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.
Use the formula M=Pm(1+m)^na / (1+m)^na -1
Calculate the total payback for a $3,500 loan at 8% annual interest with monthly payments for two years. Show all of your steps.
Use the formula M=Pm(1+m)^na / (1+m)^na -1
Answers
GPT-4o mini
Sure! Let's go through the steps to calculate the total payback for a $3,500 loan at 8% annual interest with monthly payments over two years.
### Step 1: Gather the variables
1. **Loan Amount (P)**: $3,500
2. **Annual Interest Rate**: 8% or 0.08
3. **Loan Term (n)**: 2 years
4. **Monthly Interest Rate (m)**:
- Since the interest is annual and compounded monthly, we need to divide the annual rate by 12.
\[
m = \frac{0.08}{12} = 0.00666667
\]
5. **Total Number of Payments (na)**:
- Since it's a two-year loan and payments are monthly, the total number is:
\[
na = 2 \times 12 = 24
\]
### Step 2: Plug values into the formula
Now we can put these values into the formula for monthly payment \( M \):
\[
M = \frac{P \cdot m(1 + m)^{na}}{(1 + m)^{na} - 1}
\]
Substituting the values:
\[
M = \frac{3500 \cdot 0.00666667(1 + 0.00666667)^{24}}{(1 + 0.00666667)^{24} - 1}
\]
### Step 3: Calculate \( (1 + m)^{na} \)
1. Calculate \( 1 + m \):
\[
1 + m = 1 + 0.00666667 = 1.00666667
\]
2. Raise it to the power of \( na \):
\[
(1.00666667)^{24} \approx 1.169859
\]
### Step 4: Complete the equation for M
Now substitute this back into the equation for \( M \):
- The numerator:
\[
3500 \cdot 0.00666667 \cdot 1.169859 \approx 27.5521
\]
- The denominator:
\[
1.169859 - 1 \approx 0.169859
\]
Combine these to find \( M \):
\[
M \approx \frac{27.5521}{0.169859} \approx 162.99
\]
### Step 5: Calculate Total Payback
Total payback over the two years would be \( M \times na \):
\[
\text{Total Payback} = M \cdot na = 162.99 \cdot 24 \approx 3,911.76
\]
### Final Answer
Thus, the total payback for the $3,500 loan at an 8% annual interest rate with monthly payments over two years is approximately **$3,911.76**.
Feel free to ask if you need any clarification or further assistance!
### Step 1: Gather the variables
1. **Loan Amount (P)**: $3,500
2. **Annual Interest Rate**: 8% or 0.08
3. **Loan Term (n)**: 2 years
4. **Monthly Interest Rate (m)**:
- Since the interest is annual and compounded monthly, we need to divide the annual rate by 12.
\[
m = \frac{0.08}{12} = 0.00666667
\]
5. **Total Number of Payments (na)**:
- Since it's a two-year loan and payments are monthly, the total number is:
\[
na = 2 \times 12 = 24
\]
### Step 2: Plug values into the formula
Now we can put these values into the formula for monthly payment \( M \):
\[
M = \frac{P \cdot m(1 + m)^{na}}{(1 + m)^{na} - 1}
\]
Substituting the values:
\[
M = \frac{3500 \cdot 0.00666667(1 + 0.00666667)^{24}}{(1 + 0.00666667)^{24} - 1}
\]
### Step 3: Calculate \( (1 + m)^{na} \)
1. Calculate \( 1 + m \):
\[
1 + m = 1 + 0.00666667 = 1.00666667
\]
2. Raise it to the power of \( na \):
\[
(1.00666667)^{24} \approx 1.169859
\]
### Step 4: Complete the equation for M
Now substitute this back into the equation for \( M \):
- The numerator:
\[
3500 \cdot 0.00666667 \cdot 1.169859 \approx 27.5521
\]
- The denominator:
\[
1.169859 - 1 \approx 0.169859
\]
Combine these to find \( M \):
\[
M \approx \frac{27.5521}{0.169859} \approx 162.99
\]
### Step 5: Calculate Total Payback
Total payback over the two years would be \( M \times na \):
\[
\text{Total Payback} = M \cdot na = 162.99 \cdot 24 \approx 3,911.76
\]
### Final Answer
Thus, the total payback for the $3,500 loan at an 8% annual interest rate with monthly payments over two years is approximately **$3,911.76**.
Feel free to ask if you need any clarification or further assistance!