If Ida invests \$10 at an annual interest rate of 7% compounded annually, approximately how many years will it take for her investment to double?

• A. About 5 years
• B. About 20 years
• C. About 10 years
• D. About 14 years

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem. Ida wants to know how many years it will take for her investment to double at an annual interest rate of 7% compounded annually. This involves using the formula for compound interest and solving for time.
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, and \( t \) is the time in years.
3. Step 3: Set \( A = 2P \) because the investment needs to double. Substitute \( P = 10 \), \( r = 0.07 \), and \( A = 20 \) into the formula: \( 20 = 10(1 + 0.07)^t \). Simplify to \( 2 = (1.07)^t \).
4. Step 4: Solve for \( t \) using logarithms. Take the natural logarithm (ln) of both sides: \( \ln(2) = t \cdot \ln(1.07) \). Rearrange to isolate \( t \): \( t = \frac{\ln(2)}{\ln(1.07)} \).
5. Step 5: Calculate \( \ln(2) \) and \( \ln(1.07) \) using a calculator, then divide \( \ln(2) \) by \( \ln(1.07) \) to find the approximate value of \( t \). This will give the number of years it takes for the investment to double.