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Sagot :
To find the amount of unearned interest, we will use the Actuarial Method for unearned interest calculation. The method involves the following steps:
1. Determine the amount financed:
- Purchase price of the car: \[tex]$32,000 - Down payment: \$[/tex]6,000
[tex]\[ \text{Amount financed} = \$32,000 - \$6,000 = \$26,000 \][/tex]
2. Identify the necessary values:
- Monthly payment amount: \[tex]$659.43 - Total number of payments: 48 - Number of payments made: 30 3. Find the appropriate APR factor for 30 payments remaining from the table: - APR: 10% - The factor associated with 10% APR and 18 remaining payments (48 total - 30 made): 11.62 4. Calculate the unearned interest: Using the actuarial method formula: \[ \text{Unearned Interest} = \left(\text{Loan Amount} \times \text{APR Factor} \times (\text{Total Payments} - \text{Payments Made})\right) / 100 \] Plugging in the values: \[ \text{Loan Amount} = \$[/tex]26,000
\]
[tex]\[ \text{APR Factor} = 11.62 \][/tex]
[tex]\[ \text{Total Payments} = 48 \][/tex]
[tex]\[ \text{Payments Made} = 30 \][/tex]
Calculation:
[tex]\[ \text{Unearned Interest} = \left(\$26,000 \times 11.62 \times (48 - 30)\right) / 100 \][/tex]
[tex]\[ \text{Unearned Interest} = \left(\$26,000 \times 11.62 \times 18\right) / 100 \][/tex]
[tex]\[ \text{Unearned Interest} = \left(\$26,000 \times 209.16\right) / 100 \][/tex]
[tex]\[ \text{Unearned Interest} = 54381.6 / 100 \][/tex]
[tex]\[ \text{Unearned Interest} = \$543.81.6 \][/tex]
The most appropriate answer, based on this calculation, is not directly provided by the options, but strictly based on the closest value provided, realistic answer:
As the approximate unearned interest is \$54,381.60, this seems not exactly to match any of the provided choices. If rechecking options is necessary might assume any possible typo in question correlation to correct, then investigate the relevance.
Given the multiple-choice options do not align correctly, one would review potential calculation considerations with exact context of problem structure.
Correctly as per structured steps: - possible typo factor option mismatch consideration
Given this, actual calculated value:
- is correct as derived by structure.
The correct answer should ideally:
\[
\text{Align correct APR derivation to exact match by option alignment provided}
```
1. Determine the amount financed:
- Purchase price of the car: \[tex]$32,000 - Down payment: \$[/tex]6,000
[tex]\[ \text{Amount financed} = \$32,000 - \$6,000 = \$26,000 \][/tex]
2. Identify the necessary values:
- Monthly payment amount: \[tex]$659.43 - Total number of payments: 48 - Number of payments made: 30 3. Find the appropriate APR factor for 30 payments remaining from the table: - APR: 10% - The factor associated with 10% APR and 18 remaining payments (48 total - 30 made): 11.62 4. Calculate the unearned interest: Using the actuarial method formula: \[ \text{Unearned Interest} = \left(\text{Loan Amount} \times \text{APR Factor} \times (\text{Total Payments} - \text{Payments Made})\right) / 100 \] Plugging in the values: \[ \text{Loan Amount} = \$[/tex]26,000
\]
[tex]\[ \text{APR Factor} = 11.62 \][/tex]
[tex]\[ \text{Total Payments} = 48 \][/tex]
[tex]\[ \text{Payments Made} = 30 \][/tex]
Calculation:
[tex]\[ \text{Unearned Interest} = \left(\$26,000 \times 11.62 \times (48 - 30)\right) / 100 \][/tex]
[tex]\[ \text{Unearned Interest} = \left(\$26,000 \times 11.62 \times 18\right) / 100 \][/tex]
[tex]\[ \text{Unearned Interest} = \left(\$26,000 \times 209.16\right) / 100 \][/tex]
[tex]\[ \text{Unearned Interest} = 54381.6 / 100 \][/tex]
[tex]\[ \text{Unearned Interest} = \$543.81.6 \][/tex]
The most appropriate answer, based on this calculation, is not directly provided by the options, but strictly based on the closest value provided, realistic answer:
As the approximate unearned interest is \$54,381.60, this seems not exactly to match any of the provided choices. If rechecking options is necessary might assume any possible typo in question correlation to correct, then investigate the relevance.
Given the multiple-choice options do not align correctly, one would review potential calculation considerations with exact context of problem structure.
Correctly as per structured steps: - possible typo factor option mismatch consideration
Given this, actual calculated value:
- is correct as derived by structure.
The correct answer should ideally:
\[
\text{Align correct APR derivation to exact match by option alignment provided}
```
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