To find the final balance after investing $12,000 for seven years under both simple interest and compound interest, we can use the following formulas:
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Simple Interest Formula: \[ A = P(1 + rt) \] Where:
- \( A \) = total amount after time \( t \)
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (in decimal)
- \( t \) = time in years
-
Compound Interest Formula: \[ A = P(1 + r)^t \] Where:
- \( A \) = total amount after time \( t \)
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (in decimal)
- \( t \) = time in years
Calculation for Simple Interest (5%)
Using the simple interest formula:
- \( P = 12,000 \)
- \( r = 0.05 \)
- \( t = 7 \)
Plugging in the values: \[ A = 12000(1 + 0.05 \times 7) = 12000(1 + 0.35) = 12000 \times 1.35 = 16200 \]
Total Amount with Simple Interest:
\[ A_{\text{simple}} = 16200 \]
Calculation for Compound Interest (4.5%)
Using the compound interest formula:
- \( P = 12,000 \)
- \( r = 0.045 \)
- \( t = 7 \)
Plugging in the values: \[ A = 12000(1 + 0.045)^7 \] Calculating \( (1 + 0.045)^7 \): \[ = 1.045^7 \approx 1.348850 \] Now calculate \( A \): \[ A \approx 12000 \times 1.348850 \approx 16186.20 \]
Total Amount with Compound Interest:
\[ A_{\text{compound}} \approx 16186.20 \]
Conclusion
- Final balance with Simple Interest: $16,200.00
- Final balance with Compound Interest: $16,186.20
Rounding them both to two decimal places gives:
- Simple Interest: $16,200.00
- Compound Interest: $16,186.20
Therefore, the best balance after investing the money is with Simple Interest, yielding $16,200.00.