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

Answer: (B)

Verified step by step guidance

1. 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 \( A = 2P \).
2. Step 2: Substitute the values into the formula. Since the annual interest rate is 8\%, \( r = 0.08 \). The equation becomes \( 2P = P(1 + 0.08)^t \).
3. Step 3: Simplify the equation by dividing both sides by \( P \). This gives \( 2 = (1 + 0.08)^t \), which simplifies further to \( 2 = 1.08^t \).
4. 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 \( \ln(2) = t \cdot \ln(1.08) \).
5. 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.