Brandon invested $1100 in an account paying an interest rate of 7 3/4% compounded continuously. Julian invested $1100 in an account paying an interest rate of 8 1/4% compounded monthly. After 18 years, how much more money would Julian have in his account than Brandon, to the nearest dollar?

To determine how much more money Julian would have in his account than Brandon after 18 years, we need to apply the formulas for compound interest.

### 1. **Brandon's Investment** (Continuous Compounding):
The formula for continuously compounded interest is:

\[
A = P e^{rt}
\]
Where:
- $ A $ is the amount after $ t $ years.
- $ P $ is the principal (initial investment).
- $ r $ is the annual interest rate as a decimal.
- $ t $ is the time in years.
- $ e $ is the mathematical constant (approximately 2.71828).

For Brandon:
- $ P = 1100 $
- $ r = 7 \frac{3}{4}\% = 0.0775 $
- $ t = 18 $

\[
A_{\text{Brandon}} = 1100 \times e^{0.0775 \times 18}
\]

### 2. **Julian's Investment** (Compounded Monthly):
The formula for interest compounded periodically (monthly, in this case) is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $ A $ is the amount after $ t $ years.
- $ P $ is the principal (initial investment).
- $ r $ is the annual interest rate as a decimal.
- $ t $ is the time in years.
- $ n $ is the number of times the interest is compounded per year (monthly means $ n = 12 $).

For Julian:
- $ P = 1100 $
- $ r = 8 \frac{1}{4}\% = 0.0825 $
- $ n = 12 $
- $ t = 18 $

\[
A_{\text{Julian}} = 1100 \left(1 + \frac{0.0825}{12}\right)^{12 \times 18}
\]

Now, let's calculate both amounts and find the difference.After 18 years:
- Brandon's account will have approximately \$4,438.
- Julian's account will have approximately \$4,832.

The difference between Julian's and Brandon's accounts is **\$393** to the nearest dollar.

Would you like further details or explanations on this?

Here are 5 related questions:
1. How would the difference change if both accounts had the same interest compounding method?
2. What would happen if the interest rates for both were swapped?
3. How does compounding frequency affect the final amount in an investment?
4. What would be the difference if the investment period was 25 years instead of 18?
5. How would the outcome differ if Julian's investment compounded quarterly instead of monthly?

**Tip:** The more frequently interest is compounded, the higher the final amount, as interest is being calculated and added more often.

Explore how much more money Julian would have in his account compared to Brandon after 18 years, using different compounding methods. This example compares continuous compounding at 7 3/4% and monthly compounding at 8 1/4%. Learn the key formulas and concepts behind compounding interest. Suitable for high school students (Grades 10-12).

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