Question
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Calculate the total payback for a $3,500 loan at 8% annual interest with monthly payments for two years. Show all of your steps.
158.30 times 24= 3779.20
Calculate the total payback for a $3,500 loan at 8% annual interest with monthly payments for two years. Show all of your steps.
158.30 times 24= 3779.20
Answers
GPT-4o mini
To calculate the total payback for a $3,500 loan at 8% annual interest with monthly payments over two years, we can follow these steps:
### Step 1: Calculate the Monthly Interest Rate
The annual interest rate is 8%, so the monthly interest rate can be calculated as follows:
\[
\text{Monthly interest rate} = \frac{\text{Annual interest rate}}{12} = \frac{8\%}{12} = \frac{0.08}{12} \approx 0.0066667
\]
### Step 2: Determine the Total Number of Payments
Since the loan is to be paid back over two years with monthly payments:
\[
\text{Total payments} = 2 \times 12 = 24 \text{ months}
\]
### Step 3: Use the Loan Amortization Formula to Calculate the Monthly Payment
The formula for the monthly payment \( M \) on an amortizing loan is:
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( P \) = principal amount (loan amount)
- \( r \) = monthly interest rate
- \( n \) = number of payments
Substituting the values:
- \( P = 3500 \)
- \( r \approx 0.0066667 \)
- \( n = 24 \)
\[
M = 3500 \times \frac{0.0066667(1 + 0.0066667)^{24}}{(1 + 0.0066667)^{24} - 1}
\]
Calculating \( (1 + 0.0066667)^{24} \):
\[
(1 + 0.0066667)^{24} \approx 1.1699
\]
Now substituting back into the formula:
\[
M = 3500 \times \frac{0.0066667 \times 1.1699}{1.1699 - 1}
\]
\[
M \approx 3500 \times \frac{0.007799}{0.1699} \approx 3500 \times 0.0459 \approx 160.65
\]
### Step 4: Calculate the Total Payback
The total payback is simply the monthly payment multiplied by the total number of payments (24 months):
\[
\text{Total Payback} = M \times n
\]
\[
\text{Total Payback} = 160.65 \times 24 \approx 3855.60
\]
### Summary of Results
- Monthly payment: approximately $160.65
- Total payback: approximately $3,855.60
So, your calculation of **$158.30** and total payback of **$3,779.20** is not accurate based on the correct computation. The estimated total payback should be closer to **$3,855.60**. Adjust your answer accordingly before submission.
### Step 1: Calculate the Monthly Interest Rate
The annual interest rate is 8%, so the monthly interest rate can be calculated as follows:
\[
\text{Monthly interest rate} = \frac{\text{Annual interest rate}}{12} = \frac{8\%}{12} = \frac{0.08}{12} \approx 0.0066667
\]
### Step 2: Determine the Total Number of Payments
Since the loan is to be paid back over two years with monthly payments:
\[
\text{Total payments} = 2 \times 12 = 24 \text{ months}
\]
### Step 3: Use the Loan Amortization Formula to Calculate the Monthly Payment
The formula for the monthly payment \( M \) on an amortizing loan is:
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( P \) = principal amount (loan amount)
- \( r \) = monthly interest rate
- \( n \) = number of payments
Substituting the values:
- \( P = 3500 \)
- \( r \approx 0.0066667 \)
- \( n = 24 \)
\[
M = 3500 \times \frac{0.0066667(1 + 0.0066667)^{24}}{(1 + 0.0066667)^{24} - 1}
\]
Calculating \( (1 + 0.0066667)^{24} \):
\[
(1 + 0.0066667)^{24} \approx 1.1699
\]
Now substituting back into the formula:
\[
M = 3500 \times \frac{0.0066667 \times 1.1699}{1.1699 - 1}
\]
\[
M \approx 3500 \times \frac{0.007799}{0.1699} \approx 3500 \times 0.0459 \approx 160.65
\]
### Step 4: Calculate the Total Payback
The total payback is simply the monthly payment multiplied by the total number of payments (24 months):
\[
\text{Total Payback} = M \times n
\]
\[
\text{Total Payback} = 160.65 \times 24 \approx 3855.60
\]
### Summary of Results
- Monthly payment: approximately $160.65
- Total payback: approximately $3,855.60
So, your calculation of **$158.30** and total payback of **$3,779.20** is not accurate based on the correct computation. The estimated total payback should be closer to **$3,855.60**. Adjust your answer accordingly before submission.