Suppose that $12,000
is deposited for six years at 5%
APR. Calculate the interest earned if interest is compounded quarterly. Round your answer to the nearest cent.

To calculate the interest earned when $12,000 is deposited for six years at an annual percentage rate (APR) of 5%, with interest compounded quarterly, we can use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

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 time the money is invested or borrowed for, in years.

Let's plug in the given values:
- $ P = 12,000 $ dollars
- $ r = 5\% = 0.05 $
- $ n = 4 $ (since interest is compounded quarterly)
- $ t = 6 $ years

Now, calculate $ A $:

\[
A = 12,000 \left(1 + \frac{0.05}{4}\right)^{4 \times 6}
\]

\[
A = 12,000 \left(1 + 0.0125\right)^{24}
\]

\[
A = 12,000 \left(1.0125\right)^{24}
\]

Now, calculate $ (1.0125)^{24} $:

\[
A = 12,000 \times 1.28203723171
\]

\[
A \approx 15,384.45
\]

The interest earned is the final amount minus the principal:

\[
\text{Interest Earned} = A - P = 15,384.45 - 12,000 \approx 3,384.45 \text{ dollars}
\]

So, the interest earned is **$3,384.45**.

Would you like more details or have any questions?

Here are some related questions:
1. What would the interest be if it were compounded monthly instead of quarterly?
2. How would the interest change if the APR were 6% instead of 5%?
3. What is the formula for continuous compounding, and how would it apply here?
4. How does the length of time (e.g., 10 years instead of 6) affect the compound interest?
5. Can you calculate the future value if the interest is compounded semi-annually?
6. How does compounding quarterly differ from compounding annually in terms of interest earned?
7. What is the difference between nominal interest rate and effective interest rate in this context?
8. How would you calculate the compound interest if the deposit was $15,000 instead of $12,000?

**Tip:** Remember that the more frequently interest is compounded, the more interest is earned over time.

Learn how to calculate compound interest with quarterly compounding using the given deposit amount, interest rate, and time period. Includes a step-by-step solution with detailed calculations.

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