**You wish to have $4,000 in 3 years to buy a new stereo system.**

**How much should you deposit each quarter into an account paying 6.7% compounded quarterly?**

*Do not round numbers while working on the problem.  Only round the final answer.*

*Since we are working with money, round to the cent.*

To solve this problem, we can use the formula for the future value of an annuity (since we are making regular deposits):

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

Where:
- $ FV $ is the future value (the amount you want in the future), which is $4,000.
- $ P $ is the amount deposited each period (what we need to find).
- $ r $ is the interest rate per period.
- $ n $ is the total number of periods.

Given:
- The annual interest rate is 6.7%, but since the interest is compounded quarterly, we divide this by 4 to get the quarterly interest rate:
  \[
  r = \frac{6.7\%}{4} = \frac{0.067}{4} = 0.01675
  \]
- The total number of periods $ n $ is the number of quarters in 3 years:
  \[
  n = 3 \times 4 = 12
  \]
- $ FV = 4000 $ (the future value you want).

Now, substitute the values into the formula and solve for $ P $:

\[
4000 = P \times \frac{(1 + 0.01675)^{12} - 1}{0.01675}
\]

First, let's calculate the expression:

1. $ (1 + 0.01675)^{12} $
2. Subtract 1 from the result.
3. Divide by $ r $ (which is 0.01675).
4. Finally, solve for $ P $.

Let me calculate this.The amount you should deposit each quarter is **$303.73** to have $4,000 in 3 years, given an interest rate of 6.7% compounded quarterly.

Would you like any further details or have any questions?

Here are five related questions you might find interesting:
1. How would the required quarterly deposit change if the interest rate were higher?
2. What if the compounding was done monthly instead of quarterly?
3. How much total interest would be earned over the 3 years?
4. How much would you need to deposit each month if you wanted $10,000 instead of $4,000 in 3 years?
5. What is the difference between simple interest and compound interest in this context?

**Tip:** When dealing with compound interest problems, always confirm the compounding frequency as it significantly impacts the amount needed to reach your financial goal.

Learn how to calculate the quarterly deposits needed to achieve $4,000 in 3 years with a 6.7% interest rate compounded quarterly. Get step-by-step instructions and formulas used in financial planning.

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