Question
Your teacher will grade your response to ensure you receive proper credit for your answer.
You place $4,000.00 in a bank account with an interest rate of 5.25% APR and another $2,000.00 in an account with an interest rate of 6.00% APR.
a. How much interest will be earned for each account on the original principal in 6 months?
b. What is the difference in the amount of interest earned?
c. Which account has your money working for you the best?
Be sure to include in your response:
the steps you followed to complete your computations
the final answers to each question
the mathematical reasoning for the answer to “c”
(5 points)
You place $4,000.00 in a bank account with an interest rate of 5.25% APR and another $2,000.00 in an account with an interest rate of 6.00% APR.
a. How much interest will be earned for each account on the original principal in 6 months?
b. What is the difference in the amount of interest earned?
c. Which account has your money working for you the best?
Be sure to include in your response:
the steps you followed to complete your computations
the final answers to each question
the mathematical reasoning for the answer to “c”
(5 points)
Answers
GPT-4o mini
To solve this problem, we will break it down into parts (a), (b), and (c) as requested.
### a. How much interest will be earned for each account on the original principal in 6 months?
First, let's calculate the interest earned for each account using the formula for simple interest:
\[
\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}
\]
where:
- Principal is the initial amount invested,
- Rate is the annual interest rate (as a decimal),
- Time is the period the money is invested for (in years).
#### For the first account ($4,000 at 5.25% APR):
1. Principal = $4,000
2. Rate = 5.25% = 0.0525 (as a decimal)
3. Time = 6 months = 0.5 years
Calculating the interest:
\[
\text{Interest}_1 = 4,000 \times 0.0525 \times 0.5 = 4,000 \times 0.02625 = 105
\]
So, the interest earned in the first account is **$105.00**.
#### For the second account ($2,000 at 6.00% APR):
1. Principal = $2,000
2. Rate = 6.00% = 0.06 (as a decimal)
3. Time = 6 months = 0.5 years
Calculating the interest:
\[
\text{Interest}_2 = 2,000 \times 0.06 \times 0.5 = 2,000 \times 0.03 = 60
\]
So, the interest earned in the second account is **$60.00**.
### b. What is the difference in the amount of interest earned?
Now, we will find the difference in the interest earned from both accounts:
\[
\text{Difference} = \text{Interest}_1 - \text{Interest}_2 = 105 - 60 = 45
\]
The difference in the amount of interest earned is **$45.00**.
### c. Which account has your money working for you the best?
To determine which account has your money working for you better, we should compare the effective annual interest rates by calculating their respective interest earned per $1 invested.
#### Effective Interest Rate Calculation:
1. For the first account:
\[
\text{Effective Rate}_1 = \frac{\text{Interest}_1}{\text{Principal}_1} = \frac{105}{4000} = 0.02625 \text{ or } 2.625\%
\]
2. For the second account:
\[
\text{Effective Rate}_2 = \frac{\text{Interest}_2}{\text{Principal}_2} = \frac{60}{2000} = 0.03 \text{ or } 3.00\%
\]
### Conclusion for c:
Comparing the effective interest rates:
- Account 1 has an effective rate of 2.625%.
- Account 2 has an effective rate of 3.00%.
Since **3.00%** (from the second account) is greater than **2.625%** (from the first account), the second account with an interest rate of 6.00% APR is the better option, as it provides a higher return on the investment.
### Final Answers:
a. Interest earned: $105.00 (Account 1) and $60.00 (Account 2)
b. Difference in interest earned: $45.00
c. The second account has your money working for you the best, with a higher effective interest rate of 3.00% compared to 2.625%.
### a. How much interest will be earned for each account on the original principal in 6 months?
First, let's calculate the interest earned for each account using the formula for simple interest:
\[
\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}
\]
where:
- Principal is the initial amount invested,
- Rate is the annual interest rate (as a decimal),
- Time is the period the money is invested for (in years).
#### For the first account ($4,000 at 5.25% APR):
1. Principal = $4,000
2. Rate = 5.25% = 0.0525 (as a decimal)
3. Time = 6 months = 0.5 years
Calculating the interest:
\[
\text{Interest}_1 = 4,000 \times 0.0525 \times 0.5 = 4,000 \times 0.02625 = 105
\]
So, the interest earned in the first account is **$105.00**.
#### For the second account ($2,000 at 6.00% APR):
1. Principal = $2,000
2. Rate = 6.00% = 0.06 (as a decimal)
3. Time = 6 months = 0.5 years
Calculating the interest:
\[
\text{Interest}_2 = 2,000 \times 0.06 \times 0.5 = 2,000 \times 0.03 = 60
\]
So, the interest earned in the second account is **$60.00**.
### b. What is the difference in the amount of interest earned?
Now, we will find the difference in the interest earned from both accounts:
\[
\text{Difference} = \text{Interest}_1 - \text{Interest}_2 = 105 - 60 = 45
\]
The difference in the amount of interest earned is **$45.00**.
### c. Which account has your money working for you the best?
To determine which account has your money working for you better, we should compare the effective annual interest rates by calculating their respective interest earned per $1 invested.
#### Effective Interest Rate Calculation:
1. For the first account:
\[
\text{Effective Rate}_1 = \frac{\text{Interest}_1}{\text{Principal}_1} = \frac{105}{4000} = 0.02625 \text{ or } 2.625\%
\]
2. For the second account:
\[
\text{Effective Rate}_2 = \frac{\text{Interest}_2}{\text{Principal}_2} = \frac{60}{2000} = 0.03 \text{ or } 3.00\%
\]
### Conclusion for c:
Comparing the effective interest rates:
- Account 1 has an effective rate of 2.625%.
- Account 2 has an effective rate of 3.00%.
Since **3.00%** (from the second account) is greater than **2.625%** (from the first account), the second account with an interest rate of 6.00% APR is the better option, as it provides a higher return on the investment.
### Final Answers:
a. Interest earned: $105.00 (Account 1) and $60.00 (Account 2)
b. Difference in interest earned: $45.00
c. The second account has your money working for you the best, with a higher effective interest rate of 3.00% compared to 2.625%.