To solve this problem, let's write down the formulas and the values we have:
- Principal (P): \[tex]$20,000
- Annual interest rate (r): \(6\% = 0.06\)
- Time (t): 4 years
### a. Compounded Semiannually
For interest compounded semiannually, the number of compounding periods per year (n) is 2. The formula to use is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Substituting the values:
\[ A = 20000 \left(1 + \frac{0.06}{2}\right)^{2 \times 4} \]
\[ A = 20000 \left(1 + 0.03\right)^8 \]
\[ A = 20000 \left(1.03\right)^8 \]
\[ A \approx 20000 \times 1.26677 \]
\[ A \approx 25335.4 \]
So, the accumulated value if the money is compounded semiannually is \$[/tex]25,335.40.
### b. Compounded Quarterly
For interest compounded quarterly, the number of compounding periods per year (n) is 4. The formula to use is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Substituting the values:
[tex]\[ A = 20000 \left(1 + \frac{0.06}{4}\right)^{4 \times 4} \][/tex]
[tex]\[ A = 20000 \left(1 + 0.015\right)^{16} \][/tex]
[tex]\[ A = 20000 \left(1.015\right)^{16} \][/tex]
[tex]\[ A \approx 20000 \times 1.2689855 \][/tex]
[tex]\[ A \approx 25379.71 \][/tex]
So, the accumulated value if the money is compounded quarterly is \[tex]$25,379.71.
### c. Compounded Monthly
For interest compounded monthly, the number of compounding periods per year (n) is 12. The formula to use is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Substituting the values:
\[ A = 20000 \left(1 + \frac{0.06}{12}\right)^{12 \times 4} \]
\[ A = 20000 \left(1 + 0.005\right)^{48} \]
\[ A = 20000 \left(1.005\right)^{48} \]
\[ A \approx 20000 \times 1.270389 \]
\[ A \approx 25409.78 \]
So, the accumulated value if the money is compounded monthly is \$[/tex]25,409.78.
