Final answer:
It will take approximately 31.64 years for $400 to grow to $1,000 at a 4% interest rate when compounded annually. The calculations for 8% and 16% interest rates follow the same formula, adjusting for the respective interest rates.
Step-by-step explanation:
To calculate how long it will take for $400 to grow to $1,000 at different interest rates, we can use the formula for compound interest, which is A = P(1 + r)^t. In this formula, A represents the amount of money accumulated after n years, including interest. P is the principal amount, r is the annual interest rate (as a decimal), and t is the time in years.
We're solving for t and want to find out when A reaches $1,000 from a starting point (P) of $400. Rearranging the formula gives us t = ln(A/P) / ln(1 + r). Here's how you would calculate it for each interest rate:
Let's do the math for the 4% rate as an example:
t = ln(2.5) / ln(1.04)
= ln(2.5) / 0.039220713
= 31.64
It will take approximately 31.64 years for $400 to grow to $1,000 at a 4% interest rate, compounded annually. Repeat the calculation for the other rates to find their respective times.