Final answer:
To determine the number of years required for an investment to reach a goal of $210,000 with a 11% annual return and yearly deposits of $3,000, we need to use the future value formulas for an annuity and a single sum.
Step-by-step explanation:
Calculating the Time to Reach an Investment Goal
To find out how many years it will take for the initial investment of $22,318.30 to grow to $210,000 with 11% annual interest and additional annual deposits of $3,000, we can use the future value of an annuity formula:
Future Value of Annuity (FV) = C * [((1 + r)^n - 1) / r] where C is the annual deposit, r is the annual interest rate expressed as a decimal, and n is the number of periods.
However, since the initial investment is not zero, we also need to calculate the future value of the single sum separately and add it to the future value of the annuity:
Future Value of Single Sum (FVSS) = P * (1 + r)^n where P is the initial investment.
Combining both parts into one equation gives us:
Total Future Value (TFV) = FVSS + FV
Solving this equation for n is not straightforward and typically requires a financial calculator or excel. The equation must be solved iteratively, meaning we guess and check until we find the correct number of years n that makes TFV equal to the desired goal of $210,000.
Once calculated, we round the answer to two decimal places as instructed.