Multiple Choice
If Janet increases her monthly loan payment while keeping the interest rate and loan amount constant, what will be the impact on the number of months it will take her to pay off her loan?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
Jorge takes out a loan of \$10,000 at an annual interest rate of 6%, compounded annually. If he makes equal annual payments of \$2,000 at the end of each year, approximately how many years will it take Jorge to pay off the loan?
A
7 years
B
5 years
C
6 years
D
8 years
Verified step by step guidance1
Step 1: Understand the problem. Jorge has taken a loan of \$10,000 with an annual interest rate of 6%, compounded annually. He makes equal annual payments of \$2,000 at the end of each year. The goal is to determine how many years it will take to pay off the loan.
Step 2: Use the formula for the present value of an ordinary annuity to calculate the number of years required to pay off the loan. The formula is: \( PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \), where \( PV \) is the loan amount (\(10,000), \( PMT \) is the annual payment (\)2,000), \( r \) is the annual interest rate (6% or 0.06), and \( n \) is the number of years.
Step 3: Rearrange the formula to solve for \( n \), the number of years. This involves isolating \( n \) and using logarithms to solve for the exponent. The rearranged formula is: \( n = \frac{\log(1 - \frac{PV \times r}{PMT})}{\log(1 + r)} \).
Step 4: Substitute the known values into the formula. \( PV = 10,000 \), \( PMT = 2,000 \), and \( r = 0.06 \). Perform the calculations step by step, starting with \( \frac{PV \times r}{PMT} \), then \( 1 - \frac{PV \times r}{PMT} \), and finally apply the logarithms.
Step 5: Interpret the result. The calculated \( n \) will represent the approximate number of years it will take Jorge to pay off the loan. Compare the result to the provided options (7 years, 5 years, 6 years, 8 years) to identify the correct answer.
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