harmaine borrowed \( \$ 8000 \) at a rate of \( 12.5 \% \), compcunded quarterly, Assuming she makes no payments, how much will she owe after 7 years? o not round any intermediate computations, and rouind your anower to the nearest cent. \( \square \)

To find out how much Harmaine will owe after 7 years, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \(A\) is the amount of money accumulated after n years, including interest. - \(P\) is the principal amount (\$8000). - \(r\) is the annual interest rate (decimal) (12.5% = 0.125). - \(n\) is the number of times that interest is compounded per year (4 for quarterly). - \(t\) is the number of years the money is borrowed (7). Substituting the values into the formula: \[ A = 8000 \left(1 + \frac{0.125}{4}\right)^{4 \times 7} \] \[ = 8000 \left(1 + 0.03125\right)^{28} \] \[ = 8000 \left(1.03125\right)^{28} \] Calculating \( (1.03125)^{28} \approx 2.495546 \) Now we multiply: \[ A \approx 8000 \times 2.495546 \approx 19964.37 \] So, after 7 years, Harmaine will owe approximately **\$19,964.37**. Now, onto some fun facts! If you ever wondered how compound interest can make your money work for you, consider this: Instead of paying interest, if you were to invest that same amount at 12.5%, you'd be looking at your wealth growing exponentially. It’s often called “the eighth wonder of the world” for a reason! Also, take a moment to appreciate that interest rates have a fascinating history. Back in ancient Mesopotamia, interest was first documented in transactions involving grain and silver. In today's world, the principles of compounding are crucial in finance, real estate, and even retirement planning! Understanding how money grows can empower you to make smarter financial decisions.