To find the amount owed after 7 years when [tex]$2000 is loaned at a rate of 18% compounded annually, follow these steps:
1. Identify the given values:
- Principal (P) = $[/tex]2000
- Annual interest rate (r) = 18% or 0.18
- Number of years (t) = 7
2. Understand the compound interest formula:
The formula for compound interest is:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (initial investment or loan).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years the money is invested or borrowed for.
3. Apply the formula to our specific problem:
Since the interest is compounded annually, [tex]\( n \)[/tex] = 1. Plugging in the values, we get:
[tex]\[
A = 2000 \left(1 + \frac{0.18}{1}\right)^{1 \times 7}
\][/tex]
4. Simplify inside the parentheses:
[tex]\[
A = 2000 \left(1 + 0.18\right)^7
\][/tex]
[tex]\[
A = 2000 \left(1.18\right)^7
\][/tex]
5. Calculate [tex]\( (1.18)^7 \)[/tex]:
Raising 1.18 to the 7th power, we get:
[tex]\[
(1.18)^7 \approx 3.18585
\][/tex]
6. Multiply by the principal:
[tex]\[
A = 2000 \times 3.18585
\][/tex]
[tex]\[
A \approx 6371.70
\][/tex]
7. Round to the nearest cent:
Since we need to round to the nearest cent, the amount owed after 7 years is approximately [tex]$6370.95.
Therefore, the amount owed after 7 years, rounded to the nearest cent, is $[/tex]6370.95.
