Question

A certain college graduate borrows $8000 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.

Answer 1

The payment is
`k=\$3216.92`.

Interest paid during the 3-year period is
`I_(T)=\$1651.76`

Explanation:

Hi

The Payment amount

We are going to use the formula below with
`VP=\$8000, i=10\%` and
`n=3`.

`k=(VP)/([(1-(1+i)^(-n))/(i)] ) =(8000)/([(1-(1+0.1)^(-3))/(0.1)] )=3216.92`

Interest paid during the 3 year

• First period

`I_(1)=VP*i=8000*0.1=800`

`CS_(1)=k-I_(1)=3216.92-800=2416.92`

`Balance_(1)=VP-CS_(1)=8000-2416.92=5583.08`

• Second period

`I_(2)=B_(1)*i=5583.08*0.1=558.31`

`CS_(2)=k-I_(2)=3216.92-558.31=2648.61`

`Balance_(2)=VP-CS_(2)=5583.08-2648.61=2924.47`

• Third period

`I_(3)=B_(2)*i=2924.47*0.1=292.45`

• Sum of Interest paid during the 3 years

`I_(t)=I_(1)+I_(2)+I_(3)=800+558.31+292.45=1651.76`