To determine the maturity value of [tex]$6400 invested at 3% compounded semiannually for seven years, we can follow these steps.
1. Determine the principal (P), annual interest rate (r), number of compounding periods per year (n), and the time in years (t).
- Principal (P): $[/tex]6400
- Annual interest rate (r): 3% or 0.03
- Number of compounding periods per year (n): 2 (since it is compounded semiannually)
- Time (t): 7 years
2. Calculate the rate per compounding period.
The rate per compounding period is given by dividing the annual interest rate by the number of compounding periods per year.
[tex]\[
\text{Rate per period} = \frac{r}{n} = \frac{0.03}{2} = 0.015
\][/tex]
3. Calculate the total number of compounding periods.
The total number of compounding periods is the product of the number of compounding periods per year and the number of years.
[tex]\[
\text{Total periods} = n \times t = 2 \times 7 = 14
\][/tex]
4. Apply the compound interest formula to calculate the maturity value.
The compound interest formula is:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
Substituting the values, we get:
[tex]\[
A = 6400 \left(1 + 0.015\right)^{14}
\][/tex]
5. Calculate the maturity value and round to 2 decimal places.
[tex]\[
A = 6400 \left(1.015\right)^{14} \approx 7883.24
\][/tex]
Hence, the maturity value of [tex]$6400 invested at 3% compounded semiannually for seven years is $[/tex]7883.24.
