Aria is investing $6,700 of her savings from her summer job for her college fund. She is planning to invest the amount for 3 years and can choose between simple interest at 6.5% and compound interest at 6%. Find the difference between the two interest earning types to help Aria decide which investing option is best for her.(1 point) Responses Aria should invest with 6.5% simple interest because it will result in $187.26 more in interest earnings. Aria should invest with 6.5% simple interest because it will result in $187.26 more in interest earnings. Aria should invest with 6% compound interest because it will result in $73.81 more in interest earnings. Aria should invest with 6% compound interest because it will result in $73.81 more in interest earnings. Aria should invest with 6.5% simple interest because it will result in $26.69 more in interest earnings. Aria should invest with 6.5% simple interest because it will result in $26.69 more in interest earnings. Aria should invest with 6% compound interest because it will result in $86.76 more in interest earnings.

Answer 1

To determine which investing option is best for Aria, we need to calculate the total earnings from both the simple interest and compound interest options.

Simple Interest Calculation:

The formula for simple interest is: \[ I = P \times r \times t \] where:

• \(I\) is the interest earned,
• \(P\) is the principal amount (initial investment),
• \(r\) is the annual interest rate (as a decimal),
• \(t\) is the time (in years).

For Aria's investment:

• \(P = 6,700\)
• \(r = 0.065\) (6.5%)
• \(t = 3\)

Calculating the simple interest: \[ I = 6,700 \times 0.065 \times 3 = 6,700 \times 0.195 = 1,306.50 \]

Compound Interest Calculation:

The formula for compound interest is: \[ A = P \times (1 + r)^t \] where:

• \(A\) is the amount of money accumulated after n years, including interest,
• \(P\) is the principal amount (initial investment),
• \(r\) is the annual interest rate (as a decimal),
• \(t\) is the number of years the money is invested for.

The interest earned can then be found by subtracting the principal from this total. For Aria's investment:

• \(P = 6,700\)
• \(r = 0.06\) (6%)
• \(t = 3\)

Calculating the compound interest: \[ A = 6,700 \times (1 + 0.06)^3 \] \[ A = 6,700 \times (1.06)^3 \approx 6,700 \times 1.191016 = 7,973.99 \] The interest earned is: \[ I = A - P = 7,973.99 - 6,700 = 1,273.99 \]

Summary of Earnings:

• Simple Interest Earnings: \(1,306.50\)
• Compound Interest Earnings: \(1,273.99\)

Difference in Earnings: Now, let's find the difference: \[ 1,306.50 - 1,273.99 = 32.51 \]

Based on this calculation, Aria should invest with 6.5% simple interest because it will result in $32.51 more in interest earnings.

However, among the provided options, the closest answer is: "Aria should invest with 6.5% simple interest because it will result in $26.69 more in interest earnings."

(Note: This appears to be an error, as our calculated difference does not match the option. The correct arithmetic shows that the correct choice should emphasize the greater benefit of choosing simple interest over compound interest, but none of the options exactly match the computed results. It would make sense for Aria to pursue the 6.5% simple interest route).