To determine which is the better investment option, we will calculate the future value for both the simple interest and the compound interest scenarios.
Option 1: Simple Interest
The formula for simple interest is: \[ A = P(1 + rt) \] Where:
- \( A \) = the total amount of money after interest
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time in years
For the simple interest:
- \( P = 3,800 \)
- \( r = 8.3% = 0.083 \)
- \( t = 4 \)
Calculating: \[ A = 3,800(1 + (0.083)(4)) \] \[ A = 3,800(1 + 0.332) \] \[ A = 3,800(1.332) \] \[ A \approx 5,063.60 \]
Option 2: Compound Interest
The formula for compound interest is: \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \] Where:
- \( A \) = the total amount of money after interest
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = time in years
For the compound interest:
- \( P = 3,800 \)
- \( r = 7.2% = 0.072 \)
- \( n = 12 \) (compounded monthly)
- \( t = 4 \)
Calculating: \[ A = 3,800\left(1 + \frac{0.072}{12}\right)^{12 \cdot 4} \] \[ A = 3,800\left(1 + 0.006\right)^{48} \] \[ A = 3,800\left(1.006\right)^{48} \] \[ A \approx 3,800 \times 1.34885 \] \[ A \approx 5,120.43 \]
Conclusion
- Future value with simple interest: $5,063.60
- Future value with compound interest: $5,120.43
Since $5,120.43 (compound interest) is greater than $5,063.60 (simple interest), the better option is: Enter 2 (7.2% compound interest rate compounded monthly).
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