To calculate the balance of Andrew's account after 5 years with an annual interest rate of 1.5% compounded quarterly, we can follow these steps:
1. Identify the initial amount (Principal):
- The initial amount deposited (principal) is [tex]$4,000.
2. Determine the annual interest rate:
- The annual interest rate is 1.5%, which we convert to a decimal by dividing by 100, i.e., 0.015.
3. Identify the number of times interest is compounded per year:
- Interest is compounded quarterly, which means it is compounded 4 times a year.
4. Calculate the interest rate per compounding period:
- The interest rate per quarter is the annual interest rate divided by the number of compounding periods per year:
\[
\text{Quarterly Interest Rate} = \frac{0.015}{4} = 0.00375
\]
5. Determine the total number of compounding periods:
- Over 5 years, if interest is compounded quarterly, the total number of compounding periods is:
\[
\text{Total Compounding Periods} = 4 \times 5 = 20
\]
6. Use the compound interest formula to calculate the balance:
- The compound interest formula is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount ($[/tex]4,000).
- [tex]\(r\)[/tex] is the annual interest rate (0.015).
- [tex]\(n\)[/tex] is the number of times that interest is compounded per year (4).
- [tex]\(t\)[/tex] is the time the money is invested for in years (5).
7. Plug in the values:
[tex]\[
A = 4000 \left(1 + 0.00375\right)^{20}
\][/tex]
[tex]\[
A = 4000 \left(1.00375\right)^{20}
\][/tex]
8. Calculate the final balance:
- The calculated balance after 5 years is approximately [tex]$4310.93.
Therefore, the balance of Andrew's account after 5 years will be approximately $[/tex]4310.93.
