Final answer:
To calculate the time to save $1,645 with quarterly deposits of $250 at an interest rate of 5% compounded quarterly, one would typically use the future value of an annuity formula and a financial calculator or spreadsheet for iterative calculation, as the process involves adding deposits to the balance and interest accumulation over time.
Step-by-step explanation:
To determine how long it will take to save $1,645.00 with quarterly deposits of $250.00 in an account earning 5% interest compounded quarterly, we can start by understanding the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
-
- A is the amount of money accumulated after n years, including interest.
-
- P is the principal amount (the initial amount of money).
-
- r is the annual interest rate (decimal).
-
- n is the number of times that interest is compounded per year.
-
- t is the time the money is invested for, in years.
However, since deposits are made quarterly, we need to use the future value of an annuity due formula, which is slightly more complex and takes into account regular deposits:
FV = Pmt * (((1 + r/n)^(nt) - 1) / (r/n))
To solve this problem, we would set FV to $1,645.00, Pmt to $250.00, r to 0.05 (as 5%), and n to 4 (since interest is compounded quarterly). However, the question asks for a situation with compound interest, and the formula here does not take into account the accumulated interest with each deposit. A financial calculator or spreadsheet would typically be used to solve this, as it requires iteration to consider each deposit adding to the balance and accruing interest.
Since the exact duration cannot be easily expressed in a formula, we would likely set up a table where each row represents a quarter and calculates the new balance considering the new deposit and the interest earned so far until the balance hits $1,645.00 or more.