Envy needs to have $8000 in 4 years to purchase a new sound system for the theatre they are building in their new home. How much do they need to invest in a plan that pays 2.75% compounded quarterly, to reach their goal?

To determine how much Envy needs to invest now to have $8,000 in 4 years, we use the formula for the future value of a single sum, which is:

\[
FV = PV \times (1 + r)^n
\]

Where:
- $ FV $ is the future value, which is $8,000.
- $ PV $ is the present value or the initial investment amount (this is what we need to find).
- $ r $ is the interest rate per period.
- $ n $ is the total number of compounding periods.

### Given:
- Future Value, $ FV = 8,000 $ dollars.
- Annual interest rate, $ 2.75\% $.
- Compounded quarterly.
- Time period = 4 years.

### Steps to solve:

1. **Calculate the number of periods (n):**

   Since the interest is compounded quarterly and there are 4 quarters in a year:

   \[
   n = 4 \times 4 = 16 \text{ periods}
   \]

2. **Determine the interest rate per period (r):**

   The annual interest rate is 2.75%, so the quarterly interest rate is:

   \[
   r = \frac{2.75\%}{4} = \frac{0.0275}{4} \approx 0.006875
   \]

3. **Rearrange the formula to solve for $ PV $:**

   \[
   PV = \frac{FV}{(1 + r)^n}
   \]

4. **Plug in the values and solve for $ PV $:**

   \[
   PV = \frac{8000}{(1 + 0.006875)^{16}}
   \]

Let's calculate $ PV $.Envy needs to invest approximately **$7,169.37** now in a plan that pays 2.75% interest compounded quarterly to reach their goal of $8,000 in 4 years.

Would you like more details or have any questions? Here are some related questions:

1. How much would Envy need to invest if the interest rate was 3% instead of 2.75%?
2. What if Envy wanted to reach the goal in 3 years instead of 4? How would the investment amount change?
3. How much interest would Envy earn over the 4 years?
4. How does the compounding frequency (e.g., quarterly vs. annually) affect the required initial investment?
5. What if Envy decides to invest $7,500? How much would that grow to in 4 years?

**Tip:** The frequency of compounding has a significant impact on the growth of an investment. The more frequently the interest is compounded, the more interest will be earned.

Learn how to calculate the investment needed to reach $8,000 in 4 years using compound interest. This problem involves finding the present value using the future value formula with quarterly compounding and an interest rate of 2.75% per annum.

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