객관식
Compared to a similar fixed payment loan, the total interest paid on a fixed principal loan is:
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10. Time Value of Money
Time Value of Money Equations
객관식
To the nearest year, it will take about how many years to double the original investment at an annual interest rate of 8\% compounded annually?
A
16 years
B
12 years
C
9 years
D
6 years
검증된 단계별 안내1
Step 1: Understand the formula for compound interest and doubling time. The formula for compound interest is \( A = P(1 + r)^t \), where \( A \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years. To find the doubling time, we set \( A = 2P \).
Step 2: Substitute the values into the formula. Since the annual interest rate is 8\%, \( r = 0.08 \). The equation becomes \( 2P = P(1 + 0.08)^t \).
Step 3: Simplify the equation by dividing both sides by \( P \). This gives \( 2 = (1 + 0.08)^t \), which simplifies further to \( 2 = 1.08^t \).
Step 4: Solve for \( t \) using logarithms. Take the natural logarithm (ln) of both sides: \( \ln(2) = \ln(1.08^t) \). Using the logarithmic property \( \ln(a^b) = b \cdot \ln(a) \), the equation becomes \( \ln(2) = t \cdot \ln(1.08) \).
Step 5: Rearrange the equation to isolate \( t \): \( t = \frac{\ln(2)}{\ln(1.08)} \). Calculate \( \ln(2) \) and \( \ln(1.08) \) to find the approximate value of \( t \), which represents the number of years it will take to double the investment.
