Arnett College predicts that in 18
 years it will take $350,000
 to attend the college for four years. Hannah has a substantial amount of cash and wishes to invest a lump sum of money for her child’s college fund. How much should Hannah put aside in an account with an APR of 8%
 compounded monthly in order to have $350,000
 in the account in 18
 years? Round your answer to the nearest cent, if necessary.



To solve this problem, we'll use the formula for the present value of a lump sum investment that will grow to a specific future value with compound interest:

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

Where:
- $ P $ is the present value (the amount Hannah needs to invest now).
- $ F $ is the future value (the amount needed in the future, which is $350,000).
- $ r $ is the annual interest rate (APR), expressed as a decimal (8% = 0.08).
- $ n $ is the number of times the interest is compounded per year (monthly compounding means $ n = 12 $).
- $ t $ is the number of years the money is invested (18 years).

### Step-by-Step Calculation:

1. **Given values:**
   - $ F = 350,000 $
   - $ r = 0.08 $
   - $ n = 12 $
   - $ t = 18 $

2. **Substitute the values into the formula:**

\[
P = \frac{350,000}{\left(1 + \frac{0.08}{12}\right)^{12 \times 18}}
\]

3. **Calculate the monthly interest rate:**

\[
\frac{0.08}{12} = 0.0066667
\]

4. **Calculate the exponent $ 12 \times 18 = 216 $:**

\[
P = \frac{350,000}{\left(1 + 0.0066667\right)^{216}}
\]

\[
P = \frac{350,000}{\left(1.0066667\right)^{216}}
\]

5. **Compute $ \left(1.0066667\right)^{216} $:**

\[
\left(1.0066667\right)^{216} \approx 4.037
\]

6. **Calculate the present value $ P $:**

\[
P = \frac{350,000}{4.037} \approx 86,663.57
\]

### Conclusion:

Hannah should invest approximately **$86,663.57** now in an account with an APR of 8% compounded monthly to have $350,000 in 18 years.

Would you like further details on the steps? Here are 5 related questions to deepen your understanding:

1. What if the interest was compounded quarterly instead of monthly?
2. How would the required investment change if the interest rate was 7% instead of 8%?
3. Can we solve the same problem if the future amount required was $400,000 instead of $350,000?
4. How would the result differ if Hannah started saving with a different initial amount and made regular contributions?
5. What is the significance of compounding frequency on the amount Hannah needs to invest?

**Tip:** Always check the compounding frequency when calculating future or present values, as it significantly impacts the required investment.

Arnett College predicts that in 18 years it will take $350,000 to attend the college for four years. Hannah has a substantial amount of cash and wishes to invest a lump sum of money for her child’s college fund. How much should Hannah put aside in an account with an APR of 8% compounded monthly in order to have $350,000 in the account in 18 years? Round your answer to the nearest cent, if necessary.

Find out how much Hannah needs to invest today to secure $350,000 for college in 18 years with an 8% APR compounded monthly. Learn the step-by-step solution using the present value formula. Suitable for advanced high school students.

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