To calculate the total cost of the auto loan, the fixed monthly payment, and the total interest paid, we can use the loan formula for monthly payments:
\[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \(M\) = monthly payment
- \(P\) = principal loan amount ($15,000)
- \(r\) = monthly interest rate (annual interest rate / 12 = 0.05 / 12)
- \(n\) = number of payments (loan term in months = 60)
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Calculating \(r\): \[ r = \frac{0.05}{12} \approx 0.00416667 \]
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Calculating \(M\): \[ M = 15000 \cdot \frac{0.00416667(1 + 0.00416667)^{60}}{(1 + 0.00416667)^{60} - 1} \]
First, we calculate \((1 + r)^{60}\): \[ (1 + 0.00416667)^{60} \approx 1.28368 \]
Now substituting back into the payment formula: \[ M = 15000 \cdot \frac{0.00416667 \cdot 1.28368}{1.28368 - 1} \] \[ M = 15000 \cdot \frac{0.00534749}{0.28368} \approx 15000 \cdot 0.0188346 \approx 282.52 \]
Rounding to the nearest dollar:
\[ M \approx 283 \] -
Calculating the Total Cost: The total cost of repaying the loan is the monthly payment multiplied by the number of months: \[ \text{Total Cost} = M \cdot n = 283 \cdot 60 = 16980 \]
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Calculating Total Interest Paid: The total interest paid is the total cost minus the principal: \[ \text{Total Interest} = \text{Total Cost} - P = 16980 - 15000 = 1980 \]
Final Results:
- The total cost of repaying the loan is $16,980.
- The fixed monthly payment amount is $283.
- The total amount of interest paid is $1,980.
So, the answers are:
- Total cost: $16,980
- Fixed monthly payment: $283
- Total interest paid: $1,980