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 2.7% 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.
A payment plan that accrues 2.75% interest compounded annually.
A payment plan that accrues 2.8% simple interest.
Responses
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.
A payment plan that accrues 2.75% interest compounded annually.
A payment plan that accrues 2.8% simple interest.
Answers
GPT-4o mini
To find out which payment plan will result in the least interest owed after 3 years for Marcia's loan of $7,500, we can compare the total amount of interest accumulated from each plan.
1. **Compounded Monthly (2.7%)**:
The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times that interest is compounded per year.
- \(t\) is the number of years the money is borrowed.
For this plan:
- \(P = 7500\)
- \(r = 0.027\)
- \(n = 12\)
- \(t = 3\)
\[
A = 7500 \left(1 + \frac{0.027}{12}\right)^{12 \times 3}
\]
\[
A = 7500 \left(1 + 0.00225\right)^{36} \approx 7500 \cdot (1.0851528) \approx 8138.64
\]
Interest = \(A - P = 8138.64 - 7500 \approx 638.64\)
2. **Continuous Compounding (2.7%)**:
The formula for continuous compounding is:
\[
A = Pe^{rt}
\]
Where \(e\) is the base of the natural logarithm.
\[
A = 7500 \cdot e^{0.027 \cdot 3}
\]
\[
e^{0.081} \approx 1.084
\]
\[
A \approx 7500 \cdot 1.084 \approx 8130
\]
Interest = \(A - P = 8130 - 7500 \approx 630\)
3. **Compounded Annually (2.75%)**:
For this plan, \(n = 1\):
\[
A = 7500 \left(1 + \frac{0.0275}{1}\right)^{1 \cdot 3}
\]
\[
A = 7500 \cdot (1.0275)^3 \approx 7500 \cdot (1.08506) \approx 8137.95
\]
Interest = \(A - P = 8137.95 - 7500 \approx 637.95\)
4. **Simple Interest (2.8%)**:
The formula for simple interest is:
\[
A = P(1 + rt)
\]
\[
A = 7500(1 + 0.028 \cdot 3) = 7500(1 + 0.084) = 7500(1.084) \approx 8130
\]
Interest = \(A - P = 8130 - 7500 \approx 630\)
Now let's summarize the interest amounts:
- Compounded Monthly (2.7%): Approximately \(638.64\)
- Continuous Compounding (2.7%): Approximately \(630\)
- Compounded Annually (2.75%): Approximately \(637.95\)
- Simple Interest (2.8%): Approximately \(630\)
### Conclusion:
The payment plans that result in the least interest over 3 years are:
- Continuous compounding at 2.7%: Approximately \(630\)
- Simple interest at 2.8%: Approximately \(630\)
Both the continuous compounded interest plan at 2.7% and the simple interest plan at 2.8% allow Marcia to owe the least interest after 3 years, which is approximately **$630**.
Thus, you can choose either of these two options.
1. **Compounded Monthly (2.7%)**:
The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times that interest is compounded per year.
- \(t\) is the number of years the money is borrowed.
For this plan:
- \(P = 7500\)
- \(r = 0.027\)
- \(n = 12\)
- \(t = 3\)
\[
A = 7500 \left(1 + \frac{0.027}{12}\right)^{12 \times 3}
\]
\[
A = 7500 \left(1 + 0.00225\right)^{36} \approx 7500 \cdot (1.0851528) \approx 8138.64
\]
Interest = \(A - P = 8138.64 - 7500 \approx 638.64\)
2. **Continuous Compounding (2.7%)**:
The formula for continuous compounding is:
\[
A = Pe^{rt}
\]
Where \(e\) is the base of the natural logarithm.
\[
A = 7500 \cdot e^{0.027 \cdot 3}
\]
\[
e^{0.081} \approx 1.084
\]
\[
A \approx 7500 \cdot 1.084 \approx 8130
\]
Interest = \(A - P = 8130 - 7500 \approx 630\)
3. **Compounded Annually (2.75%)**:
For this plan, \(n = 1\):
\[
A = 7500 \left(1 + \frac{0.0275}{1}\right)^{1 \cdot 3}
\]
\[
A = 7500 \cdot (1.0275)^3 \approx 7500 \cdot (1.08506) \approx 8137.95
\]
Interest = \(A - P = 8137.95 - 7500 \approx 637.95\)
4. **Simple Interest (2.8%)**:
The formula for simple interest is:
\[
A = P(1 + rt)
\]
\[
A = 7500(1 + 0.028 \cdot 3) = 7500(1 + 0.084) = 7500(1.084) \approx 8130
\]
Interest = \(A - P = 8130 - 7500 \approx 630\)
Now let's summarize the interest amounts:
- Compounded Monthly (2.7%): Approximately \(638.64\)
- Continuous Compounding (2.7%): Approximately \(630\)
- Compounded Annually (2.75%): Approximately \(637.95\)
- Simple Interest (2.8%): Approximately \(630\)
### Conclusion:
The payment plans that result in the least interest over 3 years are:
- Continuous compounding at 2.7%: Approximately \(630\)
- Simple interest at 2.8%: Approximately \(630\)
Both the continuous compounded interest plan at 2.7% and the simple interest plan at 2.8% allow Marcia to owe the least interest after 3 years, which is approximately **$630**.
Thus, you can choose either of these two options.