To determine how much you need to deposit each quarter to reach $4,500 in 4 years with an interest rate of 3.8% compounded quarterly, we use the formula for the future value of an ordinary annuity:

$FV = P \times \frac{(1 + r)^n - 1}{r}$

Where:

• $FV$ is the future value (the amount you want to have in the future, $4,500).
• $P$ is the quarterly deposit (what we want to find).
• $r$ is the quarterly interest rate.
• $n$ is the total number of periods.

Step 1: Find the values for $r$ and $n$

• The annual interest rate is 3.8%, so the quarterly interest rate is: $r = \frac{3.8\%}{4} = \frac{0.038}{4} = 0.0095$
• Since the money is deposited quarterly for 4 years, the total number of periods is: $n = 4 \text{ years} \times 4 \text{ quarters per year} = 16 \text{ quarters}$

Step 2: Rearrange the formula to solve for $P$

$P = \frac{FV}{\frac{(1 + r)^n - 1}{r}}$

Substituting the known values:

$P = \frac{4500}{\frac{(1 + 0.0095)^{16} - 1}{0.0095}}$

Step 3: Calculate the result

First, calculate the value inside the parentheses:

$(1 + 0.0095)^{16} = 1.0095^{16} \approx 1.1589$

Then, subtract 1:

$1.1589 - 1 = 0.1589$

Next, divide by the quarterly interest rate:

$\frac{0.1589}{0.0095} \approx 16.726$

Finally, divide the future value by this result:

$P = \frac{4500}{16.726} \approx 269.02$

Conclusion:

You need to deposit approximately $269.02 each quarter to reach $4,500 in 4 years.

Would you like more details on this calculation or have any questions?

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Here are 5 related questions to expand on this topic:

1. What happens if the interest rate changes to 4% compounded quarterly?
2. How much interest would you earn on your deposits over the 4 years?
3. What is the difference between monthly and quarterly compounding for this problem?
4. What if you wanted to deposit the money annually instead of quarterly?
5. How does inflation affect your goal of saving $4,500 over 4 years?

Tip: Compounding frequency can significantly impact the amount you need to save! The more frequent the compounding, the more interest you'll earn.