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)

Question

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% compound interest because it will result in $86.76 more in interest earnings.
Aria should invest with 6% compound interest because it will result in $86.76 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 6.5% 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.

GPT-4o mini · Answer

To determine which investment option is better for Aria over 3 years, we need to calculate the total interest earned from both the simple interest and the compound interest options.

Simple Interest

Simple interest is calculated using the formula: \[ I = P \times r \times t \] where:

$ I $ is the interest,
$ 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 simple interest calculation:

$ P = 6700 $,
$ r = 0.065 $ (6.5% as a decimal),
$ t = 3 $.

Now calculating the simple interest: \[ I = 6700 \times 0.065 \times 3 = 6700 \times 0.195 = 1309.50 \] So, the total interest earned with simple interest is $1309.50.

Compound Interest

Compound interest is calculated using the formula: \[ A = P \times (1 + r)^t \] where:

$ A $ is the amount after interest,
$ P $ is the principal amount,
$ r $ is the annual interest rate (as a decimal),
$ t $ is the time (in years).

The interest earned can then be calculated as: \[ I = A - P \]

For Aria's compound interest calculation:

$ P = 6700 $,
$ r = 0.06 $ (6% as a decimal),
$ t = 3 $.

Now calculating the total amount with compound interest: \[ A = 6700 \times (1 + 0.06)^3 = 6700 \times (1.06)^3 \] Calculating $ (1.06)^3 $: \[ (1.06)^3 \approx 1.191016 \] Thus, \[ A \approx 6700 \times 1.191016 \approx 7988.31 \] Now calculating the interest earned: \[ I = A - P = 7988.31 - 6700 = 1288.31 \] So, the total interest earned with compound interest is $1288.31.

Comparison

Now, let's compare the two:

Simple Interest: $ I_{simple} = 1309.50 $
Compound Interest: $ I_{compound} = 1288.31 $

Calculating the difference in interest: \[ \text{Difference} = I_{simple} - I_{compound} = 1309.50 - 1288.31 = 21.19 \]

Conclusion

Aria should invest with 6.5% simple interest because it will result in $21.19 more in interest earnings over the 3-year period compared to 6% compound interest.

None of the provided options exactly match this conclusion, as it appears there may be some discrepancies in the responses given in your query.