Final answer:
To calculate the number of years it will take for $348.00 to grow to $24,420.00 at an annual interest rate of 19.79% compounded quarterly, the compound interest formula is used, solving for the time variable t.
Step-by-step explanation:
To determine how many years it will take for an initial deposit of $348.00 to grow to $24,420.00 in an account with an annual interest rate of 19.79% compounded quarterly, we use the compound interest formula:
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.
In this case, the principal P is $348.00, the annual interest rate r is 0.1979, the interest is compounded quarterly so n is 4, and we want the final amount A to be $24,420.00. We need to solve for t.
Transforming the compound interest formula to solve for t gives us:
t = (log(A/P)) / (n * log(1 + r/n))
Substituting the given values:
t = (log(24420/348)) / (4 * log(1 + 0.1979/4))
By calculating this expression, we can find out how many years it will take for the initial deposit to grow to the desired amount.