Question
Check my work please. I need to make sure my answer is correct before I submit. Thank you!
Calculate the total payback for a $3,500 loan at 8% annual interest with monthly payments for two years. Show all of your steps.
Since the interest rate is annual we divide the 8% which 12 and that is equal to 0.0067. We will use this in our formula later, same with the information that their are 24 payments we will input all this information into our formula above.
So M=$3,500 times 0.0067(1+0.0067)^24
Over
(1+0.0067)^24 —1
When w e solve it we get the goal of 160.24 but in order to figure out the total payback amount we multiply it by the 24 and when we do that we get the answer of $3,845.76
Calculate the total payback for a $3,500 loan at 8% annual interest with monthly payments for two years. Show all of your steps.
Since the interest rate is annual we divide the 8% which 12 and that is equal to 0.0067. We will use this in our formula later, same with the information that their are 24 payments we will input all this information into our formula above.
So M=$3,500 times 0.0067(1+0.0067)^24
Over
(1+0.0067)^24 —1
When w e solve it we get the goal of 160.24 but in order to figure out the total payback amount we multiply it by the 24 and when we do that we get the answer of $3,845.76
Answers
GPT-4o mini
Let’s go through your calculations step by step to ensure accuracy in determining the total payback for a $3,500 loan at 8% annual interest with monthly payments over two years.
1. **Convert annual interest rate to monthly**:
- Annual interest rate = 8%
- Monthly interest rate = 8% / 12 = 0.6667% = 0.08 / 12 = 0.006667 (in decimal form)
2. **Calculate the number of payments**:
- Total number of payments (months) for 2 years = 2 * 12 = 24
3. **Using the formula for monthly payment (M)**:
The formula for calculating the monthly payment (M) on an amortized loan is:
\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( P \) = loan amount ($3,500)
- \( r \) = monthly interest rate (0.006667)
- \( n \) = total number of payments (24)
4. **Plugging values into the formula**:
\[
M = 3500 \times \frac{0.006667(1 + 0.006667)^{24}}{(1 + 0.006667)^{24} - 1}
\]
5. **Calculating the monthly interest factor**:
- Calculate \( (1 + 0.006667)^{24} \):
\[
(1 + 0.006667)^{24} \approx 1.170691
\]
6. **Substituting back into the equation**:
\[
M = 3500 \times \frac{0.006667 \times 1.170691}{1.170691 - 1}
\]
- Calculating the numerator:
\[
0.006667 \times 1.170691 \approx 0.00780149
\]
- For the denominator:
\[
1.170691 - 1 \approx 0.170691
\]
- Now substituting:
\[
M = 3500 \times \frac{0.00780149}{0.170691} \approx 3500 \times 0.045757 = 160.15
\]
7. **Calculating the total payback amount**:
- Multiply the monthly payment by the total number of payments:
\[
\text{Total Payback} = M \times n = 160.15 \times 24 \approx 3843.60
\]
8. **Final total payback amount**:
- Total Payback = **$3,843.60** (rounded to two decimal places).
Overall, your process was mostly correct but had some inaccuracies in the calculations, particularly with the exact monthly payment round-off. The correct total payback amount for the loan is approximately $3,843.60. Ensure that you've correctly rounded your intermediate calculations, as this can lead to final discrepancies.
1. **Convert annual interest rate to monthly**:
- Annual interest rate = 8%
- Monthly interest rate = 8% / 12 = 0.6667% = 0.08 / 12 = 0.006667 (in decimal form)
2. **Calculate the number of payments**:
- Total number of payments (months) for 2 years = 2 * 12 = 24
3. **Using the formula for monthly payment (M)**:
The formula for calculating the monthly payment (M) on an amortized loan is:
\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( P \) = loan amount ($3,500)
- \( r \) = monthly interest rate (0.006667)
- \( n \) = total number of payments (24)
4. **Plugging values into the formula**:
\[
M = 3500 \times \frac{0.006667(1 + 0.006667)^{24}}{(1 + 0.006667)^{24} - 1}
\]
5. **Calculating the monthly interest factor**:
- Calculate \( (1 + 0.006667)^{24} \):
\[
(1 + 0.006667)^{24} \approx 1.170691
\]
6. **Substituting back into the equation**:
\[
M = 3500 \times \frac{0.006667 \times 1.170691}{1.170691 - 1}
\]
- Calculating the numerator:
\[
0.006667 \times 1.170691 \approx 0.00780149
\]
- For the denominator:
\[
1.170691 - 1 \approx 0.170691
\]
- Now substituting:
\[
M = 3500 \times \frac{0.00780149}{0.170691} \approx 3500 \times 0.045757 = 160.15
\]
7. **Calculating the total payback amount**:
- Multiply the monthly payment by the total number of payments:
\[
\text{Total Payback} = M \times n = 160.15 \times 24 \approx 3843.60
\]
8. **Final total payback amount**:
- Total Payback = **$3,843.60** (rounded to two decimal places).
Overall, your process was mostly correct but had some inaccuracies in the calculations, particularly with the exact monthly payment round-off. The correct total payback amount for the loan is approximately $3,843.60. Ensure that you've correctly rounded your intermediate calculations, as this can lead to final discrepancies.