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Let's break down the problem and solve each part step by step.

### Given:
- Two investments:
  1. **6% compounded quarterly**.
  2. **5.75% compounded continuously**.
- Christine O'Brien plans to invest **$50,000**.

We will use the following formulas to solve the problem:

### Formulas:
1. **Quarterly Compounding**:
   \[
   A = P \left(1 + \frac{r}{n}\right)^{nt}
   \]
   Where:
   - $A$ = future value,
   - $P$ = initial principal (in this case, $50,000),
   - $r$ = annual interest rate (as a decimal),
   - $n$ = number of compounding periods per year,
   - $t$ = time in years.

2. **Continuous Compounding**:
   \[
   A = P e^{rt}
   \]
   Where:
   - $A$ = future value,
   - $P$ = initial principal,
   - $r$ = annual interest rate (as a decimal),
   - $t$ = time in years,
   - $e$ = Euler's number (approximately 2.71828).

### Let's calculate the answers for each part.

#### Part a: Which investment will earn the most interest in 5 years?

Let's calculate the amount for both the investments after 5 years.

1. **Quarterly Compounded Investment (6%)**:
   \[
   A = 50,000 \left(1 + \frac{0.06}{4}\right)^{4 \times 5}
   \]

2. **Continuously Compounded Investment (5.75%)**:
   \[
   A = 50,000 \cdot e^{0.0575 \times 5}
   \]

I'll calculate these now.After 5 years:

- The investment with **6% compounded quarterly** will grow to **$67,342.75**.
- The investment with **5.75% compounded continuously** will grow to **$66,654.53**.

Thus, the **quarterly compounded investment** earns more interest in 5 years.

#### Part b: How much more will the better plan earn?

To find the difference between the two plans:

\[
\text{Difference} = 67,342.75 - 66,654.53
\]

Let me calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can easily calculate the difference by subtracting the amounts:

\[
\text{Difference} = 67,342.75 - 66,654.53 = 688.22
\]

Thus, the better plan (quarterly compounded investment) earns **$688.22** more than the continuous compounding plan over 5 years.

Would you like further details or have any other questions?

Here are 5 related questions:
1. What happens if the rates are reversed (continuous is higher than quarterly)?
2. How would the results change if the investment period was 10 years instead of 5?
3. What would be the future value if Christine invested $100,000 instead of $50,000?
4. How can you calculate the equivalent interest rate if the continuous compounding is replaced by semi-annual compounding?
5. How would inflation affect the value of these investments over time?

**Tip**: Always compare effective interest rates for different compounding periods to make a better investment decision.

Christine O'Brien, who is self-employed, wants to invest $50,000 in a pension plan. One investment offers 6% compounded quarterly. Another offers 5.75% compounded continuously. Which investment will earn the most interest in 5 years? How much more will the better plan earn?

Compare the future value of a $50,000 investment over 5 years with 6% compounded quarterly vs. 5.75% compounded continuously. Learn how to calculate the interest difference and choose the best investment option.

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