Final answer:
The accumulated value of Aabha's monthly deposits with interest compounded quarterly is calculated using the future value formula for an annuity. Given the monthly deposit of $150, an annual interest compounded quarterly at 3%, and a duration of 2.5 years, we perform the calculation to find the total amount saved.
Step-by-step explanation:
To calculate the accumulated value of Aabha's monthly deposits with interest compounded quarterly, we will need to use the formula for the future value of an annuity due to the regular deposits and compounding interest. Since the interest is compounded quarterly, the annual rate must be divided by four to find the quarterly interest rate. The formula for the future value of an annuity compounded quarterly is:
FV = P × { [(1 + r/n)^(nt) - 1 } / (r/n) × (1+r/n)
Where:
- FV is the future value of the annuity.
- P is the amount of each deposit.
- r is the annual interest rate (in decimal form).
- n is the number of times the interest is compounded per year.
- t is the number of years the money is invested.
In this scenario, Aabha deposits $150 monthly for 2.5 years, the annual interest rate is 3% (0.03 in decimal form), and it is compounded quarterly (n = 4). To find the total value of Aabha's savings after 2.5 years, we need to substitute these values into the annuity formula.
However, we must first adjust the deposits to be considered at the end of each quarter for the interest to apply correctly, as it is compounded quarterly. Therefore, since there are 3 months in a quarter, Aabha effectively makes a deposit at the end of each quarter. The quarterly deposit would then be $150 × 3 = $450.
Now let's calculate the future value:
FV = 450 × { [(1 + 0.03/4)^(4 × 2.5) - 1 } / (0.03/4) × (1+0.03/4)
It's important to perform this calculation accurately to obtain the correct future value