Approximately what annual interest rate, compounded annually, is needed to double an investment in 5 years?

• A. About 10.0%
• B. About 14.9%
• C. About 20.0%
• D. About 7.2%

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to determine the annual interest rate (r) required to double an investment in 5 years, using the formula for compound interest. The formula is: \( FV = PV \times (1 + r)^t \), where FV is the future value, PV is the present value, 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, \( FV = 2 \times PV \). Substituting this into the formula gives: \( 2 \times PV = PV \times (1 + r)^5 \).
3. Step 3: Simplify the equation. Divide both sides of the equation by PV (assuming PV is not zero): \( 2 = (1 + r)^5 \).
4. Step 4: Solve for r. Take the fifth root (or raise both sides to the power of \( \frac{1}{5} \)) to isolate \( 1 + r \): \( 1 + r = 2^{\frac{1}{5}} \).
5. Step 5: Subtract 1 from both sides to find r: \( r = 2^{\frac{1}{5}} - 1 \). This will give the annual interest rate required to double the investment in 5 years.