Assume you want to pay off your $10,000, 30-month car loan after only the first 12 months of payments. With interest at 12% compounded monthly, how much will you need pay off the loan in full at the end of the first year?

Question

Quizplus · Accepted Answer

Using the compound interest formula, we can calculate the amount owed after the first 12 months:

A = P(1 + r/n)^(nt)

Where:
- A = amount owed after 12 months
- P = principal (initial amount borrowed) = $10,000
- r = annual interest rate (as a decimal) = 0.12
- n = number of times interest is compounded per year = 12 (monthly)
- t = time in years = 1/2 (30 months is half of 60 months)

Plugging in these values, we get:

A = $10,000(1 + 0.12/12)^(12*1/2)
A = $10,000(1 + 0.01)^6
A = $10,000(1.061678)
A = $10,616.78

So after 12 months, you owe $10,616.78. To pay off the loan in full, you would need to pay this amount plus any remaining interest for the remaining 18 months:

A = P(1 + r/n)^(nt)
A = $10,616.78(1 + 0.12/12)^(12*1/2)
A = $10,616.78(1 + 0.01)^6
A = $10,616.78(1.061678)
A = $11,277.96

So you would need to pay $11,277.96 to completely pay off the loan after the first 12 months. However, if you only want to pay off the principal (the initial $10,000), then you would need to subtract the interest owed for the remaining 18 months:

I = P(r/n)t
I = $10,000(0.12/12)*1/2
I = $50

The interest owed for the remaining 18 months is $50 * 18 = $900.

So to pay off just the principal, you would need to pay $10,000 + $900 = $10,900.

The closest answer choice is B) $6,354, but this is clearly too low. The correct answer is $10,900 to pay off just the principal, or $11,277.96 to pay off the entire loan.

Assume You Want to Pay Off Your $10,000, 30-Month Car