Final answer:
To find out how long it takes for $500,000 to reach $1,000,000 at a 2% interest rate, we can use the formula for compound interest. After applying the formula and calculations, it takes approximately 35 years for the money to double. This example illustrates the significant impact of compound interest over time.
Step-by-step explanation:
To calculate how long it will take for $500,000 to grow to $1,000,000 at a 2% interest rate, we need to use the formula for compound interest, which is A = P(1 + r/n)(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested or borrowed for, in years. Here we assume the interest is compounded annually (n=1). We are looking for the time t when the principal P is $500,000, the amount A is $1,000,000, and the annual interest rate r is 0.02.
First, divide both sides of the equation by P to isolate (1 + r/n)(nt) on one side. Then take the natural logarithm of both sides to solve for t. The formula thus modifies to t = ln(A/P) / n*ln(1+r/n). Plugging in the values, we get t = ln(2) / ln(1.02), which gives us 35 years.
Starting to save money early and allowing compound interest to work can significantly increase your savings, as shown in the provided examples where an investment multiplies over time. This concept underscores the importance of understanding and utilizing compound interest for long-term financial growth.