The following equation gives the amount of money owed on a loan after a certain amount of time if no payments are made. $A=P\left(1+\frac{r}{n}\right)^{n t}$ where:
$A$ = the amortized amount (total loan/investment amount over the life of the loan/investment)
$P=$ the initial amount of the loan/investment
$r=$ the annual rate of interest
$n=$ the number of times interest is compounded each year
$t=$ the time in years
Find the amount owed at the end of 3 years if $\$ 2,700$ is loaned at a rate $8 \%$ compounded semi-annually.
(Round answer to 2 decimal places. No dollar signs or commas.)

Step 1 ：Given the formula for the amount owed on a loan after a certain amount of time if no payments are made: \(A=P\left(1+\frac{r}{n}\right)^{n t}\), where:

Step 2 ：\(A\) = the amortized amount (total loan/investment amount over the life of the loan/investment)

Step 3 ：\(P\) = the initial amount of the loan/investment

Step 4 ：\(r\) = the annual rate of interest

Step 5 ：\(n\) = the number of times interest is compounded each year

Step 6 ：\(t\) = the time in years

Step 7 ：We are asked to find the amount owed at the end of 3 years if \$2700 is loaned at a rate of 8% compounded semi-annually.

Step 8 ：We substitute the given values into the formula: \(P = 2700\), \(r = 0.08\), \(n = 2\), and \(t = 3\).

Step 9 ：Substituting these values into the formula gives us: \(A = 2700\left(1+\frac{0.08}{2}\right)^{2 \times 3}\)

Step 10 ：Solving this gives us \(A = 3416.361349939201\)

Step 11 ：Rounding to two decimal places, the final answer is \(\boxed{3416.36}\)