Question
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
as needed.
Use the formula, M=Pm(1+m)na(1+m)na−1
as needed.
Answers
GPT-4o mini
To calculate the total payback for a loan using the formula provided, we will follow these steps:
### Step 1: Identify Loan Terms
- **Loan Amount (P)** = $3,500
- **Annual Interest Rate (r)** = 8% = 0.08
- **Loan Term (n)** = 2 years
- **Monthly Payments (n)** = 2 years × 12 months/year = 24 months
### Step 2: Determine the Monthly Interest Rate (m)
To find the monthly interest rate:
\[ m = \frac{r}{12} = \frac{0.08}{12} = 0.00666667 \, (\text{approximately}) \]
### Step 3: Apply the Formula
The formula provided is:
\[ M = Pm(1 + m)^{na} / ((1 + m)^{na} - 1) \]
In this formula:
- \( P \) is the loan amount ($3,500)
- \( m \) is the monthly interest rate (0.00666667)
- \( n \) is the number of payments (24 months)
- \( a \) is also the number of payments, which is 24
Substituting the values into the formula:
\[ M = 3500 \times 0.00666667 \times (1 + 0.00666667)^{24} / ((1 + 0.00666667)^{24} - 1) \]
### Step 4: Calculate \( (1 + m)^{na} \)
First, compute \( (1 + 0.00666667)^{24} \):
\[
(1 + 0.00666667)^{24} = (1.00666667)^{24} \approx 1.169859
\]
### Step 5: Substitute into the Formula
Now substitute this value back into the formula:
\[
M = 3500 \times 0.00666667 \times 1.169859 / (1.169859 - 1)
\]
\[
= 3500 \times 0.00666667 \times 1.169859 / (0.169859)
\]
### Step 6: Simplify the Calculation
Calculating the numerator:
\[
3500 \times 0.00666667 \times 1.169859 \approx 27.745387
\]
Then the denominator:
\[
1.169859 - 1 = 0.169859
\]
Dividing the two:
\[
M \approx \frac{27.745387}{0.169859} \approx 163.49
\]
### Step 7: Calculate Total Payback
To find the total payback over the 24-month period:
\[
\text{Total Payback} = M \times n = 163.49 \times 24
\]
\[
\text{Total Payback} \approx 3,923.76
\]
### Summary
- **Monthly Payment (M)** ≈ $163.49
- **Total Payback** ≈ $3,923.76
Thus, the total payback for the $3,500 loan at an 8% annual interest rate with monthly payments over two years is approximately **$3,923.76**.
### Step 1: Identify Loan Terms
- **Loan Amount (P)** = $3,500
- **Annual Interest Rate (r)** = 8% = 0.08
- **Loan Term (n)** = 2 years
- **Monthly Payments (n)** = 2 years × 12 months/year = 24 months
### Step 2: Determine the Monthly Interest Rate (m)
To find the monthly interest rate:
\[ m = \frac{r}{12} = \frac{0.08}{12} = 0.00666667 \, (\text{approximately}) \]
### Step 3: Apply the Formula
The formula provided is:
\[ M = Pm(1 + m)^{na} / ((1 + m)^{na} - 1) \]
In this formula:
- \( P \) is the loan amount ($3,500)
- \( m \) is the monthly interest rate (0.00666667)
- \( n \) is the number of payments (24 months)
- \( a \) is also the number of payments, which is 24
Substituting the values into the formula:
\[ M = 3500 \times 0.00666667 \times (1 + 0.00666667)^{24} / ((1 + 0.00666667)^{24} - 1) \]
### Step 4: Calculate \( (1 + m)^{na} \)
First, compute \( (1 + 0.00666667)^{24} \):
\[
(1 + 0.00666667)^{24} = (1.00666667)^{24} \approx 1.169859
\]
### Step 5: Substitute into the Formula
Now substitute this value back into the formula:
\[
M = 3500 \times 0.00666667 \times 1.169859 / (1.169859 - 1)
\]
\[
= 3500 \times 0.00666667 \times 1.169859 / (0.169859)
\]
### Step 6: Simplify the Calculation
Calculating the numerator:
\[
3500 \times 0.00666667 \times 1.169859 \approx 27.745387
\]
Then the denominator:
\[
1.169859 - 1 = 0.169859
\]
Dividing the two:
\[
M \approx \frac{27.745387}{0.169859} \approx 163.49
\]
### Step 7: Calculate Total Payback
To find the total payback over the 24-month period:
\[
\text{Total Payback} = M \times n = 163.49 \times 24
\]
\[
\text{Total Payback} \approx 3,923.76
\]
### Summary
- **Monthly Payment (M)** ≈ $163.49
- **Total Payback** ≈ $3,923.76
Thus, the total payback for the $3,500 loan at an 8% annual interest rate with monthly payments over two years is approximately **$3,923.76**.