Let's solve the problem step-by-step to find the value of the mutual fund after 12 years given that Aubree invested [tex]$500 into a mutual fund that paid 3% interest per year, compounded annually.
1. Identify the variables:
- Principal (P): $[/tex]500
- Interest rate (r): 3% (or 0.03 in decimal form)
- Time (t): 12 years
2. Understand the formula for compound interest:
The general formula for compound interest is:
[tex]\[
A = P (1 + r/n)^{nt}
\][/tex]
Since the interest is compounded annually, [tex]\( n \)[/tex], the number of times the interest is compounded per year, is 1. Therefore, the formula simplifies to:
[tex]\[
A = P (1 + r)^t
\][/tex]
3. Formulate the equation:
Insert the known values into the equation:
[tex]\[
A = 500 (1 + 0.03)^{12}
\][/tex]
4. Evaluate the expression:
Simplify the term inside the parentheses:
[tex]\[
1 + 0.03 = 1.03
\][/tex]
Now the equation is:
[tex]\[
A = 500 (1.03)^{12}
\][/tex]
5. Compute the final amount:
Though we won't compute it step-by-step here, this setup allows you to plug the numbers into a calculator to find:
[tex]\[
A \approx 712.8804434230897
\][/tex]
Therefore, the value of the mutual fund after 12 years is approximately [tex]\( \$712.88 \)[/tex].
