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Claire wants to take out a small personal loan to renovate her kitchen. She borrows [tex]\[tex]$ 3,000[/tex]. Her loan has an annual compound interest rate of [tex]15\%[/tex]. The loan compounds once each year.

When you calculate Claire's debt, be sure to use the formula for annual compound interest:
\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]

If Claire does not make any payments, how much will she owe after ten years?

A. [tex]\$[/tex] 12,136.67[/tex]
B. [tex]\[tex]$ 3,481.24[/tex]
C. [tex]\$[/tex] 6,090.90[/tex]
D. [tex]\$ 3,232.74[/tex]



Answer :

GinnyAnswer
To determine how much Claire will owe after ten years if she does not make any payments, we can use the formula for compound interest:

[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 (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for in years.

Given the problem:
- [tex]\( P = \$ 3,000 \)[/tex] (the principal amount)
- [tex]\( r = 0.15 \)[/tex] (annual interest rate of 15% expressed as a decimal)
- [tex]\( n = 1 \)[/tex] (interest compounds once per year)
- [tex]\( t = 10 \)[/tex] (number of years)

Substitute these values into the formula:

[tex]\[ A = 3000 \left(1 + \frac{0.15}{1}\right)^{1 \cdot 10} \][/tex]
[tex]\[ A = 3000 \left(1 + 0.15\right)^{10} \][/tex]
[tex]\[ A = 3000 \left(1.15\right)^{10} \][/tex]

Using this formula, Claire will owe \[tex]$12,136.67 after ten years. So, the correct answer is: \[ \$[/tex] 12,136.67 \]

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