How Much Does It Cost to Repay a $8000 Car Loan with 10% Interest in 3 Years?

[alane1994]

If a mod would like to change the title to something more fitting feel free.

A certain college graduate borrows 8,000 dollars to buy a car. The lender charges interest at an annual rate of 10%. Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate $k$, determine the payment rate $k$ that is required to pay off the loan in 3 years. Also determine how much interest is paid during the 3-year period.

 

[MarkFL]

Re: Not sure what to title.

Interestingly, I answered the first part of this question here, in a topic I brought here from Yahoo! Answers:

http://mathhelpboards.com/questions-other-sites-52/justcurious-question-yahoo-answers-regarding-mathematical-model-5470.html

Finding the amount of interest paid is just then the difference between the amount paid and the loan amount.

 

[alane1994]

Re: Not sure what to title.

Well, hey, would you look at that :p
I am going to look over your work, and try and make sense of it for me. Then I will attempt the second part of the problem. If I get nowhere, I shall come running to my buddies for help! :D

 

[alane1994]

Thanks very much for the help, I really appreciate it!

 

[blue_raver22]

I would approach this scenario by first understanding the concept of compound interest. Compound interest is the interest calculated on the initial principal amount as well as any accumulated interest from previous periods. This means that the interest earned in each period is added to the principal amount, and the interest for the next period is calculated on the new total.

In this case, the borrower has taken a loan of $8,000 with an annual interest rate of 10%. The interest is compounded continuously, which means that the interest is calculated and added to the principal amount infinitely many times throughout the year. To determine the payment rate $k$ required to pay off the loan in 3 years, we can use the formula for continuous compound interest:

$A = P e^{rt}$

Where A is the final amount, P is the principal amount, r is the annual interest rate, and t is the time period. In this case, we know the final amount should be $8,000$ after 3 years, so we can set up the equation as:

$8000 = 8000 e^{0.10 \cdot 3}$

Solving for $k$, we get $k = 0.3377$, which means that the borrower needs to make payments of $33.77\%$ of the principal amount annually to pay off the loan in 3 years.

To determine the amount of interest paid during the 3-year period, we can use the formula for continuous compound interest again:

$I = P(e^{rt} - 1)$

Where I is the total interest paid, P is the principal amount, r is the annual interest rate, and t is the time period. Plugging in the values, we get:

$I = 8000(e^{0.10 \cdot 3} - 1) = \$2,544.55$

Therefore, the borrower will end up paying a total of $\$2,544.55$ in interest over the 3-year period. It is important to note that this calculation assumes that the borrower makes continuous payments throughout the year. If the payments are made monthly or quarterly, the interest paid may vary slightly.