Asked by Lanie Barnhill on May 30, 2024
Verified
A $12,000 loan is repaid by semi-annual payments of $1,500 each. Interest on the loan is 10% compounded semi-annually. How long will it take to pay off the loan?
A) 5.5 years
B) 5 years
C) 21 years
D) 10 years
E) 10.5 years
Compounded Semi-annually
A method of calculating interest where the interest is added to the principal sum twice a year, resulting in interest on interest.
Semi-annual Payments
Payments that are made twice a year as a means to fulfill a financial obligation.
Loan
A borrowed amount of money that should be returned along with an additional interest payment.
- Ascertain the time needed to reach a financial objective through consistent contributions or deductions.
- Apply financial equations to address intricate issues associated with loans, savings, and investment strategies.
Verified Answer
ZK
Zybrea KnightJun 06, 2024
Final Answer :
A
Explanation :
The loan is repaid semi-annually, meaning twice a year. To find the number of payments, we use the formula for the present value of an annuity: PV=PMT×(1−(1+r)−nr)PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)PV=PMT×(r1−(1+r)−n) , where PV is the present value (loan amount), PMT is the payment amount, r is the interest rate per period, and n is the total number of payments. Rearranging to solve for n, we get n=log(1−PV×rPMT)log(1+r)n = \frac{\log(1 - \frac{PV \times r}{PMT})}{\log(1 + r)}n=log(1+r)log(1−PMTPV×r) . Plugging in the values: PV=12000PV = 12000PV=12000 , PMT=1500PMT = 1500PMT=1500 , r=0.05r = 0.05r=0.05 (since 10% annual interest compounded semi-annually gives a 5% semi-annual rate), we can solve for n. However, without needing to solve it explicitly, knowing that $1,500 is paid semi-annually means it will take 8 payments (4 years) to reach $12,000 without interest. Considering the interest, it will take more than 4 years but less than 11 years (the next possible whole number of years with semi-annual payments). The only option fitting this description is 5.5 years, implying 11 payments.
Learning Objectives
- Ascertain the time needed to reach a financial objective through consistent contributions or deductions.
- Apply financial equations to address intricate issues associated with loans, savings, and investment strategies.