To solve this problem, we need to use the formula for the future value of a series of equal payments (an ordinary annuity), since the deposits are made quarterly and the interest is compounded quarterly.

The formula is:

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

Where:

• $FV$ is the future value (the goal, $22,000)
• $P$ is the regular quarterly deposit amount (which we need to find)
• $r$ is the interest rate per period
• $n$ is the total number of periods

Step 1: Identify the known values.

• $FV = 22,000$ (the future value we want to reach)
• Interest rate per year = 4.9%, so per quarter $r = \frac{4.9\%}{4} = \frac{0.049}{4} = 0.01225$
• The number of years = 6 years, and since we are making quarterly deposits, $n = 6 \times 4 = 24$ periods

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

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

Step 3: Substitute the values into the formula.

$P = \frac{22,000 \times 0.01225}{(1 + 0.01225)^{24} - 1}$

Now, let's calculate this step-by-step.

The quarterly deposit needed to reach a future value of $22,000 over 6 years, with a 4.9% interest rate compounded quarterly, is approximately $794.07.

Would you like further details on any part of this solution? Here are some related questions you might explore:

1. What if the interest rate were higher or lower—how would the quarterly deposit change?
2. How much total interest would be earned over the 6 years?
3. What if the goal was to save in 5 years instead of 6?
4. How does changing the compounding period affect the required deposit?
5. What if instead of quarterly, the deposit was made monthly?

Tip: Compounding interest grows faster the more frequently it is compounded, making it important to understand how small changes in periods and rates affect savings.