Final answer:
By applying the amortization formula, we calculate that Jennifer can pay off a $20,000 loan with a 7.1% interest rate and $500 monthly payments in approximately 56.7 months, or 4.7 years to the nearest tenth.
Step-by-step explanation:
To determine the length of Jennifer's loan with a 7.1% interest rate and monthly payments of $500, we must use the amortization formula to find out how many months it will take for her to pay off the $20,000 loan.
The amortization formula is as follows:
M = P [i(1 + i)^n] / [(1 + i)^n - 1]
Where:
M = monthly payment
P = principal amount (initial loan balance)
i = monthly interest rate (annual rate divided by 12)
n = number of payments (total number of months)
First, we need to calculate i, the monthly interest rate.
i = annual interest rate / 12 = 7.1% / 12 = 0.0071 / 12
Now, we rearrange the formula to solve for n, the number of payments:
n = log(M / (M - P * i)) / log(1 + i)
Plugging in the values gives us:
n = log(500 / (500 - 20,000 * (0.0071 / 12))) / log(1 + (0.0071 / 12))
Using a calculator, we find that n is approximately 56.7 months.
To find the length of the loan in years, we divide the number of months by 12.
Length of loan in years = n / 12
Therefore:
Length of loan in years ≈ 56.7 / 12 ≈ 4.725 years
To the nearest tenth, the loan would last approximately 4.7 years.