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Using the chart above, find the total amount and amount of interest paid in the following compound interest problem:

[tex]$\$[/tex] 3,000[tex]$ at $[/tex]8\%[tex]$ for 5 years.

\begin{tabular}{|l|l|l|}
\hline
Compounding & Total Amount & Interest Amount \\
\hline
Annually & $[/tex]\[tex]$ & $\$[/tex] \\
\hline
Semiannually & [tex]$\$[/tex][tex]$ & $[/tex]\[tex]$ \\
\hline
Quarterly & $\$[/tex] & [tex]$\$[/tex]$ \\
\hline
\end{tabular}


Sagot :

To solve the compound interest problem for [tex]$3,000 at 8% interest for 5 years, let's compute the total amounts and the interest amounts for different compounding frequencies: annually, semiannually, and quarterly. ### Annually Compounded Interest 1. Total Amount: The formula for compound interest is given by: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] For annual compounding: - \( P = 3000 \) (principal) - \( r = 0.08 \) (annual interest rate) - \( n = 1 \) (compounding frequency per year) - \( t = 5 \) (time in years) Plugging in these values: \[ A = 3000 \left(1 + \frac{0.08}{1}\right)^{1 \times 5} = 3000 \left(1 + 0.08\right)^{5} = 3000 \times 1.4693 \approx 4407.98 \] 2. Interest Amount: The interest amount is the total amount minus the principal: \[ \text{Interest} = A - P = 4407.98 - 3000 \approx 1407.98 \] ### Semiannually Compounded Interest 1. Total Amount: For semiannual compounding: - \( n = 2 \) Plugging in the values: \[ A = 3000 \left(1 + \frac{0.08}{2}\right)^{2 \times 5} = 3000 \left(1 + 0.04\right)^{10} = 3000 \times 1.48 \approx 4440.73 \] 2. Interest Amount: \[ \text{Interest} = 4440.73 - 3000 \approx 1440.73 \] ### Quarterly Compounded Interest 1. Total Amount: For quarterly compounding: - \( n = 4 \) Plugging in the values: \[ A = 3000 \left(1 + \frac{0.08}{4}\right)^{4 \times 5} = 3000 \left(1 + 0.02\right)^{20} = 3000 \times 1.486 \approx 4457.84 \] 2. Interest Amount: \[ \text{Interest} = 4457.84 - 3000 \approx 1457.84 \] ### Summary Table Let's fill in your summary table with the calculated values: \[ \begin{tabular}{|l|l|l|} \hline Compounding & Total Amount & Interest Amount \\ \hline annually & \(\$[/tex]4407.98\) & [tex]\(\$1407.98\)[/tex] \\
\hline
semiannually & [tex]\(\$4440.73\)[/tex] & [tex]\(\$1440.73\)[/tex] \\
\hline
quarterly & [tex]\(\$4457.84\)[/tex] & [tex]\(\$1457.84\)[/tex] \\
\hline
\end{tabular}
\]

Thus, the total amounts and interest amounts for each compounding period are as listed in the table above.
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