If $1,600 is invested at an interest rate of 2.75% per year, compounded quarterly, find the value of the investment (in dollars) after 1 year, 5 years, and 10 years

To find the future value of an investment with compounded interest, we use the compound interest formula:

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

Where:
- $ A $ is the future value of the investment,
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time in years.

Given:
- $ P = 1600 $,
- $ r = 2.75\% = 0.0275 $,
- $ n = 4 $ (since it's compounded quarterly),
- $ t $ will be 1 year, 5 years, and 10 years for the three cases.

Let’s calculate the value of the investment for each time period.

### For $ t = 1 $ year:

\[
A_1 = 1600 \left(1 + \frac{0.0275}{4}\right)^{4 \times 1}
\]

### For $ t = 5 $ years:

\[
A_5 = 1600 \left(1 + \frac{0.0275}{4}\right)^{4 \times 5}
\]

### For $ t = 10 $ years:

\[
A_{10} = 1600 \left(1 + \frac{0.0275}{4}\right)^{4 \times 10}
\]

I will now calculate the future values for each time period.The value of the investment after:

1. **1 year** is approximately **$1,644.46**,
2. **5 years** is approximately **$1,834.98**,
3. **10 years** is approximately **$2,104.47**.

Would you like further details or explanations?

Here are five related questions to expand on this:
1. How does increasing the compounding frequency affect the final investment value?
2. What is the difference between simple interest and compound interest?
3. How would the final value change if the interest rate were higher?
4. Can you explain how the compound interest formula works in detail?
5. How would monthly compounding affect these calculations?

**Tip**: Always remember that compounding more frequently (e.g., monthly instead of quarterly) results in a slightly higher final value, even with the same interest rate!

If $1,600 is invested at an interest rate of 2.75% per year, compounded quarterly, find the value of the investment (in dollars) after 1 year, 5 years, and 10 years.

Learn how to calculate the future value of a $1,600 investment compounded quarterly at an annual interest rate of 2.75% for 1, 5, and 10 years. This problem uses the compound interest formula and covers key algebraic concepts related to exponential growth. Suitable for students in Grades 9-12.

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