Draper College predicts that in 18
 years it will take $300,000
 to attend the college for four years. Emma has a substantial amount of cash and wishes to invest a lump sum of money for her child’s college fund. How much should Emma put aside in an account with an APR of 12%
 compounded monthly in order to have $300,000
 in the account in 18
 years? Round your answer to the nearest cent, if necessary.



To solve this problem, we can use the formula for the future value of an investment with compound interest:

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

where:
- $ A $ is the future amount (which is $300,000 in this case),
- $ P $ is the principal (the amount Emma needs to invest today),
- $ r $ is the annual interest rate (12%, or 0.12),
- $ n $ is the number of times interest is compounded per year (monthly, so $ n = 12 $),
- $ t $ is the time in years (18 years).

Since we are solving for $ P $, we rearrange the formula:

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

### Plugging in the values:
- $ A = 300,000 $
- $ r = 0.12 $
- $ n = 12 $
- $ t = 18 $

\[
P = \frac{300,000}{\left(1 + \frac{0.12}{12}\right)^{12 \cdot 18}}
\]

Now, let's compute this step-by-step.

1. Calculate $ \frac{r}{n} $:
   \[
   \frac{0.12}{12} = 0.01
   \]

2. Calculate $ 1 + \frac{r}{n} $:
   \[
   1 + 0.01 = 1.01
   \]

3. Calculate $ nt $:
   \[
   12 \cdot 18 = 216
   \]

4. Raise $ 1.01 $ to the power of $ 216 $:
   \[
   1.01^{216} \approx 6.022575
   \]

5. Divide $ 300,000 $ by $ 6.022575 $:
   \[
   P \approx \frac{300,000}{6.022575} \approx 49,808.14
   \]

### Answer
Emma should invest **$49,808.14** today to reach $300,000 in 18 years with a 12% APR compounded monthly.

Would you like more details on any part of the solution, or have questions?

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### Related Questions
1. How would the required principal change if the APR was 10% instead of 12%?
2. What if Emma wanted to reach $400,000 instead of $300,000 in 18 years? How much should she invest now?
3. How would the answer change if the interest were compounded annually instead of monthly?
4. How does changing the compounding frequency affect the required initial investment?
5. How long would it take to reach $300,000 if she invested $49,808.14 at the same rate?

### Tip
When working with compound interest, the more frequent the compounding, the higher the final amount due to the interest-on-interest effect.

Draper College predicts that in 18 years it will take $300,000 to attend the college for four years. Emma has a substantial amount of cash and wishes to invest a lump sum of money for her child’s college fund. How much should Emma put aside in an account with an APR of 12% compounded monthly in order to have $300,000 in the account in 18 years? Round your answer to the nearest cent, if necessary.

Discover how much Emma needs to invest today to accumulate $300,000 for her child's college fund in 18 years at an annual interest rate of 12% compounded monthly. Detailed steps and calculations included.

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