Question

Continuously Compounded Interest Quick Check
4 of 54 of 5 Items




Question
Marcia wants to borrow $7,500. For which payment plan will she owe the least interest after 3 years?(1 point)
Responses

A payment plan that accrues 2.8% simple interest.
A payment plan that accrues 2.8% simple interest.

A payment plan that accrues 2.7% interest compounded monthly.
A payment plan that accrues 2.7% interest compounded monthly.

A payment plan that accrues 2.7% interest continuously compounded annually.
A payment plan that accrues 2.7% interest continuously compounded annually.

A payment plan that accrues 2.75% interest compounded annually.

Answer 1

To determine which payment plan will result in the least amount of interest owed after 3 years, we need to calculate the total interest for each option.

1. **Simple Interest at 2.8%**:
\[
I = P \times r \times t = 7500 \times 0.028 \times 3 = 630
\]

2. **Compounded Monthly at 2.7%**:
\[
A = P(1 + \frac{r}{n})^{nt}
\]
Where:
- \( P = 7500 \)
- \( r = 0.027 \)
- \( n = 12 \) (compounding monthly)
- \( t = 3 \)

\[
A = 7500 \left(1 + \frac{0.027}{12}\right)^{12 \times 3}
\]
\[
A = 7500 \left(1 + 0.00225\right)^{36} \approx 7500 \times (1.089682) \approx 8172.62
\]
Interest = \( A - P = 8172.62 - 7500 = 672.62 \)

3. **Continuously Compounded at 2.7%**:
\[
A = Pe^{rt}
\]
Where:
- \( e \) is approximately \( 2.71828 \)

\[
A = 7500 e^{0.027 \times 3} \approx 7500 e^{0.081} \approx 7500 \times 1.084 \approx 8130
\]
Interest = \( A - P = 8130 - 7500 = 630 \)

4. **Compounded Annually at 2.75%**:
\[
A = P(1 + r)^t = 7500(1 + 0.0275)^3
\]
\[
A = 7500(1.0275)^3 \approx 7500(1.0855) \approx 8141.25
\]
Interest = \( A - P = 8141.25 - 7500 = 641.25 \)

### Summary of Interest Calculated:
- **2.8% Simple Interest**: $630
- **2.7% Compounded Monthly**: $672.62
- **2.7% Continuously Compounded**: $630
- **2.75% Compounded Annually**: $641.25

### Conclusion:
Both the 2.8% simple interest and the 2.7% continuously compounded interest lead to the least interest owed after 3 years, both resulting in an interest of **$630**. If you need to choose one, you could select either. However, the continuously compounded interest is typically considered more favorable in financial contexts.