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

Answer: (B)

Verified step by step guidance

1. 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.
2. 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.
3. 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 \( t \).
4. 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 \( 12 \cdot \ln(1 + \frac{0.06}{12}) \).
5. 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.