객관식
Janet takes out a loan of \$10,000 at an annual interest rate of 6\(\text{%}\), to be repaid in equal monthly installments over 5 years. By the time Janet pays off her entire loan, how much total interest will she have paid?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
At an annual interest rate of 12.5\%, how many years will it take for an investment to triple in value if interest is compounded annually?
A
Approximately 9.4 years
B
Approximately 12.5 years
C
Approximately 6.2 years
D
Approximately 15.0 years
검증된 단계별 안내1
Understand the formula for compound interest: \( A = P(1 + r)^t \), where \( A \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate (in decimal form), and \( t \) is the time in years.
Since the investment triples in value, set \( A = 3P \). Substitute this into the formula to get \( 3P = P(1 + r)^t \). Simplify by dividing both sides by \( P \), resulting in \( 3 = (1 + r)^t \).
Convert the annual interest rate from a percentage to a decimal: \( r = 12.5\% = 0.125 \). Substitute \( r \) into the equation to get \( 3 = (1 + 0.125)^t \).
Take the natural logarithm (ln) of both sides to solve for \( t \): \( \ln(3) = t \cdot \ln(1.125) \). Rearrange to isolate \( t \): \( t = \frac{\ln(3)}{\ln(1.125)} \).
Use a calculator to compute \( \ln(3) \) and \( \ln(1.125) \), then divide the results to find \( t \), which represents the number of years it will take for the investment to triple in value.
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