Find the compound amount for the deposit and the amount of interest earned for the following account:

$23,000 at 5% compounded quarterly for 4 years

To find the compound amount and the amount of interest earned for the given account, 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 for in years.

Given:
- $ P = \$23,000 $
- $ r = 5\% = 0.05 $
- $ n = 4 $ (since the interest is compounded quarterly)
- $ t = 4 $ years

Let's plug these values into the formula to find the compound amount $ A $:

\[
A = 23000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 4}
\]

First, calculate the quarterly interest rate:

\[
\frac{0.05}{4} = 0.0125
\]

Next, calculate the exponent:

\[
4 \cdot 4 = 16
\]

Now, calculate the compound factor:

\[
\left(1 + 0.0125\right)^{16}
\]

\[
1.0125^{16} \approx 1.219
\]

Finally, calculate $ A $:

\[
A = 23000 \times 1.219 \approx 28037
\]

So, the compound amount after 4 years is approximately \$28,037.

To find the amount of interest earned, subtract the principal from the compound amount:

\[
\text{Interest Earned} = A - P
\]

\[
\text{Interest Earned} = 28037 - 23000 = 5037
\]

Therefore:
- The compound amount for the deposit is approximately \$28,037.
- The amount of interest earned is approximately \$5,037.

Would you like further details on this, or do you have any questions?

### Related Questions:
1. How does the frequency of compounding affect the compound amount?
2. What would be the compound amount if the interest rate were 6% instead of 5%?
3. How can you calculate the future value for different compounding intervals (e.g., monthly, semi-annually)?
4. What is the formula for continuous compounding, and how does it differ from periodic compounding?
5. How do you determine the effective annual rate (EAR) from a nominal interest rate compounded quarterly?
6. What is the impact of increasing the principal on the total interest earned?
7. How would the compound amount change if the investment period were 5 years instead of 4 years?
8. How do you calculate the present value given a future amount and a compounding interest rate?

### Tip:
When solving compound interest problems, always ensure you convert the interest rate and time period to match the compounding frequency for accurate calculations.

Learn how to calculate the compound amount and interest earned for $23,000 invested at 5% compounded quarterly for 4 years. Detailed step-by-step solution using the compound interest formula. Suitable for high school students and anyone interested in financial mathematics.

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