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

Answer: (A)

Verified step by step guidance

1. 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.
2. 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 \( 3 = (1 + r)^t \).
3. Convert the annual interest rate from a percentage to a decimal: \( r = 12.5\% = 0.125 \). Substitute \( r \) into the equation to get \( 3 = (1 + 0.125)^t \).
4. Take the natural logarithm (ln) of both sides to solve for \( t \): \( \ln(3) = t \cdot \ln(1.125) \). Rearrange to isolate \( t \): \( t = \frac{\ln(3)}{\ln(1.125)} \).
5. 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.