Let's break down the problem and find the future values for an account with the given parameters, both for simple interest and for interest compounded annually.
1. Simple Interest Calculation:
- Principal (P): [tex]$1600
- Annual interest rate (r): 8% or 0.08
- Time (t): 9 years
The formula for calculating the future value with simple interest is:
\[
A_{\text{simple}} = P(1 + rt)
\]
Plugging in the given values:
\[
A_{\text{simple}} = 1600 \times (1 + 0.08 \times 9)
\]
Simplifying inside the parentheses first:
\[
1 + 0.08 \times 9 = 1 + 0.72 = 1.72
\]
Now, multiplying by the principal:
\[
A_{\text{simple}} = 1600 \times 1.72 = 2752
\]
Therefore, the future value of the account with simple interest is \( \$[/tex]2752.00 \).
2. Compound Interest Calculation:
- Principal (P): [tex]$1600
- Annual interest rate (r): 8% or 0.08
- Number of times interest is compounded per year (n): 1
- Time (t): 9 years
The formula for calculating the future value with interest compounded annually is:
\[
A_{\text{compounded}} = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Plugging in the given values:
\[
A_{\text{compounded}} = 1600 \times \left(1 + \frac{0.08}{1}\right)^{1 \times 9}
\]
Simplifying inside the parentheses first:
\[
1 + 0.08 = 1.08
\]
Now, raising to the power of \( nt \) (9 in this case):
\[
(1.08)^9 \approx 1.9990046271
\]
Multiplying by the principal:
\[
A_{\text{compounded}} = 1600 \times 1.9990046271 \approx 3198.4074
\]
Therefore, the future value of the account with interest compounded annually is approximately \( \$[/tex]3198.41 \).
- So, for the simple interest, the future value is [tex]$2752.00.
- For the interest compounded annually, the future value is approximately $[/tex]3198.41.
