Question

Suppose that $2000 is loaned at a rate of 10%, compounded quarterly. Assuming that no payments are made, find the amount owed after 4 years.
Do not round any intermediate computations, and round your answer to the nearest cent.

Answer 1

To determine the total amount owed after 4 years on a loan of $2000 at a rate of 10% per annum, compounded quarterly, 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 sum 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 for.

Given values:
- \(P = 2000\)
- \(r = 0.10\) (10% as a decimal)
- \(n = 4\) (since the interest is compounded quarterly)
- \(t = 4\)

Substituting these values into the formula:

\[
A = 2000 \left(1 + \frac{0.10}{4}\right)^{4 \cdot 4}
\]

First, calculate \(\frac{0.10}{4}\):

\[
\frac{0.10}{4} = 0.025
\]

Now, substitute back into the formula:

\[
A = 2000 \left(1 + 0.025\right)^{16}
\]
\[
A = 2000 \left(1.025\right)^{16}
\]

Next, calculate \((1.025)^{16}\):

\[
(1.025)^{16} \approx 1.448882
\]

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

\[
A \approx 2000 \times 1.448882 \approx 2897.764
\]

Finally, rounding to the nearest cent:

\[
A \approx 2897.76
\]

Thus, the amount owed after 4 years is \(\boxed{2897.76}\).