Approximately what annual interest rate, compounded annually, is needed to double an investment in 4 years?

• A. 15%
• B. 17.7%
• C. 12.5%
• D. 18%

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to determine the annual interest rate, compounded annually, required to double an investment in 4 years. This involves using the formula for compound interest and solving for the interest rate.
2. Step 2: Recall the compound interest formula: \( A = P(1 + r)^t \), where \( A \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate (as a decimal), and \( t \) is the time in years. In this case, \( A = 2P \) (since the investment doubles), \( t = 4 \), and \( P \) cancels out.
3. Step 3: Simplify the formula to solve for \( r \): \( 2 = (1 + r)^4 \). This equation represents the relationship between the doubling of the investment and the annual interest rate.
4. Step 4: Take the fourth root of both sides to isolate \( 1 + r \): \( 1 + r = \sqrt[4]{2} \). Then subtract 1 from both sides to solve for \( r \): \( r = \sqrt[4]{2} - 1 \).
5. Step 5: Convert \( r \) into a percentage by multiplying the decimal result by 100. This will give the annual interest rate required to double the investment in 4 years.