Use the present value formula to determine the amount to be invested​ now, or the present value needed. The desired accumulated amount is $100,000 after 6 years invested in an account with 3​% interest compounded quarterly.

To determine the present value (the amount to be invested now) using the present value formula for compound interest, we can use the following formula:

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

Where:
- $ PV $ is the present value (the amount to invest now).
- $ A $ is the accumulated amount (future value), which is $100,000.
- $ r $ is the annual interest rate (as a decimal), which is 3% or 0.03.
- $ t $ is the time in years, which is 6 years.
- $ n $ is the number of times the interest is compounded per year, which is quarterly, so $ n = 4 $.

### Step-by-Step Calculation:
1. Substituting the values into the formula:
   \[
   PV = \frac{100,000}{\left(1 + \frac{0.03}{4}\right)^{4 \times 6}}
   \]
2. Simplifying the terms:
   \[
   PV = \frac{100,000}{\left(1 + 0.0075\right)^{24}} = \frac{100,000}{(1.0075)^{24}}
   \]
3. Calculate $ (1.0075)^{24} $.

Let me calculate the present value for you.

The present value, or the amount that needs to be invested now, is approximately **$83,583.14**.

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would the present value change if the interest rate increased to 4%?
2. What if the interest was compounded monthly instead of quarterly?
3. How much would be required if the investment period was only 5 years?
4. What would the future value be if only $80,000 was invested instead of $83,583.14?
5. How does compounding frequency affect the final accumulated amount?

**Tip:** The more frequently interest is compounded, the more the investment grows over time.

Use the present value formula to determine the amount to be invested now, or the present value needed. The desired accumulated amount is $100,000 after 6 years invested in an account with 3% interest compounded quarterly.

Calculate how much needs to be invested now to reach $100,000 in 6 years with a 3% interest rate compounded quarterly. Follow a detailed step-by-step guide to apply the present value formula for compound interest and find the answer. Ideal for high school students and financial math learners.

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