Question

an amount of $27,000 is borrowed for 5 years at 7.25% interest, compounded annually. assuming that no payments are made, find the amount owed after 5 years. use the calculator provided and round your answer to the nearest dollar.

Explanation:

Step1: Recall compound interest formula

The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, where:

• $A$ is the amount of money accumulated after $n$ years, including interest.
• $P$ is the principal amount (the initial amount of money).
• $r$ is the annual interest rate (decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time the money is invested or borrowed for, in years.

In this problem, since it's compounded annually, $n = 1$. The principal $P = 27000$, the annual interest rate $r = 7.25\% = 0.0725$, and the time $t = 5$ years.

Step2: Substitute the values into the formula

Substitute $P = 27000$, $r = 0.0725$, $n = 1$, and $t = 5$ into the formula:
$$A = 27000(1 + \frac{0.0725}{1})^{1\times5}$$
Simplify the expression inside the parentheses first: $1 + 0.0725 = 1.0725$. Then, raise this to the power of $5$: $1.0725^5$. Calculate $1.0725^5 \approx 1.414778$.

Step3: Calculate the amount $A$

Multiply the principal by this value: $A = 27000\times1.414778 \approx 38200.91$. Rounding to the nearest dollar gives $38201$.

Answer:

$\$38201$