객관식
How many years will it take for \$20 to grow to \$100 if the simple interest rate is 4\% per year?
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10. Time Value of Money
Time Value of Money Equations
객관식
How long (in years and months) will it take for an investment to double at an annual interest rate of 6\% compounded monthly?
A
Approximately 9 years and 6 months
B
Approximately 11 years and 8 months
C
Approximately 8 years
D
Exactly 12 years
검증된 단계별 안내1
Step 1: Understand the formula for compound interest. The formula is \( A = P(1 + \frac{r}{n})^{nt} \), where \( A \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) is the number of compounding periods per year, and \( t \) is the time in years.
Step 2: Set \( A \) equal to \( 2P \) because the investment needs to double. Substitute \( r = 0.06 \) (6% annual interest rate) and \( n = 12 \) (monthly compounding) into the formula.
Step 3: Rearrange the formula to solve for \( t \). Divide both sides by \( P \), resulting in \( 2 = (1 + \frac{0.06}{12})^{12t} \). Take the natural logarithm (ln) of both sides to isolate \( t \).
Step 4: Use the logarithmic property \( \ln(a^b) = b \cdot \ln(a) \) to simplify. The equation becomes \( \ln(2) = 12t \cdot \ln(1 + \frac{0.06}{12}) \). Solve for \( t \) by dividing \( \ln(2) \) by \( 12 \cdot \ln(1 + \frac{0.06}{12}) \).
Step 5: Convert the value of \( t \) into years and months. Since \( t \) represents time in years, multiply the decimal portion of \( t \) by 12 to find the number of months.
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