The formula for the present value of money is:

$PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}$

Where:

• $PV$ is the present value (the amount you need to invest now),
• $FV$ is the future value (the amount you want in the future, $25,000),
• $r$ is the annual percentage rate (APR, 6% or 0.06),
• $n$ is the number of times the interest is compounded per year (quarterly, so 4 times a year),
• $t$ is the time in years (10 years).

Let's plug in the values:

$PV = \frac{25000}{\left(1 + \frac{0.06}{4}\right)^{4 \times 10}}$

Now, I'll calculate the result.

$PV = \frac{25000}{\left(1 + 0.015\right)^{40}} = \frac{25000}{(1.015)^{40}}$

I'll compute this value now.The amount you need to invest now to have $25,000 after 10 years with an APR of 6% compounded quarterly is approximately $13,781.56.

Would you like more details or have any other questions?

Here are 5 related questions to explore:

1. How does increasing the APR affect the present value?
2. What happens to the present value if the interest is compounded monthly instead of quarterly?
3. How would the present value change if the investment period is reduced to 5 years?
4. What is the impact of continuous compounding on the present value?
5. Can the formula be applied to find how much you need to invest if you want $50,000 in 20 years?

Tip: The more frequently interest is compounded, the more you'll benefit from earning interest on interest!