Question

an amount of $31,000 is borrowed for 9 years at 8.5% interest, compounded annually. assuming that no payments are made, find the amount owed after 9 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, the interest is compounded annually, so $n = 1$. The principal $P = 31000$, the annual interest rate $r = 8.5\% = 0.085$, and the time $t = 9$ years.

Step2: Substitute the values into the formula

Substitute $P = 31000$, $r = 0.085$, $n = 1$, and $t = 9$ into the formula:
\[

$$\begin{align*} A&=31000\times(1 + \frac{0.085}{1})^{1\times9}\\ &=31000\times(1 + 0.085)^{9}\\ &=31000\times(1.085)^{9} \end{align*}$$

\]

Step3: Calculate $(1.085)^{9}$

First, calculate $(1.085)^{9}$. Using a calculator, $(1.085)^{9}\approx2.079462$.

Step4: Calculate the amount $A$

Multiply this value by the principal:
\[
A = 31000\times2.079462\approx31000\times2.079462 = 64463.322
\]

Step5: Round to the nearest dollar

Rounding $64463.322$ to the nearest dollar gives $64463$.

Answer:

$\$64463$