To calculate the future value of [tex]$9,000 earning 9% interest compounded quarterly for 6 years, we use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the future value.
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate (as a decimal).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the time in years.
Given the values:
- \( P = 9000 \) dollars
- \( r = 0.09 \) (9%)
- \( n = 4 \) (quarterly)
- \( t = 6 \) years
Let's substitute these values into the formula step by step.
1. Determine the interest rate per period:
\[
\frac{r}{n} = \frac{0.09}{4} = 0.0225
\]
2. Determine the total number of compounding periods:
\[
nt = 4 \times 6 = 24
\]
3. Calculate the base of the exponent:
\[
1 + \frac{r}{n} = 1 + 0.0225 = 1.0225
\]
4. Raise the base to the power of the total number of compounding periods:
\[
\left(1.0225\right)^{24}
\]
5. Multiply the principal by the result:
\[
A = 9000 \times (1.0225)^{24}
\]
6. Calculate the future value \(A\):
The future value comes out to approximately \$[/tex]15,351.90, when rounded to two decimal places. Thus,
[tex]\[
A \approx 15351.90
\][/tex]
So, the future value of [tex]$9,000 earning 9% interest compounded quarterly for 6 years is approximately \$[/tex]15,351.90.
