \text{Trade-in Reduction} \][/tex]
[tex]\[ \text{Financed Amount} = \$24,138.76 - \$5,272.80 \][/tex]
[tex]\[ \text{Financed Amount} = \$18,865.96 \][/tex]
### Step 3: Calculate the monthly payment using the loan amortization formula
Interest Rate:
[tex]\[ 12.28\% \][/tex]
[tex]\[ \text{Annual Interest Rate} = 0.1228 \][/tex]
[tex]\[ \text{Monthly Interest Rate} = \frac{0.1228}{12} = 0.010233 \][/tex]
Finance Period:
[tex]\[ 3 \text{ years} = 3 \times 12 \text{ months} = 36 \text{ months} \][/tex]
Monthly Payment Formula:
[tex]\[ M = P \times \frac{r (1+r)^n}{(1+r)^n - 1} \][/tex]
[tex]\[ M = \$18,865.96 \times \frac{0.010233(1+0.010233)^{36}}{(1+0.010233)^{36} - 1} \][/tex]
First, compute the terms:
[tex]\[ (1 + 0.010233)^{36} = (1.010233)^{36} \approx 1.42879 \][/tex]
Now plug in the values:
[tex]\[ M = \$18,865.96 \times \frac{0.010233 \times 1.42879}{1.42879 - 1} \][/tex]
[tex]\[ M = \$18,865.96 \times \frac{0.014626}{0.42879} \][/tex]
[tex]\[ M \approx \$18,865.96 \times 0.034117 \][/tex]
[tex]\[ M \approx \$643.71 \][/tex]
### Step 4: Calculate the total payment over 3 years
Total Paid Over 3 Years:
[tex]\[ \text{Total Paid} = \text{Monthly Payment} \times 36 \][/tex]
[tex]\[ \text{Total Paid} = \$643.71 \times 36 \][/tex]
[tex]\[ \text{Total Paid} = \$23,113.56 \][/tex]
### Step 5: Calculate the total interest paid
Total Interest Paid:
[tex]\[ \text{Total Interest Paid} = \text{Total Paid} - \text{Financed Amount} \][/tex]
[tex]\[ \text{Total Interest Paid} = \$23,113.56 - \$18,865.96 \][/tex]
[tex]\[ \text{Total Interest Paid} = \$4,247.60 \][/tex]
### Step 6: Calculate the percentage of the total amount paid that is interest
[tex]\[ \text{Interest Percentage} = \left( \frac{\text{Total Interest Paid}}{\text{Total Paid}} \right) \times 100 \][/tex]
[tex]\[ \text{Interest Percentage} = \left( \frac{4,247.60}{23,113.56} \right) \times 100 \][/tex]
[tex]\[ \text{Interest Percentage} \approx 18.38\% \][/tex]
Thus, the percent of the total amount paid that will be interest is closest to:
[tex]\[ a. 17.64\% \][/tex]
### Step 1: Calculate the total initial cost
List Price:
[tex]\[ \$21,450 \][/tex]
Sales Tax Rate:
[tex]\[ 6.88\% \][/tex]
[tex]\[ \text{Sales Tax} = 0.0688 \times \$21,450 = \$1,475.76 \][/tex]
Vehicle Registration Fee:
[tex]\[ \$1,089 \][/tex]
Documentation Fee:
[tex]\[ \$124 \][/tex]
Total Initial Cost:
[tex]\[ \text{Total Initial Cost} = \text{List Price} + \text{Sales Tax} + \text{Registration Fee} + \text{Documentation Fee} \][/tex]
[tex]\[ \text{Total Initial Cost} = \$21,450 + \$1,475.76 + \$1,089 + \$124 \][/tex]
[tex]\[ \text{Total Initial Cost} = \$24,138.76 \][/tex]
### Step 2: Calculate trade-in value reduction
Trade-in Value of 2004 Dodge Neon:
[tex]\[ \$6,591 \][/tex]
[tex]\[ 80\% \text{ of Trade-in Value} = 0.80 \times \$6,591 = \$5,272.80 \][/tex]
Financed Amount:
[tex]\[ \text{Financed Amount} = \text{Total Initial Cost} - \text{Trade-in Reduction} \][/tex]
[tex]\[ \text{Financed Amount} = \$24,138.76 - \$5,272.80 \][/tex]
[tex]\[ \text{Financed Amount} = \$18,865.96 \][/tex]
### Step 3: Calculate the monthly payment using the loan amortization formula
Interest Rate:
[tex]\[ 12.28\% \][/tex]
[tex]\[ \text{Annual Interest Rate} = 0.1228 \][/tex]
[tex]\[ \text{Monthly Interest Rate} = \frac{0.1228}{12} = 0.010233 \][/tex]
Finance Period:
[tex]\[ 3 \text{ years} = 3 \times 12 \text{ months} = 36 \text{ months} \][/tex]
Monthly Payment Formula:
[tex]\[ M = P \times \frac{r (1+r)^n}{(1+r)^n - 1} \][/tex]
[tex]\[ M = \$18,865.96 \times \frac{0.010233(1+0.010233)^{36}}{(1+0.010233)^{36} - 1} \][/tex]
First, compute the terms:
[tex]\[ (1 + 0.010233)^{36} = (1.010233)^{36} \approx 1.42879 \][/tex]
Now plug in the values:
[tex]\[ M = \$18,865.96 \times \frac{0.010233 \times 1.42879}{1.42879 - 1} \][/tex]
[tex]\[ M = \$18,865.96 \times \frac{0.014626}{0.42879} \][/tex]
[tex]\[ M \approx \$18,865.96 \times 0.034117 \][/tex]
[tex]\[ M \approx \$643.71 \][/tex]
### Step 4: Calculate the total payment over 3 years
Total Paid Over 3 Years:
[tex]\[ \text{Total Paid} = \text{Monthly Payment} \times 36 \][/tex]
[tex]\[ \text{Total Paid} = \$643.71 \times 36 \][/tex]
[tex]\[ \text{Total Paid} = \$23,113.56 \][/tex]
### Step 5: Calculate the total interest paid
Total Interest Paid:
[tex]\[ \text{Total Interest Paid} = \text{Total Paid} - \text{Financed Amount} \][/tex]
[tex]\[ \text{Total Interest Paid} = \$23,113.56 - \$18,865.96 \][/tex]
[tex]\[ \text{Total Interest Paid} = \$4,247.60 \][/tex]
### Step 6: Calculate the percentage of the total amount paid that is interest
[tex]\[ \text{Interest Percentage} = \left( \frac{\text{Total Interest Paid}}{\text{Total Paid}} \right) \times 100 \][/tex]
[tex]\[ \text{Interest Percentage} = \left( \frac{4,247.60}{23,113.56} \right) \times 100 \][/tex]
[tex]\[ \text{Interest Percentage} \approx 18.38\% \][/tex]
Thus, the percent of the total amount paid that will be interest is closest to:
[tex]\[ a. 17.64\% \][/tex]
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