If you deposit \$1,000 into a savings account that earns 10\% annual interest compounded annually, how many years will it take for your investment to grow to \$2,000?

• A. Approximately 7.00 years
• B. Exactly 10 years
• C. Approximately 7.27 years
• D. Approximately 7.27 years

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the number of years it will take for an initial deposit of \$1,000 to grow to \$2,000 in a savings account that earns 10% annual interest compounded annually. This involves using the formula for compound interest.
2. Step 2: Recall the compound interest formula: \( A = P \cdot (1 + r)^t \), 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: Substitute the known values into the formula. Here, \( A = 2000 \), \( P = 1000 \), and \( r = 0.10 \). The equation becomes \( 2000 = 1000 \cdot (1 + 0.10)^t \).
4. Step 4: Simplify the equation to isolate \( t \). Divide both sides by \( 1000 \): \( 2 = (1.10)^t \). Then, take the natural logarithm (ln) of both sides to solve for \( t \): \( \ln(2) = t \cdot \ln(1.10) \).
5. Step 5: Rearrange the equation to solve for \( t \): \( t = \frac{\ln(2)}{\ln(1.10)} \). Use this formula to calculate the approximate number of years it will take for the investment to double.