Final answer:
The annual interest rate for a $5000 deposit that compounded to $6165 after 3.5 years with monthly compounding interest is approximately 6.48%.
Step-by-step explanation:
To find the annual rate of interest the account paid, we can use 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.
In your case:
- A = $6165
- P = $5000
- n = 12 (since the interest is compounded monthly)
- t = 3.5 years
We need to find the annual interest rate, r. Rearranging the formula to solve for r, we get:
r = n[(A/P)^(1/nt) - 1]
Plugging the values into the formula, we get:
r = 12[(6165/5000)^(1/(12*3.5)) - 1]
Calculating the values inside the brackets first:
(6165/5000)^(1/(12*3.5)) = (1.233)^(1/42)
≈ 1.0054
Now, we subtract 1 and then multiply by n:
r ≈ 12 * (1.0054 - 1)
= 12 * 0.0054
r ≈ 0.0648 or 6.48%
The annual interest rate the account paid was approximately 6.48%.