Final answer:
The effective rate per quarter for an account with 3% monthly interest compounded quarterly is 9%. To determine the balance at the end of five years for a $1000 deposit, the formula A = P(1 + r/n)^(nt) is used, resulting in $1565.28.
Step-by-step explanation:
If you deposit a single payment of $1000 today into an account which pays interest at a rate of 3% per month compounded quarterly, we first need to calculate the effective rate per quarter.
The nominal quarterly interest rate is obtained by dividing the monthly interest rate by the number of months in a quarter:
Quarterly rate = 3% per month × 3 months = 9%
However, since the interest is compounded quarterly, the effective quarterly rate is a little more complicated to calculate. In one quarter, the interest is compounded only once, so in this case, the nominal rate and effective rate are the same.
Effective rate per quarter = 9%
To find out how much money you will have in the account at the end of five years, we use the compound interest formula:
A = P(1 + r/n)nt
Where:
- A = the amount of money accumulated after n years, including interest.
- P = the principal amount (the initial amount of money).
- r = the annual interest rate (decimal).
- n = the number of times that interest is compounded per year.
- t = the time the money is invested for, in years.
For our problem:
- P = $1000
- r = 9% or 0.09 (since it's quarterly)
- n = 4 (compounded quarterly)
- t = 5 years
A = $1000(1 + 0.09/4)4×5
A = $1000(1 + 0.0225)20
A = $1000(1.0225)20
A = $1000 × 1.56528
A = $1565.28
So, at the end of five years, the account balance would be $1565.28.