If you desire your savings to double in 6 years, which annual interest rate, compounded annually, is closest to what you need?

• A. 8.0%
• B. 10.0%
• C. 12.0%
• D. Approximately 12.25%

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the concept of doubling savings. This involves using the Rule of 72, which is a shortcut to estimate the time required for an investment to double given a fixed annual interest rate. The formula is: \( ext{Rate} = rac{72}{ ext{Time}} \).
2. Step 2: Apply the Rule of 72 to the problem. Since the desired doubling time is 6 years, substitute \( ext{Time} = 6 \) into the formula: \( ext{Rate} = rac{72}{6} \).
3. Step 3: Calculate the approximate interest rate using the formula. This will give you a rough estimate of the annual interest rate required for the savings to double in 6 years.
4. Step 4: Recognize that the Rule of 72 provides an approximation. For precise calculations, use the compound interest formula: \( A = P(1 + r)^t \), where \( A \) is the future value, \( P \) is the present value, \( r \) is the annual interest rate, and \( t \) is the time in years.
5. Step 5: Solve for \( r \) using the compound interest formula. Set \( A = 2P \) (since the savings need to double), \( t = 6 \), and rearrange the formula to isolate \( r \): \( r = (2)^{1/6} - 1 \). Use logarithms if necessary to simplify the calculation.