Richard purchased a car for $39,905. He made a downpayment of $15,000 and paid $614 monthly for 45 years. Find the APR.

Richard was able to payoff the loan at the end of 30 months. Using the Actuarial method find he unearned interest, and payoff amount.

My answer 39905-15000=24905
614*48=29472
29472-24905=4567
4567/24905*100=18.34
The APR will be 9%

You must have a typo. Richard spent 45 years paying for a car?

Correction not 45 yrs but 4yrs

not that simple

let the monthly rate be i
614 (1 - (1+i)^-48)/i = 24905
very tough to solve
I will let Wolfram do it

https://www.wolframalpha.com/input/?i=sove+614+(1+-+(1%2Bx)%5E-48)%2Fx+%3D+24905

i= .00709283
12i = .08511... or 8.511% per annum compounded monthly

To find the APR (Annual Percentage Rate), we need to calculate the effective interest rate on the loan. We can do this by finding the total amount paid in interest over the loan term, and then dividing it by the principal borrowed.

First, let's calculate the total amount paid in interest over the loan term. We know Richard paid $614 monthly for 45 years (which is 540 months), and he made a downpayment of $15,000. So, the total amount paid towards the car would be:

Total amount paid towards the car = (Monthly payment * Number of months) + Downpayment
Total amount paid towards the car = ($614 * 540) + $15,000

Now, let's calculate the total amount paid in interest by subtracting the principal borrowed from the total amount paid towards the car:

Total amount paid in interest = Total amount paid towards the car - Principal borrowed
Total amount paid in interest = (Total amount paid towards the car) - $39,905

To find the APR, we need to divide the total amount paid in interest by the principal borrowed, and then multiply it by 100 to get a percentage:

APR = (Total amount paid in interest / Principal borrowed) * 100

Now, let's plug in the values:

Total amount paid towards the car = ($614 * 540) + $15,000
Total amount paid towards the car = $331,560 + $15,000
Total amount paid towards the car = $346,560

Total amount paid in interest = ($346,560) - $39,905
Total amount paid in interest = $306,655

APR = ($306,655 / $39,905) * 100
APR ≈ 768.98%

So, the APR for this car loan is approximately 768.98%.

Now, let's move on to the second part of the question, finding the unearned interest and payoff amount using the Actuarial method.

To find the unearned interest, we need to calculate the interest accrued on the remaining balance of the loan at the end of 30 months. We can do this by subtracting the principal paid off from the total interest paid over the loan term.

Principal paid off = Monthly payment * Number of months paid
Principal paid off = $614 * 30

Now, let's calculate the remaining balance by subtracting the principal paid off from the principal borrowed:

Remaining balance = Principal borrowed - Principal paid off
Remaining balance = $39,905 - (614 * 30)

To find the unearned interest, we need to calculate the interest accrued on the remaining balance. We'll use the formula for simple interest:

Interest accrued = Remaining balance * (APR / 100) * (Time / 12)

Where:
- APR is the annual interest rate (which we calculated above as 768.98%)
- Time is the remaining time in months (which is 540 - 30)

Now, let's plug in the values:

Remaining balance = $39,905 - (614 * 30)
Remaining balance ≈ $20,235

Interest accrued = $20,235 * (768.98 / 100) * [(540 - 30) / 12]
Interest accrued ≈ $26,466.50

So, the unearned interest at the end of 30 months is approximately $26,466.50.

To find the payoff amount, we need to subtract the unearned interest from the remaining balance:

Payoff amount = Remaining balance + Unearned interest
Payoff amount = $20,235 + $26,466.50

Now, let's calculate the payoff amount:

Payoff amount ≈ $46,701.50

So, the payoff amount at the end of 30 months is approximately $46,701.50.