Question

Charmaine borrowed $8000 at a rate of 7%, compounded semiannually. Assuming she makes no payments, how much will she owe after 8 years?
Do not round any intermediate computations, and round your answer to the nearest cent.

Answer 1

To calculate how much Charmaine will owe after 8 years on her loan of $8000 at a rate of 7% compounded semiannually, 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 (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 number of years the money is invested or borrowed.

In this case:
- \( P = 8000 \)
- \( r = 0.07 \) (7%)
- \( n = 2 \) (since the interest is compounded semiannually)
- \( t = 8 \)

Now we can substitute these values into the formula:

\[
A = 8000 \left(1 + \frac{0.07}{2}\right)^{2 \times 8}
\]

Calculating inside the parentheses first:

\[
A = 8000 \left(1 + 0.035\right)^{16}
\]
\[
A = 8000 \left(1.035\right)^{16}
\]

Now we need to calculate \( (1.035)^{16} \):

\[
(1.035)^{16} \approx 1.747422
\]

Now substituting this value back into the equation for \( A \):

\[
A \approx 8000 \times 1.747422
\]
\[
A \approx 13979.376
\]

Finally, rounding to the nearest cent:

\[
A \approx 13979.38
\]

Therefore, Charmaine will owe approximately **$13,979.38** after 8 years.