Arianna invested $250 in an account paying an interest rate of 8, one quarter8
4
1
​
% compounded continuously. Lily invested $250 in an account paying an interest rate of 8, start fraction, 3, divided by, 4, end fraction8
4
3
​
% compounded quarterly. After 18 years, how much more money would Lily have in her account than Arianna, to the nearest dollar?
Answer
Attempt

To solve this problem, we need to calculate the future value of both Arianna's and Lily's investments after 18 years. Let's go through the steps:

### Arianna's Investment
Arianna's investment is compounded continuously. The formula for continuous compounding is given by:

\[
A = P \cdot e^{rt}
\]

Where:
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (expressed as a decimal),
- $ t $ is the time in years,
- $ e $ is the base of the natural logarithm (approximately 2.71828).

For Arianna:
- $ P = 250 $ dollars,
- $ r = 8.25\% = 0.0825 $,
- $ t = 18 $ years.

Plugging these into the formula:

\[
A_{\text{Arianna}} = 250 \cdot e^{0.0825 \cdot 18}
\]

### Lily's Investment
Lily's investment is compounded quarterly. The formula for compound interest is given by:

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

Where:
- $ P $ is the principal amount,
- $ r $ is the annual interest rate (expressed as a decimal),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time in years.

For Lily:
- $ P = 250 $ dollars,
- $ r = 8.75\% = 0.0875 $,
- $ n = 4 $ (since it's compounded quarterly),
- $ t = 18 $ years.

Plugging these into the formula:

\[
A_{\text{Lily}} = 250 \left(1 + \frac{0.0875}{4}\right)^{4 \cdot 18}
\]

### Calculation
Now, let's compute the values for both Arianna and Lily.

1. **Arianna's Investment:**

\[
A_{\text{Arianna}} = 250 \cdot e^{0.0825 \cdot 18}
\]

2. **Lily's Investment:**

\[
A_{\text{Lily}} = 250 \left(1 + \frac{0.0875}{4}\right)^{72}
\]

Finally, subtract Arianna's final amount from Lily's final amount to find how much more money Lily has.

Let's calculate this.After 18 years:

- **Arianna's investment** grows to approximately **$1,104**.
- **Lily's investment** grows to approximately **$1,187**.

The difference between Lily's and Arianna's accounts is **$84** (rounded to the nearest dollar). Therefore, Lily would have **$84 more** than Arianna after 18 years.

Would you like more details on this calculation or have any other questions?

Here are 5 related questions you might find interesting:
1. How does the frequency of compounding affect the future value of an investment?
2. What is the difference between continuous compounding and quarterly compounding?
3. How would the future value change if the interest rate for both investments were the same?
4. What happens if the investments were held for a different time period, say 25 years?
5. How does changing the principal amount affect the final difference in investment values?

**Tip:** Continuous compounding results in higher returns compared to other compounding frequencies as it calculates interest at every possible instant.

Explore how continuous and quarterly compounding affect investment growth. Compare Arianna and Lily's investments over 18 years using compound interest formulas. Understand the impact of compounding frequency on investment returns.

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