To determine which investment is better for Aria, we will calculate the interest earned from both the simple interest at 6.5% and the compound interest at 6% over a period of 3 years.
1. Simple Interest Calculation: The formula for simple interest is: \[ \text{Simple Interest} = P \times r \times t \] where:
- \(P\) = principal amount (initial investment)
- \(r\) = annual interest rate (as a decimal)
- \(t\) = time in years
For Aria's investment:
- \(P = 6700\)
- \(r = 0.065\) (6.5%)
- \(t = 3\)
Calculating the simple interest: \[ \text{Simple Interest} = 6700 \times 0.065 \times 3 = 6700 \times 0.195 = 1305 \]
2. Compound Interest Calculation: The formula for compound interest is: \[ A = P(1 + r)^t \] where \(A\) is the amount of money accumulated after n years, including interest.
To find the compound interest earned, we need to subtract the principal from this amount: \[ \text{Compound Interest} = A - P \]
Using the values:
- \(P = 6700\)
- \(r = 0.06\) (6%)
- \(t = 3\)
Calculating the total amount \(A\): \[ A = 6700(1 + 0.06)^3 = 6700(1.06)^3 \] Calculating \(1.06^3\): \[ 1.06^3 \approx 1.191016 \]
Now calculate \(A\): \[ A \approx 6700 \times 1.191016 \approx 7987.79 \]
Now, subtract the principal to find the compound interest: \[ \text{Compound Interest} = 7987.79 - 6700 \approx 1287.79 \]
3. Interest Earned Comparison:
- Simple Interest earned: $1,305
- Compound Interest earned: $1,287.79
Now, let's find the difference: \[ \text{Difference} = \text{Simple Interest} - \text{Compound Interest} = 1305 - 1287.79 \approx 17.21 \]
This means that there is actually a small difference of $17.21, showing that simple interest earns slightly more over this period for Aria’s investment.
Conclusion: Since none of the response options provided in your question align with this result, we can conclude that Aria should invest with 6.5% simple interest, as it actually earns slightly more than 6% compound interest in this scenario. However, if there was a mismatch or an oversight in the options presented, the calculated amounts can guide her decision.