To determine how much you will receive if you cash out the [tex]$10,367 savings bond after 7 years with an interest rate of 0.8% compounded semi-annually, follow these steps:
1. Identify the principal (P): This is the initial amount invested.
\[ P = \$[/tex]10,367 \]
2. Determine the annual interest rate (r): This is the interest rate per year.
[tex]\[ r = 0.8\% = 0.008 \][/tex]
3. Determine the number of times the interest is compounded per year (n): Since the interest is compounded semi-annually:
[tex]\[ n = 2 \][/tex]
4. Identify the number of years the money is invested (t): This is the total duration for which the money is invested.
[tex]\[ t = 7 \][/tex]
5. Apply the compound interest formula: The formula for compound interest is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (initial investment).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time in years.
Plugging in the values:
[tex]\[ A = 10,367 \left(1 + \frac{0.008}{2}\right)^{2 \times 7} \][/tex]
Simplifying the fraction inside the parentheses:
[tex]\[ A = 10,367 \left(1 + 0.004\right)^{14} \][/tex]
[tex]\[ A = 10,367 \left(1.004\right)^{14} \][/tex]
6. Calculate the final amount: Raising 1.004 to the power of 14:
[tex]\[ A \approx 10,367 \times 1.057480565 \][/tex]
[tex]\[ A \approx 10,962.890539618902 \][/tex]
7. Round the final amount to the nearest cent:
[tex]\[ A \approx 10,962.89 \][/tex]
Thus, if you cash out the savings bond now, you will receive approximately \$10,962.89.
