Multiple Choice
When interest accrues and is added to the principal balance of a student loan, which of the following best describes the result?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
Inga has an outstanding loan with a remaining balance of \$12,000. She makes fixed annual payments of \$3,000 at an annual interest rate of 5\%. Using the time value of money equations, how many more years will it take Inga to pay off the loan?
A
6 years
B
3 years
C
4 years
D
2 years
Verified step by step guidance1
Identify the key variables in the problem: the loan balance (Present Value, PV) is \$12,000, the fixed annual payment (PMT) is \$3,000, the annual interest rate (r) is 5% or 0.05, and the unknown is the number of years (n) required to pay off the loan.
Use the Present Value of an Ordinary Annuity formula to solve for the number of years (n): \( PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \). Rearrange this formula to isolate \( n \).
Rearranging the formula: \( \frac{PV \times r}{PMT} = 1 - (1 + r)^{-n} \). Then, isolate \( (1 + r)^{-n} \): \( (1 + r)^{-n} = 1 - \frac{PV \times r}{PMT} \).
Take the natural logarithm (ln) of both sides to solve for \( n \): \( -n \times \ln(1 + r) = \ln(1 - \frac{PV \times r}{PMT}) \). Finally, solve for \( n \): \( n = -\frac{\ln(1 - \frac{PV \times r}{PMT})}{\ln(1 + r)} \).
Substitute the known values into the formula: \( PV = 12000 \), \( PMT = 3000 \), \( r = 0.05 \). Perform the calculations step by step to determine \( n \), the number of years required to pay off the loan.
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