Asked by Michael Ariestan on May 25, 2024
Verified
You have borrowed $130,000 to buy a new motor home. Your loan is to be repaid over 15 years at 8% compounded monthly. If you pay an extra $200 per month on the motor home, how many years will it take to pay off the loan?
A) 10.3 years
B) 11.5 years
C) 12.8 years
D) 13.3 years
Compounded Monthly
The process where interest earned is added to the principal, so that from that moment on, the interest that has been added also earns interest.
Motor Home
A type of self-propelled recreational vehicle (RV) that offers living accommodation combined with a vehicle engine.
Extra Payments
Payments made on a loan that exceed the minimum required payment, often used to reduce the loan balance more quickly.
- Learn to quantify and comprehend the dues on mortgages, loans, and annuities, with an emphasis on scenarios that incorporate additional payments.
- Discover and critique the financial positives and negatives of loan endeavors, along with calculating the complete interest payout over the term of the loan.
Verified Answer
KG
Katie GombertMay 28, 2024
Final Answer :
B
Explanation :
First, we need to calculate the monthly payment for the original loan amount of $130,000 over 15 years at 8% compounded monthly. Using the formula:
P = (PV * r) / (1 - (1+r)^(-n))
where P is the monthly payment, PV is the present value (loan amount), r is the monthly interest rate (8%/12), and n is the total number of months (15*12).
We get:
P = (130000 * (8%/12)) / (1 - (1+(8%/12))^(-15*12))
P = $1243.96
This means that without any extra payments, it would take 15 years to pay off the motor home loan.
However, with an extra $200 per month, the new monthly payment becomes $1243.96 + $200 = $1443.96.
Using the same formula above, but using this new monthly payment, we can solve for the number of months it will take to pay off the loan:
P = (130000 * (8%/12)) / (1 - (1+(8%/12))^(-n)) + 200
$1443.96 = ($1083.33 / (1 - (1+(8%/12))^(-n))) + 200
$1260.63 = ($1083.33 / (1+(8%/12))^(-n))
(1+(8%/12))^(-n) = $1083.33 / $1260.63
(1+(8%/12))^n = $1260.63 / $1083.33
n = ln($1260.63 / $1083.33) / ln(1+(8%/12))
n = 138.28 months
n = 11.52 years
Therefore, it will take approximately 11.5 years to pay off the loan with the extra $200 payments.
P = (PV * r) / (1 - (1+r)^(-n))
where P is the monthly payment, PV is the present value (loan amount), r is the monthly interest rate (8%/12), and n is the total number of months (15*12).
We get:
P = (130000 * (8%/12)) / (1 - (1+(8%/12))^(-15*12))
P = $1243.96
This means that without any extra payments, it would take 15 years to pay off the motor home loan.
However, with an extra $200 per month, the new monthly payment becomes $1243.96 + $200 = $1443.96.
Using the same formula above, but using this new monthly payment, we can solve for the number of months it will take to pay off the loan:
P = (130000 * (8%/12)) / (1 - (1+(8%/12))^(-n)) + 200
$1443.96 = ($1083.33 / (1 - (1+(8%/12))^(-n))) + 200
$1260.63 = ($1083.33 / (1+(8%/12))^(-n))
(1+(8%/12))^(-n) = $1083.33 / $1260.63
(1+(8%/12))^n = $1260.63 / $1083.33
n = ln($1260.63 / $1083.33) / ln(1+(8%/12))
n = 138.28 months
n = 11.52 years
Therefore, it will take approximately 11.5 years to pay off the loan with the extra $200 payments.
Learning Objectives
- Learn to quantify and comprehend the dues on mortgages, loans, and annuities, with an emphasis on scenarios that incorporate additional payments.
- Discover and critique the financial positives and negatives of loan endeavors, along with calculating the complete interest payout over the term of the loan.