To find the best balance after investing $12,000 for 7 years at both simple interest and compound interest, we will calculate the final amounts for each.
Simple Interest Calculation
The formula for simple interest is:
\[ A = P + (P \times r \times t) \]
Where:
- \(A\) = total amount after interest
- \(P\) = principal amount (initial investment)
- \(r\) = interest rate (as a decimal)
- \(t\) = time (in years)
For the simple interest with:
- \(P = 12,000\)
- \(r = 0.05\) (5%)
- \(t = 7\)
Substituting the values:
\[ A = 12000 + (12000 \times 0.05 \times 7) \] \[ A = 12000 + (12000 \times 0.35) \] \[ A = 12000 + 4200 \] \[ A = 16200 \]
Compound Interest Calculation
The formula for compound interest is:
\[ A = P(1 + r)^t \]
Where:
- \(A\) = total amount after interest
- \(P\) = principal amount
- \(r\) = interest rate (as a decimal)
- \(t\) = time (in years)
For the compound interest with:
- \(P = 12,000\)
- \(r = 0.045\) (4.5%)
- \(t = 7\)
Substituting the values:
\[ A = 12000(1 + 0.045)^7 \] \[ A = 12000(1.045)^7 \]
Calculating \( (1.045)^7 \):
\[ (1.045)^7 \approx 1.348850 \]
Now calculating \(A\):
\[ A \approx 12000 \times 1.348850 \approx 16186.20 \]
Conclusion
Now we compare both amounts:
- Simple Interest: $16,200.00
- Compound Interest: $16,186.20
The best balance after 7 years, rounded to two decimal places, is:
\[ \boxed{16200.00} \] (from simple interest).