Hale College predicts that in 10 years it will take $200,000 to attend the college for four years. Hailey has a substantial amount of cash and wishes to invest a lump sum of money for her child’s college fund. How much should Hailey put aside in an account with an APR of 12% compounded monthly in order to have $200,000 in the account in 10 years? Round your answer to the nearest cent, if necessary.

This problem involves finding the present value of an investment, given a future value, an interest rate, and a time period. To solve it, we'll use the formula for the **present value** $ P $ of a sum given compound interest:

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

Where:
- $ A $ is the future value (\$200,000),
- $ r $ is the annual interest rate (12% or 0.12),
- $ n $ is the number of times the interest is compounded per year (monthly compounding, so $ n = 12 $),
- $ t $ is the time in years (10 years),
- $ P $ is the present value (the amount Hailey needs to invest today).

Let's plug in the numbers:

\[
P = \frac{200,000}{(1 + \frac{0.12}{12})^{12 \times 10}}
\]

I will now calculate the value of $ P $.Hailey should invest approximately **$60,598.96** today in an account with an APR of 12% compounded monthly to have $200,000 in 10 years.

Would you like more details or have any other questions? 

Here are some related questions to explore:

1. What if the interest rate was compounded quarterly instead of monthly?
2. How much would the final amount be after 10 years if she invested $70,000 instead?
3. What happens if Hailey can only invest $50,000? How much will she have in 10 years?
4. How does the investment amount change with a lower interest rate, such as 8%?
5. How long would it take to reach $200,000 if the initial investment was $60,598.96 but with a 10% interest rate?

**Tip**: Higher compounding frequency (e.g., monthly vs. yearly) leads to a higher future value for the same interest rate.

Learn how to calculate how much Hailey should invest now to reach $200,000 in 10 years with a 12% APR compounded monthly. This financial math problem covers key concepts like compound interest, present value, and exponential growth, suitable for students in Grades 10-12.

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