Let's break down the problem using the compound interest formula to determine how much the investment will be worth after 12 years. 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 (the initial amount of money).
- [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 number of years the money is invested for.
Given values:
- [tex]\( P = \$300 \)[/tex]
- [tex]\( r = 6\% = 0.06 \)[/tex]
- [tex]\( n = 4 \)[/tex] (since the interest is compounded quarterly)
- [tex]\( t = 12 \)[/tex] years
Now, plug in the values into the formula:
[tex]\[ A = 300 \left(1 + \frac{0.06}{4}\right)^{4 \cdot 12} \][/tex]
[tex]\[ A = 300 \left(1 + 0.015\right)^{48} \][/tex]
[tex]\[ A = 300 (1.015)^{48} \][/tex]
Calculating the term inside the parentheses first:
[tex]\[ 1.015^{48} \approx 2.0434786 \][/tex]
Now multiply by the principal amount:
[tex]\[ A = 300 \times 2.0434786 \approx 613.0434867938941 \][/tex]
So, the investment will be worth approximately \[tex]$613.04 after 12 years.
In summary, by investing \$[/tex]300 at an annual interest rate of 6%, compounded quarterly, the amount will grow to about \$613.04 in 12 years.
