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

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 6.5% 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.
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 $73.81 more in interest earnings.

Answer 1

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

1. Simple Interest Calculation

The formula for simple interest is:

\[ I = P \times r \times t \]

Where:

• \( I \) = Interest earned
• \( P \) = Principal amount (initial investment)
• \( r \) = Interest rate (as a decimal)
• \( t \) = Time (in years)

For Aria:

• \( P = 6700 \)
• \( r = 0.065 \) (which is 6.5% as a decimal)
• \( t = 3 \)

Calculating simple interest:

\[ I = 6700 \times 0.065 \times 3 \] \[ I = 6700 \times 0.195 \] \[ I = 1306.50 \]

So, the total simple interest earned over 3 years is $1306.50.

2. Compound Interest Calculation

The formula for compound interest is:

\[ A = P(1 + r)^t \]

Where:

• \( A \) = Amount after time \( t \)
• \( P \) = Principal amount (initial investment)
• \( r \) = Interest rate (as a decimal)
• \( t \) = Time (in years)

First, we need to calculate the total amount \( A \):

For Aria:

• \( P = 6700 \)
• \( r = 0.06 \) (which is 6% as a decimal)
• \( t = 3 \)

Calculating for compound interest:

\[ A = 6700(1 + 0.06)^3 \] \[ A = 6700(1.06)^3 \] \[ A = 6700 \times 1.191016 \] \[ A \approx 8079.67 \]

Next, we calculate the total compound interest earned:

\[ \text{Compound Interest} = A - P \] \[ \text{Compound Interest} = 8079.67 - 6700 \] \[ \text{Compound Interest} \approx 1379.67 \]

3. Comparing the Two Options

• Simple Interest earned: $1306.50
• Compound Interest earned: $1379.67

Now, we find the difference between the two interest amounts:

\[ \text{Difference} = 1379.67 - 1306.50 \approx 73.17 \]

Conclusion

Aria should invest with 6% compound interest because it will result in approximately $73.17 more in interest earnings compared to 6.5% simple interest. However, since this isn't directly one of the options, the closest option responding to this is:

• Aria should invest with 6% compound interest because it will result in $73.81 more in interest earnings.

The correct choice is slightly off due to rounding or calculation error, but it's the closest available in your responses.