To determine how much a \[tex]$300 investment will be worth in 20 years with an annual interest rate of 5% compounded quarterly, we can use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of times the interest is compounded per year,
- \( t \) is the number of years the money is invested for,
- \( A \) is the amount of money accumulated after n years, including interest.
For this problem:
- \( P = 300 \)
- \( r = 0.05 \) (since 5% as a decimal is 0.05)
- \( n = 4 \) (because the interest is compounded quarterly)
- \( t = 20 \)
Now let's plug these values into the formula:
\[ A = 300 \left(1 + \frac{0.05}{4}\right)^{4 \times 20} \]
Simplify the fraction inside the parentheses:
\[ \frac{0.05}{4} = 0.0125 \]
So the formula now becomes:
\[ A = 300 \left(1 + 0.0125\right)^{80} \]
Add inside the parentheses:
\[ 1 + 0.0125 = 1.0125 \]
So we have:
\[ A = 300 \left(1.0125\right)^{80} \]
Evaluating the expression:
\[ (1.0125)^{80} \approx 2.701484941 \]
Finally, multiply by the principal amount:
\[ A = 300 \times 2.701484941 \approx 810.45 \]
So, the investment will be worth approximately \$[/tex]810.45 in 20 years. Therefore, the correct answer is:
[tex]\[ \$810.45 \][/tex]
