An investment of $6000 was placed in an account for 3 years at 4.25% p.a. compounded annually. How much more money would be collected if the investment was compounded quarterly?

To determine how much more money would be collected if the investment were compounded quarterly rather than annually, we need to calculate the future value of the investment under both compounding conditions and compare the results.

### Future Value with Annual Compounding

The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Where:
- $ P $ is the principal amount (\$6000)
- $ r $ is the annual interest rate (4.25% or 0.0425)
- $ n $ is the number of times the interest is compounded per year
- $ t $ is the number of years

For annual compounding ($ n = 1 $):
\[ A_{\text{annual}} = 6000 \left(1 + \frac{0.0425}{1}\right)^{1 \cdot 3} \]
\[ A_{\text{annual}} = 6000 \left(1 + 0.0425\right)^3 \]
\[ A_{\text{annual}} = 6000 \left(1.0425\right)^3 \]
\[ A_{\text{annual}} = 6000 \times 1.1322 \]
\[ A_{\text{annual}} = 6793.20 \]

### Future Value with Quarterly Compounding

For quarterly compounding ($ n = 4 $):
\[ A_{\text{quarterly}} = 6000 \left(1 + \frac{0.0425}{4}\right)^{4 \cdot 3} \]
\[ A_{\text{quarterly}} = 6000 \left(1 + 0.010625\right)^{12} \]
\[ A_{\text{quarterly}} = 6000 \left(1.010625\right)^{12} \]

Using a calculator:
\[ A_{\text{quarterly}} \approx 6000 \times 1.13509 \]
\[ A_{\text{quarterly}} \approx 6810.54 \]

### Difference in Amounts

\[ \text{Difference} = A_{\text{quarterly}} - A_{\text{annual}} \]
\[ \text{Difference} = 6810.54 - 6793.20 \]
\[ \text{Difference} = 17.34 \]

Therefore, the amount collected with quarterly compounding would be \$17.34 more than with annual compounding.

Would you like more details or have any questions?

Here are some related questions:
1. What is the formula for compound interest?
2. How does the frequency of compounding affect the future value of an investment?
3. What is the effect of increasing the interest rate on compound interest?
4. How would the future value change if the investment period was 5 years instead of 3 years?
5. What is continuous compounding, and how is it calculated?
6. How does quarterly compounding compare to monthly compounding?
7. What is the difference between simple interest and compound interest?
8. How does inflation affect the real value of an investment with compound interest?

**Tip:** When comparing investments, always consider the compounding frequency as it can significantly impact the future value.

Learn how to calculate compound interest with different compounding frequencies using a detailed example. Compare the future value of an investment compounded annually versus quarterly. Understand the impact of compounding frequency on investment returns.

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