To calculate the amount Yasmin will pay back, we can use the formula for compound interest, which is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after \( n \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form).
- \( n \) 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 Yasmin's case, the principal, \( P \), is $4,000, the annual interest rate, \( r \), is 6% (or 0.06 in decimal form), the number of times the interest is compounded per year, \( n \), is once since it's compounded annually, and the time, \( t \), is 7 years.
We can plug the values into the formula:
\[ A = 4000 \left(1 + \frac{0.06}{1}\right)^{(1)(7)} \]
\[ A = 4000 \left(1 + 0.06\right)^7 \]
\[ A = 4000 \times (1.06)^7 \]
We can then calculate the expression step-by-step:
\[ A = 4000 \times 1.06^7 \]
\[ A = 4000 \times 1.418519 \] (Calculating \( 1.06^7 \) using a calculator)
Lastly, multiply the principal by the compounding factor:
\[ A \approx 4000 \times 1.418519 \]
\[ A \approx 5674.076 \]
Yasmin will pay back approximately $5,674.08 after 7 years, when we round the amount to the nearest cent.
