Approximately what annual interest rate, compounded annually, is needed to double an investment over eight years?

• A. 12%
• B. 6.5%
• C. 8.7%
• D. 9%

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to determine the annual interest rate required to double an investment over eight years using the formula for compound interest. The formula 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.
2. Step 2: Set up the equation based on the problem. Since the investment needs to double, \( A = 2P \). Substitute \( A = 2P \), \( t = 8 \), and \( P \) into the formula: \( 2P = P(1 + r)^8 \).
3. Step 3: Simplify the equation. Divide both sides of the equation by \( P \) (assuming \( P \neq 0 \)): \( 2 = (1 + r)^8 \).
4. Step 4: Solve for \( r \). Take the eighth root of both sides to isolate \( 1 + r \): \( 1 + r = \sqrt[8]{2} \). Then subtract 1 from both sides: \( r = \sqrt[8]{2} - 1 \).
5. Step 5: Interpret the result. The value of \( r \) represents the annual interest rate needed to double the investment over eight years. Use a calculator or logarithmic methods to approximate \( \sqrt[8]{2} \) if needed.