Question

How long will it take to pay off a loan of $55,000 at an annual rate of 9 percent compounded monthly if you make monthly payments of $800? Use five decimal places for the monthly percentage rate in your calculations.

The number of years it takes to pay off the loan is __ years?

Please show your work.

Answer 1

Loan repayment can be calculated using the formula for present value of annuity:

PV of annuity = P*[(1-(1+r)^(-n)) / r]

P - Periodic payment = 800

r - rate per period = 0.09/12 = 0.0075

n - number of periods = ?

55000 = 800*((1-(1+0.0075)^(-n)) / 0.0075)

68.75*0.0075 = 1-1.0075^(-n)

1 - 0.515625 = 1.0075^(-n)

0.484375 = 1.0075^(-n)

1/1.0075^n = 0.484375

1.0075^n = 1/0.484375 = 2.06452

Taking log on both sides

n * log(1.0075) = log(2.06452)

n = 0.31482/0.003245 = 97.02 months

no. of years = 97.02/12 = 8.08 years