Joseph buys a Jeep for $60,000, financing it with a five-year 6.5% APR loan paid monthly. He decides to pay an extra $50 per month in addition to his monthly payments. Approximately how long will he take to pay off the loan under these conditions?

Question

Quizplus · Accepted Answer

First, let's calculate Joseph's monthly payment using the loan formula:

PV = [PMT x (1 - (1 + r)^-n)] / r

Where:
PV = loan amount ($60,000)
PMT = monthly payment
r = monthly interest rate (6.5% / 12 = 0.0054167)
n = total number of payments (5 years x 12 months = 60)

Substituting the values, we get:

$60,000 = [PMT x (1 - (1 + 0.0054167)^-60)] / 0.0054167

Solving for PMT, we get:

PMT = $1,154.57

This is the monthly payment Joseph needs to make to pay off the loan in 5 years.

However, Joseph decides to pay an extra $50 per month, so his actual monthly payment is:

$1,154.57 + $50 = $1,204.57

Now we can calculate how long it will take Joseph to pay off the loan with this payment amount. We'll use the same loan formula, but solve for n this time:

PV = [PMT x (1 - (1 + r)^-n)] / r

Where:
PV = loan amount ($60,000)
PMT = monthly payment ($1,204.57)
r = monthly interest rate (6.5% / 12 = 0.0054167)
n = total number of payments (what we need to calculate)

Substituting the values, we get:

$60,000 = [$1,204.57 x (1 - (1 + 0.0054167)^-n)] / 0.0054167

Multiplying both sides by 0.0054167 and simplifying, we get:

$324.00 = [$1,204.57 - $1,204.57 / (1 + 0.0054167)^n]

We can solve for n by trial and error, or by using a calculator or spreadsheet. Using the latter method, we get:

n = 57.52 months

Therefore, Joseph will take approximately 4 years and 9 months to pay off the loan with the extra $50 payment. The answer is B.

Joseph Buys a Jeep for $60,000, Financing It with a Five-Year