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.
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Simple Interest at 2.8%: \[ I = P \times r \times t = 7500 \times 0.028 \times 3 = 630 \]
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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 = \( A - P = 8172.62 - 7500 = 672.62 \)
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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 = \( A - P = 8130 - 7500 = 630 \)
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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.
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