To determine how much [tex]$500 will be worth in 10 years with an annual interest rate of 2.5% compounded semiannually, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (the initial sum of money), which is $[/tex]500.
- [tex]\( r \)[/tex] is the annual interest rate (decimal), which is 0.025.
- [tex]\( n \)[/tex] is the number of times interest is compounded per year, which is 2 (since it's compounded semiannually).
- [tex]\( t \)[/tex] is the time the money is invested for in years, which is 10 years.
Following the steps:
1. Principal (P): [tex]$500
2. Annual Rate (r): 2.5% or 0.025 in decimal
3. Number of times compounded per year (n): 2
4. Time in years (t): 10
Plug these values into the formula:
\[ A = 500 \left(1 + \frac{0.025}{2}\right)^{2 \times 10} \]
First, calculate the rate per compounding period:
\[ \frac{0.025}{2} = 0.0125 \]
Next, calculate the number of compounding periods:
\[ 2 \times 10 = 20 \]
Now, apply these values to the formula:
\[ A = 500 \left(1 + 0.0125\right)^{20} \]
\[ A = 500 \left(1.0125\right)^{20} \]
Calculating the value inside the parentheses to the power of 20:
\[ 1.0125^{20} \approx 1.2820372317085844 \]
Now multiply this by the principal amount to find the accumulated amount:
\[ A \approx 500 \times 1.2820372317085844 \]
\[ A \approx 641.0186158542922 \]
Finally, round to the nearest cent:
\[ A \approx 641.02 \]
Therefore, after 10 years, $[/tex]500 invested at an annual interest rate of 2.5%, compounded semiannually, will be worth $641.02.
