Answer: To find the monthly payment R necessary to pay off a loan, we can use the following formula:
R = P * (r/12) / (1 - (1 + r/12)^(-12n))
where P is the loan amount, r is the annual interest rate (as a decimal), and n is the number of years of the loan.
To complete parts a through c, we'll need to make some assumptions about the loan. Let's assume:
P = $10,000
r = 5% (0.05 as a decimal)
n = 3 years
a) Calculate the number of monthly payments.
Since the loan is for 3 years, the number of monthly payments will be:
n = 3 * 12 = 36
So there will be 36 monthly payments.
b) Calculate the monthly payment.
Substituting the given values into the formula, we get:
R = 10000 * (0.05/12) / (1 - (1 + 0.05/12)^(-12*3))
R = $299.71 (rounded to the nearest cent)
So the monthly payment necessary to pay off the loan is $299.71.
c) Verify that the loan is paid off in 3 years.
To verify that the loan is paid off in 3 years, we can multiply the number of monthly payments by the monthly payment amount:
36 * $299.71 = $10,789.56
Since the original loan amount was $10,000, and the total paid over 36 months is $10,789.56, the loan is paid off in 3 years (with a bit extra paid due to the interest).
Explanation: