Decide which of the two plans will provide a better yield.
Plan​ A: ​$40 comma 000 invested for 3 years at​ 1.5%, compounded quarterly
Plan​ B: ​$40 comma 000 invested for 3 years at​ 1.4%, compounded continuously
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.
Plan B is better by ​$

enter your response here.
​(Simplify your answer. Type an integer or a decimal. Round to the nearest cent as​ needed.)
B.
Plan A is better by ​$

enter your response here.
​(Simplify your answer. Type an integer or a decimal. Round to the nearest cent as​ needed.)
C.
Both plans are same.

To determine which plan provides a better yield, we need to calculate the future value for both Plan A and Plan B, then compare them.

### Plan A: $40,000 at 1.5%, compounded quarterly for 3 years

The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $A$ is the amount of money after interest,
- $P$ is the principal (initial investment) = 40,000,
- $r$ is the annual interest rate = 1.5% or 0.015,
- $n$ is the number of compounding periods per year = 4 (quarterly),
- $t$ is the number of years = 3.

Substitute the values:
\[
A = 40,000 \left(1 + \frac{0.015}{4}\right)^{4 \times 3}
\]

### Plan B: $40,000 at 1.4%, compounded continuously for 3 years

The formula for continuous compounding is:
\[
A = P e^{rt}
\]
where:
- $P$ is the principal = 40,000,
- $r$ is the annual interest rate = 1.4% or 0.014,
- $t$ is the number of years = 3,
- $e$ is Euler's number (approximately 2.71828).

Substitute the values:
\[
A = 40,000 \times e^{0.014 \times 3}
\]

I'll now calculate both future values.### Results:
- **Plan A** (compounded quarterly at 1.5%) results in: \$41,837.59
- **Plan B** (compounded continuously at 1.4%) results in: \$41,715.78

Since Plan A yields more than Plan B by approximately **\$121.81**, the better option is **Plan A**.

Thus, the correct choice is:
**B. Plan A is better by \$121.81.**

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What is the formula for continuous compounding and its applications?
2. How does quarterly compounding compare to monthly compounding over 3 years?
3. How would the outcome change if the rates were higher?
4. What happens if the time period increases to 5 years?
5. Can you explain the impact of compounding frequency on investment returns?

**Tip:** Always pay attention to the compounding frequency, as it can significantly affect the outcome of interest calculations.

Decide which of the two plans will provide a better yield.
Plan A: $40,000 invested for 3 years at 1.5%, compounded quarterly
Plan B: $40,000 invested for 3 years at 1.4%, compounded continuously

Learn how to compare two investment plans: $40,000 at 1.5% compounded quarterly and $40,000 at 1.4% compounded continuously for 3 years. This problem explores compound interest and continuous compounding, with step-by-step calculations. Ideal for Grades 10-12.

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